ABSTRACT
 Title of dissertation: Remote and Local
 Entanglement of Ions
 using Photons and Phonons
 David Lee Hayes, Doctor of Philosophy, 2012
 Dissertation directed by: Professor Christopher Monroe
 Joint Quantum Institute,
 University of Maryland Department of Physics and
 National Institute of Standards and Technology
 The scaling of controlled quantum systems to large numbers of degrees of free-
 dom is one of the long term goals of experimental quantum information science.
 Trapped-ion systems are one of the most promising platforms for building a quan-
 tum information processor with enough complexity to enable novel computational
 power, but face serious challenges in scaling up to the necessary numbers of qubits.
 In this thesis, I present both technical and operational advancements in the con-
 trol of trapped-ion systems and their juxtaposition with photonic modes used for
 quantum networking. After reviewing the basic physics behind ion trapping, I then
 describe in detail a new method of implementing Raman transitions in atomic sys-
 tems using optical frequency combs. Several di erent experimental setups along
 with simple theoretical models are reviewed and the system is shown to be capable
 of full control of the qubit-oscillator system. Two-ion entangling operations using
 optical frequency combs are demonstrated along with an extension of the operation
designed to suppress certain experimental errors. I then give an overview of how spa-
 tially separated ions can be entangled using a photonic interconnect. Experimental
 results show that pulsed excitation of trapped ions provide an excellent single pho-
 ton source that can be used as a heralded entangling gate between macroscopically
 separated systems. This heralded entangling gate is used to show a violation of a
 Bell inequality while keeping the detection loophole closed and can be used a source
 private random numbers. Finally, the coherent Coulomb force-based gates are com-
 bined with the probabilistic photon-based gates in a proof of concept experiment
 that shows the feasibility of a distributed ion-photon network.
Remote and Local Entanglement of Ions using Photons and Phonons
 by
 David Lee Hayes
 Dissertation submitted to the Faculty of the Graduate School of the
 University of Maryland, College Park in partial ful llment
 of the requirements for the degree of
 Doctor of Philosophy
 2012
 Advisory Committee:
 Professor Chris Monroe, Chair/Advisor
 Professor Steve Rolston, Co-Advisor
 Professor Luis Orozco
 Professor Eite Tiesinga
 Professor Edo Waks
c Copyright by
 David Lee Hayes
 2012
To Kate
 ii
Graduate school has been a challenging time for me and I would like to thank
 the many people that have helped me along the way. First, I need to thank my
 parents for making sure that I was able to eventually  nd my way onto this road.
 Thanks for the math tutor Mom. Once I found myself buried deep in the physics
 clan, there were many people I worked with who have helped me see the light
 that I needed to either clarify my thinking or just brighten my day. While I was
 in graduate school in New Mexico, I had some luck in  nding my way into the
 quantum information group headed by Ivan Deutsch. While it was quickly apparent
 to me that Ivan was an incredible physicist, I eventually came to know that I could
 always trust that he always had my well being in mind and couldn?t have been
 a better advisor. Once I got to Maryland, I started working closely with what
 would become a long list of great people. Steve Olmschenk, Peter Maunz, Dzmitry
 Matsukevich, Dave Hucul, Qudsia Quraishi, Kenny Lee, Susan Clark, Shantanu
 Debnath, Daniel Brennan and  nally Volkan Inlek have all worked closely with me
 and have all been instrumental in my success. Although I didn?t work on the same
 projects as the other members of the group, I immensely enjoyed the company of
 Ming-Shien Chang, Brian Fields, Rajibul Islam, Kihwan Kim, Andrew Manning,
 Emily Edwards, Simcha Korenblit, Le Lou, Jonathan Mizrahi, Crystal Senko, Jon
 Sterk, Chenglin Cao, Taeyoung Choi, Kale Johnson and Jake Smith. I would like to
 especially thank Wes Campbell for letting me come into his o ce to bug him and
 answering late night emails about weird ideas of his or mine and showing me a good
 time when I needed it. But most importantly, I want to thank my advisor Chris
 Monroe. Chris has been a great mentor while I?ve been here. He has the wisdom to
 iii
know when to speak and when not to. By that, I mean that there has been times
 when the best thing for me was to go o on my own and think about something,
 but there are also times for brainstorming or a little reassurance and Chris always
 seems to know what time it is.
 iv
Table of Contents
 List of Tables vii
 List of Figures viii
 1 Introduction 1
 1.1 Ion-photon networks . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
 2 Ytterbium ions 9
 2.1 The Paul Trap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
 2.1.1 Trapping charged particles . . . . . . . . . . . . . . . . . . . . 9
 2.1.2 Dynamics in the Paul trap . . . . . . . . . . . . . . . . . . . . 12
 2.2 Trapping Yb+ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
 2.2.1 The 171Yb+ hyper ne qubit . . . . . . . . . . . . . . . . . . . 29
 2.2.2 State initialization and detection . . . . . . . . . . . . . . . . 30
 2.2.3 Qubit rotations with resonant microwaves . . . . . . . . . . . 39
 2.3 Multiple Ions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
 3 Control through Raman transitions 50
 3.1 Stimulated Raman transitions driven by an optical frequency comb . 50
 3.1.1 Raman Transitions in a three-level atom using a single pulse
 train . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
 3.1.2 Control of the 171Yb+ clock qubit . . . . . . . . . . . . . . . . 63
 3.1.3 Control of the qubit-oscillator system . . . . . . . . . . . . . . 66
 3.1.4 Error Suppression in Raman Transitions . . . . . . . . . . . . 83
 3.2 Light Shifts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
 4 Phonon mediated entanglement of ions 98
 4.1 Spin-Dependent Forces . . . . . . . . . . . . . . . . . . . . . . . . . . 98
 4.2 Two Atom Entanglement . . . . . . . . . . . . . . . . . . . . . . . . . 105
 4.2.1 Experimental Implementation of the M lmer-S rensen Gate . 108
 4.3 Coherent Error Suppression in M lmer-S renson Operations . . . . . 117
 4.4 Two Ion Walsh Gates . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
 5 Photon mediated entanglement of ions 132
 5.1 Entangling atoms by detecting a single photon . . . . . . . . . . . . . 133
 5.2 Entangling atoms through the detection of two photons . . . . . . . . 143
 5.2.1 Errors in the Type II protocol . . . . . . . . . . . . . . . . . . 153
 5.3 Bell?s theorem and private random numbers . . . . . . . . . . . . . . 156
 6 Integrating Phonon and Photon based entangling operations 164
 6.1 The Quantum Repeater . . . . . . . . . . . . . . . . . . . . . . . . . 164
 6.1.1 2 + 1 Quantum Repeater Elementary Unit . . . . . . . . . . . 167
 6.1.2 2 + 1 Experiment . . . . . . . . . . . . . . . . . . . . . . . . . 168
 6.2 Conclusions and Outlook . . . . . . . . . . . . . . . . . . . . . . . . . 178
 v
A Paraboloidal Ion Trap 181
 B Probability amplitude for Raman transitions in a three-level atom using an
 optical frequency comb 187
 C Walsh function Identity 191
 Bibliography 194
 vi
List of Tables
 2.1 Resonant transition frequencies for neutral Ytterbium isotopes . . . . 20
 2.2 Relevant parameters for 171Yb+ . . . . . . . . . . . . . . . . . . . . . 49
 3.1 Rabi frequencies for di erent pulse shapes . . . . . . . . . . . . . . . 57
 5.1 Photonic gate experiment results . . . . . . . . . . . . . . . . . . . . 151
 5.2 Bell inequality experiment results . . . . . . . . . . . . . . . . . . . . 161
 vii
List of Figures
 1.1 Quantum version of Moore?s law . . . . . . . . . . . . . . . . . . . . . 2
 1.2 Concept drawing of an ion-photon network . . . . . . . . . . . . . . . 4
 2.1 Schematic of the four rod trap . . . . . . . . . . . . . . . . . . . . . . 11
 2.2 Comparison of a numerical integration of the Matheiu equation with
 an approximate analytic solution . . . . . . . . . . . . . . . . . . . . 15
 2.3 RF resonator can . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
 2.4 Neutral Ytterbium Energy Diagram . . . . . . . . . . . . . . . . . . . 19
 2.5 Ionized Ytterbium Energy Diagram . . . . . . . . . . . . . . . . . . . 22
 2.6 Doppler-free saturated absorption laser lock . . . . . . . . . . . . . . 24
 2.7 Five-ion crystal ICCD camera image . . . . . . . . . . . . . . . . . . 27
 2.8 Axial trap frequency as a function of endcap voltages . . . . . . . . . 29
 2.9 Asymptotic optical pumping e ciencies. . . . . . . . . . . . . . . . . 33
 2.10 Single ion state detection . . . . . . . . . . . . . . . . . . . . . . . . . 34
 2.11 Cooling a PMT to suppress dark counts. . . . . . . . . . . . . . . . . 34
 2.12 Diagrams showing which laser frequencies are present for Doppler
 cooling, optical pumping and state detection protocols . . . . . . . . 35
 2.13 Bayesian inference state detection with multiple ions and multiple
 detectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
 2.14 Single ion rotations using resonant microwaves. . . . . . . . . . . . . 40
 2.15 Single ion microwave Ramsey experiment . . . . . . . . . . . . . . . . 43
 2.16 Two ion motional mode illustration . . . . . . . . . . . . . . . . . . . 47
 2.17 Motional mode structure for 10 ions . . . . . . . . . . . . . . . . . . . 48
 3.1 An illustration of a Raman transition . . . . . . . . . . . . . . . . . . 51
 3.2 An energy diagram illustrating a stimulated Raman transition driven
 by an optical frequency comb . . . . . . . . . . . . . . . . . . . . . . 58
 3.3 Driving Rabi oscillations using a single pulse train . . . . . . . . . . . 60
 3.4 The relevant Clebsch-Gordan coe cients for Raman transitions in
 171Yb+ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
 3.5 Co-propagating Raman beams setup . . . . . . . . . . . . . . . . . . 68
 3.6 Carrier transition line shape . . . . . . . . . . . . . . . . . . . . . . . 69
 3.7 Driving Raman transitions with counter-propagating beams . . . . . 70
 3.8 An illustration of the resonance condition for Raman transitions using
 frequency shifted frequency combs . . . . . . . . . . . . . . . . . . . . 75
 3.9 Using Raman transitions for an autocorrelation measurement . . . . . 77
 3.10 Raman spectroscopy of the qubit-oscillator system . . . . . . . . . . . 84
 3.11 Ramsey experiments with and without a beat note lock . . . . . . . . 86
 3.12 Noise eater data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
 3.13 Suppressing errors due to laser intensity  uctuations . . . . . . . . . . 88
 3.14 Sideband cooling measurement . . . . . . . . . . . . . . . . . . . . . . 90
 3.15 Stark shift to Rabi frequency measurement with the femtosecond laser 95
 3.16 Intensity autocorrelation for the femtosecond laser . . . . . . . . . . . 96
 viii
4.1 Illustration of a spin-dependent force acting on a single ion . . . . . . 103
 4.2 Spin-dependent force experiments with a single ion . . . . . . . . . . 106
 4.3 Spin-dependent force experiments with a single ion . . . . . . . . . . 108
 4.4 Symmetric detuning scan for calibrating the M lmer-S rensen Gate . 112
 4.5 Time scan for calibrating the M lmer-S rensen Gate . . . . . . . . . 113
 4.6 Parity measurement of two entangled ions . . . . . . . . . . . . . . . 116
 4.7 Phase space picture of detuning errors in the M lmer-S rensen gate . 118
 4.8 Illustration of the error suppression in the M lmer-S rensen gate . . . 119
 4.9 Some Walsh functions . . . . . . . . . . . . . . . . . . . . . . . . . . 123
 4.10 Walsh modulated spin-dependent force operations with a single ion . 125
 4.11 Walsh modulated M lmer S rensen gates with two ions . . . . . . . . 130
 5.1 W-state scattering pattern . . . . . . . . . . . . . . . . . . . . . . . . 141
 5.2 A schematic of the type II experiment . . . . . . . . . . . . . . . . . 145
 5.3 An illustration of the protocol to entangle the clock qubit with pho-
 tonic frequency qubits . . . . . . . . . . . . . . . . . . . . . . . . . . 147
 5.4 Bounds on the randomness generated in a Bell Inequality Experiment 160
 5.5 Bounds on the randomness generated by remote entangled ions . . . . 162
 6.1 Repeater Architecture . . . . . . . . . . . . . . . . . . . . . . . . . . 165
 6.2 Schematic of the blade trap . . . . . . . . . . . . . . . . . . . . . . . 170
 6.3 Flowchart for the 2 + 1 experiment . . . . . . . . . . . . . . . . . . . 171
 6.4 Schematics showing the optical setup for the 2 + 1 experiment . . . . 173
 6.5 Individual addressing data . . . . . . . . . . . . . . . . . . . . . . . . 174
 6.6 Measured populations of the 2 + 1 experiment . . . . . . . . . . . . . 177
 A.1 Paraboloidal ion trap schematic . . . . . . . . . . . . . . . . . . . . . 182
 A.2 Parabolic coordinate system . . . . . . . . . . . . . . . . . . . . . . . 182
 A.3 Graphical analysis of paraboloidal ion trap design parameters . . . . 186
 ix
Chapter 1
 Introduction
 1.1 Ion-photon networks
 The emergence of quantum information science has spurred the development
 of quantum mechanical devices with ever increasing degrees of complexity and con-
 trol. With the long term goal of building a quantum computer, researchers have
 made tremendous progress since the  rst demonstration of an entangling logic gate
 [1] to recent experiments demonstrating \a combination of all of the fundamental
 elements required to perform scalable quantum computing" [2] and small scale quan-
 tum simulations of magnetism [3, 4] and  eld theories [5]. The number of qubits
 that experimentalists have been able to entangle has been obeying a quantum ver-
 sion of Moore?s law, e ectively doubling every 5:7 years as shown in Fig. 1.1 with
 the current record being 14 qubits [6]. A wide variety of quantum information pro-
 cessing platforms are being developed simultaneously including neutral atoms [7],
 electron spins [8], superconductor qubits [9] and optical  elds [10] to name a few and
 techniques developed for quantum information processing have started being used
 in other  elds such as metrology [11], imaging [12], communication networks [13]
 and condensed matter theory [14]. Despite all of the progress going on around the
 world, it is generally thought that fault-tolerant quantum computing is years, if not
 decades away because of the challenges foreseen in scaling the prospective systems
 1
 
 
 
 
 
 
 
1996 1998 2000 2002 2004 2006 2008 2010
 1.0
 10.0
 5.0
 2.0
 3.0
 1.5
 7.0
 Year
 Numbe
 ro
 fEntangle
 d
 Qubit
 s
 Figure 1.1: This log plot shows that number of qubits are able to be faithfully
 entangled in the laboratory is roughly doubling every 5:7 years. This data was
 shared courtesy of [15].
 to large numbers of qubits. The minimum requirements for a universal quantum
 computer have been compiled into the now famous DiVincenzo requirements [16]
 which state that the system needs,
  well characterized qubits which can be easily scaled to large numbers
  the ability to initialize the qubits to a simple  ducial state
  long coherence times compared to the time scale of gate operations
  a universal set of quantum gates
  a qubit-speci c measurement capability
 As will be discussed in this thesis, trapped ions are close to demonstrating all of
 these criteria with outstanding exception being the scalability issue. In order for a
 quantum computer to outperform a classical computer and do something \interest-
 ing," it is generally accepted that at least 1000 logical qubits will be needed. Because
 2
a quantum computer will require error correction [17] in order to  ght decoherence,
 there will be additional overhead and likely require several, if not many, physical
 qubits for every logical qubit. Given the enormity of the task, it will be crucial that
 physicists and engineers think hard about how to optimize both the control and the
 layout of such a device.
 At its most basic level, a quantum computer will require a quantum memory
 for storage and manipulation of information and a quantum data bus for transfer
 of information. The internal atomic states of trapped ions have been shown to pro-
 vide an excellent quantum memory with coherence times of several minutes being
 demonstrated [18] and systems of many trapped ions have demonstrated the use of
 phonons as a natural quantum data bus for entangling operations [19, 1, 20, 21].
 The  rst proposal for trapped ion quantum computing envisioned a large ion crys-
 tal that was controlled through laser interactions with individual ions [19]. This
 approach runs into several complicating issues when scaled to very large numbers
 of ions. One problem with using a single ion crystal of N qubits is the increased
 mass leads to a decrease in the motional coupling parameter  / 1p
 N
 which leads
 to slower gate speeds. Other problems include a decrease in the inter-ion spacing
 resulting in addressability problems and the increased stray excitations from the
 spectral crowding of the motional-mode structure. It might, therefore, be advanta-
 geous to design a complex trapped ion system around a modular design philosophy
 in which the system is made up of separated ion traps of manageable complexity
 that are connected through a quantum channel that can be turned on and o like a
 switch. One possible way to do this is by building an ion trap that consists of many
 3
(a) 
(b) 
Figure 1.2: A concept drawing of a proposed ion-photon network. (a) The ele-
 mentary logic unit (ELU) represents a single trapped ion register where quantum
 information is stored and manipulated through Coulomb force mediated gates. (b)
 Using an optical cross-connect switch, the di erent registers can be connected using
 ion-photon entangling operations along with entanglement swapping measurements
 on the photons.
 trapping regions that can be connected by shuttling individual ions using dynamic
 control voltages [22] and there has been a lot of progress made in recent years to-
 ward realizing this architecture [23, 24]. Another possible way to do this is to use
 a photonic interconnect between the ion registers as depicted in Fig. 1.2 [25]. In
 this modular architecture, the elementary logic units (ELUs) contain trapped ion
 registers where information is stored and manipulated using coherent interactions
 while the interconnections are established through an entanglement swapping appa-
 ratus that involves a measurement of photon pairs that are entangled with respective
 registers.
 A closely related emerging quantum technology is the quantum communication
 network. Not long after the power of quantum computers had been shown to be
 capable of cracking state of the art encryption codes, [26], it was realized that
 4
privacy among communicating parties could be guaranteed by the laws of physics
 if one has access to a quantum communication network [27]. It wasn?t long after
 the community became familiar with DiVincenzo?s criteria that he added two more
 requirements to his list for the construction of a quantum network [16];
  the ability to interconvert stationary and  ying qubits
  the ability to faithfully transmit  ying qubits between speci ed locations
 These additional criteria are not only crucial to quantum communication networks,
 but will likely be needed for distributed quantum computing and for error correc-
 tion which might require communication between di erent pieces of the computer.
 Photons, being the fastest way to transmit information, are one obvious choice for
  ying qubits and researchers are currently studying di erent methods of interfac-
 ing stationary and  ying qubits. In tradition with the conventional circuit model
 of quantum computing [28], much of the e ort is geared toward engineering de-
 terministic interactions between stationary and  ying qubits with the goal being
 the production of a single photon in the desired mode on demand. This approach
 implies that an atom trapped in free space will not su ce and one must resort to
 engineering a strong coupling environment through the interaction with a cavity
 [29, 30, 31], with re ective optics [32] or with high numerical aperture lenses [33].
 Another approach, however, has shown that it possible to establish interconnects
 between free space atoms through a heralded probabilistic photonic channel [34],
 albeit at the cost of lower e ciency.
 5
Besides the di culties involved with interfacing photonic and matter qubits,
 photons will be subject to a host of destructive processes in transit between di er-
 ent quantum memories. For example, any optical  ber used as a quantum channel
 for photons has an attenuation length associated with it that causes the amount of
 transmitted light to be depleted in an exponential fashion. Classical communication
 networks also su er from attenuation of signals and use repeater stations to amplify
 signals before the attenuation leads to any serious deterioration of the signal. A
 quantum communication network?s advantage, however, stems from the inability
 of an eavesdropper to gain information about the transmission without being de-
 tected [35] as enforced by the quantum no-cloning theorem [36] which has the added
 consequence that ampli cation of the signal is not possible. Nonetheless, it is still
 possible to construct a quantum repeater that is capable of faithful transmission of
 quantum information in the presence of attenuation through the use of distributed
 entanglement [37]. As discussed in more detail in Chapter 6, an ion-photon quantum
 repeater can be constructed through a series of memory nodes that are connected
 through photonic channels.
 While all of the necessary components of such an architecture have been ex-
 tensively studied independently, the complexities of merging these technologies are
 only beginning to be understood and dealt with. This thesis aims to show experi-
 mental advances in both the coherent control of individual trapped ion registers and
 their integration with a photonic interconnect channel.
 The thesis is organized as follows;
 6
Chapter 2 begins with a brief review of basic ion trapping physics along with
 a detailed description of the experimental apparatus. The procedure for loading
 Ytterbium ions is detailed along with a description of the relevant atomic structure of
 171Yb+ and basic characteristics of the system. The de nition of the qubit subspace
 is given along with an overview of state initialization, manipulation and detection
 techniques.
 Chapter 3 describes a new technique of controlling trapped ion hyper ne qubits
 using mode locked pulsed lasers. Simple models are developed for the description
 of the interaction between the atomic system and the pulsed laser and experimental
 data showing full control over the system is shown.
 Chapter 4 gives an overview of the technique commonly used to implement
 coherent interactions between neighboring ions. Experimental data is presented
 demonstrating the entanglement of two trapped 171Yb+ ions. An extension of this
 entangling gate that is designed to be more resistant to certain errors is described
 and experimentally demonstrated.
 Chapter 5 is an overview of photon mediated entangling operations amongst
 stationary matter qubits. Two basic types of photonic gates are described along
 with their inherent advantages and disadvantages and some relevant scaling laws. A
 Bell inequality experiment is demonstrated and used as random number generator.
 Chapter 6 presents our progress toward the construction of an ion-photon
 network. Using the coherent entangling operations among neighboring ions in one
 trap along with a photonic connection that is shared with another distant trap, we
 demonstrate a correlation between the qubit state in one register and collective state
 7
of another register.
 8
Chapter 2
 Ytterbium ions
 2.1 The Paul Trap
 Trapped ions have been at the forefront of experimental quantum information
 science since at least 1995 [1] when the  rst demonstration of an entangling gate
 between two qubits took place. The trapped ion platform was well positioned to take
 the lead in experimental quantum information science after having bene ted from
 decades of development by researchers interested in mass spectrometry and atomic
 clocks. The development of ion traps was largely motivated by the long con nement
 times and the extreme isolation of particles that is desired by precision measurement
 experiments. The overlap of these desiderata in precision measurements and the
 search for the perfect qubit is what led to the ion trap being the most successful
 platform in these early days of quantum information science. In this chapter, some
 basic tools of an ion trap laboratory are reviewed.
 2.1.1 Trapping charged particles
 One consequence of Maxwell?s laws of electromagnetism is that static electric
  elds can not produce a stable trap for a charged particle. This statement can be
 derived using the fact that electric  elds are divergenceless in free space, r  ~E = 0,
 as shown by Earnshaw?s theorem. Because the  eld can be written as the gradient
 9
of a conservative  eld that satis es Laplace?s equation, the potential can possess no
 maxima or minima, only saddle points, implying that a  eld that generates a trap
 in one-dimension will necessarily create an anti-trap in another dimension. But
 by considering dynamic  elds, a stable trap can be generated in a con guration
 now referred to a Paul trap [38, 39]. Most experiments described in this thesis
 were performed using a four rod trap like the one shown in Fig. 2.1. In the simplest
 con guration, a trap is formed by applying DC voltages to the needle shaped endcap
 electrodes and an oscillating radio frequency voltage to two opposing rods, with the
 other two rods and the endcap electrodes being RF grounded through shunting
 capacitors. The DC voltages provide a restoring force for a charged particle in the z
 direction, but as dictated by Maxwell?s equation form an anti-trap in the x y plane.
 Near the center of the trap, the  eld can be linearized and the electric potential can
 be approximated by,
  (x; y; z) = V0
  rf
 2
 x2  y2
 R2
 cos T t+ U0
  dc
 R2
  
z2  
1
 2
 ( dcx
 2 + (1  dc)y
 2)
  
; (2.1)
 where V0cos T t is the applied RF voltage, U0 is the DC voltage, R is the distance
 from the electrode to the trapping region and  rf ;  dc;  dc are geometric factors that
 must be found through experiment or simulation. In the ideal case where the RF
 electrodes are hyperbolic and the DC endcaps are far enough away as to be negligible,
 the factor  rf limits to one. We have ignored the small RF  eld in the z direction due
 to the endcaps which allows the motion along the z direction to be approximated
 10
Figure 2.1: A schematic of the four rod trap is shown indicating where the static
 and time-dependent voltages are applied.
 as purely harmonic motion with a secular frequency,
 !z =
 p
 eU0 dc=R2m: (2.2)
 The motion in the x direction is described by the Mathieu equation,
 m
 d2x
 dt2
 = eV0 rf
 x
 R2
 cos T t eU0
  dc
 R2
  dcx; (2.3)
 with the motion along y obeying a similar di erential equation. To understand how
 a force like the one described in Eq. (2.3) creates a stable trap, consider a particle
 interacting with the  eld,
 E(x; t) = E0(x)sin( T t): (2.4)
 For simplicity, assume the form E0(x) = kx and that at t = 0 the particle is located
 x0 > 0. During the  rst half of the  eld cycle, 0 < t < 2 = T , a positively charged
 particle will feel a positive force and accelerate away from the origin. However,
 11
during the second half of the cycle the particle will feel a negative force and accelerate
 toward the origin. Since the gradient of E0(x) is positive, the particle feels a larger
 impulse in the second half of the cycle and therefore acquires a net negative change
 in momentum over the entire cycle. This e ectively creates a restoring force in the
 x direction. This idea is implemented in a variety of designs in di erent ion traps
 around the world.
 2.1.2 Dynamics in the Paul trap
 In this section, I will review the classical version of this problem, but will omit
 many of the mathematical details as they are extensively covered in many other
 texts [40]. Since the potential in Eq. (2.1) is separable in x; y and z, and z is well
 approximated by purely harmonic motion, the rest of the picture can be  lled in by
 considering the e ect of the  eld in the x direction. Eq. (2.3) can be rewritten in
 dimensionless form as,
 d2x
 d 2
 + (a+ 2qcos2 )x = 0; (2.5)
 where we have de ned the following parameters;  =  T t=2; a =
 2eU0 dc dc
 m 2TR
 2 and q =
 2eV0 rf
 m 2TR
 2 . Solutions to the Matheiu equation can be found using Floquet theory, which
 is familiar to most physicists in the form of Bloch?s theorem concerning solutions
 to the time-independent Schr odinger equation with a periodic potential. Floquet?s
 theorem states that the di erential equation (2.5) has solutions of the form, x( ) =
  ( )ei  where  ( +  ) =  ( ) and  is a constant. In analogy with the solutions
 to Bloch?s problem, the solution is the product of a function  that has the same
 12
periodicity as the drive function and time dependent phase with the quantity  
sometimes referred to as the quasi-energy. The general solution to Eq. (2.5) is
 expressed as,
 x( ) = e i   ( ): (2.6)
 Because  ( ) is periodic, the function admits a Fourier decomposition,
  ( ) =
 1X
 n= 1
 cne
 in : (2.7)
 Plugging Eq. (2.7) into the di erential equation (2.5) and using the orthogonality
 of the di erent Fourier components yields an expression for  ,
 (  n)2 =
 a+ q (cn+2 + cn 2)
 cn
 : (2.8)
 The parameter a comes from the static voltages in the z direction and in our case
 is quite small compared to the e ect of the RF voltages. If a is assumed to be
 small compared to q, the expression in Eq. (2.8) becomes  2 = q
  
c2
 c0
 + c 2c0
  
. The
 fractions c2c0 and
 c 2
 c0
 can be found to lowest order by rewriting Eq. (2.8) as,
 cn
 cn+2
 =
 1
 (  n)2
 q  
cn 2
 cn
 =
 1
 (  n)2
 q  
 
1
 (  n+2)2
 q
  cn 4cn 2
  : (2.9)
 The second equality is reached by using the expression for cncn+2 with the substitutions
 n! n 2 and n 2! n 4. This process can be continued to yield an expression
 in terms of a continued fraction, and similar expressions can be derived for cncn 2 ,
 13
which to lowest order yield,
 cn
 cn 2
  
q
 (  n)2
 : (2.10)
 Putting it all together implies,
   
 
q
  
c2
 c0
 +
 c 2
 c0
   1=2
 (2.11)
  
 
q
  
q
 (  2)2
 +
 q
 (  2)2
   1=2
 (2.12)
  
q
 p
 2
 : (2.13)
 An approximate solution can then be written down by ignoring Fourier components
 where jnj > 2 and using c2  c0
 q
 2 to write,
 x( )  x0e
 i(  + )
  
1 +
 q
 2
 cos2 
 
: (2.14)
 We see that the motion is comprised of pieces, one that describes an oscillation at
 the frequency !s =   =t =
   T
 2 with an amplitude x0 and another that describes
 an oscillation at the frequency ! = 2 =t =  T with an amplitude x0
 q
 2 . Remem-
 bering that both   1 and q  1 means that the motion can be thought of as
 harmonic motion together with small fast oscillations at the RF drive frequency  T
 as illustrated in Fig. 2.2.
 A complimentary analysis [38] that considers the force on the charged particle
 averaged over the time 2 = T , leads to the following expression for the average force
 14
(a)
 (b)
 Figure 2.2: The approximate solution to the Matheiu equation (2.14), (red), is
 compared to a numerical integration of Eq. (2.3), (blue). In  gure (a) the value of
 a = 0 and q = 0:1 and in  gure (b) the values are a = 0 and q = 0:3.
 15
felt by the particle,
  F =  
e2
 4m 2T
 @
 @x
 E20(x): (2.15)
 This implies that the particle exhibits nearly harmonic motion when it is trapped in
 a region where the  eld E0(x) is well described by a linear function. This analysis
 provides a useful picture in which the ion is trapped by a potential well that is
 proportional to the square of the electric  eld, with the proportionality factor being
 determined by the charge and mass of the particle and the frequency of the RF
 drive. By assuming the potential in Eq. (2.1) and that the geometric factor is
 approximately 1, the transverse secular frequency of Yb+ is,
 !x
 2 
 10MHz
 V0(kV)
  T=2 (MHz)R2(mm)
 (2.16)
 !(4 rod)x
 2 
 48MHz
 V0(kV)
  T=2 (MHz)
 ; (2.17)
 with R  0:46 mm in the four-rod trap. The expression in Eq. (2.17) is not always
 valid since the Matheiu equation solutions are only stable in limited regimes, with
 the case of a = 0 yielding stable orbits when 0 < q < qmax = 0:93 [39]. Using Eq.
 (2.15), one  nds that the secular frequency is related to the RF drive frequency as,
 !x =
  T
 2
 p
 2
 q; (2.18)
 meaning that the secular frequency must be approximately 2
 p
 2 times smaller than
 the RF drive frequency in order to create a stable trap. If the voltage applied to the
 electrodes is limited to 1 kV, this restriction, along with Eq. (2.17), dictates that
 16
the RF drive frequency must be no less than  10 MHz. In the lab, the RF potential
 is typically generated using drive frequencies of 10 40 MHz with the large voltage
 being generated through the use a helical resonator can [41] like the one shown in
  g. 2.3. The voltage on the trap electrodes is given by V0 =  
p
 PQ, where   20
 is determined by the geometry of the helical resonator, P is the input RF power
 and Q is the quality factor of the resonator. The trap was successfully loaded using
 a can with a resonance frequency of 35 MHz, Q = 270 and driven with P = 9 W,
 resulting a trapping frequency of 1:3 MHz. A second can was constructed in hopes
 of achieving a higher trapping frequency by using a lower RF drive frequency of 12
 MHz, but it was found that the trap was not stable for more than one ion unless
 the RF input power was reduced to P = 2 W, resulting in a trapping frequency
 of 1:8 MHz. The trap is made complete by the application of the DC voltages to
 the needles, which is typically on the order of 10  600 V. These parameters yield
 trapping frequencies on the order of 1  3 MHz in the x  y plane and 0:01  1
 MHz in the z direction. Numerical simulations of the trapping environment reveal
 that the depth of the four-rod trap is on the order of 10 eV, or 105 K [42]. This ion
 trap resides in an ultra-high vacuum environment whose pressure is measured to be
  10 11 Torr by an ion gauge and has been observed to hold a single ion for several
 weeks.
 17
Figure 2.3: A helical resonator can is shown schematically to illustrate the inductive
 coupling between the RF input antenna and the trap electrodes.
 2.2 Trapping Yb+
 To load the trap, an atomic oven source is placed a few centimeters away from
 the center of the trap and resistively heated for 1  5 minutes. Neutral Ytterbium
 is ionized in a two-photon process in which an atom that is initially in the ground
 1S0 state is excited to the 1P1 state and then to a state where one of the valence
 electrons becomes unbound. The 1S0 ,1 P1 transition has a resonance frequency
 of c=(398:9nm) and the transition from the excited 1P1 state to the continuum
 requires a photon with an energy of at least ch=(394:1nm). As will be discussed
 later, the main dipole transition used in the Yb+ system has a resonant frequency
 of c=(369:5nm) which corresponds to a photon energy that is more than enough to
 free the valence electron. The photoionization procedure is therefore achieved by
 focusing two laser beams, 398:9 nm and 369:5 nm, in the center of the trap. Since
 the atomic oven generates a thermal cloud of atoms streaming toward the center of
 the trap, it important to take into account Doppler shifts seen by the atom if isotope
 selectivity is desired. The chamber holds two atomic oven sources, one being packed
 18
Figure 2.4: The relevant energy level structure for ionizing neutral Ytterbium. Yt-
 terbium is ionized in a two-photon process with the  rst photon resonantly driving
 the 1S0 ,1 P1 and the second photon liberating the electron from the atom.
 with all the di erent isotopes of Ytterbium in their natural abundance ratios and
 the other being a puri ed source with higher than 90% being 171Yb. A list showing
 the resonances of the 1S0 ,1 P1 for the di erent Ytterbium isotopes is shown in
 table 2.1.
 The ions are imaged using a NA = 0:23 triplet lens that has a working distance
 of 13 mm. The lens is placed  3 mm from the vacuum chamber window which is
 3:3 mm thick and whose vacuum side surface is 8:5 mm from the center of the ion
 trap. The triplet lens stack generates an image plane  300 mm from the ion where
 19
Isotope Shift from 174Yb (MHz) absolute  in nm
 176Yb -509.310(50) 398.91190
 173Yb(F = 5=2) -253.418(50) 398.91177
 173Yb(F = 3=2) 515.975(200) 398.91136
 172Yb 533.309(53) 398.91135
 173Yb(F = 7=2) 587.986(56) 398.91132
 171Yb(F = 3=2) 832.436(50) 398.91119
 171Yb(F = 1=2) 1153.696(61) 398.91102
 170Yb 1192.393(66) 398.91100
 168Yb 1887.400(50) 398.91063
 173Yb(centroid) 291.516(54) 398.91147
 171Yb(centroid) 939.523(39) 398.91113
 Table 2.1: This shows the resonant frequencies for the 1S0,1P1 transition for the
 di erent neutral Ytterbium isotopes as taken from [43]. The resonance frequency
 for 174Yb is 751; 525; 987:761(60) MHz or 398:91163 nm.
 20
a 500  m pinhole is placed to reduce the background light created by light scattered
 from the trap electrodes. After the pinhole, a doublet lens stack refocuses the image
 onto either a photo-multiplier tube (PMT) or an intensi ed CCD camera (ICCD)
 depending on the position of  ip mirror. The PMTs used in these experiments
 are Hammamatsu PMTs with quantum e ciencies ranging from 0:2 0:4 and dark
 count rates on the order of 10 Hz and the camera is the Princeton Instruments
 PI-MAX ICCD. The camera is only used for imaging and trouble-shooting and not
 for quantitative measurements as the read out time and noise characteristics are
 inferior to that of the PMT. In addition to the pinhole, interference  lters designed
 to pass 369 nm light made by Semrock inc. are placed in front of the PMT and
 camera.
 The ion is localized by being Doppler cooled on the 2S1=2,2P1=2 transition.
 The D1 transition is not a closed cycling transition since the atom has a 0:5% chance
 of decaying to the low lying 2D3=2 state from the 2P1=2 state. Because of the relatively
 long lifetime of the 2D3=2 state, 53 ms, a second laser is used to depopulate this
 state through the 3 [3=2]1=2 state with resonant 935:2 nm light.
  The combination
 of the 369:5 nm and 935:2 nm light results in a closed bow-tie con guration cycling
 transition as depicted in  g. 2.5. An added complication comes from the presence
  The bracket notation used for the 3 [3=2]1=2 state is a reference to an angular momentum
 coupling scheme that di ers from the usual Russell-Saunders (L-S) coupling. When the atom is
 excited to the 3 [3=2]1=2 state, a second electron is excited to the valence shell from the f shell,
 resulting in a hole that carries angular momentum. The L-S coupled core has a total angular
 momentum Jc which is then coupled to the angular momentum of the two valence electrons to give
 the angular momentum K = L+ Jc, which is the value contained in the brackets. The spin of the
 two valence electrons are coupled together and denoted by the multiplicity superscript. Finally
 the spin is coupled to K to give the total angular momentum J = K +S, which is denoted by the
 subscript [44].
 21
Figure 2.5: The energy level structure for Yb+.
 22
of the low lying 2F7=2 state that has a lifetime of approximately 6 years [45]. When
 the ion is being cooled, it is possible for the atom to make the transition to the 2F7=2
 state through collisions with the background gas, which is observed to occur on the
 timescale of tens of minutes. For this reason, an addition laser tuned near 638:6 nm
 is directed toward the ion to depopulate the 2F7=2 state through 1 [5=2]5=2 which has
 a decay channel to 2D3=2.
 This cooling scheme is most easily implemented using a Yb+ isotope that has
 no nuclear spin and, therefore, a simple energy level structure. In this case, both the
 2S1=2 and 2P1=2 manifolds consist of only two Zeeman levels split by  5 MHz by the
 presence of a small magnetic  eld of  4 G. In the case of 171Yb+, which has a spin
 1=2 nucleus, the hyper ne structure of the
 n
 2S1=2;2 P1=2;2 D3=2;3 [3=2]1=2
 o
 manifolds
 must be considered to avoid optically pumping the ion into a dark state. The 171Yb+
 ion is cooled with light that is nearly resonant with the 2S1=2jF = 1i,2P1=2jF =
 0i transition, but o -resonant scattering from the F = 1 manifold in the excited
 state will eventually pump the atom to the F = 0 ground state. To clean out the
 population in the F = 0 state, the 369:5 nm light is phase modulated with an EOM
 at 7:35 GHz so that the second sideband at 14:4 GHz will be nearly resonant with
 the 2S1=2jF = 0i,2P1=2jF = 1i transition. Likewise, the 935:2 nm light is also
 phase modulated at 3:07 GHZ, which is the sum of the hyper ne splittings of the
 2D3=2 and 3 [3=2]1=2 states. The cooling beam is positioned so as to have a projection
 on all three principal axes of the trap so that the ion is cooled along all directions
 and a biasing DC voltage is applied to two of the electrodes to ensure that there are
 no degeneracies in the trapping frequencies.
 23
?/4PBS cavity
 rf-PD
 739.05 nm
 Bias T
 120 MHz
 PI
 D
 to di
 od
 e
 cu
 rr
 en
 t
 error sig
 to cavity piezo
 PBS
 ?/2
 25 mW
 100 kHz
 low passPI
 D
 to g
 ra
 tin
 g
 -10 d
 B
 iodine
 setup
 (b)
 to frequency doubler
 and experiment
 10 MHz
 low pass
 (a)
 lock-in
 amp
 VCO PID
 to cavity
 piezo
 in (a)
 iodine cell
 photo-
 receiverAOM
 fiber EOM
 25 mW
 739.05 nm
 from (a)
 probe
 ref.
 cold finger
 5 kHz
 200 MHz
 pum
 p
 probe
 referen
 ce
 10 dBm
 10 kHz - 
20 GHz
 (b)
 Figure 2.6: (a) The laser is locked to an external cavity using a Pound-Drever-Hall
 scheme [46]. (b) The cavity is locked to the Doppler free saturated absorption signal
 inside an iodine cell.
 24
Because of the small branching ratios in the bow-tie cycling transition, the
 maximum scatter rate is well approximated by neglecting the time the atom spends
 in the 2D3=2 and 3 [3=2]1=2 states. The maximum scattering rate of
 171Yb+ is di-
 minished due to the presence of coherent dark states. Populating the coherent dark
 states is avoided by the application of the small magnetic  eld which has the ad-
 verse e ect of making the cooling laser slightly o resonant for the   transitions
 in the D1 line. The optimization of this e ect versus the desire to avoid population
 trapping in the coherent dark states results in a reduction of the scatter rate by
 about a factor of 3.
 The minimum temperature of the Yb+ ion is found by considering the steady
 state solution of the optical Bloch equations and the theory of Doppler cooling.
 The optical Bloch equations can be derived from a master equation describing the
 coherent interaction of a two-level atom with a monochromatic light source and
 e ects of spontaneous emission,
 d 
dt
 =  
i
 ~
 [H;  ] 
 
2
  
 +   +   +  
 
+      
+; (2.19)
 where H includes the unperturbed two-level atom Hamiltonian and the interaction
 with the laser and  is the spontaneous emission rate. If the Rabi frequency of
 the dipole transition is  and the detuning of the laser from resonance is  , the
 steady-state solution of Eq. (2.19) shows that the population in the excited state is
 given by,
  ee =
 s0=2
 1 + s0 + (2 = )
 2 ; (2.20)
 25
where the on-resonance saturation parameter s0  2 2= 2. Because the scattering
 rate is proportional to the population of the excited state, the scatter rate is,
  p =   ee: (2.21)
 In the case of a traveling wave, the optical Bloch equations can be modi ed to
 estimate the optical force on a atom moving with velocity v,
 F = ~k
 s =2
 1 + s
  
1 +
 2 vk
 (1 + s) ( 2 +  2=4)
  
 F0   v; (2.22)
 where the generalized saturation parameter s  s0
 1+(2 = )2
 and the damping coe cient
 for the velocity dependent part of the optical force is,
  =  ~k2
 4s0( = )
 (1 + s0 + (2 = )2)
 2 : (2.23)
 The maximum damping coe cient is achieved when  =   =2 and s0 = 2, which
 yields  max = ~k2=4. The constant force F0 in Eq. (2.22) can be removed by
 considering the e ect of an externally applied trapping potential and in the case of
 low intensities, s  1, one can estimate a minimum temperature achievable using
 Doppler cooling. By  rst de ning the recoil energy, Er, as the amount of kinetic
 energy an atom at rest would gain through spontaneous emission as,
 Er =
 ~2k2
 2M
 = ~!r; (2.24)
 26
Figure 2.7: An image taken with an intensi ed CCD camera showing a 5 ion crystal
 being Doppler cooled. The ion spacing is approximately 10  m, showing that our
 imaging resolution is on the order of 1  m.
 the average frequency of each absorbed photon is !abs = !a+!r whereas the average
 frequency of each emitted photon is !abs = !a  !r. The atom, therefore, gains an
 average energy of 2~!r from each scattering induced recoil, which occurs at a rate  p.
 By equating the cooling rate ~F  ~v with the heating rate  p2~!r, one  nds that the
 minimum kinetic energy is achieved when  =   =2 and a minimum temperature
 TD = ~ =2kB. For Yb, the maximum scatter rate and minimum temperature are
  =2  61 MHz and 470  K. Once the ion is cooled to Doppler temperature they can
 easily be seen on the camera as shown in  g. 2.7. The localization of the particle can
 be estimated as
 p
 hx2i for a harmonic oscillator obeying a Boltzmann distribution.
 27
For the Hamiltonian H = ~!(aya+ 1=2), the partition function is,
 Z =
 1X
 n=0
 e ~!(n+1=2)=kBT = e  ~!=2
 1
 1 e  ~!
 : (2.25)
 The density matrix is then used to calculate hx2i,
  
x2
  
= Tr[x^2 ] (2.26)
 =
 1
 Z
 ~
 2m!
 1X
 n=0
 e  ~!(n+1=2) hnj (a+ ay)2 jni (2.27)
 =
 2
 Z
 ~
 2m!
 e  ~!=2(1 + e  ~!)
 2(1 e  ~!)2
 (2.28)
 = x20coth( ~!=2); (2.29)
 where x0 =
 p
 ~=2m! is the characteristic length scale and is equal to the spread in
 the ground state wavefunction. Substituting the Doppler temperature into the last
 equation, we can estimate the localization of a trapped ion,
  x =
 r
 ~
 2m!
 coth(!= ): (2.30)
 The experiments described in this text were all done with trap frequencies ranging
 from 300kHz to 4MHz which corresponds to a localization ranging from 81nm to
 6nm, a range which beyond the precision of our imaging system (1  m).
 Once an ion is trapped and laser cooled so as to be su ciently localized for
 observation on a camera, the secular frequency of the trap can be measured directly.
 One method of measuring this quantity is to apply a relatively small oscillating
 28
 
 
 
 
 
 
60 80 100 120 140
 Needle Voltage HVL250
 300
 350
 400
 Axial Frequency HkHzL
 Figure 2.8: A measurement of the axial frequency as a function of the DC voltage U0
 that is applied the endcap needles. The function  t to the data is  axial = m
 p
 U0 +b,
 where the  t parameters are m = 31:8kHz=V1=2 and b = 20:0kHz. The small
 constant o set is likely due to the weak RF trap that exists in the axial direction
 due to the endcap needles being RF grounded. Note the error bars for the data
 points are small enough to be obscured by the size of the points.
 voltage, (1-10 V), to one of the trap electrodes while observing the response of
 the ion on the camera. If the frequency of this applied voltage is brought into
 resonance with the secular motion, the ion will be parametrically driven leading
 to an observable delocalization of the ion. The resonance frequency?s dependence
 on the applied DC voltages is observed to scale like the square root of the needle
 voltage,  g. 2.8, as is to be expected as shown by Eq. (2.2).
 2.2.1 The 171Yb+ hyper ne qubit
 We have so far limited our discussion of the control of the Ytterbium ion to that
 of its motional degrees of freedom and its electronic structure. The 171Yb+ isotope
 possesses a spin 1=2 nucleus and, therefore, hyper ne structure within the electronic
 structure. In particular, 2S1=2 jF = 0;mf = 0i  j0i and 2S1=2 jF = 1;mf = 0i  j1i
 29
form an especially attractive qubit. As we now begin to discuss in detail, this system
 has been shown to ful ll all of DiVincenzo?s criteria for a suitable qubit.
 The mf = 0 states in the 2S1=2 manifold are primarily chosen for the exception-
 ally long coherence time that is observed. The frequency splitting for these two states
 has been measured to be  0 = 12:642812118466 +  2z GHz, where  2z = (310:8)B2
 Hz with B in Gauss [18].
 2.2.2 State initialization and detection
 The Ytterbium ion lends itself to simple state initialization and detection
 protocols as described in [47]. State initialization is achieved through an optical
 pumping procedure that prepares the ion in j0i. This is done by illuminating the
 ion with laser light that is resonant with 2S1=2jF = 1i $2 P1=2jF = 1i transition. If
 the polarization of the laser has  ^;  ^+ and  ^ components, the ion will eventually
 by pumped to j0i since every state in the 2P1=2jF = 1i manifold decays to j0i with
 a probability of 1=3. This procedure is ultimately limited by an o -resonant process
 that happens when an ion in j0i absorbs a photon from the pumping beam and
 then falls into the 2S1=2jF = 1i manifold. The hyper ne splitting of the 2S1=2 results
 in the optical pumping beam being 12:6 GHz detuned from the resonance of this
 process. An estimate of the theoretical limit of the state initialization protocol can
 be made by looking at the steady state solution of a simple rate equation for the
 population of the bright state. If we consider all of the population in 2S1=2jF = 1i
 to be represented by P1(t) and the population of the 2S1=2jF = 0i state to be P0(t),
 30
then the state initialization protocol can be modeled with the following di erential
 equation,
 _P1(t) =   1;0P1(t) +  0;1P0(t); (2.31)
 where  i;j is the pumping rate from Pi to Pj. If the laser is tuned to be on resonance
 with the 2S1=2jF = 1i  !2P1=2jF = 1i transition, then the pumping rates can be
 estimated to be,
  1;0 =
 1
 3
  
2
 s
 1 + s
 (2.32)
  0;1 =
 2
 3
  
2
 s
 1 + s+ 4
  !hf
  
 2 ; (2.33)
 where the factors of 1=3 and 2=3 account for the Clebsch-Gordan coe cients. Eq.
 (2.31) can be simpli ed by using the normalization condition P0 + P1 = 1. If the
 di erential equation is then recast in terms of the dimensionless time  =  0;1t and
 the ratio of the pumping rates R   1;0= 0;1, and the ion is assumed to be initially
 in the 2S1=2jF = 1i manifold, the solution to Eq. (2.31) is,
 P1( ) =
 1
 1 +R
  
1 +Re (1+R) 
 
: (2.34)
 This leads to the steady state solution,
 P1(1) =
 1
 1 +R
 : (2.35)
 31
Because of power broadening, the optimal state preparation is in the limit s ! 0,
 giving a minimum of lims!0 P1(1) = 2
 3+4(!hf= )
 2 . The actual 171Yb
 + hyper ne split-
 ting and D1 linewidth give an estimate of  10 6 for the minimum state preparation
 error. Fig. 2.9 shows how the state preparation error scales with the saturation pa-
 rameter and how long the optical pumping light is on.
 The qubit state is read out using a state-dependent  uorescent technique which
 takes advantage of the fact that 2S1=2jF = 0i$2P1=2jF = 0i is a forbidden tran-
 sition. Therefore, light that is resonant with the 2S1=2jF = 1i$2P1=2jF = 0i will
 cause an ion in j1i to scatter many photons, while an ion in j0i will scatter very
 few photons as shown in  g. 2.10. The large di erence in the number of photons
 scattered means that a simple discriminator method can be used for state detec-
 tion. Because of a  nite background, we ascribe the events where 0 or 1 photons
 are detected to a projection onto j0i and the events where more than 1 photon is
 detected to a projection onto j1i. This protocol is limited by the o -resonant process
 that populates the 2P1=2jF = 1i manifold which can decay into j0i. See [48] for a
 detailed description of the limits of this protocol. Another source of noise in this
 protocol comes from background light and dark counts on the PMT. Background
 light is minimized by using a well focused beam to avoid unwanted scatter o of the
 trap electrodes in combination with an optical  lter from Semrock inc. that blocks
 light that has a larger wavelength than 370nm. Dark counts on the PMT can be
 minimized by cooling the PMT module as shown in  g. 2.12.
 The fast readout of the PMT makes it ideal for state detection of a single ion,
 but unless individual addressing is used, a multi-qubit register requires more than
 32
10-4 0.01 1 100 104 106s
 10-5
 10-4
 0.001
 0.01
 0.1
 1
 1
 1 +W1,0  W0,1
 10 100 1000 104 Gt
 10-5
 10-4
 0.001
 0.01
 0.1
 1
 P1HtL
 Figure 2.9: The  rst plot shows the steady state solution of Eq. (2.31) as a function
 of the saturation parameter. The second plot shows P1(t) for di erent intensities;
 s = 0:01(blue), s = 0:5(red), s = 4(green), s = 5000(black)
 33
Figure 2.10: The histograms show the probability of detecting di erent numbers of
 photons for the two di erent qubit states. If the ion is optically pumped to j0i,
 we usually only detect one or two photons upon shining laser light onto the ion. If
 instead, we apply a  pulse before applying the detection beam, we see an average
 of 10 photons. If we ascribe the events where we detect zero or one photons to a
 projection onto j0i and more than one photon to a projection onto j1i, then our
 state detection  delity is 98%.
  
 
 
 
 
  
 
 
  
   
 
 
 
 
 
  
  
 
 
  
 
 
  
 
   
 
  
 
  
2 4 6 8
 24.3 C-Temp
 2
 4
 6
 8
 10
 Dark Counts per second
 Figure 2.11: The data shows the minimization of dark counts on the PMT being
 used for state detection. As the module is cooled with a thermo-electric cooler by
 about 10 C the dark counts are seen to decrease by about a factor of 4. The PMT
 was operating in the dark and dark counts were recorded for a minute and then
 rounded to the nearest integer to calculate the counts per second. The temperature
 of the module was recorded by attaching a thermocouple to the outside casing. The
 temperature was not taken lower than shown in order to avoid condensation on the
 electronics.
 34
S2 1/2
 P2 1/2
 S2 1/2
 P2 1/2
 S2 1/2
 P2 1/2
 (a) (b) (c)
 Cooling Optical Pumping State Detection
 Figure 2.12: The laser frequencies present for Doppler cooling, optical pumping and
 state detection protocols are shown. (a) The solid lines show the carrier frequency
 of c=(369:5261 nm) and the dashed line shows the carrier plus the sum of the two
 hyper ne splittings, (14:7 GHz), imparted by passing the beam through an EOM.
 (b) For the optical pumping routine, the laser is modulated at 2:1 GHz to couple
 the two F = 1 manifolds. We do not shown the carrier frequency here. (c) The
 state detection protocol used an unmodulated beam.
 one detector for readout. This can be done with either a ICCD camera or an array
 of PMTs, but one must take care to minimize the cross-talk between detectors.
 One way to do this is to use a Bayesian inference method which takes into account
 information from neighboring detectors. For example, if detector A sees one photon
 and the neighboring detector B sees many photons, depending on the amount of
 cross-talk, the photon seen by detector A might be ascribed to the ion that is lined
 up with detector B instead of assuming that ion A emitted a photon. The number
 of photons detected by detector A and B will be labeled da and db. Using Bayes?
 rule, the probability that a certain distribution of detected photons, fda; dbg, was
 produced by a pair of ions in the state j i is,
 P (j ijda; db) =
 P (da; dbjj i)P (j i)
 P
  ; P (da; dbjj  i)P (j  i)
 ; (2.36)
 35
where the sum is over all possible spin states of the two ions. In order to calculate
 P (da; dbjj i) we need to know the probability of a single ion emitting N photons and
 the probability that a single photon emitted by ion A will be detected by detector
 B. We de ne the probability of a single ion in the state j i emitting N photons as
 P ( ;N), the probability of a detected photon being seen by the wrong detector as
 Pw and the probability of a detected photon being seen by the correct detector as
 Pc. If j i = j  i, then,
 P (da; dbjj  i) =
 da+dbX
 j=0
 jX
 i=0
 P ( ; j)P ( ; da+db j)P
 da+2i j
 w P
 db 2i+j
 c
  
j
 i
   
da + db  j
 db + i j
  
:
 (2.37)
 The index j represents how many photons ion A contributed to the total number
 seen and the index i represents how many of those were detected by the detector
 aligned with ion B. Using these de nitions, one can easily see that the  rst binomial
 coe cient comes from the mulitiplicity of the photons emitted from ion A seen by
 detector B, while the second binomial coe cient is the multiplicity of the photons
 from ion B seen by detector A.
 As a numerical test of this detection method, we can calculate the expected
 state detection error for an ideal system of two ions and two detectors. For simplicity,
 we will assume that an ion generates a Poisson distribution in the number of detected
 photons in an ensemble of experiments with the average number of photons for the
 dark state being 0:01 and 10 for the bright state, which resembles the situation in
 for 171Yb+ in the sense that the optimal discriminator choice is to assign j0i to the
 events where 0 or 1 photons are detected and j1i to all other events. By choosing
 36
Poisson distributions, the overlap of the two distributions is smaller than what
 is actually the case and therefore the single ion state detection is more e cient,
 but it should be su cient for demonstrating the power of Bayesian inference in
 suppressing crosstalk errors. For the numerical simulation, we assume that the
 system is perfectly calibrated in the sense that the probability of a photon emitted
 from ion A was detected by the wrong detector is known and equal to the probability
 of a photon from ion B being detected by the wrong detector. As an example, we
 can calculate the likelihood that one dark ion and one bright ion would result in
 n photons being seen by detector A and 10 photons being seen by detector B. As
 shown in  g. 2.13, in the case of 10% detector crosstalk, the Bayesian method tells
 us that unless detector A sees 5 or more photons, the best bet is to say that the
 measurement record was generated by ion A being in the dark state and ion B
 being in the bright state. This particular case would contribute to the overall state
 detection error by 14(P (2; 5jj01i)+P (3; 5jj01i)+P (4; 5jj01i)) = 0:004 with the factor
 of 1=4 coming from there being 4 basis states in the case of two qubits. However,
 this calculation of the error for the discriminator method is an overestimate since
 the optimal cuto for a single ion is not necessarily the same as the optimal cuto 
for two ions in the presence of cross talk. By summing over all possible photon count
 records and picking out the events that result in a state detection error, the optimal
 discriminator can be chosen. In the case of 10% crosstalk, the state detection is
 optimized when j0i is assigned to all events that register less than 4 photons. By
 summing over all possible photon records, the optimized discriminator method is
 found to result in an error rate of 2:5% while the Bayesian inference method results
 37
   
 
 
 
     
2 4 6 8 10 n
 0.2
 0.4
 0.6
 0.8
 1.0
 P H 01? n,5L
 Figure 2.13: A plot of P (j01ijn; 5) as de ned by Eq. (2.36) and Eq. (2.37). This
 calculation assumes Poisson distributions for the individual ion photon distributions
 and that 10% of the photons generated by each ion are detected by the wrong
 detector. As seen in the plot, when detector B registers 5 photons, the best bet for
 the state of the ions is j01i when detector A registers fewer than 5 photons.
 38
in an error rate of 1:6%. See [49, 50, 51] for related methods of state detection using
 spatio and temporal photon distribution information.
 2.2.3 Qubit rotations with resonant microwaves
 Single qubit rotations can easily be achieved through the application of mi-
 crowaves near the resonant frequency 12:6 GHz. If we consider a single qubit in-
 teracting with a monochromatic source of microwaves, the Hamiltonian is given
 by,
 H0 =  
!0
 2
  z +  Bcos(! t+  )(j0i h1j+ j1i h0j); (2.38)
 where the units are chosen such that ~ = 1. By transforming to the interaction
 picture, the Hamiltonian becomes,
 H1 =
  B
 2
 (e i((!0 ! )t+ ) j0i h1j+ ei((!0 ! )t+ ) j1i h0j); (2.39)
 where we have taken advantage of the rotating wave approximation. The dynam-
 ics can now be solved for easily by transforming to yet another frame where the
 Hamiltonian takes the time-independent form,
 H2 =  
!0  ! 
2
  z +
  B
 2
 (cos( ) x  sin( ) y) (2.40)
 39
  
 
  
  
  
 
  
 
 
  
 
 
 
 
 
  
 
 
 
 
     
 
 
 
 
 
  
 
 
 
 
    
 
  
 
 
 
 
 
 
 
 
   
  
 
 
 
 
 
 
 
 
 
 
   
  
 
 
 
 
 
 
 
     
 
 
 
 
 
  
 
 
 
 
0.1 0.2 0.3 0.4 0.5
 tHmsecL
 0.2
 0.4
 0.6
 0.8
 1.0
 PH?? flL
 Figure 2.14: The data shows the evolution of hj "ih" ji as a function of the time
 resonant microwaves are directed toward trapped 171Yb+ after initializing the state
 to j #ih# j.
 De ning   !0  ! ,    B and    cos( ) x  sin( ) y, the rotation axis  r
 and Rabi frequency ~ , is given by,
  r =
   z +    p
  2 +  2
 (2.41)
 ~ =
 p
  2 +  2; (2.42)
 and the evolution operation is then given by,
 U2 = e
  i
 ~ t
 2  r : (2.43)
 The decoherence of the qubit states can be related to  uctuations in the local
 40
magnetic  eld by considering the Hamiltonian H =  !0+! (t)2  z where the con-
 stant frequency !0 = 2  0 and the  uctuating ! (t) is determined by the magnetic
  eld. Since the change in the splitting is proportional to the square of the mag-
 netic  eld, we can estimate the change in the splitting as ! (t)=2 = 310B2 Hz
  310(B20 + 2B0 B(t)) where we have ignored the presumably small term propor-
 tional to  B(t)2. To keep the notation simple, we can add the constant contribution
 from the magnetic  eld to !0 and rede ne ! (t) to only include the time dependent
  uctuations of the magnetic  eld. Because the Hamiltonian describing this situation
 commutes with itself at di erent times, only one term in the Magnus expansion is
 non-zero and the time evolution operator is given by,
 U = e
 i
 2
 R t
 0 dt
 0(!0+ !(t0)) z : (2.44)
 If the qubit is prepared in superposition state j i = (j0i + j1i)=
 p
 2, then the time
 evolved state is given by,
  (t) = U j i h jU y (2.45)
 =
 1
 2
  
j0i h0j+ j1i h1j (2.46)
 +e i(!0t+
 R t
 0 dt
 0 !(t0)) j0i h1j+ ei(!0t+
 R t
 0 dt
 0 !(t0)) j1i h0j
  
(2.47)
 The last expression is only true in the context of a single experiment. In order to
 calculate a density matrix that is useful for describing the state made in the lab,
 we need to calculate an ensemble average. Before taking the ensemble average, the
 41
expression can be simpli ed by going to a frame that rotates at !0 through the
 unitary U0 = ei
 !0
 2  z . In the rotating frame the density matrix is given by,
 ~ =
 1
 2
  
j0i h0j+ j1i h1j+
 D
 e i
 R t
 0 dt
 0 !(t0)
 E
 j0i h1j+
 D
 ei
 R t
 0 dt
 0 !(t0)
 E
 j1i h0j
  
(2.48)
 If the random acquired phase  (t) =
 R t
 0 dt
 0 !(t0) has a Gaussian distribution, then
 the density matrix can be written as,
 ~ =
 1
 2
  
j0i h0j+ j1i h1j+ e 
1
 2h (t)2i (j0i h1j+ j1i h0j)
  
(2.49)
 If the noise is Gaussian, the ensemble average is given by [52],
  
 (t)2
  
=
 Z t
 0
 dt1
 Z t
 0
 dt2 h !(t1) !(t2)i : (2.50)
 In the case of a white noise spectrum, the noise is delta correlated, i.e. h !(t1) !(t2)i =
   (t1 t2), where  is the standard deviation of  ! from zero. This implies that the
 coherence decays exponentially in time with a rate given by the standard deviation
 of  !.
 The coherence time of our clock qubit is measured by performing a Ramsey
 experiment using coherent microwaves. If two near resonant microwave pulses are
 applied sequentially with a time delay between them, the pulses will add construc-
 tively when the time delay is equal to an integer multiple of the inverse of the
 detuning. However, the qubit splitting or the phase of the microwaves are  uctuat-
 ing randomly, this Ramsey oscillation will decay. Fig. 2.15 shows that a microwave
 42
 
 
 
 
 
  
 
 
 
  
 
 
 
 
 
 
 
 
50 100 150 200tH msecL
 0.4
 0.6
 0.8
 1.0
 P H ?? flL
 Figure 2.15: By applying two  =2 microwave pulses near resonance with a variable
 delay t between them reveals a coherence time of approximately 200 msec. This
 coherence time is likely due to phase noise on the microwave source a opposed to
  uctuations in the qubit splitting [47]. The function that is  t to the data assumes
 an exponential decay.
 Ramsey experiment with a single ion reveals a coherence time of approximately 200
 msec. As shown in [47], these measurements are typically limited by phase noise in
 the microwave source which can be eliminated in an experiment that uses two ions
 in separate traps and compares the parity of the two ions to reveal a coherence time
 of 2:5 sec, which is probably limited by di erential magnetic  eld noise.
 2.3 Multiple Ions
 When multiple ions are in the trap, the mutual repulsive Coulomb interactions
 must be considered in the potential energy of the system,
 V (ri) =
 NX
 i=1
 3X
 j=1
 m!2j
 2
 x2j +
 1
 2
 NX
 i;j=1;i 6=j
 e2
 jri  rjj
 : (2.51)
 43
The motion of the collection of particles is commonly described by assuming the
 ions can be cooled down to a temperature such that the ions do not move far from
 the positions that minimize the energy of the system. In that regime, the forces
 felt by the ions can be linearized by expanding the potential about the equilibrium
 positions. If !x;y!z > 0:73N
 0:86, where N is the number of ions in the trap, the ions
 will tend to line up in the z direction when they are cooled to low temperatures
 [53]. For the case of two Yb+ ions, the equilibrium positions are (0; 0; l), where
 we have introduced the length scale l3 = e2=4m!2z . This implies that two ions in a
 trap with a 300 kHz axial frequency will be about 7:7  m apart and about 3:4  m
 apart for a 1 MHz trap. The ions? displacement coordinates from these equilibrium
 positions, ~ri, are de ned as,
 ~R1(t) = (0; 0; l) + ~r1(t) (2.52)
 ~R2(t) = (0; 0; l) + ~r2(t) (2.53)
 ~ri = xix^+ yiy^ + ziz^: (2.54)
 We then linearize the Coulomb interaction by considering the force of repulsion
 between the two ions,
 ~Fc =
 e2
 (~R1  ~R2)3
 (~R1  ~R2) (2.55)
 44
Now we introduce the vector ~D = ~R1  ~R2 = (Dx; Dy; Dz) and expand the force
 components around the point ~D0 = (0; 0; 2l).
 F xc =
 e2
 (D2x +D2y +D2z)3=2
 Dx (2.56)
 =
 e2
 (2l)3
 (x1  x2) + ::::: (2.57)
 F yc =
 e2
 (D2x +D2y +D2z)3=2
 Dy (2.58)
 =
 e2
 (2l)3
 (y1  y2) + ::::: (2.59)
 F zc =
 e2
 (D2x +D2y +D2z)3=2
 Dz (2.60)
 =
 e2
 (2l)2
  2
 e2
 (2l)3
 (z1  z2)::::: (2.61)
 We see that to lowest order the three spatial dimensions are not coupled to each
 other and can be analyzed separately. If we transform to a coordinate system such
 that ~z1(t) =  l+ z1(t) and ~z2(t) = l+ z2(t) so that z1 and z2 are just the excursions
 from the equilibrium points, then we can write Newton?s second law in simple matrix
 form,
 m
 d2
 dt2
 0
 B
 B
 @
 z1
 z2
 1
 C
 C
 A =
 0
 B
 B
 @
  m!2z  
e2
 4l3
 e2
 4l3
 e2
 4l3  m!
 2
 z  
e2
 4l3
 1
 C
 C
 A
 0
 B
 B
 @
 z1
 z2
 1
 C
 C
 A : (2.62)
 The normal modes, bji , can be found by assuming their dynamics are described by
 simple harmonic motion so that they have the form,
 bjz = e
  i!z;jt
 0
 B
 B
 @
  j
  j
 1
 C
 C
 A ; (2.63)
 45
where  j and  j are constants. If this simple time dependence is assumed then Eq.
 (2.62) can be recast as an eigenvalue equation. Remembering that e2=4ml3 = !2z ,
 the eigensystem is found to be,
 b1z =
 e i!ztp
 2
 0
 B
 B
 @
 1
 1
 1
 C
 C
 A b
 2
 z =
 e i
 p
 3!zt
 p
 2
 0
 B
 B
 @
 1
  1
 1
 C
 C
 A : (2.64)
 In general, the solution is described by superposing the  solutions in order to  t
 the initial conditions. The equations for x are given by,
 m
 d2
 dt2
 0
 B
 B
 @
 x1
 x2
 1
 C
 C
 A =
 0
 B
 B
 @
  m!2x +
 e2
 8l3  
e2
 8l3
  e
 2
 8l3  m!
 2
 x +
 e2
 8l3
 1
 C
 C
 A
 0
 B
 B
 @
 x1
 x2
 1
 C
 C
 A ; (2.65)
 and the eigensystem is found to be,
 b1x =
 e i!x;1tp
 2
 0
 B
 B
 @
 1
 1
 1
 C
 C
 A b
 2
 x =
 e i!x;2tp
 2
 0
 B
 B
 @
 1
  1
 1
 C
 C
 A ; (2.66)
 where !x;1 = !x and !x;2 =
 p
 !2x  !2z . The eigenmodes of motion in the y direction
 are identical in form to that of the x direction. The symmetric mode is referred
 to as the center-of-mass mode, while the antisymmetric mode is referred to as the
 breathing mode when describing the axial motion and is referred to as the tilt mode
 when describing the transverse motion. See  g. 2.16 for an illustration of these
 eigenmodes.
 In  nding the general mode structure of a linear chain of N ions, we follow
 46
Transverse Tilt mode
 ?tiltt ? ?x??z
 Axial Center of Mass mode
 ?coma ??z Axial Breathingmode
 ?bra ? 3?z
 Transverse Center of Mass mode
 ?comt ??x
 Figure 2.16: The illustrations show the relative motion of two ions in their normal
 modes of motion along with the characteristic frequencies of those modes.
 the notation used in [54] and start by solving for the equilibrium positions given
 the potential in Eq. (2.51). By assuming that the ions line up along the z direction
 in their ground state, the gradient of the potential with respect to each ion?s z
 coordinate gives N expressions for the net force felt by each ion. By setting the net
 force of each ion to be zero, the set of N coupled algebraic equations that must be
 satis ed is,
 uj  
j 1X
 n=1
 1
 (uj  un)2
 +
 NX
 n=j+1
 1
 (uj  un)2
 = 0; (2.67)
 where uj is the position of the jth ion in units of l = (e2=4m!2z)
 1=3. The normal modes
 of motion can then be found by expanding the potential about these equilibrium
 47
0 20 40 60 80 100
 w Z  w Y
 Figure 2.17: The  gure shows the mode structure for 10 ions with the axial modes
 represented in red and the transverse modes represented in blue. The lowest fre-
 quency mode is the axial center-of-mass mode, while the highest frequency mode is
 the transverse center-of-mass mode.
 positions and  nding the eigenvectors and eigenvalues of the matrix de ned by,
 A n;j =
 (
  2 +
 PN
 p=1;p 6=j a = juj  upj
 3 (n = j)
  a = juj  unj
 3 (n 6= j)
 ; (2.68)
 where   = ! =!z, ax = ay =  1, and az = 2 and the eigenfrequencies being
 ! ;k =
 p
   ;k! where   ;k are the eigenvalues of A . The mode structure for a 10
 ion crystal is shown in Fig. 2.17.
 The description of the ion crystal through the normal modes of motion provides
 an elegant means for understanding how control over a large collection of ions might
 be possible. However, it also presents a challenge in terms of the scalability of ion
 crystals to the large numbers that are needed for quantum information processors.
 As can be seen in Fig. 2.17, the addition of more ions will lead to spectral crowding
 48
that will eventually make addressing single modes of motion di cult from a practical
 point of view. This problem is, in principle, able to be circumvented by using a
 modular architecture as described in the  rst chapter of this thesis.
 We conclude this chapter with a table useful experimental parameters for
 171Yb+ shown in table 2.2.
 Parameter Expression Value
 Doppler temperature ~ 2kb 470  K
 Recoil energy ~k
 2
 4 m 8.5 kHz
 Saturation Intensity  hc 2 3Rbr 51 mW=cm
 2
 Ground state localization
 q
 ~
 2m!
 5 nmp
 !=2 (MHz)
 Thermal state localization
 q
 ~ 
2m!2
 24 nm
 !=2 (MHz)
 Ion spacing 2
  
e2
 4m!2
  1=3
 3:5  m
 (!=2 (MHz))2=3
 Table 2.2: A table of relevant parameters for 171Yb+. Note that the parameter
 Rbr that appears in the saturation intensity is the branching ratio from 2P1=2 to
 2S1=2, which is  0:995. It should also be noted that this formula for the saturation
 intensity assumes a two level atom and ignores the hyper ne and Zeeman structure.
 The formula given for the localization of an ion in a thermal state is derived by
 assuming the Doppler temperature, expanding Eq. (2.29) around  ~! = 0 and
 keeping only the leading term.
 49
Chapter 3
 Control through Raman transitions
 3.1 Stimulated Raman transitions driven by an optical frequency
 comb
 A natural approach toward coherent entangling interactions among neighbor-
 ing ions is to use the strong Coulomb force shared between them. In order to
 generate entanglement between spins using the position dependent Coulomb inter-
 action, one must engineer a control set capable of addressing and entangling the
 spin and motional degrees of freedom. This approach was pioneered in [19] Cirac
 and Zoller in which they proposed using the shared collective motion of the ions as
 a quantum data bus between separated qubits. In this chapter, I will review how
 these forces can be made using stimulated Raman transitions and how they can be
 implemented using a mode-locked pulsed laser. Many of the results reviewed in this
 chapter were presented in [55].
 3.1.1 Raman Transitions in a three-level atom using a single pulse
 train
 As shown in the previous chapter, the qubit state can be controlled through the
 application of the electromagnetic radiation tuned to the resonance frequency of the
 50
2P1/2
 2S1/2
 0 1
 kD
 Figure 3.1: The two hyper ne levels in the ground state can be coupled by two
 phase locked laser beams whose beat note is equal to the qubit energy splitting.
 The momentum transfer in such a transition is proportional to the di erence in the
 two wavevectors  k.
 qubit splitting. The absorption of microwave photons does impart a momentum kick
 to the ion, but the coupling as quanti ed by the Lamb-Dicke parameter,  = kx0 =
 2 
 
q
 ~
 2m! is on the order of 10
  6 for a 1 MHz trap. The coupling to the motional
 degree of freedom can be enhanced by many orders of magnitude by using a two
 photon transition that employs optical photons. In a lambda system, a coupling
 between the two ground states can be engineered by applying two phase locked
 laser beams that have a beat note equal to the ground state energy splitting as
 depicted in  g. 3.1. Because this is a two photon process, the coupling parameter  
is proportional to the di erence in wave vectors of the two photons. If, for example,
 the two beams are counter propagating and have a wavelength that is close to the
 2S1=2$2P1=2 transition, the Lamb-Dicke parameter is on the order of 10 1. High
  delity qubit operations through Raman transitions are typically achieved by either
 phase locking two monochromatic lasers or by modulating a single cw laser with
 and AOM or an EOM. However, the technical demands of phase-locked lasers and
 the limited bandwidths of the modulators hinder their application to experiments.
 51
By using a pulsed laser, we can exploit the large bandwidth to easily bridge large
 frequency gaps to gain control over complex atomic systems.
 In driving transitions between hyper ne levels using optical  elds, one must
 consider the coupling to the excited 2P1=2 and 2P3=2 states. However, most of the
 important physics can be gleaned from a much simpler analysis which only considers
 the coupling to one of these excited states. We start this chapter with a derivation
 of the e ective Hamiltonian that arises from the mutual coupling of the qubit states
 to only one excited state through an interaction with a mode-locked pulsed laser.
 In the regime where a single pulse has a negligible e ect on the ion, we consider
 an ideal train of N pulses that produce an electric  eld at a  xed point in space
 described by,
 E(t) =
 N 1X
 n=0
 f(t nT  t0)e
 i!ct: (3.1)
 We have ignored any carrier-envelope phase shift [56] since, as we will see, this phase
 is irrelevant for a two photon transition. The frequency domain picture is captured
 by the non-unitary Fourier transform of Eq. (3.1).
 F [f(t)] =
 Z 1
  1
 dte i!tf(t) (3.2)
 ~E(!) =
 Z 1
  1
 dte i!t
 N 1X
 n=0
 f(t nT  t0)e
 i!ct (3.3)
 =
 N 1X
 n=0
 e i(! !c)(nT+t0) ~f(!  !c) (3.4)
 In the limit of an in nite number of pulses, a simple picture arises with the help of
 52
the following identity,
 1X
 n= 1
 e in!T = !r
 1X
 k= 1
  (!  k!r) (3.5)
 !r =
 2 
T
 (3.6)
  r =
 1
 T
 ; (3.7)
 where  r is the repeating frequency of the laser. Therefore, the pulsed laser produces
 the following spectrum;
 ~E(0)(!) = lim
 N!1
 ~E(!) = ~f(!  !c)!r
 1X
 k= 1
  (!  !c  k!r) (3.8)
 Eq. (3.8) describes a frequency comb whose envelope function ~f(!) is centered
 around the carrier frequency !c. Note that there is an o set frequency given by
 !cmod!r. The inverse Fourier transform of Eq. (3.8) gives,
 F 1[ ~f(!)] =
 1
 2 
Z 1
  1
 d!ei!t ~f(!) (3.9)
 E(0)(t) =
 1X
 k= 1
 Eke
 i!kte i(!k !c)t0 (3.10)
 Ek  
!r
 2 
~f(!k  !c) (3.11)
 !k  !c + k!r (3.12)
 Eq. (3.10) is just the in nite pulse train written as a Fourier series. This expression
 can now be used to describe the dynamics in a lambda system driven by a long pulse
 train. Consider the Hamiltonian describing a three-level atom interacting with the
 53
pulsed laser,
 i~
 @
 @t
  = H (3.13)
 H = H0 +Hint(t) (3.14)
 H0 = ~
 0
 B
 B
 B
 B
 B
 B
 @
 !1
 !2
 !3
 1
 C
 C
 C
 C
 C
 C
 A
 (3.15)
 Hint(t) =   (r)  E(t) (3.16)
 H = H0  
X
 k
   EkRe
  
ei!kte i(!k !c)t0
  
; (3.17)
 where !1 < !2 << !3 and  (r) is the atomic dipole operator. A complete solution
 to the Schr odinger equation with this Hamiltonian can be studied using multi-mode
 Floquet theory [57], but I will restrict the following analysis to the perturbative
 regime. Since the Raman transitions use the excited P states to establish a coupling
 between the qubit states, it is important to use a far-detuned light source in order
 to avoid building up signi cant population in the unstable electronic level where
 spontaneous emission of a photon can erase the quantum information in the qubit
 and even carry the atom out of the computational states. While detuning the laser
 from resonance results in a weaker coupling between the qubit states, it will be shown
 that the relative strength of this coupling as compared to the rate of spontaneous
 emission grows favorably with larger detunings. In the regime where the optical
  elds are tuned far enough o resonance as to only establish a minuscule population
 in the excited electronic state, the dynamics are accurately described by an e ective
 54
Hamiltonian that couples the qubit states directly. In this section, I will follow the
 method outlined in ref. [58]. For a Hamiltonian that can be written in the following
 way,
 HI =
 MX
 n=1
 h^ne
  i!nt + h^yne
  i!nt; (3.18)
 it can be shown that second order perturbation theory predicts the following e ective
 Hamiltonian,
 Heff =
 MX
 n;m=1
 1
 ~ !m;n
 h
 h^ym; h^n
 i
 ei(!m !n)t (3.19)
 1
  !m;n
 =
 1
 2
  
1
 !m
 +
 1
 !n
  
: (3.20)
 Using the Hamiltonian describing the interaction between the frequency comb
 and the three level atomic system, the commutators in Eq. (3.19) generate four
 distinct operators, j3ih3j j1ih1j, j3ih3j j2ih2j,j2ih1j and j1ih2j. The  rst and second
 operators describe Stark shifts while the third and fourth describe the coupling
 between the qubit states. If we ignore the Stark shift terms for now, the e ective
 Hamiltonian is,
 Heff =  
 2
 4
 M=2X
 m;n=1
 1
 ~ !m;n
 ~Em ~En(j2ih1je
 i(!m !n)te i(m n)!rt0 (3.21)
 + j1ih2je i(!m !n)tei(m n)!rt0) (3.22)
 !m =   !r(k0  m+ 1) (3.23)
 !n =   !hf  !r(k0  n+ 1); (3.24)
 55
where  = !3  !c and k0 is an arti cial cut o imposed by the bandwidth of
 the laser. In writing these expressions, I have set !1 to be \sea-level," i.e. !1 = 0
 and !2 = !hf . The sum in Eq. (3.18) is assumed to be  nite in the derivation of
 Eq. (3.19), which is naturally imposed by the  nite bandwidth of the laser that
 restricts the number of signi cant comb teeth. Even this e ective Hamiltonian
 would require multi-mode Floquet theory for an exact solution, but an approximate
 solution can be found by considering the resonant terms where the time-dependence
 in the exponential functions vanishes. These so called stationary terms are picked
 out by requiring !m  !n = 0, which implies that the ratio
 !hf
 !r
  q is an integer.
 This is simply the mathematical description of the physical requirement that the
 two-photon Raman transition can only be driven if there exists optical frequency
 di erences that match the energy di erence of the qubit states. The stationary
 terms can then be picked out of the sum by inserting a Kronecker delta function
  m;n q, in which case the e ective Hamiltonian is reduced to,
 Heff   (j2ih1je
 iq!rt0 + j1ih2jeiq!rt0)
 M=2X
 n
  
4~ !n;n q
 ~En ~En q: (3.25)
 Eq. (3.25) shows that the e ective Hamiltonian is a coupling between the qubit
 states, with a Rabi frequency that depends on the bandwidth of the laser and the
 detuning  while the phase of the induced rotation is determined by the arrival
 time of the pulse train. The sum in Eq. (3.25) is di cult to evaluate analytically,
 but it can be approximated by an integral if the repetition rate is small compared
 to the bandwidth of a single pulse. In general, the integral is di cult to evaluate
 56
f(t)  
p
  
2E0sech( t= )
  r j j
 2E20
  
!hf  
e!hf  =2 e !hf  =2
 E0
 (4 )1=4
 e 
1
 2 (
 t
  )
 2  r j j
 2E20
  e
  (!hf  )2=4
 E0p
 2 
sinc(t= )  r j j
 2E20
  tri(!hf =2)
 Table 3.1: This table shows di erent expressions for the Rabi frequency when dif-
 ferent functional forms are chosen for the pulse envelope when evaluating Eq. (3.26)
 within the approximation that the sum can be replaced by an integral. The function
 tri(t)  max(1 jtj; 0) is the triangular distribution and  is the characteristic time
 of the pulse.
 for any realistic pulse shape with the exception  at-top spectrum. To simplify the
 expression, we can use the fact that the detuning  is usually large compared to
 both the hyper ne splitting and the bandwidth of the frequency comb and use an
 expansion of 1 !n;n q in !HF= and !r= . The lowest order approximation is simply
 1
  !n;n q
  1= , allowing the following simple expression for the Rabi frequency,
  
2
  
j j2
 P
 k
 ~Ek ~Ek q
 2 
; (3.26)
 where  =2 is de ned in the same way as in Eq. (2.40). The expression for the
 Rabi frequency invites a simple picture of a three level system interacting with an
 ensemble of continuous wave lasers as illustrated in  g. 3.2. If the sum in Eq. (3.26)
 is approximated by an integral, analytic expressions for the Rabi frequency can be
 calculated in the case of several common spectra found in mode-locked pulsed lasers,
 some of which are shown in table 3.1. This result predicts Rabi oscillations when
 the repetition rate of the laser is a multiple of the qubit splitting. As an alternative
 to the frequency domain picture where photons are absorbed from one comb tooth
 57
Figure 3.2: The Stokes Raman process driven by frequency combs is shown here
 schematically. An atom starting in j #i can be excited to a virtual level by absorbing
 a photon from the blue comb and then driven to the j "i state by emitting a photon
 into the red comb. Although drawn here as two di erent combs, if the pulsed laser?s
 repetition rate or one of its harmonics is in resonance with the hyper ne frequency,
 the absorption and emission can both be stimulated by the same frequency comb.
 Because of the even spacing of the frequency comb, all of the comb teeth contribute
 through di erent virtual states which result in indistinguishable paths that add
 constructively.
 58
and emitted into another, one can also think of the time domain picture. If the
 dynamics are con ned to the qubit space, one can imagine the Bloch vector rotating
 on a sphere and is periodically kicked by sequential laser pulses. If the q parameter
 is an integer, then the Bloch vector will have the same azimuthal angle every time
 it is kicked by another laser pulse, allowing all of the kicks to add constructively.
 This e ect was demonstrated using a Ti:Sapph laser that is frequency doubled
 and detuned from the D1 resonance by 9 THz so as to have a center wavelength of
 373:5 nm. A thin- lm polarizer followed by a  =4 waveplate was used to prepare
 the beam with circular polarization so that the qubit states were coupled through
 the 2P1=2jF = 1;mf = 1i state. The repetition rate of the laser was measured to
 be 80:78 MHz, making the q parameter of the pulsed laser almost exactly a half
 integer. The pulse picker was then used to extinguish every other pulse, at which
 point the q parameter was nearly an integer and the qubit state was observed to
 oscillate as illustrated by the data in  g. 3.3. To tune the q parameter to be exactly
 on resonance, a mirror on a translation stage inside the laser cavity was adjusted by
 a few millimeters.
 The Rabi frequency can be written in terms of the intensity of the pulse train
 by starting with the formula for the intensity of a monochromatic wave,
 I = c 0
  
jE(t)j2
  
=
 c 0
 2
 jE0j
 2 ; (3.27)
 where the average is done over one or many optical cycles and the factor of 1=2
 comes from the average of cos2. If the envelope of the pulse varies slowly compared
 59
 
  
 
 
 
 
 
 
 
 
   
 
 
 
 
 
 
 
 
  
 
 
 
 
 
 
 
 
     
 
 
 
 
 
 
 
   
 
 
  
 
 
  
   
 
 
  
  
 
 
 
  
 
 
 
   
  
 
 
  
  
 
 
 
  
   
 
 
 
 
  
 
 
 
 
                     
0.00 0.01 0.02 0.03 0.04 0.05
 0.0
 0.2
 0.4
 0.6
 0.8
 1.0
 t 1ms
 PH ? L
 Ti:Sapph
 ~747nm BIBO
 PID
 PD
 BP
 LO
 PP
 Figure 3.3: (a) The data shows the qubit evolution due to a single pulse train. The
 pulsed laser used in the experiment has a repetition rate of 80:78 MHz, meaning the
 parameter q = 156:5 and does not meet the resonance condition. By using a pulse
 picker, (PP), to extinguish every other pulse, the parameter q = 313 and ful lls the
 resonance condition and causes the qubit state to oscillate as shown by the blue
 circles. By only letting every third pulse through, the parameter q = 469:5 and is
 again far from resonance causing the qubit to stay in the initialized state as shown
 by the red squares. Each data point is the result of 300 experiments, giving a typical
 statistical error of  0:03. (b) A Ti:Sapph laser with a center wavelength of 747 nm
 is doubled using a non-linear crystal and directed onto a single ion. By adjusting
 the cavity length inside the laser and using a pulse picker, the repetition rate is
 made commensurate with the qubit splitting to drive coherent Raman transitions.
 A small amount of the beam is picked o and directed onto a fast photodiode to
 generate an RF comb which is then band-pass  ltered to select a single comb tooth.
 The comb tooth is then mixed with a stable local oscillator to generate an error
 signal which is then fed to a PID loop to stabilize the repetition rate of the laser
 via a piezo mounted behind one of the cavity mirrors inside the laser cavity. When
 locked, the repetition rate is stable to within 1 Hz for more than an hour.
 60
to the optical frequency, then we should be able to write the average intensity as,
  I =  R
 c 0
 2
 Z 1
  1
 dt jf(t)j2 : (3.28)
 Now we can write a formula for the base Rabi frequency in terms of the saturation
 parameter s = I=Isat,
 Isat =
  2 0c
 4 2
 (3.29)
  0 =  R 
jE0j
 2  2
  
(3.30)
 =
  2s
 2 
; (3.31)
 where  is the characteristic time of the pulse as shown in table 3.1. The Rabi
 frequency of oscillations in the qubit driven by the comb is a factor of two larger
 than the frequency of oscillations for the case of two cw lasers. In the cw case, if
 the qubit is initialized to the j#i state and hit with a  pulse, one of the lasers
 drives the atom to the excited state while the other laser drives the atom from the
 excited state back down to the j"i state. In the case of the frequency comb that
 is on resonance, each comb tooth can participate in both of those processes, since
 there is always another comb tooth that is the right distance away because of the
 even spacing. The frequency comb, therefore, uses the power twice as e ciently
 when !hf!r is an integer.
 An estimate of the o -resonant scattering rate can be made by  rst calculat-
 ing the population of the eliminated excited state. An alternative to the e ective
 61
Hamiltonian theory is to calculate the di erential equations governing the evolution
 of the di erent states? probability amplitudes and adiabatically eliminate the ex-
 cited state from the system. This approach is more cumbersome than the e ective
 Hamiltonian theory but allows for an estimation of the excited state population. As
 shown in App. B, the excited state probability amplitude is approximately given
 by,
 ~c3  
 
 
X
 k
 ~Ek
  
ei!kt + e i!kt
   
~c1e
 i!ct + ~c2e
 i(!c !HF )t
  
(3.32)
  
 
 
X
 k
 ~Ek
  
ei!kt + e i!kt
  
ei t
   
A1e
 i 2 t +B1e
  i 2 t
  
ei!ct (3.33)
 +
  
A2e
 i 2 t +B2e
  i 2 t
  
ei(!c !HF )t
  
(3.34)
 Now assume the pulses have very little overlap in order to write down an expression
 for the probability of being in the excited state as,
 j~c3j
 2  2
  
 
 
 
 
 
 
 
2X
 n
 f 2(t nT ) [1 + cos (2!ct)] [1 + sin ( t) sin (!HF t)] (3.35)
 The average of this expression will be dominated by the \DC" term,
 hj~c3j
 2i = 2
  
 
 
 
 
 
 
 
2
 *
 X
 n
 f 2(t nT )
 +
 (3.36)
 = 2
  
 
 
 
 
 
 
 
2
  r
 Z 1
  1
 dtf 2(t) (3.37)
 = 2
  
 
 
 
 
 
 
 
2
  r jE0j
 2  (3.38)
 = 2
  0
  
(3.39)
 62
This result suggests a scattering rate of,
 Rs =
 2 
 
 0 (3.40)
  =
  2
  
X
 k
 EkEk q (3.41)
 =
  2 R
  
~f 2(!HF ) (3.42)
 Rs
  
=
 2 
 
~f 2(0)
 ~f 2(!HF )
 (3.43)
 For the hyperbolic secant pulse,
 Rs
  
=
 2 
 
sinh
  
!HF  
2
  
!HF  =2
 : (3.44)
 For a Gaussian pulse,
 E(t) = E0e
  (t= )2 (3.45)
 Rs
  
=
 2 
 
e(!HF  )
 2=8: (3.46)
 This result shows that the decoherence induced by o -resonant scattering can be
 suppressed by detuning the laser farther from resonance while increasing the power
 to maintain a given Rabi frequency.
 3.1.2 Control of the 171Yb+ clock qubit
 The previous section used a simpli ed model in order to build intuition about
 the physics behind Raman transitions driven by an optical frequency comb and we
 63
now seek to provide a more accurate description that includes the e ects of both 2P
 states and the embedded Clebsch-Gordan structure. The Clebsch-Gordan algebra
 that must be considered will only result in overall scaling factors for the previously
 derived Rabi frequency and spontaneous scattering rate, but the inclusion of the
 second excited state leads to quantum interference between the di erent processes.
 The main result from the previous section is that the Rabi frequency resulting from
 the optical frequency comb is essentially that of the continuous wave laser case
 with the amplitude being proportional to the autocorrelation of the pulse spectrum
 evaluated at the qubit splitting. For this section, we will assume that the bandwidth
 of the pulse is large compared to the qubit splitting, meaning that this prefactor
 can be approximated as the average power in the beams. In this approximation, the
 Rabi frequency, Stark shifts and scattering rates for the Raman transition control
 using a single pulse train that is circularly polarized are given by [59],
  
2
 =
 1
 3
 g2
  
1
  
+
 1
  fs   
 
(3.47)
  0 =
 g2
 12
  
1
  
 
2
  fs   
 
(3.48)
  1 =
 g2
 12
  
1
  +  hf
  
2
  fs     hf
  
(3.49)
 Rs =  
g2
 3
  
1
  2
 +
 2
 ( fs   )
 2
 !
 ; (3.50)
 where g is equal to the Rabi frequency of the resonant one-photon 2S1=2()2P3=2
 transition and  fs is the  ne-structure splitting. The ratio of this Rabi frequency
 to scattering rate is identical to the ratio of Eq. (3.44) and Eq. (3.46) in the
 64
P2
 S2 1/2
 P2 3/2
 3
 1
 3
 1?
 6
 1?
 6
 1
 2
 1
 3
 2
 3
 1
 3
 1
 3
 2
 2
 1
 1/2
 Figure 3.4: Using these Clebsch-Gordon coe cients, one can easily understand the
 relative weights for the di erent excited state contributions appearing in the expres-
 sions for the Rabi frequency and Stark shifts in Eq. (3.47,3.48,3.49). The contribu-
 tions from the two 2P states are equal because
  q
 1
 3
  2
 =
 q
 2
 3
 q
 1
 6 , noting that the
 F = 2 states do not contribute since they do not couple to j0i. The Stark shift terms
 from the 2P1=2 level are proportional to
  q
 1
 3
  2
 for both qubit states whereas the
 contributions from the 2P3=2 level are proportional to
  q
 2
 3
  2
 and
  q
 1
 6
  2
 +
  q
 1
 2
  2
 for the qubit states j0i and j1i respectively.
 limits  fs   and !hf  1. The di erent weights appearing in the equations
 for the Rabi frequency, Stark shifts and scattering rates originate from the relevant
 Clebsch-Gordon coe cients that are shown in  g. 3.4.
 The experiments done using the Ti:Sapph laser that was detuned from the D1
 line by 9 THz are expected to have a scattering rate to Rabi frequency ratio at the
 10 4 level. By going to a 355 nm pulsed laser system, where the laser is blue detuned
 from the 2P1=2 state by 33 THz and red detuned from the 2P3=2 state, the ratio of
 the of the scattering rate to Rabi frequency is approximately 10 6, meaning that
 65
spontaneous emission induced decoherence is a negligible e ect in our experiments.
 Also, the di erential AC Stark shift to Rabi frequency ratio for the 355 nm system
 is approximately 10 4, meaning that frequency  uctuations due to changes in the
 laser intensity are also negligible.
 3.1.3 Control of the qubit-oscillator system
 Starting with the optical frequency comb provided by a mode-locked laser,
 linear optical elements can be used to shape the spectrum into one that is suitable
 to gain full control over the qubit-oscillator system. With the four di erent hyper ne
 levels in the ground state of the Ytterbium ion and the three di erent degrees of
 freedom of the oscillator system, come many di erent transitions that are available
 depending on the beam geometry. In the previous section, it was shown that Raman
 transitions can be driven when the repetition rate was commensurate with the energy
 splitting of the two states of interest. Since the repetition rate of a laser is not
 capable of being changed on a fast time scale, full control over the qubit-oscillator
 system is simplest when the optical spectrum is shaped by linear elements outside
 of the laser cavity. In contrast to the previous section, it will now be desirable that
 the repetition rate be incommensurate with any transition energy in order to isolate
 di erent transitions spectroscopically.
 The simplest way to resolve the di erent transitions in the system is to use a
 single AOM that is driven at two di erent frequencies as shown in  g. 3.5, thereby
 generating two co-propagating pulse trains. In this case, the two frequency combs
 66
seen by the ion are frequency shifted relative to one another so that the beat note
 between the di erent combs is resonant with the energy splitting of interest, meaning
 that the new requirement for driving a transition between the qubit states is given
 by jn!r +  !aomj = !hf where n is an integer. This method allows for more control
 over the qubit since the phase of the AOM drive can now be used to choose the
 rotation axis for the qubit instead of the arrival time of the pulse train. However,
 this method can be impractical for some setups given the  nite bandwidth of the
 AOM being used. Depending on the repetition rate, one might have to use a AOM
 that has a bandwidth comparable to !r=2. To demonstrate this control setup, we
 use a 355 nm pulsed laser system called the Vanguard, which was manufactured
 by Spectra-Physics. The Vanguard laser used in this setup has a repetition rate of
 80:566 MHz and is capable of producing 3:5 W of average power. The repetition rate
 of this laser is such that the single AOM can be driven near 77:58 MHz and 71:48
 MHz to generate the correct beat note since 157 80:566+(71:48 77:58) = !hf=2 .
 As shown in  g. 3.6, the carrier transition can then be brought into resonance with
 the beat note of the frequency combs by tuning one of the drive frequencies of the
 AOM.
 The small angle between the two beams in the co-propagating geometry results
 in a negligible momentum transfer meaning that the coupling to the motional degree
 of freedom is highly suppressed. In order to address the motional states of the ion,
 the beam is sent through a beam splitter and then each arm is sent through separate
 AOM?s and directed onto the ion from di erent angles as shown in  g. 3.7. The
 two beams are directed onto the ion from orthogonal directions and both beams are
 67
Vanguard laser
     355 nm AOM
 PD
 PID
 S&HVCAfast
 PD
 Band
 Pass
 LO
 AMP
 HP8672
 Low
 Pass
 HP8640
 Low
 Pass
 PID
 S&H
 l/4TFP
 Figure 3.5: A co-propagating beam geometry setup is shown. Since the repetition
 rate of the Vanguard laser is not able to be locked, a feed-forward locking scheme is
 used instead in which the quantity n2 frep  !aom is stabilized. The 532 nm light
 that is generated by the Vanguard system is sent onto a fast photodetector in order
 to generate an RF comb which is then bandpass  ltered and ampli ed to generate
 a single comb tooth. A local oscillator, (HP8672), is detuned from the frequency
 nfrep by  and mixed with the comb tooth. The mixed signal is low-pass  ltered
 and mixed with the HP8640 which is tuned to !1 to generate an error signal that
 is fed to a PID to stabilize nfrep  !1. This signal is then summed with a second
 oscillator that can be tuned so that jn2 frep   !aomj = !HF where  !aom is the
 di erence between the two RF frequencies driving the AOM. A thin- lm-polarizer,
 (THP), is inserted after the AOM to polarize the beam and the de ected beam
 is used to stabilize the intensity of the beam with a PID sample and hold circuit.
 The sample and hold circuit is implemented digitally with an FPGA board that
 is triggered right before any experiment and shut o before the experiment begins
 in order to avoid any phase shifts coming from the voltage controlled attenuator,
 (VCA). A  =4 waveplate is inserted after the TFP in order to couple to the mf = 1
 Zeeman states in the excited P manifolds.
 68
f (MHz)
 71.45 71.50 71.55
 1.0
 0.8
 0.6
 0.4
 0.2
 P(     )
 Figure 3.6: By scanning the frequency of the local oscillator, (LO), in the co-
 propagating geometry shown in  g. 3.5, the line shape of the carrier transi-
 tion is seen. The oscillator that is locked to the frequency comb is run near
 77:58 MHz and the repetition rate of the laser is measured to be 80:566 MHz,
 so that when the second generator is tuned to 71:48 MHz, the resonance condition
 j157 80:566 77:58 + 71:48j = !hf=2 is satis ed. The typical statiscal error for
 the data is  0:03.
 69
Vanguard laser
     355 nm AOM1
 AOM2
 Delay stage
 2f
 2f
 B
 x?
 y?
 z?
 Figure 3.7: By splitting the pulsed laser beam and sending each arm through dif-
 ferent AOMs, we can address the motional degree of freedom of the ion. A delay
 stage is inserted into one of the arms in order to ensure that the two path lengths
 are matched to better than 0:1 mm. As shown in  g. 3.5, one of the AOM?s is
 driven with an RF generator that is locked to one of the RF comb teeth generated
 by the pulsed laser while the other AOM frequency can be tuned in order to single
 out a desired transition. The power in both arms are stabilized using digital sample
 and hold circuits as shown in  g. 3.5. In order to combat frequency dependent
 beam steering, lens of focal length f are placed before and after the AOMs so that
 a collimated beam focuses inside the AOM and then gets recollimated. The focal
 length of the lens used is in principle irrelevant, but in practice needs to be long
 enough so that the  rst order di racted beam can separated.
 70
linearly polarized and mutually orthogonally to the small magnetic  eld, (lin-perp-
 lin). Each beam is passed through a separate AOM to shift its carrier frequency so
 that the electric  elds from the beams at the location of the ion can be written as,
 ~E1(t) =
 x^
 p
 2
 X
 n
 f(t nT )e i(
 ~k1 ~x (!c+!ao1)t) (3.51)
 ~E2(t) =
 y^
 p
 2
 X
 n
 f(t nT   T )e i(
 ~k2 ~x (!c+!ao2)(t  T )); (3.52)
 where  T accounts for any delay between the two pulse trains. Because of the
 Clebsch-Gordan coe cients, x^ couples j#i to the symmetric superposition of the
 2P1=2 jF = 1;mf = 1i and jF = 1;mf =  1i states and couples j"i to the antisym-
 metric combination. The opposite is true for y^. The same is true for the coupling to
 the 2P3=2 states, but their contribution will be neglected in the following analysis.
 We call the symmetric superposition j+i and the antisymmetric superposition j i.
 In this case, the interaction Hamiltonian is,
 Hint(t) =   ^  Re
 h
 ~E1(t) + ~E2(t)
 i
 (3.53)
 =  
 
p
 2
 X
 n
 f(t nT )cos
  
~k1  ~x (!c + !ao1)t
  
 
(j#i h+j+ j+i h#j+ j"i h j+ j i h"j)
  
 
p
 2
 X
 n
 f(t nT   T )cos
  
~k2  ~x (!c + !ao2)(t  T )
  
(j"i h+j+ j+i h"j+ j#i h j+ j i h#j) (3.54)
 If the position of the ion is considered to be a classical variable for now, the wave-
 function can be expanded in terms of the eigenstates of the unperturbed atomic
 71
Hamiltonian and one  nds the following set of di erential equations for the evolu-
 tion of the probability amplitudes,
 E(t)  
 
p
 2
 X
 n
 f(t nT ) (3.55)
 _c# + i!#c# = iE(t)cos
  
~k1  ~x (!c + !ao1)t
  
c+
 +iE(t+  T )cos
  
~k2  ~x (!c + !ao2)(t+  T )
  
c (3.56)
 _c" + i!"c" = iE(t)cos
  
~k1  ~x (!c + !ao1)t
  
c 
+iE(t+  T )cos
  
~k2  ~x (!c + !ao2)(t+  T )
  
c+ (3.57)
 _c+ + i!ec+ = iE(t)cos
  
~k1  ~x (!c + !ao1)t
  
c#
 +iE(t+  T )cos
  
~k2  ~x (!c + !ao2)(t+  T )
  
c" (3.58)
 _c + i!ec = iE(t)cos
  
~k1  ~x (!c + !ao1)t
  
c"
 +iE(t+  T )cos
  
~k2  ~x (!c + !ao2)(t+  T )
  
c#; (3.59)
 where I?ve set both of the energies of the symmetric and antisymmetric states equal
 to ~!e. By transforming to rotating frame de ned as,
 ~c# = c# (3.60)
 ~c" = c"e
 i!HF t (3.61)
 ~c+ = c+e
 i!ct (3.62)
 ~c = c e
 i!ct; (3.63)
 72
the equations of motion become,
 _c# + i!#c# = iE(t)cos
  
~k1  ~x (!c + !ao1)t
  
c+e
  i!ct
 + iE(t+  T )cos
  
~k2  ~x (!c + !ao2)(t+  T )
  
c e
  i!ct
 (3.64)
 _c" + i(!"  !HF )c" = iE(t)cos
  
~k1  ~x (!c + !ao1)t
  
c e
  i(!c !HF )t
 + iE(t+  T )cos
  
~k2  ~x (!c + !ao2)(t+  T )
  
c+e
  i(!c !HF )t
 (3.65)
 _c+ + i(!e  !c)c+ = iE(t)cos
  
~k1  ~x (!c + !ao1)t
  
c#e
 i!ct
 + iE(t+  T )cos
  
~k2  ~x (!c + !ao2)(t+  T )
  
c"e
 i(!c !HF )t
 (3.66)
 _c + i(!e  !c)c = iE(t)cos
  
~k1  ~x (!c + !ao1)t
  
c"e
 i(!c !HF )t
 + iE(t+  T )cos
  
~k2  ~x (!c + !ao2)(t+  T )
  
c#e
 i!ct:
 (3.67)
 If the lowest energy state is de ned to have zero energy, !# = 0, the adiabatic
 approximation _c+  _c  0 gives the following set of expressions for the excited
 73
state amplitudes,
 c+  
E(t)
  
cos
  
~k1  ~x (!c + !ao1)t
  
c#e
 i!ct
 +
 E(t+  T )
  
cos
  
~k2  ~x (!c + !ao2)(t+  T )
  
c"e
 i(!c !HF )t (3.68)
 c  
E(t)
  
cos
  
~k1  ~x (!c + !ao1)t
  
c"e
 i(!c !HF )t
 +
 E(t+  T )
  
cos
  
~k2  ~x (!c + !ao2)(t+  T )
  
c#e
 i!ct (3.69)
 Plugging these into the di erential equations for the qubit states and making a
 rotating wave approximation with respect to the optical frequency yields the e ective
 equations of motion for the qubit states,
 _c# = i
 E(t)E(t+  T )
  
h
 c# + cos
  
~ k  ~x  !t+ !a02 T
  
c"e
  i!HF t
 i
 (3.70)
 _c" = i
 E(t)E(t+  T )
  
h
 c" + cos
  
~ k  ~x  !t+ !a02 T
  
c#e
 i!HF t
 i
 ; (3.71)
 where  !  !ao1  !ao2. The factors of !ao2 T inside of the cosine functions can
 be ignored since they act like a phase shift on one the AOMs which just acts to shift
 the rotation axis of the operation. Starting from Eq. (3.71),
 _c# = i
  2
 4 
X
 k;k0
 EkEk0e
 ik!rteik
 0!r(t+ T )
 h
 2c# +
  
ei(
 ~ k ~x  !t) + e i(
 ~ k ~x  !t)
  
c"e
  i!HF t
 i
 :
 (3.72)
 The spatial dependence can be ignored for the calculation of the carrier frequency.
 The resonant terms are picked out by requiring (k+k0)!r+ ! !HF = 0, (assuming
 74
Dwaom
 whf
 wrep
 Figure 3.8: The two photon Raman transition using frequency shifted combs is
 illustrated. If the repetition rate of the laser is not commensurate with the desired
 transition frequency, the beam can be split into two and sent through separate AOM
 frequency shifters, ( g. 3.7), in order in order to satisfy the resonance condition in
 Eq. (3.75).
 (k + k0)!r   !  !HF 6= 0). The resonant o -diagonal coupling is then given by,
  
2
 =
  2
 4 
X
 k;k0
 EkEk0 k;q0 k0e
 ik!r T (3.73)
 =
  2
 4 
X
 k
 EkEq0 ke
 ik!r T (3.74)
 q0  
!hf   !
 !r
 : (3.75)
 When the delay  T = 0 and the parameter q0 is an integer, the beat note between
 the two frequency combs resonantly drives the Raman transition as illustrated in  g.
 3.8. As before, the sum can then be approximated by an integral if the repetition
 75
rate is small compared to the bandwidth and 1=T . The extra exponential factor in
 this expression makes the integral more di cult to evaluate analytically, but yields
 a simple expression for the case of Gaussian pulses as de ned in table 3.1, which
 case the Rabi frequency is proportional to,
  / e 
1
 4( T )
 2
 ; (3.76)
 which states that the pulses must arrive at the ion simultaneously to drive the
 Raman transition. Eq. (3.76) is only an approximation and does not re ect the
 fact that the Rabi frequency should revive when the delay becomes equal to the
 spacing between pulses. But since the disparity between the repetition rate and the
 bandwidth of typical mode locked lasers is usually several orders of magnitude, Eq.
 (3.76) can be used to measure the bandwidth of a pulse through the measurement
 of the Rabi frequency as a function of the delay between the two arms as shown in
  g. 3.9.
 Equations (3.71) can also be derived from the following two-level Hamiltonian,
 He =  
jE(t)j2
  
cos
  
~ k  ~x  !t
  
(  e
  i!HF t +  +e
 i!HF t) (3.77)
 where I?ve dropped the terms that describe the di erential AC Stark shift. Plugging
 in the pulse train for the time-dependent electric  eld amplitude gives,
 He   
 2
 4 
X
 n
 jf(t nT )j2 cos
  
~ k  ~x  !t
  
(  e
  i!HF t +  +e
 i!HF t) (3.78)
 76
0 5 10 15-5-10-15-20
 Time Delay (psec)
 700
 600
 500
 400
 300
 200
 100
 0
 2p
  x 
Rabi 
Fr
 equen
 cy (k
 Hz
 )
 Figure 3.9: The Rabi frequency of the Raman transition on the carrier as a function
 of the delay between the two pulse trains introduced by a retrore ector on a movable
 stage as shown in  g. 3.7. The data was taken using the 355 nm Vanguard laser. A
  t to the data using the function in Eq. (3.76) suggests the laser produces a pulse
 with a bandwidth given by   6psec.
 77
where the approximation is made by assuming the pulses have very little overlap.
 The time dependent intensity envelope
 P
 n jf(t nT )j
 2 has a frequency spectrum
 of an RF comb. By assuming that q0 is nearly an integer, one of the RF comb teeth
 can be used to make a rotating wave approximation and rewrite the Hamiltonian
 as,
 He   
 2
 4 
X
 n
 jf(t nT )j2 (  e
  i( ~ k ~x ( ! !HF )t) +  +e
 i( ~ k ~x ( ! !HF )t)): (3.79)
 By de ning the parameter  as the Bloch rotation angle due to a single pulse,
  =  
 2
 2 
Z 1
  1
 jf(t)j2 (3.80)
 =  
 2
 2 
jE0j
 2  (3.81)
 =  
 0T
 2
 ; (3.82)
 and changing the classical position variable to a time-dependent position operator,
 the Hamiltonian in Eq. (3.79) can be transformed to the Schr odinger picture to
 give,
 Hs = !t
  
aya+ 1=2
  
 
!0
 2
  z +
  
2
 X
 n
  (t nT )
  
  e
  ikx^ +  +e
 ikx^
  
; (3.83)
 where we?ve assumed  k is along the x direction and that the pulses are short
 compared to the hyper ne time scale so that they can be modeled as delta pulses.
 To  nd the time evolution operator, we  rst write down the time evolution operator
 78
for one delta kick at t = 0.
 i
 @
 @t
  =
  
H0 +
  
2
  (t)
  
  e
  ikx^ +  +e
 ikx^
  
 
 (3.84)
 i
 Z  
  
dt
 @
 @t
  =
  
2
 Z  
  
dt (t)
  
  e
  ikx^ +  +e
 ikx^
  
 (3.85)
 If the delta function is modeled as a square pulse of width  and height 1= in the
 limit  ! 0, then its easy to see that the time evolved state due to one kick is given
 by,
  (0 +  ) = e i
  
2Q0 (0) (3.86)
 Q0    e
  ikx^ +  +e
 ikx^: (3.87)
 Between delta kicks, the system evolves according to H0 = !t
  
aya+ 1=2
  
 !02  z.
 The time evolved state after N pulses is then,
 j (NT )i =
  
e iH0T e iQ0T
  N
 j (0)i : (3.88)
 79
We take the example of three pulses,
 j (3T )i = e iH0T e i
  
2Q0T e iH0T e i
  
2Q0T e iH0T e i
  
2Q0T j (0)i (3.89)
 = e iH0T e i2H0T ei2H0T e i
  
2Q0T e iH0T e iH0T eiH0T e i
  
2Q0T e iH0T e i
  
2Q0T j (0)i
 (3.90)
 = e i3H0T e i
  
2Q2T e i
  
2Q1T e i
  
2Q0T j (0)i (3.91)
 Qn  e
 inH0tQ0e
  inH0t (3.92)
 Now the pattern is clear and we can write down the time evolved state after N
 pulses,
 j (NT )i = e iH0NT
 n=N 1Y
 n=0
 e i
  
2QnT (3.93)
 = e iH0NT
 X
 kN 1
 ( i  2QN 1)
 kN 1
 kN 1!
 ::::
 X
 k0
 ( i  2Q0)
 k0
 k0!
 j (0)i (3.94)
 = e iH0NT
 X
 kN 1;::::;k0
 ( i  2)
 kN 1+:::+k0
 kN 1!:::k0!
 QkN 1N 1 :::Q
 k0
 0 j (0)i (3.95)
 With this formula, it is easy to write down the time evolution operator to  rst order
 in  ,
 U(NT; 0) = 1 i
  
2
 N 1X
 n=0
 Qn +O( 
2): (3.96)
 To carry out the sum, we take advantage of being in the Lamb-Dicke regime and
 80
write,
 eikx^(nT )  1 + i (ae i!nT + ayei!nT ) (3.97)
 X
 n
 Qn  
X
 n
   e
  i!0nT (1 i (ae i!nT + ayei!nT )) (3.98)
 + +e
 i!HFnT (1 + i (ae i!nT + ayei!nT )) (3.99)
 All the sums in the  rst order term are geometric and can be done using the formula,
 N 1X
 n=0
 ei!nT =
 sinN!T=2
 sin!T=2
 ei(N 1)!T=2 =
 1 eiN!T
 1 ei!T
 : (3.100)
 For example, if the o set frequency between the combs  ! is tuned to satisfy the
 resonance condition for the red sideband, #r  (!hf +  !  !t)T = 2 j, where j is
 an integer, then the sum in Eq. (3.99) is approximately given by,
 N 1X
 n=0
 Qn  i 
sinN#r=2
 sin#r=2
 ei#r(N 1)=2 +a+ h:c:: (3.101)
 The coe cient in Eq. (3.101) has an amplitude  N on resonance and the other
 terms in Eq. (3.99) that drive the carrier and other sideband transitions can be
 neglected when N  (!tT )
  1 as we now show. If we require the sideband transition
 amplitude to dominate the carrier then we can write,
  N  
1 eiN!0T
 1 ei!0T
 : (3.102)
 81
But the red sideband is in resonance, then !0T = 2 q + !tT giving,
  N  
1 eiN!tT
 1 ei!tT
  
1 ei!tNT
  i!tT
 ; (3.103)
 where the last step was taken by assuming !tT  1. The factor 1  ei!tNT is of
 order unity, so we can write the resolved sideband limit criterion as a condition on
 the number of pulses,
 N  ( !tT )
  1: (3.104)
 This result can also be derived by a more heuristic method. In order to resolve
 the sideband transition, it should be required that the spectral power resonant with
 the sideband transition is large compared to the spectral power resonant with the
 carrier transition. This requirement must be even more stringent given that the
 sideband transition is naturally weaker than the carrier transition be a factor of  .
 If the beat note between the comb teeth is assumed to be resonant with the sideband
 transition, then the spectral power that is resonant with the carrier transition can
 be estimated by considering a  nite train of N pulses,
  (t) /
 N 1X
 n=0
 jf(t nT )j2 (3.105)
 =
 1X
 n= 1
 jf(t nT )j2 rect
  
t NT=2
 NT
  
; (3.106)
 where the second expression uses the rectangle function in order eliminate the pulses
 outside of the window of interest. The Fourier transform of this pulse train can be
 82
calculated using the property that products are transformed to convolutions. As
 shown earlier, the transform of the in nite pulse train is a sum of delta functions
 spaced by the repetition rate with an envelope function given by the transform of a
 single pulse. The transform of the rectangle function is a sinc function, meaning that
 the transform of a  nite train is a sum of sinc functions spaced by the repetition rate
 with an envelope function. Considering a normalized comb tooth centered about
 the sideband transition frequency, the sideband transition dominates when,
     sinc (!tNT ) (3.107)
  
 
!tNT
 : (3.108)
 After canceling the common factors of  , Eq. (3.104) is recovered. As shown in the
 arguments above, scanning the di erence frequency of the AOM?s that are used will
 reveal the di erent transitions that are possible in the qubit-oscillator system. The
 result from such a scan is shown in  g. 3.10.
 3.1.4 Error Suppression in Raman Transitions
 One source of error in the control  elds is due to a  uctuating repetition rate
 of the pulsed laser due to small changes in the laser cavity length. This noise source
 can be suppressed by either a feedback method or by a feedforward method. Some
 pulsed lasers can have a piezo mounted behind one of the cavity mirrors, in which
 case monitoring of the repetition rate can be fed back to this piezo to directly
 stabilize the rep rate. The bandwidth of this type of lock will likely be limited to a
 83
200 202 204 206
 AOM1 Frequency in MHz
 0.6
 0.3
 0.4
 0.5
 0.2
 0.1
 P( | )
 (a,-1,0)
 (a,0,-1) (a,-1,1)
 (a,0,0)
 (a,1,-1) (a,0,1)
 (a,1,0) (b,-1,0)
 (b,-1,0) (a,0,2)
 (a,2,0) (b,1,0)
 (b,0,1)
 (b,1,-1)
 (b,0,0)
 (b,-1,1)
 Figure 3.10: The probability of changing the qubit state is plotted as a function of
 the frequency of one the AOMs in the Raman beam set up shown in  g. 3.7. Using a
 Raman probe duration of 80  s, we can see resolved carrier and sideband transitions
 of a single trapped ion. The transitions are labeled, (j; nx; ny), to indicate the
 carrier index and the change in the number of phonons in the two transverse modes
 that accompany a spin  ip. The carrier indicies, a = 158 and b =  155, are a
 re ection of the span between the di erent comb teeth being used as dictated by
 the resonance condition, j2 j r +  !aomj = !hf .
 84
few kHz by the response of the piezo mounted to the massive mirror. This type of
 lock was used in Ti:Sapph laser that is mentioned in this section. Many commercial
 laser systems do not allow for a piezo to mounted behind a mirror, as in the case
 of the 355nm Vanguard laser. In this case, a feedforward method can be used to
 compensate for  uctuations in the repetition rate. This method focuses on the fact
 that the beat note between comb teeth is crucial for clean Raman transitions and
 not the repetition rate alone. In order to stabilize this beat note, the repetition
 rate can be monitored and any deviations can be fed forward to one of the AOMs
 as shown in  g. 3.5. As shown in  g. 3.11, a Ramsey experiment on the carrier
 transition without a beat note lock in place shows a coherence time of only 2 ms,
 and a coherence time of 150 ms when the phase-lock-loop is switched on. The
 bandwidth of this lock will likely be limited by the rise time of the AOM being used
 which is usually on the order of half a microsecond which should be fast enough to
 suppress repetition rate noise caused by small changes in the laser cavity length.
 Another important source of noise stems from power  uctuations in the beam.
 This can be corrected by taking a sample of the beam after the AOMs and feeding
 back to the RF drive power as shown in  g. 3.5. Because this AOM is used as an
 optical switch for the Raman beams, an analog PID is not used in this feedback loop.
 Instead, a digital PID is used so that the lock point can be stored in memory while
 the beam is turned o . When the beam is on, the voltage from the photodetector is
 sent to an FPGA chip which outputs a PID voltage that is then applied to a voltage
 controlled attenuator that the RF passes through. Figure 3.12 shows the suppression
 in intensity  uctuations on the photodetector used in the feedback loop. Because
 85
(a)
 1.0
 0.8
 0.6
 0.4
 0.2
 5 10 15 20 25 t(msec)
 P(   )
 (b)
 0.8
 0.6
 0.4
 0.2
 50 100 150 200 t(msec)
 P(   )
 Figure 3.11: A set of Ramsey experiments are performed in a co-propagating geom-
 etry as shown in  g. 3.5. (a) The Ramsey experiment was  rst performed without
 the beat-note lock switched on and shows a coherence time of only 2 msec, implying
 that the noise on the rep rate of the laser has a bandwidth of approximately 1 kHz.
 (b) When the beat-note lock is turned on, the coherence time is seen to be extended
 by almost two orders of magnitude.
 86
0 100 200 300 400
 Hz
 -90
 -80
 -70
 -60
 -50
 -40
 rms V HdBmL
 Figure 3.12: The data shows the di erence in power  uctuations in the Raman beam
 when a noise eater is used. The sample of the beam is directed onto a photodetector
 that outputs a voltage that is proportional to the optical power which was plugged
 into a spectrum analyzer to generate the root mean square of the  uctuations in
 dBm. The two curves show that the noise eater reduces the low frequency intensity
 noise by about 20 dBm.
 the voltage controlled attenuator imparts a phase shift on the RF, the noise eater is
 never used during the control operations but is brie y turned on immediately before
 the experiments begin. This system, is therefore unable to suppress noise that is
 fast compared to the timescale of a single experiment which is on the order of 1 ms.
 Another source of noise for the control  eld comes from air currents which
 result in pressure gradients in the optical beam. This can lead to beam steering
 which is seen as an intensity  uctuation by the ion. This can be suppressed to some
 degree by using an enclosure around the Raman beams. The data shown in  g. 3.13
 show the improvement in the  delity of Rabi oscillations on the carrier transition
 when using the noise eater circuit and an enclosure around the Raman beam setup.
 87
(a)
 0.8
 0.6
 0.4
 0.2
 0.5 1.0 1.5 2.0
 t(msec)
 P(   )
 (b)
 0.8
 0.6
 0.4
 0.2
 0.5 1.0 1.5 2.0
 P(   )
 t(msec)
 Figure 3.13: Rabi  opping on the carrier transition with and without intensity noise
 suppression. (a) Using a co-propagating beam geometry like the one shown in  g.
 3.5, the Rabi oscillation is observed to decay substantially after approximately a
 40 pulse. (b) With the noise eater circuit turned on, the HEPA  lter fans over the
 Raman laser turned o and an enclosure around the Raman beam setup, the Rabi
 oscillations are seen to persist over a much longer time scale.
 88
Besides  uctuating control  elds, the other main source of error in Raman
 operations comes from imperfect state preparation. As discussed earlier, the ini-
 tial purity of the internal state of the ion as prepared by optical pumping can be
 quite good, but the preparation of the motional degree of freedom is not as simple.
 As shown earlier, the qubit-oscillator dynamics are well described by the e ective
 Hamiltonian in Eq. (3.77) which implies that the Rabi frequency for qubit rotations
 are position dependent. The most straight forward way to get around this issue is
 to prepare a nearly pure motional state using laser cooling. By  rst Doppler cooling
 and then sideband cooling [60], the ion can be prepared in the vibrational ground
 state with near unit  delity as shown in  g. 3.14. To sideband cool to the ground
 state e ciently, one must take into account the vibrational level dependence of the
 sideband transition frequency given by,
  n;n 1 =  0e
   2=2 1p
 n
  L1n 1( 
2)   0 
p
 n; (3.109)
 where  0 is the base Rabi frequency, L n is the generalized Laguerre polynomial [61]
 and the approximation is valid when  
p
 n < 1 . The algorithm that is used to cool
 to the ground state then starts by optically pumping the ion(s) to j0i, driving a  
pulse on the red sideband n! n 1 transition with time being determined by Eq.
 (3.109), then optically pumping to j0i, driving n  1 ! n  2 and repeating this
 procedure n times. If the initial  time is optimized, an ion in a 2 MHz trap can
 usually be cooled to the ground state in about 20 cycles as shown in  g. 3.14.
  The expression for the blue sideband transition is  n;n+1 =  0e  
2=2 1p
 n+1
  L1n( 
2)   
p
 n+ 1
 and the expression for the carrier transition is  n;n =  0e  
2=2L0n( 
2).
 89
7.1
 Detuning from carrier transition (MHz) 
P( ) 
0.7 
0.0 
65.1 6.1 6.1 65.1
 7.1
 (b) 
Figure 3.14: The data shows a probe of the  rst order sideband transitions before
 and after sideband cooling. The red open circles show the probability of  ipping the
 spin on the sideband transitions after Doppler cooling. The closed blue circles show
 the probability of  ipping the spin on the sideband transitions after sideband cooling
 as described in the text. After sideband cooling to the ground state, it is not possible
 to  ip the spin on the red sideband transition and the blue sideband transition is
 signi cantly narrowed. After starting with  n  10 after Doppler cooling, we are
 able to cool the ion to  n  0:03, a measurement which is limited by the accuracy of
 the state detection protocol.
 90
3.2 Light Shifts
 The AC Stark shift, or light shift, is a well known e ect on atomic energy
 levels that comes from the application of an o resonant optical  eld. This level
 shift can be calculated using second order perturbation theory and in the case of a
 single monochromatic beam is given by [62],
 h Ei =
 jE0j
 2
 2~
 X
 k 6=g
 j gkj
 2 !gk
 !2gk  !
 2
 ; (3.110)
 where the sum is taken over all atomic energy levels that are connected to the level
 in question through a dipole transition,  gk is the dipole matrix element, !gk is the
 transition frequency and E0 and ! are the amplitude and frequency of the applied
 optical  eld. In this section I will show how to derive a similar expression for the
 Stark shift in the case of the optical frequency comb and show that it yields Eq.
 (3.110) in the appropriate limit. Take a pulse train with a carrier frequency !c and
 an electric  eld envelope for a single pulse E0f(t).
 H(t) =  
 ^E0
 T
 cos(!ct)
 1X
 j= 1
 ~f(!j)e
 i!jt (3.111)
 !j  j!r (3.112)
 The energy shift due to these o -resonant comb teeth can be calculated using per-
 turbation theory as shown below. First, expand the wavefunction in terms of the
 states  k(x) which are eigenstates of the unperturbed Hamiltonian with energy Ek.
 91
Consider an atom initially in the state jgi and the Dyson series evolution,
 j (t)i =
  
1 +
 1
 i~
 Z t
 t0
 dt1V (t1) +
  
1
 i~
  2 Z t
 t0
 dt1
 Z t1
 t0
 dt2V (t1)V (t2) + ::::
 !
 j (0)i
 (3.113)
 c(2)g (t) = 1 + hgj
  
1
 i~
  2 Z t
 t0
 dt1
 Z t1
 t0
 dt2V (t1)V (t2) j (0)i ; (3.114)
 where V is the Hamiltonian in the interaction picture with respect to the unper-
 turbed atomic Hamiltonian and V gg = 0. Using a resolution of the identity,
 c(2)g (t) = 1 +
 X
 k 6=g
  
1
 i~
  2 Z t
 t0
 dt1
 Z t1
 t0
 dt2V
 g;k(t1)V
 k;g(t2) (3.115)
 c(2)g (t) = 1 +
 X
 k 6=g
  
1
 i~
  2 Z t
 t0
 dt1
 Z t1
 t0
 dt2H
 g;k
 I (t1)e
  i!kgt1Hk;gI (t2)e
  i!gkt2(3.116)
 If the Stark shift is denoted by  E,
 cg(t) = jcg(t)j e
  i
 R t
 t0
 dt0 E(t0)=~ (3.117)
  1 
1
 i~
 Z t
 t0
 dt0 E(t0) (3.118)
  E(t) =  
1
 i~
 X
 k 6=g
 Z t
 0
 dt0Hg;kI (t)e
  i!kgtHk;gI (t
 0)e i!gkt
 0
 (3.119)
 Now we can plug in the Hamiltonian describing the interaction with the pulse train.
  E(t) =  
jE0j
 2
 i~T
 X
 k 6=g
 j kgj
 2
 X
 j;n
 ~f(!j)f(t nT )cos(!ct)e
  i!kgt  (3.120)
 i
 2
  
e i(!gk !j+!c)t  1
 !gk  !j + !c
 +
 e i(!gk !j !c)t  1
 !gk  !j  !c
  
(3.121)
 92
Now we can  nd the average Stark shift by integrating over a single pulse and
 multiplying by the repetition rate. If we integrate over a single pulse, we end up
 with 8 di erent terms which are all just the spectrum of the pulse evaluated at
 di erent frequencies.
 h Ei =  
 r jE0j
 2
 4~T
 X
 k 6=g
 j kgj
 2
 X
 j
 ~f(!j) 
 ~f(2!c  !j) + ~f(!j) ~f(!c  !kg) ~f(!c + !kg)
 !gk  !j + !c
 +
 ~f(!j) + ~f(2!c  !j) ~f(!c  !kg) ~f(!c + !kg)
 !gk  !j  !c
  
(3.122)
 De ning !c  !kg   and noting that terms like ~f(!c)  0 yields,
 h Ei  
 r jE0j
 2
 2~T
 X
 k 6=g
 j kgj
 2
 X
 j
 ~f(!j)( ~f(!j) ~f( ))
 !gk  !j
 (!gk  !j)2  !2c
 ; (3.123)
 which limits to Eq. (3.110) in the case of a cw laser. The di erential AC Stark Shift
 can then be calculated to be,
    
 r jE0j
 2
 4~2T
 X
 k 6=g
 j kgj
 2
 X
 j
 ~f(!j)
  
~f(!j) ~f( )
 !j +  
 
~f(!j) ~f( + !hf )
 !j +  + !hf
 !
 :
 (3.124)
 This expression can be simpli ed further by assuming that the detuning  is large
 compared to the bandwidth of the laser, in which case ~f( )  0.
 This e ect can be observed but is di cult to directly compare to theory since
 the shift is proportional to the intensity of the laser at the ion. A more convenient
 quantity to measure is the ratio of the Stark shift to the Rabi frequency since in
 93
the case of both beams having the same intensity, both the Stark shift and Rabi
 frequency are directly proportional to the intensity. Using a home built Ti:Sapphire
 laser, we measured the di erential AC Stark shift as function of the carrier frequency
 in a region close enough to the D1 line that the e ect of the 2P3=2 could be ignored.
 In order to measure the Stark shift to Rabi frequency ratio at certain wavelength,
  rst the carrier transition is driven and the frequency of oscillations as a function of
 time is measured. To measure the Stark shift, two Ramsey experiments are done.
 The  rst Ramsey experiment consists of a  =2 microwave pulse, then a delay  t,
 then a second  =2 microwave pulse is applied. If the microwave frequency is slightly
 detuned from resonance, the  nal state of the qubit will oscillate as a function of the
 delay time between the two pulses with a frequency that is equal to the detuning
 from resonance. In the second Ramsey experiment, the laser was turned on during
 the delay time but was prepared with a polarization that does not drive Raman
 transitions as the Stark shift is independent of the polarization of light [59]. The
 di erence in the two Ramsey oscillation frequencies is then interpreted as the Stark
 shift.
 In order to calculate the theoretical curves shown in  g. 3.15, one needs to
 know spectrum of the pulse train being used. The center wavelength of the laser was
 measured using an optical spectrum analyzer and the width of a single pulse was
 estimated using an intensity autocorrelation measurement. As shown in  g. 3.16,
 the autocorrelation measurement data is well described by assuming a sinc(t= ) pulse
 with  = 90 fsec, a pulse that corresponds to a  at-top spectrum in the frequency
 domain which is close to what was observed on the optical spectrum analyzer. The
 94
-10 -5 0 5 10 15
 -0.006
 -0.004
 -0.002
 0.000
 0.002
 0.004
 DH THz L
 Dn
 W
 Figure 3.15: The data show the ratio of the di erential Stark shift and the Rabi
 frequency for Raman transitions. The theory curve shown is a function computed
 using Eq. (3.131) with a  t parameter determined by an autocorrelation measure-
 ment. The red dotted line shows the location of a two photon resonance that can
 change the principal quantum number of the S shell valence electron and drive the
 transition [Xe]4f 146s$ [Xe]4f 146s. When the laser was tuned near this frequency,
 the ion was observed to go dark for periods of time ranging from 1 minute to 1 hour.
 95
    
  
 
 
 
 
 
 
 
  
  
 
 
 
 
 
 
 
            
-500 500 1000 1500 2000
 Delay HfsecL
 0.2
 0.4
 0.6
 0.8
 1.0
 I
 Figure 3.16: The data shows the normalized intensity autocorrelation of a single
 pulse produced by a Ti:Sapph laser that is then doubled using an BBO critically
 phase matched crystal. The function that is  t to the data is the autocorrelation of
 a sinc pulse as shown in Eq. (3.125).
 function in the  gure is given by,
 Z 1
  1
 dt sinc
  
t
  
 
sinc
  
t T
  
 
=
   
 
T=  cos
  
T
  
 
sin
  
T
  
  
 
T
  
 3 : (3.125)
 If a square distribution in frequency space is assumed, which corresponds to a sinc
 pulse in time,
 E(t) = E0cos(!ct)
 X
 n
 sinc
  
t nT
  
 
(3.126)
 f(t) = E0sinc(
 t
  
) (3.127)
 ~f(!) = E0 
 
2
 rect(! =2) (3.128)
 where rect(! =2) is the rectangular function centered around ! = 0 with a width
 96
of 2= . Using the expressions that were derived for a 3-level atom the ratio of the
 Stark shift to Rabi frequency is,
  =
 j j2
 P
 k EkEk q
  
(3.129)
 Ek =  r ~f(k!r) (3.130)
   
 
=
 1
 4
 P
 j
 ~f(!j)
  
~f(!j) ~f( )
 !j+ 
 
~f(!j) ~f( +!hf )
 !j+ +!hf
  
P
 k
 ~f(k!r) ~f((k q)!r)
  
(3.131)
 If we use the  at-top spectrum, the sums in Eq. (3.131) can be approximated as
 integrals and then done analytically as in the following example;
 X
 j
 rect( j!r 2 )rect(
 j!r 
2 )
 j!r +  
=
 X
 j
 rect( j!r 2 )
 j!r +  
(3.132)
  
Z 1
  1
 dj
 rect( j!r 2 )
 j!r +  
(3.133)
 =
 Z 1=!r 
 1=!r 
dj
 1
 j!r +  
(3.134)
 =
 1
 !r
 ln
  + 1 
  1 
(3.135)
 Note that these expressions do not include e ects from unequal intensities in the
 two beams, but nonetheless  t to the observed data nicely as shown in  g. 3.15.
 97
Chapter 4
 Phonon mediated entanglement of ions
 4.1 Spin-Dependent Forces
 As mentioned in a previous chapter, engineering a spin-dependent force allows
 for the controlled entanglement of ions. In this chapter, I will begin by showing how
 to generate a spin-dependent force using stimulated Raman beams and how these
 forces can be studied using a single trapped ion. I will then describe the evolution
 of two ions under the the in uence of these forces and show how to measure the
  delity of these states. A variation on the standard pulse sequence will be described
 and shown to provide protection against uncontrolled  uctuations in the trapping
 frequency.
 The discussion in the previous chapter shows that the pulsed laser can be used
 to generate e ective Hamiltonians that largely con ne the dynamics to the Hilbert
 space spanned by the tensor product of the qubit and a single mode of harmonic
 motion. For most of this chapter, I will assume that the ion?s evolution can be
 described in this restricted space and that the resolved sideband limit is valid. All
 of the operations referred to will be engineered through combinations of the three
 primitive operators that are called the carrier, the red-sideband and blue-sideband.
 Starting with the e ective Hamiltonian that drives Raman transitions in the lin-
 98
perp-lin con guration with  elds,
 ~E1(t) = x^E1(t)e
  i(~k1 ~x (!c+!ao1)t+ 1) (4.1)
 ~E2(t) = y^E2(t)e
  i(~k2 ~x (!c+!ao2)t+ 2); (4.2)
 the Hamiltonian is,
 He =  
jg(t)j2
  
cos
  
 ~k  ~x  !t+   1;2
   
 +e
 i!hf t +   e
  i!hf t
  
; (4.3)
 where,
 g(t)  
E(t) g;e
 ~
 (4.4)
  !  !ao1  !a02; (4.5)
 where I assumed E1 = E2 = E . If the two beams being used are derived from a single
 beam and a beam splitter in a Mach-Zender con guration as shown in  g. 3.7 then
 the relative phase   1;2 is set by the relative phase of the AOM drive frequencies.
 If  kx0
 p
  n+ 1   
p
  n+ 1 1, the exponential operators can be linearized to give,
 He   
jg(t)j2
 2 
  
1 + i 
 
ae i!tt + ayei!tt
   
e i !tei  1;2
 +
  
1 i 
 
ae i!tt + ayei!tt
   
ei !te i  1;2
   
 +e
 i!hf t +   e
  i!hf t
  
(4.6)
 In order to generate a Hamiltonian that drives a speci c transition, the AOM
 frequencies are tuned to bring the corresponding operator into resonance with a
 99
Fourier component of jg(t)j2. If jg(t)j2 has a Fourier component at !0 of strength
 g0, then tuning the AOM frequencies to be near the carrier resonance such that
 !0 +  !  !HF =  , where   !t, generates the e ective Hamiltonian,
 He  C^(t)   
 
2
  
 +e
  i tei c +   e
 i te i c
  
(4.7)
 where   g
 2
 0
  , which is the same Hamiltonian used in the previous chapter describing
 the interaction of a two level system interacting with a classical monochromatic
 source. When exciting the red or blue sideband, we get the following Hamiltonians,
 R^(t)   i
   
2
  
 +ae
  i rtei r    a
 yei rte i r
  
(4.8)
 B^(t)   i
   
2
  
 +a
 ye i btei b    ae
 i bte i b
  
: (4.9)
 As shown in [20, 63, 64], a spin-dependent force can be generated by applying
 the red and blue sideband operators simultaneously with a symmetric detuning
  r =   b   . In this case the time evolution operator can be solved for easily
 by considering the Magnus expansion. This expansion allows the time evolution
 operator to be written as the exponentiation of an operator
 P
 k O^k so that,
 U(t; t0) = exp
 "
 X
 k
 O^k
 #
 (4.10)
 O^1 =  
i
 ~
 Z t
 t0
 dt0H(t0) (4.11)
 O^2 =  
1
 2~2
 Z t
 t0
 Z t0
 t0
 dt0dt00[H(t0); H(t00)]: (4.12)
 100
In general, the Magnus expansion has an in nite number of terms, but as will
 be shown, the series terminates after the second term in the case of the M lmer-
 S rensen interaction. The M lmer-S rensen Hamiltonian will be de ned as the sum
 of the red and blue sideband interactions with the symmetric detuning,
 Hms(t) = R^(t) + B^(t) (4.13)
 =  i
   
2
  
 +ae
  i rtei r    a
 yei rte i r
 + +a
 ye i btei b    ae
 i bte i b
  
(4.14)
 =  i
   
2
  
 +e
 i s    e
  i s
   
ae i tei m + ayei te i m
  
(4.15)
 where the spin phase  s = ( r +  b)=2 and the motional phase  m = ( r   b)=2.
 De ning the spin operator S^   i
  
 +ei s    e i s
  
, the  rst term in the Magnus
 expansion is,
 O^1 =
   
2
 S^
 Z t
 t0
 dt0
  
ae i t
 0
 ei m + ayei t
 0
 e i m
  
(4.16)
 = S^
  
 (t)ay    (t)a
  
(4.17)
  (t) =
   
2
 ei t  ei t0
 i 
e i m : (4.18)
 101
The second term in the Magnus expansion is given by,
 O^2 =  
1
 2
 Z t
 t0
 Z t0
 t0
 dt00dt0[H(t0); H(t00)] (4.19)
 =  
1
 2
  
  
2
  2
 S^2
 Z t
 t0
 Z t0
 t0
 dt00dt0(e i (t
 0 t00)  ei (t
 0 t00)) (4.20)
 =  
 
  
2
  2
 S^2
 1
 i 
(t t0  
sin (t t0)
  
): (4.21)
 All of the higher order terms in the Magnus expansion are made up of nested com-
 mutators of the Hamiltonian at di erent times. For example, the third order term
 is calculated by integrating terms like [H(t); [H(t0); H(t00)]]. But because the spin
 operator in the M lmer-S rensen interaction is constant and the time dependent
 harmonic oscillator operators have a scalar commutator, all higher order terms in
 the Magnus expansion are zero. This allows for the time evolution operator to be
 written in a simple form;
 Ums(t) = e
  i
 R
 dt0H(t0) 12
 R R
 dt0dt00[H(t0);H(t00)] (4.22)
 = exp
 h
 S^(  (t)ay   (t)a) + iS^2 0(t)
 i
 (4.23)
  0(t)   
 
  
2 
 2
 ( (t t0) sin (t t0)): (4.24)
 The e ect of the M lmer-S rensen operation can be understood as implement-
 ing a spin-dependent force which is made obvious by expressing the spin operator
 in its eigenbasis and examining its action on a quantum state that is routinely pre-
 pared in the laboratory, ( g. 4.1),  =
 P
  P j #z;  ih#z;  j, where  describes the
 102
p
 x
 s??
 s??
 ?
 ?
 2
 ?
 Figure 4.1: When a single ion prepared in j #zi = j # si  j " si and subjected
 to the spin-dependent force, the motional wavepacket becomes entangled with the
 spin. For a  nite detuning, the motional wavepackets execute circular trajectories
 in phase space and disentangle from the spin when t = 2 = .
 motional state of the ion. The spin operator  i( +ei s    e i s) has eigenstates
 1p
 2
 ( ie i s j#zi+ j"zi) with eigenvalues  1. Rewriting the time evolution operator
 in this basis gives,
 Ums(t) = exp
  
(  (t)ay   (t)a)(j " sih" s j  j # sih# s j)
 +i 0(t)(j " sih" s j+ j # sih# s j)
  
; (4.25)
 and its action on the state j #z;  i is,
 Ums(t)j #z;  i = UMS(t)
  
 iei sj " s ;  i+ ie
 i sj # s ;  i
  
=
 p
 2 (4.26)
 =  
i
 p
 2
 ei 0(t)ei s
  
D^( (t))j " s ;  i
  D^(  (t))j # s ;  i
  
=
 p
 2; (4.27)
 103
where the displacement operator is de ned as D^( (t)) = exp[  (t)ay   (t)a]. The
 action of the displacement operator is simple in the coherent state basis as de ned
 by j i = D^( )j0i, with j0i being the ground state of the harmonic oscillator. Note
 that in the case of one ion, the phase  iei 0(t)ei s is global and can, therefore, be
 ignored. Using the identity, D^( )D^( ) = ei? ; D^( +  ), where ? ;  Im[   ],we
 see that the action of the spin-dependent force is expressed as,
 Ums(t)j #z;  i =
 1
 p
 2
  
ei? ; j " s ;  +  (t)i  e
  i? ; j # s ;    (t)i
  
: (4.28)
 To compare this theory to an observable in the lab, we can calculate the probability
 of  nding the the ion in j1i after preparing it in j0i and applying the spin-dependent
 force operation,
 P"(t;  ;  ) = Tr[j"zi h"zj  ] (4.29)
 = h (t)j "zi h"z j (t)i (4.30)
 =
 1
 4
  
ei? (t); h   (t)j  e i? (t); h +  (t)j
  
 
 
e i? (t); j   (t)i  ei? (t); j +  (t)i
  
(4.31)
 =
 1
 4
  
2 2Re[e2i? (t); h   (t)j +  (t)i]
  
: (4.32)
 Using the expression for the overlap of two coherent states, h j i = e 
1
 2 (j j
 2+j j2 2   ),
 the expression for the ion brightness becomes,
 P"(t;  ;  ) =
 1
 2
  
1 cos(4? (t); )e
  2j (t; )j2
  
: (4.33)
 104
Now to consider a thermal distribution as the initial condition,
 P"(t;  ) =
 Z
 d2 P ( )P"(t;  ;  ) (4.34)
 P ( ) =
 1
   n
 e j j
 2= n (4.35)
 Z 1
  1
 d2 e j j
 2= nei4Im[  
 ] =   ne 4 nj j
 2
 (4.36)
 P"(t;  ) =
 1
 2
  
1 e ( n+1=2)j2 (t; )j
 2
  
(4.37)
 In order to test this theory, we prepared a single 171Yb+ ion in the four-rod trap
 and generated the spin-dependent force using the Ti:Sapph Raman laser set up
 discussed in the previous chapter. To generate the bichromatic force, one of the
 comb frequency shifters was simultaneously driven with two modulation frequencies.
 When these combs are tuned to drive the red and blue sidebands (in conjunction
 with the third frequency comb in the other beam), the ion experiences the spin-
 dependent force. The theoretical curve in Eq. (4.37) is compared with experimental
 data in  g. 4.2 and is seen to agree quite well. As seen in the  gure, the periodic
 decoupling of the spin and motional degrees or freedom occurs with high  delity,
 but is much more sensitive to the laser frequency in the case of an ion in a thermal
 state as compared to an ion that has been cooled to its ground state.
 4.2 Two Atom Entanglement
 These spin dependent forces can be used to entangle two ions. First, I?ll just
 assume that we can isolate a single mode of motion and work with the following
 105
(a)
 (b)
 0.5
 0.4
 0.3
 0.2
 0.1
 0.0
 0.5
 0.4
 0.3
 0.2
 0.1
 0.0
 -0.02 -0.01 0.00 0.01 0.02
 -0.02 -0.01 0.00 0.01 0.02
 P(    )
 P(    )
 d/2p(MHz)
 d/2p(MHz)
 Figure 4.2: The probability of detecting j1i given the preparation of j0i and the
 application of the bichromatic force as function of the symmetric detuning  . In
 both cases, the pulse was applied for tg = 172  sec. The revival of the initial
 spin state is seen when the symmetric detuning is equal to an integer times 1=tg
 as evidenced by the narrow regions where the probability of detecting j1i is nearly
 zero. (a) The  rst data set was taken with an ion that was Doppler cooled and the
 theory was plotted assuming an average phonon number of  n = 8. (b) The second
 data set was taken with an ion that was sideband cooled to the ground state and
 the theory was plotted assuming an average phonon number of  n = 0:01.
 106
Hamiltonian,
 H = !ta
 ya 
!hf
 2
 ( (1)z + 
(2)
 z )  cos(~k ~x  !t  )
  
 (1)+ +  
(2)
 + +  
(1)
  +  
(2)
  
 
(4.38)
 A RWA can be made in the interaction picture to produce the red and blue sideband
 Hamiltonians just like the one ion Hamiltonians,
 R^(2)   i
   
2
  
( (1)+ +  
(2)
 + )ae
 i rte i r  ( (1) +  
(2)
  )a
 ye i rtei r
  
(4.39)
 B^(2)   i
   
2
  
( (1)+ +  
(2)
 + )a
 yei bte i b  ( (1) +  
(2)
  )ae
  i btei b
  
(4.40)
 So now the M lmer-S rensen interaction,
 H(2)ms = R^
 (2) + B^(2) (4.41)
 =  i
   
2
  
 +e
 i s    e
  i s
   
ae i tei m + ayei te i m
  
; (4.42)
 where     
(1)
  +  
(2)
  is the total spin operator. If the atoms are prepared in the
 state, j ##zi, the ideal evolution of the state is most easily calculated by assuming
 that the phase space trajectories close at the end of the gate operation, ( g. 4.3).
 In that case, the displacement operators in Eq. (4.27) limit to the identity operator
 so that the  nal state is given by,
 Ums(t)j ##zi = e
 iS^2 0(t) (j ## si+ j "" si  j #" si  j "# si)
 2
 (4.43)
 =
 ei4 0(t) (j ## si+ j "" si) j #" si  j "# si
 2
 : (4.44)
 107
p
 x
 ?
 ??
 2
 ?
 s???
 s???
 s???
 s???
 Figure 4.3: When two ions are prepared in j ##zi = j ## si+j "" si j #" si j "# si
 and subjected to the spin-dependent force, the motional wavepacket splits into three
 components. The spin states with non-zero eigenvalues of the total spin operator
 execute circular trajectories in phase space and then disentangle from the motion
 when t = 2 = . The total area swept out in phase space is equal to the relative spin
 phase  .
 Using Eq. (4.24), it is easy to see that choosing a gate time tg = 2  and laser power
 such that  = 2  , the accumulated phase is ei4 0(tg) =  i and the  nal state is the
 maximally entangled state,
 Ums(tg)j ##zi = e
  i =4 j ##zi  ie
 i2 ms j ""zip
 2
 : (4.45)
 4.2.1 Experimental Implementation of the M lmer-S rensen Gate
 There are many di erent experimental parameters that must be calibrated in
 order to achieve the ideal evolution described in the previous section and we now
 describe some methods used in the laboratory to  ne tune the system. Once two ions
 have been loaded into the trap, one should  rst make sure that the detection beam
 is properly adjusted. Usually the power in the detection beam is set so that a single
 ion prepared in j1i scatters an average of 10 photons, so that two ions prepared
 in j11i should scatter an average of 20 photons. To make sure that both ions are
 108
contributing equally to the number of photons observed, one can use integrated
 images taken with the ICCD camera. The next step is to equalize the power being
 used in the red and blue sidebands. This step is complicated by the fact that most
 commercial combiners used to combine the two RF signals have a non-linear output
 whose combined frequency components usually do not have the same power ratios
 as the inputs . The situation is further complicated by the frequency dependent
 di raction e ciency of the AOM that is used and the di erent di raction angles
 for the two di erent frequency components. In order to mitigate these e ects, the
 optical system is designed to image the input beam of the AOM onto the ions so
 that the red and blue sideband beams focus at the same spot inside the vacuum
 chamber. The di raction e ciency of the AOM driven with two frequencies during
 the operation was measured as a function of the drive frequency so that that the red
 and blue sideband frequencies could have equal di raction e ciencies. By  nding
 the peak in di raction e ciency, the red and blue sideband frequencies can be made
 to straddle this peak by choosing the proper drive frequency for the other AOM
 that is used in the second Raman beam. Next, one should make sure that both
 ions will be equally illuminated by the Raman beams, which is most easily done by
 driving the carrier transition and mapping out the Rabi frequency as a function of
 the position of the  nal lens in the system used to focus the beam on the ion. Lastly,
 the power in the RF drives being used is  ne tuned by measuring the red and blue
 sideband transition frequencies, which is most easily done with a single ion. During
  This problem might be eliminated if the two frequencies were generated together using an
 arbitrary waveform generator.
 109
this last step, it is important to have both RF drives on during the measurement of
 the sideband transitions since the RF combiner output is non-linear in the inputs.
 In order to calibrate only the red sideband, the generator used to drive the blue
 sideband is set to a frequency that does not drive any transition but is still in the
 neighborhood of the blue sideband and similarly for calibrating the blue sideband
 transition.
 Once the beams are aligned, the proper power, detuning and timing must be
 found. The  rst parameter to be calibrated is the precise frequency of the carrier
 transition. It is usually not su cient to simply  nd the center of the carrier feature
 in a frequency scan such as the one shown in  g. 3.10 since the linewidth of the
 transition is so large compared to the necessary precision. In principle, one can
 decrease the power in the Raman beams in order to narrow the transition to gain
 more accuracy, but this method incurs the additional complexity of power dependent
 AC Stark shifts that must taken into account. The easiest method found is to
 keep one of the RF drive frequencies  xed and detuned several linewidths red of
 the red sideband and then scan the other frequency blue of the blue sideband. A
 sharp peak should be visible in the frequency scan when the two detunings are
 exactly symmetric about the carrier transition. This peak is in fact where the slow
 version of the M lmer-S rensen gate would operate. Then both RF frequencies are
 scanned simultaneously, always staying symmetric about the carrier transition to
  nd the correct detuning. For the symmetric detuning scan, the gate time is set to
 tg =  =  as measured by the sideband transition frequency. As shown in  g. 4.4,
 the symmetric detuning scan features reveal where the RF frequencies should be in
 110
order to implement the ideal gate. The  nal step is to  x the symmetric detuning
 as shown in  g. 4.4 and scan the gate time in order for  ne tuning as shown in  g.
 4.5. These last two steps can then be iterated until the maximum gate  delity is
 achieved. The theoretical predictions for the di erent populations are calculated in
 the same way as Eq. (4.37) and given by,
 P##(t;  ) =
 1
 4
  
1 + 2e j2 (t; )j
 2( n+1=2)cos(4 0(t;  )) +
 1
 2
  
1 + ej4 (t; )j
 2( n+1=2)
   
(4.46)
 P#"(t;  ) = P"#(t;  ) =
 1
 8
  
1 + ej4 (t; )j
 2( n+1=2)
  
(4.47)
 P""(t;  ) =
 1
 4
  
1 2e j2 (t; )j
 2( n+1=2)cos(4 0(t;  )) +
 1
 2
  
1 + ej4 (t; )j
 2( n+1=2)
   
:
 (4.48)
 The gate performance is quanti ed by a measurement of the state  delity. If
 the ideal state is j i and the state created in the experiment is  ^, then the  delity
 is de ned as F = h j ^j i. This quantity can be found by measuring the density
 matrix and then calculating the overlap with j i, but a little algebra shows which
 measurements contain the relevant information. To see this, consider the expression
 for the  delity,
 F = h j ^j i (4.49)
 = Tr[j ih j ^] (4.50)
 =
 1
 2
 Tr[(j ##i+ ei j ""i)(h## j+ e i h"" j) ^] (4.51)
 111
1.5
 1.0
 0.5
 0.0
 -0.010 -0.005 0.000 0.005 0.010
 Tw
 o Ion B
 rig
 htnes
 s
 d/2p  (MHz)
 (a)
 1.0
 0.8
 0.6
 0.4
 0.2
 0.0
 -0.010 -0.005 0.000 0.005 0.010
 Popul
 ation
 s
 d/2p (MHz)
 (b)
 Figure 4.4: The detuning from the red and blue sidebands is symmetrically scanned
 using a gate time of 360  s. (a) The total ion brightness is plotted, i.e. P (#"
 ) + P ("#) + 2P (""). At approximately  2:8 kHz the total ion brightness is seen
 to approach 1, signaling the creation of the maximally entangled state. (b) Using
 a  tting protocol, the di erent populations are plotted with P (##) represented as
 diamonds, P ("#) + P (#") as squares and P ("") as circles. At approximately  2:8
 kHz the odd parity populations are seen to approach zero while the even parity
 populations both approach 0:5, signaling the creation of the state j ##i + ei j ""i.
 The theoretical curves plotted with the data are shown in Eqs. (4.46,4.47,4.48). The
 deviation from theory near  =2 = 0:01 is likely due to an o -resonant coupling to
 another motional mode.
 112
Figure 4.5: After  nding the approximate ideal symmetric detuning  , the gate
 time is scanned. Using a  tting protocol, the di erent populations are plotted with
 P (##) represented as diamonds, P ("#) + P (#") as squares and P ("") as circles. At
 approximately 280  s the odd parity populations are seen to approach zero while
 the even parity populations both approach 0:5, signaling the creation of the state
 j ##i+ ei j ""i.
 113
The operator j ih j can be expanded in the Pauli operator basis,
 j ##ih## j+ j ""ih"" j =
 1
 2
 (I(1)I(2) + Z(1)Z(2)) (4.52)
 ei j ##ih"" j+ e i j ""ih## j = cos 
X(1)X(2)  Y (1)Y (2)
 2
 + sin 
X(1)Y (2) + Y (1)X(2)
 2
 : (4.53)
 This implies that the  delity is measured as the following expectation value,
 F =
 1
 4
  
I(1)I(2) + Z(1)Z(2) + cos (X(1)X(2)  Y (1)Y (2)) + sin (X(1)Y (2) + Y (1)X(2))
  
;
 (4.54)
 where we are using the notation f x;  y;  zg = fX; Y; Zg. The  rst term two terms
 are found by measuring in the standard basis and then calculating the probability of
  nding the two ions in the same state, i.e. 14
  
I(1)I(2) + Z(1)Z(2)
  
= 12( ##;##+ "";"").
 The second two terms are found by applying a global  =2 pulse before making
 a measurement. To see this, consider the  =2 rotation operator R^( =2; ?) which
 rotates the qubit about the cos?X + sin?Y axis. If two qubits are subject to this
 rotation before being measured in the standard basis, then the measurement result
 114
is given by,
 Tr[Z(1)Z(2)R^(1)( =2; ?)R^(2)( =2; ?) R^y(1)( =2; ?)R^y(2)( =2; ?)] (4.55)
 =
 D
 R^y(1)( =2; ?)R^y(2)( =2; ?)Z(1)Z(2)R^(1)( =2; ?)R^(2)( =2; ?)
 E
 (4.56)
 =
   
cos?Y (1)  sin?X(1)
   
cos?Y (2)  sin?X(2)
   
(4.57)
 =
 1 cos2?
 2
  
X(1)X(2)
  
+
 1 + cos2?
 2
  
Y (1)Y (2)
  
(4.58)
  
sin2?
 2
  
X(1)Y (2) + Y (1)X(2)
  
: (4.59)
 The last expression shows that the measurement result as a function of ? has an
 amplitude that is equal to twice that of the last two terms in Eq. (4.54). We call
 this curve the parity,  (?). Because the parity curve has twice the amplitude of the
 of the o -diagonal elements in the  delity in Eq. (4.54), the contrast of the curve,
  c, is four times bigger than the o -diagonal  delity terms. Putting this all together
 implies that the  delity is measured by,
 F =
 1
 2
 ( ##;## +  "";"") +
 1
 4
  c (4.60)
 The data shown in  g. 4.6 shows a measurement of this operator after the prepara-
 tion of an entangled state using the bichromatic force described above. Note that is
 crucial that that the RF used for the  =2 pulse have a well de ned phase relationship
 to the phase  ms. In our case this is assured by using a set of phase locked DDSs to
 generate the red and blue sideband and carrier drives, which we respectively denote
 as f r;  b;  cg. Since the DDSs are digitally programmed, a stable phase relationship
 115
P( )?
 Pc
 Figure 4.6: The contrast of this curve,  ( ), together with a measurement in the
 standard basis is a measurement of the state  delity after applying the M lmer-
 S rensen interaction. The contrast of this curve together with a measurement of
 the populations shows that state  delity was F = 0:97 0:01.
 is achieved by programming the carrier frequency to be  c =
  r+ b
 2 . If this frequency
 is not chosen precisely, the phase relationship between the gate and the analyzing
 will drift over the time that is needed to take the parity curve, e ectively washing
 out the signal. The data shown in this section demonstrates that a 355 nm pulsed
 laser system is an e ective means of executing high  delity entangling operations
 between hyper ne qubits in 171Yb+. The  delity can likely be increased further by
 going to higher trapping frequencies, where the gate time can be reduced so as to
 further suppress heating and other noise sources.
 116
4.3 Coherent Error Suppression in M lmer-S renson Operations
 Like any quantum operation, the bichromatic gate is sensitive to many po-
 tential sources of error. One source of error that is observed in the laboratory is a
  uctuating detuning  that can arise from a  uctuating trapping frequency. Because
 the spin and motion disentangle at times that are multiples of 2 = , any  uctuation
 in the trapping frequency will lead to residual entanglement between the spin and
 motion, thereby leaving the spin in a mixed state. The required level of precision
 grows with higher temperatures since the overlap between two states separated in
 phase decreases exponentially with temperature. To see this, consider a qubit under
 the in uence of the spin-dependent force. If the initial motional state is assumed
 to be a Gaussian state,  (x), with an uncertainty in position,  x, and we describe
 a small detuning or timing error in the gate operation as an unintentional momen-
 tum displacement, ~q, then the overlap between the two motional states is given by
 R1
  1 dx 
 (x)e iqx (x) = exp
  
 12 (q x)
 2 . For a harmonic oscillator in a thermal
 state,  x increases approximately as
 p
 T for kBT > ~!, meaning that the overlap
 between the two states decreases exponentially. As I will show in this section, a
 composite pulse can be designed that suppresses this type of error to any desired
 order [65].
 If the trapping frequency  uctuates on a time scale that is slow compared
 to the gate time, then the detuning can be considered as static during a single
 experiment. If the detuning is given by 2 =tg +  , where  is the  uctuation
 in the trap frequency, then the  nal displacement of the di erent wavepackets is
 117
p
 x
 |
 |
 Figure 4.7: This  gure is an illustration of the e ect of a  uctuating trap frequency
 on the bichromatic force operation. In this case, the detuning,  , is too small for
 the chosen gate time. This error implies that the orbits executed in phase space do
 not close and thereby leave the spin and motion entangled to some degree.
   (t) =    2(2 =tg+ )(e
 i t  1) as illustrated in Fig. 4.7. The fact that  (t) is non-
 zero at the end of the operation implies some residual entanglement between the
 spin and motional degrees of freedom. If the parameter  is small compared to
 2 =tg, then the  delity of the operation can be expanded using Eq. (4.37) so that,
 F = 1 P"j#(tg;  +  ) (4.61)
 =
 1
 2
  
1 + e ( n+1=2)
  
 
 
  
(2 =tg+ )
 (ei t 1)
  
 
 
2
  
(4.62)
 = 1 
2 2 (  )2 ( n+ 1=2)
  4
  2 +O( 3): (4.63)
 The  delity for the one-ion operation is calculated by assuming a target state
 where the spin and motion are completely decoupled, which in the case of a single
 118
p
 x
 |
 |
 Figure 4.8: The e ect of adding a second pulse to the bichromatic gate is shown
 here. The solid and dashed curves show the two di erent trajectories taken by the
 two di erent wave packets associated with spin up and spin down in the S^ basis
 with only the spin up trajectory being labeled for clarity. When the initial phase
 of the second pulse is chosen to be opposite that of the  rst pulse, it is clear that
 the  nal displacement from the origin is signi cantly reduced, thereby reducing the
 amount of residual entanglement between the spin and motion.
 ion, implies a restoration of the initial spin state. If a second phase space trajectory
 is added to the operation where the initial phase of the force is opposite of the  rst
 trajectory, then the error incurred in the  rst operation will largely be canceled by
 the second operation as illustrated in  g. 4.8. Note that the direction of the spin-
 dependent force can be inverted by the mapping H )  H which can be a ected
 by either changing the spin phase  s =
  r+ b
 2 by  or by changing the motional
 phase  m =
  r  b
 2 by  . The error suppression of this composite operation can
 be quanti ed by considering two separate spin dependent force operations. If two
 spin dependent force operations are applied with the  rst parameterized by the
 119
displacement  1 and the second by  2, then the  nal state is given by,
 U^2U^1j #z;  i =
 1
 p
 2
 D^2
  
D^ ( 1) j " si  D^ (  1) j # si
  
j i (4.64)
 =
 eiIm[ 2 
 
1]
 p
 2
  
D^ ( 1 +  2) j " si  D^ (  1   2) j # si
  
j i:
 (4.65)
 If the  rst pulse is executed using  s = 0 and the second is executed using  s =
  , the  rst displacement is  1 =
   
2(2 =tg+ )
 (ei 2 =  1) and the second is  2 =
    2(2 =tg+ )(e
 i 4 =  ei 2 = ) and the  delity is,
 F = 1 P"j#(tg;  +  ) (4.66)
 =
 1
 2
  
1 + e ( n+1=2)
  
 
 
  
(2 =tg+ )
 (ei 2 =  1 (ei 4 =  ei 2 = ))
  
 
 
2
  
(4.67)
 = 1 
8 4 (  )2 ( n+ 1=2)
  6
  4 +O( 5); (4.68)
 showing that the error has been pushed to higher order in  . It is straightforward to
 generalize this calculation to an arbitrary number of pulses. Using a thermal state
 for the initial motional state, the probability P"j#( 1; ::::;  n) for n displacement
 operations is given by,
 hj "ih" ji =
 1
 2
 0
 @1 exp
 2
 4 
 
 
 
 
 
2
 nX
 j=1
  j
  
 
 
 
 
2
 ( n+ 1=2)
 3
 5
 1
 A (4.69)
 120
The sum in Eq. (4.69) can be rewritten as,
 nX
 j=1
  j =
   
2
 nX
 j=1
 Z tj
 tj 1
 dte i 
(i)
 m ei( + )t ; (4.70)
 where t0 = 0 and tn = tg. If the phase  s switches between 0 and  , then the function
 ei s that appears in the time evolution operator can be replaced by a time dependent
 function which switches between  1. Because the phase  ips occur when the wave
 packets are near the origin of phase space, they will occur at integer multiples of
 2 = . This allows the time dependent phase to be represented by Walsh functions,
 WAL(k; t=tg), so that Eq. (4.70) can be rewritten as,
   
2
 nX
 j=1
 Z tj
 tj 1
 dte i 
(j)
 m ei( + )t =
   
2
 Z tg
 0
 dtWAL(k; t=tg)e
 i( + )t; (4.71)
 where the index k speci es the times at which phase  ips occur. Walsh functions are
 piecewise constant functions that form a complete orthonormal set that alternate
 between  1 at times that are speci ed by the index k, see  g. 4.9. I will now
 show that by choosing speci c Walsh functions for the modulation of the phase  m,
 the integral in Eq. (4.71) can be made arbitrarily small, thereby eliminating the
 residual spin-motion entanglement. The proof of this requires a precise de nition
 of the Walsh functions. Several ways of ordering Walsh functions exist and here
 we use the dyadic ordering, or Payley ordering [66]. This particular ordering allows
 for the Walsh functions to be de ned in terms of Rademacher functions, which are
 square waves de ned as R(n; x)  sign[sin(2n x)]. These functions allow the Walsh
 121
functions to be de ned as,
 WAL(k; x)  
m+1Y
 i=1
 R(i; x)bi 1 ; (4.72)
 where k is expressed as a binary number k = bm2m + ::::: + b020 and bi = 0 or 1.
 With this de nition it can be shown that,
 Z 1
 0
 dxWAL(2n  1; x)ei2
 n+1 x
 nX
 l=0
 alx
 l = 0; (4.73)
 where al are constant coe cients. A proof of Eq. (4.73) is given in appendix C.
 Putting this all together gives an expression for the  delity of a composite pulse
 using the Walsh function of index k = 2n  1,
 Fn = 1 hj "ih" ji (4.74)
 =
 1
 2
 0
 @1 + exp
 2
 4 
 
 
 
 
 
2
 nX
 i=1
  i
  
 
 
 
 
2
 ( n+ 1=2)
 3
 5
 1
 A (4.75)
 =
 1
 2
  
1 + exp
 h
  4
  
 O( n+1)
  
 2 ( n+ 1=2)
 i 
: (4.76)
 The scaling of the in delity of the operation scales with the error  is found by
 choosing the Walsh index dependent detuning that optimizes the two ion entangling
 gate for speed  k =
 p
 k + 1 = 2n=2  , (shown in the next section), and expanding
 122
Figure 4.9: The Walsh functions WAL(1; x); WAL(3; x) and WAL(7; x) are shown.
 Notice that WAL(3; x) can be constructed as two sequential WAL(1; x) functions
 with a phase  ip on the second pulse. Likewise, WAL(7; x) can be constructed as
 sequential WAL(3; x) pulses with a phase  ip on the second pulse.
 123
around  = 0 for the  rst few Walsh functions,
 F0 = 1 
2 2 (  )2 ( n+ 1=2)
  40
  2 +O( 3) (4.77)
 = 1 ( n+ 1=2)
  
  
p
 2  
 2
 +O( 3) (4.78)
 F1 = 1 
8 4 (  )2 ( n+ 1=2)
  61
  4 +O( 5) (4.79)
 = 1 ( n+ 1=2)
  
  
p
 2  
 4
 +O( 5) (4.80)
 F3 = 1 
128 6 (  )2 ( n+ 1=2)
  83
  6 +O( 7) (4.81)
 = 1 ( n+ 1=2)
  
  
p
 2  
 6
 +O( 7): (4.82)
 This e ect was demonstrated using the Vanguard 355 nm laser system with
 a single 171Yb+ ion in the four-rod trap with a 1:8 MHz trapping frequency. To
 impart the spin-dependent force, we drive one of the AOMs with two frequencies
 tuned close to the red and blue  rst order sidebands. The phase of these RF drives
 can be shifted on demand as triggered by a single TTL pulse. In order to change
 the phase of the spin-dependent force by  , without changing the spin basis that
 the force is diagonal in, we change both of these phases by  such that   s =  
and   m = 0. As can be seen in the data in  g. 4.10, the higher order Walsh
 sequences exhibit spin revivals of high purity over a much larger range of detunings
 in the neighborhood of  = 2 =tg.
 124
 
 
 
  
  
 
         
 
  
 
 
 
      
  
    
     
 
  
 
 
  
 
 
  
 
 
  
  
 
 
 
 
 
  
  
 
 
 
 
 
 
 
 
  
    
 
 
 
  
 
 
 
 
 
 
 
   
 
 
 
 
 
 
  
     
 
 
  
 
 
 
 
 
 
 
 
  
 
  
 
 
 
 
  
 
 
  
   
 
 
 
  
  
 
 
 
 
  
 
 
  
 
 
 
 
 
 
  
  
  
 
 
    
 
 
 
 
 
   
   
  
 
 
 
  
   
 
 
    
 
   
 
    
 
 
   
 
 
 
 
 
 
   
 
        
 
  
  
   
 
(a)
 -4 -2 0 2 40.
 0.2
 0.4
 0.6
 d H2p  t0L
 PH? L
  
 
         
   
 
  
 
 
  
 
 
  
 
  
 
 
 
 
 
 
  
 
 
 
   
  
     
     
 
 
 
 
  
 
 
 
 
 
 
 
 
 
  
 
 
 
  
 
 
 
 
 
 
 
 
 
 
 
 
 
     
 
 
 
 
 
 
  
 
 
 
 
 
 
  
 
 
 
 
   
 
 
 
 
 
 
 
 
 
 
 
 
 
 
  
 
 
 
 
 
 
  
  
   
 
 
 
 
  
  
  
 
 
 
 
 
   
 
 
 
 
 
 
 
     
 
 
 
  
 
 
 
 
 
   
 
 
  
 
    
  
 
 
 
 
 
 
 
 
  
   
   
 
    
 
  
 
 
 
 
 
 
   
 
 
 
   
 
 
   
 
 
 
 
 
   
       
 
(b)
 -6 -4 -2 0 2 4 60.
 0.2
 0.4
 0.6
 d H2p  t1L
 PH? L
    
  
 
 
 
 
              
  
   
  
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
   
 
 
 
 
     
 
 
 
  
 
 
 
 
 
     
     
 
 
 
 
 
 
 
  
 
 
  
 
 
 
   
    
 
 
 
 
 
   
 
 
 
 
    
 
    
 
 
 
 
 
 
 
 
   
 
 
 
 
 
 
 
 
 
 
 
 
 
    
  
 
  
 
 
 
 
 
 
 
 
 
 
 
  
  
 
 
  
 
 
 
    
 
 
 
  
     
    
   
 
 
 
 
  
   
  
 
 
 
 
 
  
 
 
 
 
 
 
 
 
 
 
 
    
 
 
 
 
 
   
 
   
  
          
    
 
(c)
 -6 -4 -2 0 2 4 60.
 0.2
 0.4
 0.6
 d H2p  t3L
 PH? L
 Figure 4.10: An ion prepared in j #i and  n  8 is subjected to the standard and
 composite spin-dependent force operations. The data shown are plotted together
 with theoretical curves assuming an initial thermal state of motion. (a) The data
 show the probability of  nding the ion in j "i as a function of the symmetric detuning
  for t0 = 100  s. (b) The operation is implemented using WAL(1; t=t1) for the phase
  s(t) with t1 =
 p
 2t0. (c) WAL(3; t=t3) is used for  s(t) with t3 = 2t0. Note the
 narrow resonance at  t3=2 = 2 corresponds to a trajectory where the phase  ips
 occur when the motional wave packet is not at the origin of phase space.
 125
4.4 Two Ion Walsh Gates
 For two ions, the relative phase term  k(t) must be included. We will be
 interested in  nding the the evolution operator comprised of multiple pulses, with
 the evolution from a single pulse beginning at t0 given by,
 U( ; t; t0) = D(S^  m( ; t; t0))exp[
 i 2S^2
 4 
(t t0  
1
  
sin( (t t0)))]: (4.83)
 The evolution due to two sequential operations is,
 U1U0 = exp[ 
i 2S^2
 4 
(tf  t0  
1
  
sin( (tf  t1)) 
1
  
sin( (t1  t0)))]D^1D^0
 (4.84)
 D^1D^0 = D
  
S^ 1
  
D
  
S^ 0
  
(4.85)
 = D
  
S^( 0 +  1)
  
exp[
 1
 2
 S^2(  1 0   1 
 
0)] (4.86)
 For three pulses, the three displacement operators multiply together to give
 D2D1D0 = D^(S^
 2X
 i=0
  i)exp[
 1
 2
 S^2(  2 1  2 
 
1 + 
 
1 0 + 
 
2 0  1 
 
0  2 
 
0)] (4.87)
 126
Notice that all the di erent phase terms add up so that the real parts cancel allowing
 us rede ne the spin-dependent phase for N pulses as,
 U = e iS^
 2 N (t)D^
  
S^
 NX
 i=0
  i
 !
 (4.88)
  N(t)  
NX
 i>j=0
 Im[  i j] 
 
  
2 
 2
 ( (tN+1  t0) 
NX
 i=0
 sin( (ti+1  ti)))
 (4.89)
 Using this expression for the evolution operator, we can calculate the state  delity for
 the gate assuming a detuning error  . We?ll denote the perfect evolution operator
 as U and the one that describes the error as U  U( +  ) and write the  delity
 as,
 F( ) =
 X
  
P Tr
 h
 U j ##;  ih##;  jU
 y
  U j ##;  ih##;  jU
 y
 i
 (4.90)
 After some algebra the  delity can be written as,
 F( ) =
 1
 4
 X
  
P 
1
 4
  
 ie i4 N h jD( 
NX
 i=0
  i)j i  ie
  i4 N h jD(
 NX
 i=0
  i)j i+ 2
 !
  
iei4 N h jD(
 NX
 i=0
  i)j i+ ie
 i4 N h jD( 
NX
 i=0
  i)j i+ 2
 !
 (4.91)
 =
 1
 4
  
 
 e j
 PN
 i=0  ij
 2
 ( n+1=2) + iei4 N
  
 
 
2
 ; (4.92)
 where both the  nal displacement
 PN
 i=0  i and the phase  N depend on the detuning
  +  . The temperature dependence in the expression for the  delity in Eq. (4.92)
 is scaled by the  nal displacement, which becomes more resistant to detuning errors
 for higher order Walsh functions as shown in the discussion of single ion composite
 127
pulses. The other term in the expression for the  delity that comes from the spin-
 dependent phase  N in the evolution operator does not obey the same scaling law
 as the displacement term. Therefore, we are prevented from writing down a simple
 expression for how the  delity scales with detuning errors as we did in the case of
 one ion. However, numerical evaluation of the phase  N shows that it too becomes
 more robust to detuning errors for higher order sequences, albeit not as strongly as
 the displacement term. By plotting the  delity as a function of the detuning error
  , as in  g. 4.11, it is clear that the higher order Walsh sequences are e ective in
 suppressing the coherent errors stemming from detuning errors.
 The gate times for the composite entangling operations are not equal to the
 gate time for the standard operation. The spin dependent phase  N is proportional
 to the area swept out in phase space and the number of closed circles executed when
 using WAL(2n  1) is equal to 2n = k + 1. The form of the identity in Eq. (4.73)
 implies that the total gate time for the gate using the Walsh function of index k is
 t(k)g = (k + 1)2 = k. To see how the gate is to be altered, note that the total phase
 accumulated during the gate should be independent of which sequence is used. So
 for a given Rabi frequency, we can write,
  0 =  k (4.93)
 t(0)g
  0
 =
 t(k)g
  k
 (4.94)
 2 
 20
 =
 (k + 1)2 
 2k
 (4.95)
  k =
 p
 k + 1 0 = 2
 n=2+1  : (4.96)
 128
This implies that the composite pulse gate time must be at least tg = 2n=2 =  with
 the detuning being optimized at  = 2n=2+1  . While the exponential nature of the
 gate time becomes daunting for large n, small errors can easily be corrected with a
 modest increase in the gate time. The increasing gate time means that increasingly
 complex gate sequences will eventually perform worse than simpler ones as the
 gate becomes sensitive to other noise sources such as laser power  uctuations or
 spontaneous scattering events. It is, however, worth noting that the extra time
 needed for performing the modulated sequence might be partially o set by the
 decreased sensitivity to the initial to the initial temperature of the oscillator, thereby
 reducing the amount of resource intensive cooling that may be needed.
 The two ion Walsh gate was tested using two 171Yb+ ions in a 1:8 MHz trap.
 The beat note was set to be near resonant with the tilt mode since it is less sus-
 ceptible to heating than the COM mode [67]. The state  delity measurement for
 the MS gate is compared for the di erent pulse sequences in Fig. 4.11. The in-
 creased robustness to detuning errors can be quanti ed by de ning a characteristic
 width of the high  delity region, which is referred to as the passband Bk. Since
 the smallest in delity in the data sets is of order 0:1, we choose to de ne the pass-
 band as the range of detunings where the in delity is always observed to be lower
 than 0:2 and estimate B0  0:5 kHz, B1  0:7 kHz, and B3  1:5 kHz, demon-
 strating the composite sequences? tendency to suppress symmetric detuning errors.
 The maximum  delities observed for the sequences k = f0; 1; 3g are, respectively,
 F2 = f0:91 0:02; 0:92 0:02; 0:95 0:01g. The relative modest increase in the
 maximum  delity achieved indicates that the trapping frequency is not changing
 129
0
 B
 1
 B
 3
 B
 (d)
 0
 3
 7
 1
 15
 1.0 1.5 2.0 2.5 3.0
 0.2
 0.4
 0.6
 0.8
 1.0
 d  W-2n 2+1+2
 F
 Figure 4.11: The state  delity of a two ion MS gate as a function of the symmetric
 detuning  is compared for the  rst three Walsh gates. In all three data sets, the ions
 were sideband cooled to the motional ground state before implementing the gate. In
  gures (a)-(c), the measured  delity as a function of the detuning  is compared with
 the theoretical curves. (d) shows the ideal  delity curves for W(0)(blue), W(1)(red),
 W(3)(green), W(7)(purple) and W(15)(cyan). The curves are shifted in frequency
 in order to facilitate comparison and highlight the increasingly large regions of high
  delity for the higher order sequences. As a guide to the eye, the ideal curves shown
 in (d) are  t to the data by varying   and including an overall scale factor to
 account for additional experimental imperfections. The three di erent  t values of
   agree with each other to within 10%. The estimates for the characteristic widths
 of the high  delity regions, Bk, are shown to increase rapidly with the higher order
 sequences.
 130
signi cantly, (1 kHz), on a time scale of less than the time between recalibrations
 of the detuning  , which is approximately 5 min. We observed trap frequency  uc-
 tuations of 1 kHz on a time scale of about 1 hour.
 In this chapter, I have demonstrated high  delity entangling operations be-
 tween neighboring ions using spin-dependent forces generated by Raman transitions.
 An extension of the standard entangling operation was shown to provide protection
 against trap frequency  uctuations. This dynamic error suppression might prove to
 be crucial to the long term stability of a quantum information processor based on
 qubits coupled through collective harmonic oscillator modes.
 131
Chapter 5
 Photon mediated entanglement of ions
 The most obvious photonic degree of freedom for encoding quantum informa-
 tion is the polarization [68], which forms a natural qubit, but one can also choose the
 frequency [69], orbital angular momentum [70] or time-bin degree of freedom[71].
 There are many proposals for quantum computing with optical  elds and a lot of
 progress has been made in demonstrating the necessary primitives including one
 and two qubit gates[10], but a quantum memory based on photonic qubits faces
 di cult challenges. In fact, many research groups are working toward photon stor-
 age inside material quantum systems [72, 73, 74]. The experiments described in this
 thesis do not address manipulation of quantum information stored in photon qubits,
 but rather they are used as an intermediary agent to create e ective interactions be-
 tween atomic qubits. The experiments described in this thesis use frequency encoded
 qubits, but a good review of how di erent encodings can be used in conjunction with
 trapped ions can be found in [75].
 The underlying principle allowing the entanglement of spatially disparate atomic
 qubits is known as entanglement swapping. Entanglement generally requires an in-
 teraction between constituent entities, but not necessarily between every member
 of the group. If, for example, one can entangle two spatially separated ions with
 respective optical modes and then measure the state of the photons in an entangled
 132
basis, the ions may be projected into an entangled state without ever having directly
 interacted with one another. The basic idea is to set up the measuring apparatus
 so as to erase any information that might betray the origin of the photon(s). While
 these protocols are extremely useful for generating remote entanglement between
 disparate atomic qubits, they are inherently probabilistic. Here, I review two di er-
 ent protocols for entangling ions using scattered photons. The  rst type is explained
 only in a theoretical setting as we have not carried out any experiments using the
 protocol, whereas the second protocol is described in the context of the experiments
 that were performed in our laboratory.
 5.1 Entangling atoms by detecting a single photon
 The  rst method of entangling ions through scattered photons, (type I), uses
 only one photon and a single photon detector that is incapable of discerning which
 atom emitted the photon [76]. In the simplest case, we consider two three-level
 atoms that are both initialized to their internal ground states j0i. We then imagine
 simultaneously exciting the atoms with a weak laser pulse that is resonant with the
 transition j0i , jei and generates the following transformation,
  0 = j00i )  1 =
 p
 (1 Pe)j00i+
 p
 (Pe) (j0ei+ je0i) +O(P
 2
 e ); (5.1)
 where Pe is the probability of a single atom absorbing a photon from the laser pulse.
 The state jei is unstable and can decay back to j0i or to a third state j1i. The
 detector will be placed somewhere so that the polarization of the spontaneously
 133
emitted photon associated with the jei ! j0i decay channel is orthogonal to the
 polarization of a photon resulting from a jei ! j1i transition. If a polarizer is placed
 in front of the detector so as to block the photons resulting from the jei ! j0i
 transition, the detection of a scattered photon heralds the creation of the state,
  2 =
 p
 (Pe)
  
ei 1j01i+ ei 2j10i
  
+O(P 2e ): (5.2)
 If the excitation pulse has an intensity such that Pe << 1, then the state is approx-
 imately given by,
  2  e
 i 1 j01i+ ei 2j10i; (5.3)
 showing that the detection of a single photon yields a maximally entangled state to
 O(P 2e ). If the probability of detecting a single photon is Pdet, then the probability
 of successfully generating the entangled state is Psucc  PePdet. If the phase ei 1 is
 considered to be global and factored out of the state, the relative phase e i( 1  2)
 is given by the relative path lengths of the two possible photon paths in units of
 the transition wavelength  e;1, i.e. e i( 1  2) = e i(ke;1x1 ke;1x2) = e ike;1(x1 x2). The
 interferometric stability of the path lengths required for the phase stability presents
 a technical challenge for the type I protocol in general. This protocol was recently
 demonstrated and reported on in [77].
 The type I protocol can not only be used to connect distant ions, but can also
 be used to generate large entangled states of ions that reside in a single trap. This
 protocol has the advantages that large entangled states can be generated with a
 134
single laser pulse and if the ions are in a single trap, the interferometric stability
 issue would largely be mitigated. In the case of N ions, the type I protocol is
 capable of generating what is known as a W-state, which has a single excitation
 that is symmetrically distributed throughout the system,
 j ~WNi =
 1
 p
 N
  
ei 1j00:::001i+ ei 2j00:::010i+ :::+ ei N j10:::000i
  
: (5.4)
 The notation j ~WNi is used for the generalized W-state where the di erent phase
 factors present in the state are arbitrary, while the notation jWNi will be used for
 a state where all of the phase factors are equal. While generating j ~WNi sounds
 relatively simple in this protocol, proving that the state generated in the labora-
 tory is actually the multipartite entangled W state can be quite di cult [78]. To
 measure the density matrix of an N qubit state requires 22N measurement settings
 and individual addressing in general, even though in some cases these requirements
 can be signi cantly loosened using compressed sensing [79] or other related tech-
 niques. It was, however, shown in [80] that the multipartite entanglement of the
 W-state can be detected by observing a negative expectation value of the witness
 operator N 1N I^  jWNihWN j. This implies that one simply has to measure the  -
 delity F = Tr [ jWNihWN j] and get a value above N 1N to show that the state is
 multipartite entangled, a task that becomes prohibitive for large numbers of ions.
 By exploiting the symmetry of jWNi, it can be shown [81] that the witness operator
 can be measured by only 2N  1 measurement settings without the need for indi-
 vidual addressing, thereby signi cantly reducing the complexity of con rming the
 135
multipartite entanglement of the state. The trade-o in using the witness operator
 instead of state tomography is that a witness operator might fail to detect multipar-
 tite entanglement if the  delity is too low, i.e. a negative expectation value of the
 witness operator is a su cient but not a necessary requirement for the veri cation
 of multipartite entanglement.
 The  delity of the W-state that can be generated in a typical ion trap setup can
 be calculated by drawing a parallel between the storage of coherence in the atoms
 through an inelastic scattering event and the coherence in the light  eld through
 an elastic scattering event. A scattering event is called elastic if the atom returns
 to the original internal energy state and inelastic if it ends up in a di erent state.
 The experiment described in [82], where two mercury ions in a single trap were
 weakly excited by a resonant laser pulse, the position where scattered photons were
 detected was shown to exhibit an interference pattern in elastic scattering events
 but no interference in inelastic scattering events. In the case of elastic scattering,
 because both ions will be in the same state after the scattering event, no information
 about which ion scattered the photon will exist, allowing the di erent optical paths
 to interfere [82, 83]. In an inelastic scattering event, one of the ions will be found in
 a di erent state upon measurement, thereby betraying the path taken by the photon
 and inhibiting the interference of the two possible paths. This is reminiscent of the
 famous discussions between Einstein and Bohr about Young?s double slit experiment.
 In a gedanken experiment, Einstein imagined a small measuring apparatus that
 was capable of determining which of the two slits that a single photon had passed
 through, thereby showing that quantum theory must be wrong in predicting that
 136
even single photons were somehow passing through both slits. Of course, it was
 subsequently shown that any such measuring apparatus would have a back-action
 e ect on the photon in question that resulted in the interference pattern being
 washed out, thereby keeping quantum theory safely intact. The mercury experiment
 in [82] de nitively demonstrates the precision of Bohr?s conclusions.
 The elastic scattering events yield an interference pattern with the regions
 of high intensity being where the photon phases e i~k ~R add constructively. In the
 case of inelastic scattering, the photon phase e i~k ~R gets imprinted on the ions upon
 detection of a photon. The ions will also pick up a dynamical phase ei!t, where t
 is the time it takes for the photon to reach the detector, and ! is the frequency
 di erence between j0i and j1i. If the time that a photon takes to traverse the full
 length of the ion crystal is short compared to ! 1, this dynamical factor can be
 ignored. If we use the jF = 1;mf = 0i and jF = 1;mf = 1i states in 171Yb
 +,
 the Zeeman splitting with a magnetic  eld on the order of a few Gauss makes
 this approximation valid for ion crystals much smaller than one meter. When this
 approximation is valid, the regions of high intensity in the elastic case are the same
 points at which the detection of a photon in the inelastic case signals the creation of
 a W-state with all the phases being equal. Therefore, in order to identify scattering
 regions that herald a high  delity W-state, it should su ce to calculate the elastic
 scattering cross-section for N ions in a trap and identify the high intensity region.
 In order to  nd the points of high intensity in the radiation pattern, we generalize
 the derivation of the scattering cross-section for two ions in a single trap [84] to N
 ions. Starting with the di erential scattering cross-section as given by the electric
 137
dipole Hamiltonian in second-order perturbation theory
 d 
d 
=
 X
 f
  
 
 
 
X
 p;j
 h f jDp   ^oute i
 ~kout Rp j jih jjDp   ^ine i
 ~kin Rp j ii
 !0  !in + (Ej  Ei)=~ i =2
  
 
 
 
2
 ; (5.5)
 where Dp and Rp are the dipole and position operators for the pth ion,  ^ is a
 polarization vector, ~kin and ~kout are incoming and outgoing wavevectors and the
 indices i and f represent the initial and  nal state of the ions. It is important to
 note that this expression only applies to elastic scattering events, since in this case
 the probability amplitudes for the photon scattering o di erent ions are added
 together because these processes are indistinguishable. The quantum states in the
 perturbation expansion are taken to be eigenvectors of the unperturbed Hamiltonian
 of N ions in a harmonic trapping potential and are therefore product states of the
 ions internal degrees of freedom and the motional state of the system.
 We now examine the special case of the ion crystal axis lying in the plane
 de ned by the incoming and outgoing wavevectors and the quantization axis being
 perpendicular to that plane. In this case, the dipole operator only contributes an
 overall scaling factor to the scattering cross-section and is therefore ignored in this
 calculation. As explained in [84], the denominators in Eq. (5.5) are nearly constant
 for the 171Yb+ ion cooled near the Doppler limit on the 370 nm line as a consequence
 of the recoil frequency, 8.5 kHz, being small compared to the linewidth of 20 MHz.
 By considering the denominators to be nearly constant, they can be factored out of
 the sum allowing us to do the sums over the j and f indices via resolutions of the
 138
identity operator, giving
 d 
d 
= hfnHOgi j
 X
 p;p0
 e i(
 ~kout ~kin) (Rp Rp0)j fnHOgii; (5.6)
 where fnHOgi denotes the initial harmonic oscillator quantum numbers of the ions.
 Eq. (5.6) might be seen as a classical interference pattern from a di raction grating
 comprised of slits that oscillate around  xed points and recoil upon de ection of
 light quanta. After taking the expectation value, the  nal scattering cross-section
 is given by
 d 
d 
=
 X
 p;p0
 ei  (Up Up0) k^ x^
 NY
 m=1
 e [(Ap;m Ap0;m) 
m
 a  k^ x^]
 2
 ( nma +1=2) [(Tp;m Tp0;m) mt  k^ y^]
 2
 ( nmt +1=2):
 (5.7)
 In the above equation,  nma ( n
 m
 t ) are the average number of axial (transverse) thermal
 quanta in mode m and    
 
 
 ~k
  
 
 d. U is a vector containing the equilibrium positions
 of the ions in units of length d = (e2=m(!1a)
 2)1=3. The Lamb-Dicke parameter for
 the mth mode in the axial (transverse) direction is  ma(t) =
  
 
 ~k
  
 
 (~=(2m!ma(t)))
 1=2. The
 di erence between incoming and outgoing wavevectors is  k^ = k^out k^in, and Ap;m,
 Tp;m are elements of the transformation matrices from position coordinates to axial
 and transverse normal coordinates, respectively.
 The  delity of the entangled state with arbitrary phase of two ions in separate
 isotropic harmonic traps (frequency  and average thermal index  n) as derived in
 [76] is given by
 F ( ) =
 Z 1
 0
 d e  e 4 
2( n+1=2)(1 cos( )cos(    )); (5.8)
 139
with  = k
 p
 ~=2m and  being the angle between the excitation beam and the
 emission direction. In the limit of weak con nement,  <<  , and because the
 integrand decays exponentially in  , we can approximate cos
  
  
 
 
 1 allowing us
 to carry out the integration and arrive at
 F ( )  e 4 
2( n+1=2)[1 cos ] = e 8 
2( n+1=2)
 h
 ( k^ x^)2+( k^ y^)2
 i
 (5.9)
 This shows the relationship between Eq. (5.8) and the contrast of the fringes in
 Eq. (5.7). This suggests that the fringe contrast of Eq. (5.7) for N = 2 might be
 interpreted as the  delity of the entangled state with arbitrary phase when two ions
 are in the same trap. Moreover, because the fringe peaks correspond to points of
 common phase, the full expression, (Eq. (5.7)), for N = 2 might be interpreted as
 the  delity of the state where the relative phase is equal to zero. We contend that
 Eq. (5.7) should be a valid prediction of the  delity of an N qubit W-state in the
 weak con nement regime, with   >> 1 ensuring no ion-ion photon exchange.
 The emission pattern for three ions is plotted in  g. 5.1(a), clearly showing
 the degrading e ect of recoil due to a large scattering angle. As a consequence of the
 ions not being evenly spaced, the scattering pro le from ten ions in  g. 5.1(b) shows
 that the only points where radiation adds up in phase is in the forward scattering
 direction. This implies that in an inelastic scattering event, the only points where
 light detection will yield a W-state with all the terms having the same phase is in
 the forward scattering direction. The angular size of this spot,   , in the case where
 the excitation pulse is along the crystal axis can be estimated by  rst normalizing
 140
?
 for 3 ions ? for 10 ions
 1.0
 -1.0 -0.5
 0.5
 1.00.5
 -0.5
 -1.0
 ??? ???
 axis of linear ion crystal
 (a) (b)
 Figure 5.1: (a) The di erential scattering cross section, in the plane de ned by
 the ion crystal axis and the excitation vector, is shown in the polar plot with the
 excitation beam coming in along the ion crystal axis at  = 0. The dashed line
 represents the normalized intensity (2=3) required for a negative expectation value
 of the entanglement witness. For the example shown here, realistic trap parameters
   = 600,
 q
 ~k2 
2m!2x
 = 1 and !x!y = 10 were chosen. (b) The scattering cross section for
 ten ions in a harmonic trap is plotted together with the required normalized intensity
 (0:9, dashed circle) for a negative expectation value of the witness operator.
 141
Eq. (5.7) by dividing by N2, ignoring the Debye-Waller factors and making the
 small scattering angle approximation, giving
 d 
d 
 1 
  4 2 
4N2
 X
 p>p0
 (Up  Up0)
 2 : (5.10)
 The sum in Eq. (5.10) can be approximated by numerically solving for the equi-
 librium positions for di erent numbers of ions in a harmonic trap, which yields
 P
 p>p0 (Up  Up0)
 2  0:45N2:87. We de ne the spot size to be the region where the
 intensity is at least f times the maximum and  nd the angular size of the spot to
 be approximately given by
 2   2
 1:7(1 f)1=4
  1=2 
N 0:21 (5.11)
 Remembering that the entanglement witness demands that the  delity of the W-
 state be bounded by F  N 1N , we set f =
 N 1
 N . With the interpretation that the
 elastic scattering cross-section represents the  delity of the W-state for an inelastic
 scattering event, the fraction of photons scattered into the plane of interest that will
 yield an N particle W-state is,
 2  
2 
 
0:55
  1=2 
N 0:46: (5.12)
 Eq. (5.12) allows the estimation of an upper bound on the angle subtended by
 the detector being used to signal the creation of a multipartite entangled state.
 This scaling law shows that the e ciency with which one can create multipartite
 142
entanglement decreases rather slowly with the number of entangled ions.
 5.2 Entangling atoms through the detection of two photons
 The second type of photonic interconnect, (type II), uses an entanglement
 swapping measurement that is triggered by the detection of two photons [85, 86].
 The protocol works by  rst generating two photons, each of which is entangled
 with the internal state of a respective ion. These photons are then measured in an
 entangled basis which projects the ions into an entangled state. Here we describe a
 protocol that uses the frequency of the emitted light as the photon qubit.
 Imagine two ions that are trapped in di erent vacuum chambers with the
 di erent positions x1 and x2. The ions are  rst Doppler cooled and optically pumped
 to the state 2S1=2jF = 0;mf = 0i, so that the state of the two ions is given by,
  0 = j00i (x1) (x2); (5.13)
 where the functions  (xi) represent the motional state of the ions. Once the ions?
 spin state has been initialized, resonant microwaves can be used to generate arbitrary
 superposition spin states,
  1 = ( j0i+  j1i) ( j0i+  j1i) (x1) (x2): (5.14)
 In the next step, each ion is excited with an ultra-fast laser pulse whose center wave-
 length is resonant with the 2S1=2 $2 P1=2 transition. This laser pulse is generated by
 143
a Ti:Sapphire mode-locked laser that generates picosecond pulses at a repetition rate
  rep = 80 MHz, a center wavelength near 739:05 nm and an average power of  1 W.
 A fast EOM is used pick a out a single pulse upon being triggered by an external
 TTL pulse. The single pulse is then directed through a critically phase matched
 BIBO crystal to generate a 369:5 nm pulse which is then directed toward one of the
 ions. The remaining unconverted 739 nm light is picked o using a dichroic mirror
 and sent through a second non-linear crystal to generate a resonant pulse for the
 second ion. A delay stage is inserted into the path of one of the pulses to ensure
 that the two pulses arrive at the two ions nearly simultaneously. In order to adjust
 the power and polarization of the pulses, a  =2 waveplate followed by thin- lm po-
 larizer and another  =2 waveplate is placed in the beam path. The  rst waveplate
 in conjunction with the polarizer is used to adjust the intensity of the pulse while
 the second waveplate is adjusted so that the pulse is  ^ polarized. The intensity is
 set so that the light coherently transfers the population in the 2S1=2jmf = 0i states
 to the 2P1=2jmf = 0i states so that the state of the two ions becomes,
  2 = ( j1
 0i+  j00i) ( j10i+  j00i) (x1) (x2) (5.15)
 where j00i and j10i are the mf = 0 hyper ne levels in the 2P1=2 manifold. The lifetime
 of the excited 2P1=2 state is 8:12 nsec [87], which implies that after about 30 nsec,
 the ions will have spontaneously emitted photons in the random directions ~ki; ~kj.
 Since the excitation pulse is only 1 ps long, the probability of the ion absorbing a
 second photon is approximately 1 ps8 ns  10
  4.
 144
Tsunam
 i
 739n
 m
 PP
 Figure 5.2: The  gure shows two trapped ions that are separated by approximately
 1 meter. A Ti:Sapph laser is tuned to 739 nm and directly through a pulse picker,
 (PP), to let through a single pulse on demand. The single pulse is frequency dou-
 bled by non-linear crystal and then sent through a dichroic mirror to pick o the
 remaining 739 nm light which is then sent through a second doubling crystal. The
 resonant pulses are directed onto the two ions and the path lengths are adjusted so
 that the photons from di erent ions arrive at the beam splitter within about 100
 psec of each other. The lens on the right hand side of the ion traps collect the single
 photons resulting from the resonant excitation from the Ti:Sapph and direct them
 into the interferometer. The entanglement of the two ions is heralded when both
 PMTs inside the interferometer detect a photon. The state of the ions is then read
 out using the PMTs on the left hand side of the ion traps.
 145
Once the two ions have decayed, they will have produced photons whose fre-
 quency and polarization are entangled with the internal state of the ions. If the
 photons are then measured in an entangled basis, the ions will be projected into
 an entangled state. As shown in  g. 5.2, a Bell basis measurement is made on the
 photons by setting up collection optics near the trap which couple the scattered light
 into single mode  bers which are directed toward a beam splitter that erases the
 which-way information contained in the momentum vectors of the photons. Two
 PMTs are placed behind the beam splitter and, as we will see, the ions will be
 projected into an entangled state when both PMTs detect a photon. We refer to
 these double detection events as a coincident detection event or the heralding event.
 Because of the di erent decay channels in 171Yb+, polarizers are used to select the
 emitted photons that are  ^ polarized, so that the heralding event only occurs when
 the ions have decayed back into the qubit space instead of into a mixture that in-
 cludes the mf =  1 states as illustrated in  g. 5.3. In the type II protocol, both
 ions recoil from photon emission and therefore the motional state of the ions after
 a scattering event does not reveal any information about which ion emitted which
 photon. This implies that the motional state of the ions is irrelevant and will be
 ignored from this point on. The single mode  bers are aligned as to select out
 photons in the modes ~k1 and ~k2 and due to the selection rules of the D1 line, the
 frequency of the photons can either be equal to the splitting between 2S1=2jF = 0i
 and 2P1=2jF = 1i or the splitting between 2S1=2jF = 1i and 2P1=2jF = 0i. The dif-
 ference between these two frequencies is equal to the sum of the hyper ne splitting
 for the two electronic levels, 12:6 + 2:1 = 14:7 GHz, which is large compared to
 146
0
 1 2
 1/2S
 2
 1/2P
 Figure 5.3: After preparing the ion in an equal superposition of the qubit states,
 the atom is excited to the 2P1=2 state using a  ^ polarized resonant laser pulse with
 su cient bandwidth as to span the hyper ne frequency. If the spontaneously emit-
 ted photon is captured in the single mode  ber, it will pass through a polarizer to
  lter out the   transition photons, thereby created the ion-photon entangled pair
 j0bi+ j1ri.
 147
the natural linewidth of the transition meaning that these two frequencies are well
 resolved and should form a good qubit. We refer to the larger frequency state as
 jbi and the smaller frequency state as jri. Remembering that speci c wavevectors
 and polarizations are singled out for the operation of the gate, the post-scattering
 ion-photon state is,
  3 =  ^
  
ei b;1 j0ijb1i+ e
 i r;1 j1ijr1i
   
ei b;2 j0ijb2i+ e
 i r;2 j1ijr2i
  
; (5.16)
 where the phases included will be determined by the photon energy and the  nal
 path length and the subscripts on the photon modes refers the ~k degree of freedom.
 The operator  ^ is a symmetry operator for the photons that is needed since they
 will be directed through a beam splitter that couples the modes ~k1 and ~k2, thereby
 creating a situation where the indistinguishability of the photons must be taken into
 account. An example of how the symmetry operator enforces the Bose statistics of
 the photons is,
  ^jr1ijb2i =
 1
 p
 2
 (jr1ijb2i+ jb2ijr1i) : (5.17)
 The action of the beam splitter is that of a rotation operator, B^, in the
 n
 ~k1; ~k2
 o
 space and makes the following transformations,
 B^jr1i =
 1
 p
 2
 (jr1i+ jr2i) (5.18)
 B^jr2i =
 1
 p
 2
 (jr1i  jr2i) ; (5.19)
 with the frequency of the photon being irrelevant. We can now calculate the ion-
 148
photon state prior to the photon detection,
  4 = B^ ^
  
~ ~ j00ijb1b2i+ ~ ~ j01ijb1r2i+ ~ ~ j10ijr1b2i+ ~ ~ j11ijr1r2i
  
; (5.20)
 where we have rede ned the coe cients by absorbing the phases present in  3, i.
 e. ~ = ei b;1 . It is straightforward to show that the action of the beam splitter
 ensures that half of the terms in  4 do not contribute to the coincident detection
 events. For example,
 B^ ^jb1b2i =
 1
 p
 2
 (jb1b1i  jb2b2i) ; (5.21)
 showing that either both photons will be found to have momentum ~k1 or they will
 both be found to have momentum ~k2, meaning that the two photons will always
 exit the same port of the beam splitter if they have the same frequency [88, 89]. A
 similar calculation shows,
 B^ ^jr1b2i =  ^
 (jr1i+ jr2i)p
 2
 (jb1i  jb2i)p
 2
 ; (5.22)
 showing that the symmetry operator and the action of the beam splitter commute
 when the photons are not identical. We can then calculate the state after a coincident
 detection event, which we represent with a projection operator P^ that picks out the
 components where the two photons have di erent momenta. The  nal state is then,
  5 = P^ 4 =
  
~ ~ j01i  ~ ~ j10i
  
 ^ (jr1b2i  jb1r2i) ; (5.23)
 149
showing that, when both PMTs detect a photon, the ions are projected into an
 entangled state in general. By carefully keeping track of the excitation time of the
 ions, the phase acquired while in the excited state and the path lengths from the
 ions to the beam splitter, it can be shown [42] that up to an overall phase factor the
  nal state of the ions is,
  5 =   j01i  e
 ic k(te1 te2)ei k x  j10i (5.24)
 where  k = kblue  kred = 2 =(2cm) and tei is the excitation time of ion i. Because
 the length scale of  k is so large, phase stability is not an issue in the type II
 protocol.
 This mapping from  1 to  5 can be represented by the non-unitary operator
  (1)z (I   
(1)
 z  
(2)
 z ) and was shown to be useful for the creation of cluster states that
 might be used as a resource in one-way quantum computation [90] or a quantum
 repeater [37]. This gate was demonstrated using two 171Yb+ ions separated by ap-
 proximately one meter, ( g. 5.2). In order to characterize the gate, we prepared
 a representative set of input states as shown in table 5.1 and measured an average
 output state  delity of 0:89(2) [86]. In that case, the atoms were prepared and hit
 with excitation pulses at a rate of 70 kHz with a coincident event being observed
 approximately every 11 minutes, meaning that the success probability was approxi-
 mately 2 10 8. The low success probability of the gate, Psucc, is a consequence of
 150
Input State Expected State Fidelity Events
 j0 + 1i  j0 + 1i j01i  j10i 0.89(2) 210
 j0 + i1i  j0 + 1i j01i  ij10i 0.86(2) 179
 j0 1i  j0 + 1i j01i+ j10i 0.85(1) 178
 j0 i1i  j0 + 1i j01i+ ij10i 0.81(2) 188
 j0 + 1i  j1i j01i 0.86(5) 42
 j0i  j0 + 1i j01i 0.91(4) 52
 j0i  j1i j01i 0.98(2) 48
 Table 5.1: Results of the remote quantum gate process. Listed are the input and
 expected output states of the gate, the overlap of the measured and ideal state and
 the number of heralding events. The success probability is 2 10 8 and the average
 state  delity is F = 0:89(2).
 several factors,
 Psucc = PBellP
 2
 det = PBell (p  TfiberToptics (  =4 ))
 2 ; (5.25)
 where PBell is the probability that the photons are found to be in the correct Bell
 state and depends on the initialization coe cients  ;  ;  ;  and Pdet is the proba-
 bility of detecting a single photon emitted by an ion. p is the probability that ion
 emitted a photon with the correct polarization and is equal to 1=2 for the 171Yb+;
  = 0:15 is the quantum e ciency of each PMT; Tfiber is the coupling e ciency of the
 single mode  ber to the light that is collected from the ion was estimated to be 20%;
 Toptics = 0:95 is the transmission through the other optical elements and  = 0:995
 accounts for the branching ratio to the 2D3=2 state from 2P1=2;   =4 = 0:02 is
 the fractional solid angle subtended by the collection lens, NA = 0:23. These fac-
 tors multiply together to give Psucc  PBell8:5  10 8. The repetition rate of the
 151
experiment was limited by the microwave initialization pulse of duration 30  sec.
 Using these results, we can estimate how far this system could be taken toward
 developing novel quantum information processors. For example, a quantum repeater
 architecture [37] using deterministic gates between ions within a single node can use
 the type II photonic gate to connect di erent nodes in a time approximately given by
 T  Tsuccln(N 1) with Tsucc being the average time needed to see a coincident event.
 Given that the attenuation in optical  bers for 369 nm limits the distance between
 nodes to a distance of about 10 meters, a 10 km network can be expected to be fully
 connected once every 1:3 hours. Even if the attenuation length is increased by a
 factor of 100, the connection time is still on the order of 24 minutes for this success
 rate, implying that dramatic improvements are needed in both the repetition rate of
 the experiment and the success probability before this platform can be considered
 to be a viable candidate for building a useful quantum communication network.
 We now explore some estimates on what connection time might be possible for this
 system.
 The initialization of the ions begins with an optical pumping routine in which
 an average of 15 photons should be scattered in order to adequately purify the
 spin state, requiring approximately 1  sec. It was recently shown that high power
 o -resonant laser pulses can be used to enact ultrafast single qubit rotations [91],
 which would reduce the time needed to generate a coherent superposition of the
 qubit states to about 10 ps. The biggest uncertainty in the achievable repetition
 rate might come from the time that must be dedicated to keeping the ions at a
 reasonable temperature. For the four-rod trap, the heating rate was measured to be
 152
7 quanta per second. If the ion can be cooled to the Doppler temperature in about
 1 ms and the operations done during the experiment do not add signi cantly to the
 heating rate, then it might be possible to only dedicate 0:1% of the time to cooling.
 These numbers imply that the repetition rate will most likely be limited to 1 MHz
 as limited by the optical pumping time.
 The success probability can also be improved in several ways. Most impor-
 tantly, the fractional solid angle subtended by the collection lens can be increased by
 using a higher numerical aperture lens, re ective optics such as a parabolic mirror
 (see Appendix A) or high  nesse cavity, or refractive optics. If for example, the
 numerical aperture of the lens being used is increased by a factor of 3, the fractional
 solid angle increases by a factor of 9, which in turn increases the success probability
 by a factor of 81. One might also assume an improvement in the e ciencies of
 PMTs by a factor of 2, as was recently reported by Hammamatsu inc., which gives
 an overall factor of 4 improvement in the success probability. These improvements
 would yield a connection time of 1 sec for a N = 1; 000 node quantum repeater
 that could stretch over 10 km.
 5.2.1 Errors in the Type II protocol
 As discussed in [42], two known sources of error in the type II protocol are po-
 larization mixing of the emitted photons and the mode mismatch of the interferome-
 ter. Assuming that the only error is due to the mode mismatch in the interferometer,
 153
it can be shown [42] that the state  delity is given by,
 Fmm =
 1
 2 V 2
 ; (5.26)
 where V is the visibility of the fringe contrast in the interferometer. The visibility
 is de ned in terms of the intensities of incident light within the interferometer as,
 V = Imax IminImax+Imin , where Imax = jE1 + E2j
 2 and Imin = jE1  E2j
 2. This quantity
 is measured by coupling laser light into the single mode  bers used for photon
 collection and measured to be V > 0:98 implying that the mode-mismatch results
 in a loss of  delity of approximately 4%.
 Polarization mixing errors are a result of  + and   transitions, that when
 viewed from an angle that is not in the plane perpendicular to the quantization axis,
 produce photons with a polarization having a component in the same direction as
 photons produced by  transitions. In spherical coordinates,  ^ photons are always
 polarized along the  ^ direction, no matter what the viewing angle is. If a photon
 with  ^ polarization stemming from a  transition is captured by the single mode  ber
 coupled to the ion, it will not be  ltered out by the polarizers in the interferometer
 and could trigger a coincident event with one of the ions being in one of the mf =  1
 states. A detailed calculation shows that the probability of   transition producing
 a  ^ photon is given by [42],
 P ; ( ) =
 R  cos(arcsin( 1 (
  
2  )))
 0 d 
R  =2+ 
 =2 d sin 
3
 8 cos
 2 
R  cos(arcsin( 1 (
  
2  )))
 0 d 
R  =2+ 
 =2 d sin 
 
3
 8 cos
 2 + 38 sin
 2 
 : (5.27)
 154
This equation is the time averaged power of a radiating dipole per solid angle for
   transitions projected onto  ^ integrated over of the solid angle subtended by a
 lens with a numerical aperture NA = sin normalized by the total radiated power
 projected onto  ^. Because the type II protocol heralds the creation of entanglement
 upon detection of two photons, the probability of getting a coincident event that
 stems from the wrong atomic transitions is approximately 2P ; ( ). Using the nu-
 merical aperture NA = 0:23 of the CVI lens used in the experiment, Eq. (5.27)
 can be numerically evaluated to show that the probability of erroneous coincident
 counts is approximately 3%.
 Another source of error comes from the two photons arriving at the beam
 splitter at slightly di erent times. When the photons arrive at di erent times, the
 destructive interference associated with the photons of the same frequency is not
 perfect and can therefore lead to both PMTs in the interferometer registering a
 detection event. This e ect can be quanti ed by considering a modi cation to the
 state in Eq. (5.16), where we now imagine that the photon coming from the second
 ion is delayed with respect to the  rst ion by a time T . We describe the photon
 wavepacket with momentum k^1 as,
  1(t) =  (t)e
   t=2e i!tk^1; (5.28)
 where   1 is the lifetime and ! is the transition frequency. If these wave functions
 are used in the above analysis, one  nds that the probability of making an error due
 155
a temporal o set of the two photon wavepackets is given by,
 P (T ) =
 1 e  T
 2 e  T
  
1 1
 1+(
 !r;b
  )
 2
  ; (5.29)
 where !r;b is the frequency di erence between the red and blue Because the two
 excitation pulses used to excite the two ions in our experiment are derived from the
 same pulse, this equation is more conveniently expressed as a path length di erence
 instead of a temporal di erence. If the expression is then expanded in terms of this
 path length di erence L,
 P (L) =
  
 
c
 +
  3
 c!2r;b
 !
 L+O(L2) (5.30)
 =
 L
 2:4m
 +O(L2): (5.31)
 The path length di erence is matched to approximately 1 cm in the laboratory,
 meaning that the error due to temporal mismatch of the photon modes comes in at
 the 1% level.
 5.3 Bell?s theorem and private random numbers
 The type II experiment with distant ions is not only useful for quantum net-
 working, but may also be used for the generation of private random numbers. The
 notion of private random numbers is a reference to the fact that a person in posses-
 sion of a black box device that is supposed to generate random numbers can never
 156
be absolutely sure that the device is truly random. Even if the numbers produced
 by the device pass every known randomness test available [92], one can never ex-
 clude the possibility that the numbers are stored in a memory inside the device and
 known by an adversary, making them deterministic. It can be shown, however, that
 a device that is capable of violating a Bell inequality [93] can be used to generate
 genuine private random numbers [94, 95]. A Bell inequality is a limit on correlations
 between two or more systems that is imposed by an assumption of local realism. Any
 system that can be described as one that obeys a hidden variable model, like a black
 box with a memory, will not violate a Bell inequality. A particularly simple and
 important Bell inequality is referred to as the CHSH inequality [96], which is named
 after John Clauser, Michael Horne, Abner Shimony and Richard Holt. To formulate
 the inequality we imagine two systems are measured by detectors A and B with
 the measurement settings a and b respectively. If the systems are Boolean-valued
 upon measurement, then we can de ne the two di erent measurement outcomes
 to have numerical values of  1. If this is true, then at least one of the quantities
 B(b;  ) + B(b0;  ) or B(b0;  ) B(b0;  ) should be equal to zero if the measurement
 outcomes only depend on the measurement settings and the possibly hidden vari-
 able  , while the other quantity should be equal to 2. If this is the case, then the
 correlation function,
 A(a;  )B(b;  ) + A(a;  )B(b0;  ) + A(a0;  )B(b;  ) A(a0;  )B(b0;  )
 = A(a;  ) (B(b;  ) +B(b0;  )) + A(a0;  ) (B(b;  ) B(b0;  ))  2: (5.32)
 157
If the expectation value of a random variable X is E(X) =
 R
  X( ) ( )d where  
is the probability density on the hidden parameter space  , then,
 E(A(a)B(b)) + E(A(a)B(b0)) + E(A(a0)B(b)) E(A(a0)B(b0))  2: (5.33)
 Quantum mechanics, however, admits states that are capable of violating this bound
 on local hidden variable models. In particular, if the singlet state j "#i  j #"i is
 measured with the following measurement settings,
 A(a) =  z  I (5.34)
 A(a0) =  x  I (5.35)
 B(b) =
 1
 p
 2
 I  ( z +  x) (5.36)
 B(b0) =
 1
 p
 2
 I  ( z   x); (5.37)
 a simple calculation shows that the expectation value of the correlation function in
 Eq. (5.33) is equal to 2
 p
 2. As shown in [95], the degree to which this inequality
 is violated can be used to quantify the amount of private randomness that is being
 generated by the system. If the two observed systems are assumed to satisfy the
 laws of quantum theory, separated and non-interacting during each measurement
 and the two measurement settings (x; y) are generated by a random process then
 the randomness of the measurement outcomes as quanti ed by the min-entropy is
 bounded by,
 H1  nf(I^   ); (5.38)
 158
with probability greater than 1   where  = O(
 p
  log =q2n) is a statistical pa-
 rameter and q = minx;y P (x; y) is the probability of the least probable measurement
 setting pair. The parameter n is the number of measurements made on the system,
 the value I^ is the observed expectation value of the correlation function in Eq. (5.33)
 and the function f is found using semi-de nite programming and shown in  g. 5.4.
 The experimental setup described in Sec. 5.2 satis es all the requirements
 needed to realize this random number generator experimentally; namely that the
 two entangled particles can be assumed to be non-interacting during the detection
 process and near perfect detection of every event, i.e. the so-called detection loop-
 hole is closed. Note that the particles do not need to be space-like separated and
 the user must only be able to determine that the two subsystems are not capa-
 ble of interacting. Because the ions are separated by such a large distance in the
 experiment, their mutual interaction can be safely assumed to be negligible.
 The random numbers are then generated by using two 171Yb+ ions along with
 the type II heralded entanglement protocol. The measurement settings are set by
 applying coherent microwave rotations after the heralding event and before the
 detection process. By using random number generators, the measurement settings
 were chosen from an even distribution, (P (x; y) = 1=4), so as to minimize the
 number of runs required to obtain a meaningful bound on the output entropy. After
 running the experiment for approximately one month, a total of 3; 016 entanglement
 events were observed, resulting in a measured value for the correlation function of
 I^ = 2:414  0:058, see table 5.2 for a summary of the measurement results. The
 observed CHSH violation implies that at least H1 > 42 new random bits were
 159
1.2
 1.0
 0.33
 M
 in-
 en
 tro
 py bound f(I) (bits
 )
 CHSH Expection I
 2.1         2.2         2.3         2.4         2.5         2.6         2.7        2.8
 Figure 5.4: A plot of the function f(I) which bounds the amount of randomness
 generated in a device that violates a Bell inequality. The function f(I) (curve
 a) is found using semide nite programming, which is a numerical method that is
 guaranteed to converge to the exact result. Curve b is an analytic lower-bound
 derived in [95] and has the form f(I)   log2
  
1 log2
  
1 +
 q
 2 I
 2
 4
   
. Curve c
 is bound on the entropy implied solely by the no-signalling principle and has the
 form f(I)   log2 (3=2 I=4). By assuming the no-signalling principle, we assume
 that the devices cannot be used for arbitrary fast communication. The function
 f(I) starts at zero at the local threshold I = 2 and implies a positive min-entropy
 for systems that violate the CHSH inequality.
 160
generated in the experiment with a 99% con dence level.
 (x; y)  x;  y N(0; 0) N(0; 1) N(1; 0) N(1; 1) Total P (a = bjxy)
 0,0 0 ; 45 293 94 70 295 752 0.782
 0,1 0 ; 135 298 70 74 309 751 0.808
 1,0 90 ; 45 283 69 64 291 707 0.812
 1,1 90 ; 135 68 340 309 89 806 0.195
 Table 5.2: Observed number of events N(a; b) for which the measurement on one
 atom gave outcome a and the measurement on the other atom gave outcome b,
 given the binary choices of the measurement bases (x; y) corresponding to  =2 qubit
 rotations with phase angles ( x;  y) on the equator of the Bloch sphere. The last
 column gives the fraction of events where a = b given each input. If the experiment
 is interpreted as consisting of identical and independent realizations, the data then
 indicate a CHSH observable of I^ = 2:414 0:058:
 While it may sound circular to require that the measurement settings be chosen
 randomly, the small random seed that is used for the input measurement settings can
 be concatenated with the output of the machine, thereby establishing a randomness
 expansion scheme. When the number of measurements becomes large enough, a
 catalysis e ect is possible wherein a seed string of length O(
 p
 nlog
 p
 n) produces a
 much longer random output string of entropy O(n).
 As a  nal remark, since the publication of Bell?s theorem in 1964, experimen-
 talists have been hard at work trying to con rm that the universe does not obey
 the traditional notion of local realism [97, 98, 99, 100, 101, 102, 103, 104, 105]. All
 of the experiments cited here, including our experiment, have indeed violated a Bell
 inequality, but none have ever been able to do so in a loop-hole free manner. Most
 physicists consider the matter to be settled in light of the di erent loop-holes being
 closed in di erent instances and have accepted that local realism is in fact untenable
 161
M
 in
 -e
 n
 tr
 o
 py boun
 d, H (bits
 )
 200
 100
 120
 140
 160
 180
 80
 60
 40
 20
 Number of runs, n
 2,000 2,500 3,000 3,500 4,000
 Figure 5.5: The plot shows a bound on the minimum entropy produce versus the
 number of trials n for an observed CHSH violation of I^ = 2:414 and con dence
 level 1   = 99%. The bound on the entropy implied by quantum theory is for a
 uniform choice of inputs [P (x; y) = 1=4] (curve a) and for a biased choice of inputs
 P (00) = 1 3q, P (01) = P (10) = P (11) = q, where q =  n 1=2 with  = 11 (curve
 b). For a given number n of uses of the device, the uniform choice of inputs leads
 to more randomness in the outputs. On the other hand, biased inputs require less
 randomness to be generated, and the net amount of randomness produced (given
 by the di erence between the output and the input entropy) becomes positive for
 su ciently large n. Curve c is the bound on the entropy implied by the no-signalling
 principle alone for a uniform choice of inputs. The inset shows the net amount of
 entropy produced for the biased choice of inputs with the observed CHSH violation.
 162
with the laws governing the universe. On the other hand, it has been pointed out
 [106] that the lack of a loop-hole free violation of Bell?s inequality after 40 years
 of trying might be akin to the search for perpetual motion machines in the 18th
 and 19th centuries. In that case, decades of trial and failure eventually led to the
 discovery of the second law of thermodynamics. Could it be that the mounting
 decades of searching for a system capable of violating the principle of local realism
 will eventually be considered as empirical evidence of the principle?s sanctity and
 that quantum theory will have to be tweaked? Until the data comes in, the question
 remains open.
 163
Chapter 6
 Integrating Phonon and Photon based entangling operations
 6.1 The Quantum Repeater
 A quantum network that is spatially distributed will su er from in- ight de-
 coherence. If photons are used as  ying qubits, absorption and scattering will cer-
 tainly be sources of decoherence. If two quantum registers are connected through
 a photonic channel that has an attenuation length  , (given in dB/meter), then
 the probability of successfully transmitting a photon through the channel is e  l0
 where l0 is the length of the channel, a problem that will have to be addressed when
 constructing a large scale quantum communication network. A quantum repeater
 operates by building up entanglement between separate nodes and then using en-
 tanglement swapping measurements to connect the ends of the chain. A three node
 quantum repeater comprised of trapped ions and photons is shown schematically
 in  g. 6.1. As described in [37] the two most distant ions in the chain, (L1 and
 R2), can be be entangled using a combination of the probabilistic photonic gate dis-
 cussed in chapter 5 and the deterministic gate described in chapter 4. The protocol
 begins by preparing all of the ions in 50-50 superposition states and then exciting
 all of them simultaneously with resonant laser pulses. In the event that both PMTs
 behind one of the beam splitters register an event, two of the three nodes will then
 be connected. Because of the low success probability of the type II photonic en-
 164
Resonant Pulse O-Resonant
 Raman Beams
 L 1 L 2 M1 M2 R1 R2
 BS BS
 Figure 6.1: A quantum repeater architecture using ion and photons. In this hybrid
 approach, di erent nodes of the repeater are connected using a probabilistic photonic
 connection and the entanglement within the nodes is generated using deterministic
 phonon interactions.
 165
tanglement protocol, it is extremely unlikely that both pairs of PMTs will register
 coincident detection events at the same time. But assuming that the other ions
 can be reinitialized without disturbing the entangled pair, the photonic gate can be
 repeatedly attempted on the unentangled ions until they are also connected. In the
 case that the initial  =2 pulses are done with the same phase, the resulting state of
 ions L1;M1;M2; and R2 after two coincident detection events is ideally,
  1 = (j "#i  j #"i) (j "#i  j #"i) ; (6.1)
 where the ordering of the qubits is jL1;M1;M2; R2i. If an entangling gate of the
 M lmer-S rensen type is then applied to the middle qubits, the resulting state is,
  2 = j "L1iU
 M1;M2
 ms j #M1"M2ij #R2i
  j "L1iU
 M1;M2
 ms j #M1#M2ij "R2i
  j #L1iU
 M1;M2
 ms j "M1"M2ij #R2i
 + j #L1iU
 M1;M2
 ms j "M1#M2ij "R2i (6.2)
 = j "L1i
  
ei =4 (j #M1"M2i  ij "M1#M2i)
  
j #R2i
  j "L1i
  
ei =4
  
j #M1#M2i  ie
 i2 m j "M1"M2i
   
j "R2i
  j #L1i
  
ei =4
  
j "M1"M2i  ie
  i2 mj #M1#M2i
   
j #R2i
 + j #L1i
  
ei =4 (j "M1#M2i  ij #M1"M2i)
  
j "R2i: (6.3)
 166
Ignoring the global phase ei =4 allows the state to rewritten as,
  2 =  j #M1#M2i
  
j "L1"R2i  ie
  i2 mj #L1#R2i
  
+ j #M1"M2i (j "L1#R2i  ij #L1"R2i)
  ij "M1#M2i (j "L1#R2i+ ij #L1"R2i)
 + iei2 m j "M1"M2i
  
j "L1"R2i+ ie
  i2 mj #L1#R2i
  
; (6.4)
 showing that upon measurement of M1 and M2 the ions L1 and R2 are projected
 into a maximally entangled state.
 6.1.1 2 + 1 Quantum Repeater Elementary Unit
 As discussed in Sec. 6.1, an ion-photon based quantum repeater may be con-
 structed through the combination of probabilistic photonic gates and deterministic
 Coulomb interaction based gates. The minimum setup required to demonstrate the
 feasibility of such a setup is with two ion traps, one containing one ion and the
 other containing two ions, which we refer to as the 2 + 1 experiment. We refer to
 the two traps as the east and west trap respectively. All three ions are initially
 Doppler cooled and then initialized to the state j000i, where the ordering of the
 ions is fWest; East1; East2g. All three ions are then subjected to  =2 pulses, fol-
 lowed by a single resonant pulse to excite the atoms to 2P1=2. All three atoms will
 then spontaneously emit photons, but only one of the two ions, say E1, in the east
 trap is coupled to a single mode  ber that directs the photon to the Bell state mea-
 167
surement apparatus described in chapter 5. Therefore, upon coincident detection of
 two photons, the the W and E1 ions are projected into an entangled state and the
 state of the third ion, E2, is mixed since the photon it emitted was not detected
 and is likely of the wrong polarization. Therefore, in order to purify the state, E2
 is then optically pumped to the j0i state with a tightly focused beam that does not
 signi cantly e ect the state of E1. After these steps, the ions should ideally be in
 the state,
  1 =
  
j "#i  ei j #"i
  
j #i; (6.5)
 where  is determined by the relative phase of the  =2 pulses applied in the initial-
 ization stage. In the next step, the M lmer-S rensen gate is applied to the two ions
 in the east trap, which should ideally produce the state,
  2 = j "i
  
j ##i  iei2 mj ""i
  
 ei j #i (j "#i  ij #i) ; (6.6)
 showing that a measurement in the  z basis should show a correlation of the spin
 of the single ion in the west trap with the parity of the two ions in the east trap.
 6.1.2 2 + 1 Experiment
 This correlation between the spin of one ion and the parity of two distant
 ions was demonstrated using two ions in a four-rod trap and one ion in a blade trap
 approximately one meter away. As depicted in  g. 6.2, the blade trap is constructed
 from four alumina blades that are gold coated. Similar to the four-rod trap, two
 168
opposing blades have a large RF voltages applied to them while the other two are
 RF grounded through shunting capacitors. The center segments of the RF grounded
 blades are held at a lower voltage compared to the outer segments to provide a trap
 in the z direction, as is done with the needles in the four-rod trap.
 The experiment consists of two loop sequences which are referred to here as
 the fast loop and the slow loop. As depicted in the  ow diagram in  g. 6.3, the
 fast loop?s aim is to initialize the ions and attempt to establish the photonic link
 between the two traps. This is done by  rst Doppler cooling all three ions for
 50  sec, optically pumping the state to j000i, applying a  =2 pulse on all three ions
 and then applying a resonant 369:5 nm pulse to both traps simultaneously. The
 single mode  ber is only coupled to one of the ions in the four-rod trap, meaning
 that a coincident event registered in the interferometer heralds the entanglement of
 two of the three ions while leaving the third ion in a highly mixed state. In the event
 that no coincident count occurs, the ions are repumped to j000i, rotated by  =2 and
 excited with another resonant pulse. After repeating this sequence 20 times, the
 ions are recooled for 50  s. A coincident event triggers the exit of the loop sequence
 and begins the sequence for the implementation of the M lmer-S rensen gate.
 After exiting the loop sequence, the  rst step is to purify the spin state of
 the third ion that is left in a mixed state after being excited by the resonant pico-
 second pulse. This is done using an optical pumping beam that enters through the
 objective lens as shown in  g. 6.4. It is crucial that this beam be focused as tightly
 as possible as to not disturb the spin state of the other ion in the trap. This was
 done by using light from a single mode  ber and a high quality output coupler from
 169
Figure 6.2: The blade trap mounted in the vacuum chamber and an up close drawing
 of the four blades. The high voltage RF is applied to the blades that are completely
 coated with gold with the other two blades being RF grounded and attached to
 DC voltage supplies. The  ve di erent electrodes on the DC blades allow for the
 possibility of creating anharmonic trapping potentials that could be used for splitting
 ion chains or to arrange ion crystals to have an even spacing.
 170
D
 op
 pl
 er
  C
 oo
 lin
 g
 50
   
sec?
 O
 pt
 ic
 al
  P
 um
 pi
 ng
 5
 sec?
 P
 ic
 os
 ec
 on
 d
 P
 ul
 se
 C
 oi
 nc
 id
 en
 ce
 ev
 en
 t?
 no
 ye
 s
 M
 ol
 m
 er
 ga
 te
 25
 0
 sec?
 D
 et
 ec
 tio
 n
 1
 msec
 sec?
 8
 pu
 ls
 e
 2/?
 x 
20
 In
 di
 vi
 du
 al
  P
 um
 pi
 ng
 20
 sec?
 fa
 st
  lo
 op
 sl
 ow
  lo
 op
 Figure 6.3: This  owchart shows how the 2 + 1 experiment is done. Because of the
 low photon collection e ciency, the fast loop is repeated approximately 108 times
 before a coincident event is registered.
 171
Micro Laser Systems, Inc. The divergence of the output mode was measured and
 used in optical optimization software in order to calculate how far from the objective
 lens the  ber should reside. In order to test this individual optical pumping setup,
 two ions loaded in the trap were prepared in the state j11i and subjected to this
 tightly focused pumping beam. As shown in Fig. 6.5, the ion that is targeted
 by the beam pumps to the dark state within 50  s while the second ion pumps
 down in 50 ms. In order to purify the spin state to the 1% level, the ion must
 scatter about 10 photons. Given that the pumping rates of the two ions in the
 trap di er by a factor of 1000, the ion that is not targeted will scatter a photon
 in about 1  2% of the experiments. The axial secular frequency was measured to
 be 300 kHz meaning that the separation between the two ions is ( e
 2
 2m!2z
 )1=3 = 5  m.
 Assuming that the targeted ion sits at the center of a Gaussian pumping beam with
 an electric  eld pro le proportional to e 
1
 2 (
 x
  )
 2
 , we can infer that the width of the
  eld is approximately   5p
 2ln(1000)
 = 1:3  m. Given that the intensity and not
 the electric  eld is usually the measured quantity, we say that the beam waist is
 approximately 2  m.
 The M lmer-S rensen gate is then applied in a way that we have not described
 yet. All of the discussion of the MS gate have thus far assumed the excitation of a
 single mode of motion, which is only valid if the gate speed is limited to be larger
 than  ! 1, where  ! = !com  !tilt is the frequency di erence between the two
 transverse modes of motion. As discussed in chapter 2, the frequency di erence
 between these two modes increases with the axial con nement frequency, implying
 that the gate speed can be increased by pushing the ions closer together which is
 172
Vanguard
 355nm Tsunam
 i
 739n
 m
 PP
 (a)
 St
 at
 e de
 te
 ctio
 n
  
 
PM
 Ts
 Interferometer
 369 nm optical 
pumping light
 (b)
 5%
 Figure 6.4: (a) A schematic of the optical setup for the 2 + 1 experiment. One
 ion is loaded into the blade trap and two are loaded into the four rod trap that
 is approximately 1 meter away. The Vanguard laser is split in half and used in a
 co-propagating geometry for the West trap, ( g. 3.5), and in a counter-propagating
 geometry for the east trap, ( g. 3.7). The entanglement swapping measurement is
 heralded when a coincident detection events is registered on the two PMTs inside
 the interferometer and the state of the of the ions is read out on the other PMTs. (b)
 The individual addressing optical pumping beam is brought in through a single mode
  ber and directed onto the imaging lens using a 5% beam sampler and focused onto
 one of the ions in the four-rod trap. A 5% beam sampler is used so as to minimize
 the amount of light from the ion that is lost during state detection.
 173
 
 
 
 
 
 
 
    
  
             
 
  
  
        
 
    
 
0.00 0.02 0.04 0.06 0.08 0.10Time H msecL
 0.5
 1.0
 1.5
 2.0
 Ion Brightness
  
   
 
 
  
 
 
 
 
  
 
 
 
 
  
 
 
 
 
 
 
   
 
 
 
  
 
 
  
  
 
 
      
  
 
  
  
   
  
       
 
    
  
 
  
 
 
           
 
        
 
0 10 20 30 40 50 Time H msecL
 0.5
 1.0
 1.5
 2.0
 Ion Brightness
 Figure 6.5: The two data sets show identical experiments done over di erent scales.
 Two ions are loaded into the four-rod trap and prepared in j11i. The individual
 addressing optical pumping beam is then turned on for a variable amount of time
 before detecting the state of the two ions using a simple discriminator method.
 Notice that the  rst data set shows that one of the ion pumps to the dark state in
 approximately 40  s while the second data set shows that the second ion eventually
 pumps to the dark state in approximately 40 ms, (note the  rst data point in the
 second set). With the ions being 5  m apart, the 3 orders of magnitude discrepancy
 in the pumping times for the two ions implies a beam waist of approximately 1:3  m.
 174
at odds with the desire for the ions to be far apart so that the individual pumping
 beam does not decohere the spin that is entangled with the ion in the other trap.
 Because of these opposing constraints, we implement the M lmer-S rensen gate
 in a regime that excites both the COM and the tilt mode simultaneously. If the
 frequency di erence between the two modes is  , then one can drive the COM
 mode and the tilt mode o -resonantly with symmetric but opposite detunings so
 that both phase space trajectories close at the end of the gate sequence. If the
 di erence frequency between the two modes is  , then the symmetric detuning  that
 appears in the MS time-evolution operator is set to  com =  =2 and  tilt =   =2.
 Besides allowing for further separated ions, this con guration has the added bene t
 of being able to entangle two ions in a shorter time for a given laser power. This
 is because the accumulated spin-dependent phase is proportional to the motional
 phase space area swept out and this con guration essentially doubles the area since
 two motional modes are involved. It should be noted, however, that the center of
 mass mode is generally more susceptible to heating [67]. In the experiment, the
 frequency di erence between the two modes was set to 16 kHz, and a composite
 pulse of two 125  s pulses is used to for the MS gate. To keep the repetition
 rate of the experiment high, high  delity in the gate operation is sacri ced by not
 sideband cooling. When the ions are sideband cooled to the ground state, M lmer-
 S rensen state  delities of & 0:9 are routinely seen and  delities as high as 0:97
 have been recorded. When the ions are only Doppler cooled the  delity is usually
 limited to . 0:8 and can sometimes drift to lower values near 0:7 on the time scale
 of minutes. The decrease in the maximum  delity for the Doppler cooled ions is
 175
likely due to the e ective increase in the Lamb-Dicke parameter  
p
  n+ 1 and a
 higher sensitivity to  uctuations in the trapping frequency. In the case of a single
 mode MS interaction, the decrease in  delity due to the deviation from the Lamb-
 Dicke regime is  4  
2
 4 Var(n1) [107]. For an exponential distribution, the variance is
 equal to the square of the mean, implying that this error comes in at the 2  3%
 level for the Doppler temperature but is likely to be much higher in the 2 + 1
 experiment due to the lack of su cient laser cooling power, the cooling duty cycle
 of the slow loop and additional heating from the optical pumping and ps pulses. A
 second additional source of in delity is likely due to an increased sensitivity to trap
 frequency  uctuations when the ions are not cooled to the ground state.
 The experiment was run for approximately 30 hours, not including time that
 is taken to recalibrate the system and to reload ions after a loss, and a total of
 228 coincident events were seen. As shown in  g. 6.6, the single-ion spin in the
 blade trap was found to be correlated with the parity of the two ions in the four-
 rod trap. By using a discriminator method for the state detection of the single
 ion in the west trap, the measurement record for the other two ions was separated
 into two distributions depending on the state of the west ion. By  tting sample
 detection histograms to the measurement records for the two cases, we can estimate
 the populations of the two ions and  nd that when the west ion is bright, the two
 ions in the east trap are found in an even parity state with a probability of 0:63 0:04
 and when the west ion is dark, the two ions are found in an odd parity state with
 a probability of 0:67 0:04.
 The 65% spin-parity correlation that was observed is consistent with the main
 176
Figure 6.6: The results of the 2+1 data are separated into two sets according to the
 observed state of the West ion. When the west ion was found in the bright state,
 the two east ions were found to be in an even parity state with 63(4)% probability
 and when the west ion was found in the dark state, the two east ions were found to
 be in an odd parity state with 67(4)% probability.
 known sources of error. Some preliminary photon interference data was taken with
 one ion in the blade trap and one in the four-rod trap and the ions were found
 to be anti-correlated in 85% of the coincident events. This  gure is slightly lower
 than the 90% anti-correlation that was reported in [86], which likely the result
 of two problems. First, the pulse-picker used in the 2 + 1 experiment was not
 correctly calibrated and is observed to oscillate after being opened on the timescale
 of microseconds. Secondly, the micro-motion of the ion in the blade trap could not
 be fully nulled which was likely due to a shorted electrode in the trap. In order to
 have good photon interference at the beam splitter, it is crucial that the ion not
 have any micromotion in the direction in which it must emit light to couple to the
  ber. Otherwise, the emitted photons will be Doppler shifted and aquire sidebands
 at the RF trapping frequency. We were able to null the ion?s micromotion in the
 direction of the photon emission, but only at the cost of increasing the micromotion
 177
amplitude in the direction of the beam that is used for state detection. This led to
 the state detection error in the west trap to be approximately 5%. The M lmer-
 S rensen gate  delity for the bimodal setup without sideband cooling was usually
 observed to be 75% 80% while the experiment was running. Assuming these errors
 are uncorrelated, we expect the  delity of the 2 + 1 experiment to be approximately
 0:8 0:85 = 0:68 if no other errors are present.
 6.2 Conclusions and Outlook
 The data shown in this  nal chapter represent the  rst steps toward connecting
 two distant trapped ion registers in an architecture that could be useful for scalable
 quantum computation. In order to fully demonstrate the coherence between the
 two registers, it will be necessary for the control  elds of the two registers to be
 phase locked, requiring either interferometric stability of the di erent laser paths in
 the experiment, or phase insensitive gate schemes coupled with microwave controls
 [108]. However, for the system to be seriously considered for a distributed quantum
 computing network, there are two things that need to be greatly improved. First,
 the  delity of the two gates that are used must be improved. The M lmer-S rensen
 gate performance can most likely be greatly improved by moving to a stronger
 trapping potential. A stronger trapping potential will result in a smaller variance
 in the initial motional excitation number and provide greater protection from stray
 couplings to other transitions. The main sources of error in the photonic gate were
 light leakage through the pulse-picker and mode mismatch in the interferometer. A
 178
better extinction ratio for the pulse-picker might be achieved using a lock similar
 to the one demonstrated in [109]. The mode mismatch in the interferometer might
 be addressed using adaptive optics such a deformable mirrors in combination with
 wavefront imaging to optimize the system [110, 111, 112]. The second outstanding
 issue that needs to be addressed is the success rate of the photonic gate. This
 issue can be addressed on several fronts, including photon collection e ciency which
 is most easily improved by moving to high numerical aperture lens, but must be
 considered in the context of polarization mixing. Increasing the repetition rate of the
 experiment is currently limited by the optical pumping time which can not be much
 shorter than the microsecond time scale. One possible way around this limitation is
 to use many ions that are prepared o -line and then sequentially shuttled into the
 trapping zone that is coupled to the interferometer. Because the type II photonic
 gate is nearly insensitive the motional state of the ion, the shuttling does not have
 to be limited by any adiabatic time scales imposed by the trap.
 Assuming these improvements can be made, one must eventually confront
 another technical challenge stemming from the addressing errors in the application
 of the resonant pulses used to generate single photons. In order to allow for long
 computations in a distributed ion-trap processor, the quantum information stored
 in the registers must be protected from the decoherence that would be induced by
 the resonant pulses used to generated the single photons. Because the success rate
 is so low and we require many resonant pulses to establish the photonic link, any
 cross-talk to untargeted ions will destroy the stored information. One attractive
 method around this problem is to use one ion for the memory qubits and a second
 179
ionic species for establishing a photonic link. Since the transition frequencies can
 vastly di erent, the absorption probability for the memory ions will be drastically
 reduced. Phonon based quantum logic gates between di erent ion species have
 been already been demonstrated using 27Al+ and 9Be+ in spectroscopy experiments
 [113]. Although the quantum logic used in these experiments is more similar to that
 proposed by Cirac and Zoller [19], it should also be possible to implement the more
 robust M lmer-S rensen gate between di erent species. While there may be pitfalls
 in the road ahead, the road exists and there is a visible horizon. The construction
 of a fully functional ion-photon network will be a di cult task and may require a
 little luck, but as Thomas Je erson said, \I?m a big believer in luck, and I  nd the
 harder I work the more I have of it."
 180
Appendix A
 Paraboloidal Ion Trap
 This appendix shows that a paraboloidal electrode con guration is capable of
 creating a trapping region located at the focus of a paraboloid.
 The trap design to be described is perhaps closest in spirit to a ring-and-fork
 quadrapole trap where an AC voltage is applied to a ring electrode with cylindrical
 symmetry and capping DC voltages are applied to electrodes on the top and bottom
 of the ring. Our design shown in Fig. A.1 consists of a paraboloidal surface seg-
 mented as to form three distinct electrodes with cylindrical symmetry. Similar to
 the ring-and-fork trap, an AC voltage is applied to the ring shaped middle electrode
 and then DC capping voltages are applied to the electrodes above and below the
 ring. Unlike the ring-and-fork trap, there is an asymmetry in the direction of the
 axis of symmetry which will be re ected in the form of the potential. This implies
 that the minimum of the trapping potential will not in general be located at the
 center of the AC electrode.
 Constraints on the dimensions of the di erent segments of the trap can be
 found by requiring the trapping potential minimum to be located at the focus of
 the paraboloid. These constraints are found by solving Laplace?s equation with the
 181
RF DC
 Figure A.1: The proposed paraboloidal ion trap design is shown schematically, with
 the DC voltages being applied to the top and bottom regions and an AC voltage
 being applied to the middle region.
 potential being piece-wise constant on the conducting surface de ned by,
 z( ; f) =
  2
 4f
  f: (A.1)
 Laplace?s equation is separable in paraboloidal coordinates de ned as,
 x =   cos (A.2)
 y =   sin (A.3)
 z =
 1
 2
 ( 2   2); (A.4)
 with 0   <1, 0   <1,   <    . In these coordinates, paraboloids with
 a1
 a2
 b0
 z
 r
 - 1.5 - 1.0 - 0.5 0.5 1.0 1.5
 - 1.5
 - 1.0
 - 0.5
 0.5
 1.0
 Figure A.2: The parabolic coordinates  0,  1 and  2 are shown.  0 de nes the
 surface of the ion trap while  1 and  2 de ne the segmentation dimensions. The
 points of intersection de ne the Cartesian coordinates z1 and z2.
 182
an axis of symmetry along the z axis are described by surfaces of constant  or  .
 If the electrodes are symmetric with respect to the polar angle  , the potential does
 not depend on  and the general solution to Laplace?s equation is given by,
  int( ;  ) = M J0(  )I0(  ) (A.5)
  ext( ;  ) = M J0(  )K0(  ) (A.6)
 with M being an arbitrary non-negative constant determined by the boundary
 conditions [114]. J0 is a Bessel function of the  rst kind and I0 and K0 are modi ed
 Bessel functions of the  rst and second kind. In general, the potential will be a
 super-position of the these and for simplicity we only write the solution for the
 interior region,
  ( ;  ) =
 Z 1
 0
 d M J0(  )I0(  ): (A.7)
 The average potential seen by the ion is calculated by separately considering
 the contributions of the AC and DC potentials. On the boundary  =  0 =
 p
 2f ,
 the application of the AC voltage to the central segment while grounding the top
 and bottom and segments gives a boundary potential of the form,
  AC( ;  0) = V ( (   1)  (   2))cos t; (A.8)
 where V is a voltage,  is the drive frequency,  is the Heaviside function and  1
 and  2 are the positions of the segment cuts in parabolic coordinates as shown in
 183
 g.A.2. To solve for M , we use Hankle?s integral relation,
 f(r) =
 Z 1
 0
 J ( r) d 
Z 1
 0
 f( )J (  ) d : (A.9)
 Application of Eq. (A.9) with the boundary conditions stated in Eq. (A.8) gives,
 M = V cos t
 Z  2
  1
 d   
J0(  )
 I0(  0)
 : (A.10)
 The integral over  can be done, giving the following exact formula for the amplitude
 of the AC potential,
  AC( ;  ) = V
 Z 1
 0
 d  
J0(  )I0(  )
 I0(  0)
 (  1J1(  1) +  2J1(  2)): (A.11)
 If the DC potential is de ned by the application of a common voltage on the top
 and bottom of the paraboloid and grounding the center segment so that the DC
 boundary potential is given by,
  DC( ;  0) = U( ( 1   ) +  (   2)); (A.12)
 then a similar analysis shows that the potential in the interior region due to the DC
 voltages is given by,
  DC( ;  ) = U
 Z 1
 0
 d  
J0(  )I0(  )
 I0(  0)
 ( 1J1(  1)  2J1(  2)): (A.13)
 184
We now solve for the conditions on  1 and  2 to ensure the ion is trapped at
 the focus of the paraboloid. For the pseudo-potential to have a minimum at the
 focus, the amplitude of the AC potential must have an extremum at the focus [38].
 Setting x = y = 0 and requiring the derivative with respect to z to be zero in the
 limit of z ! 0 gives,
 Z 1
 0
 d  2
  1J1(  1)  2J1(  2)
 I0(  0)
 = 0: (A.14)
 Eq. (A.14) gives the required relationship between  1 and  2 in order for the
 parabolic ion trap to have a minimum at the optical focus. This transcendental
 equation must be solved numerically or graphically, with the former method being
 illustrated in  gure A.3. To satisfy Eq. (A.14), one can plot the function s(z)
 de ned as,
 s(z) =
 Z 1
 0
 d  2
  J1(  )
 I0(  0)
  
 
 
 
 =
 p
 2z+ 20
 ; (A.15)
 and then graphically  nd two segment positions z1 and z2 by  nding two intersection
 points of s(z) and a horizontal line. As Fig. A.3 implies, there are an in nite number
 of ways to segment the paraboloidal ion trap so as to give a potential minimum at
 the optical focus. This allows one to choose a segmentation that optimizes the
 characteristics of the potential and geometry of the electrodes according to any
 desiderata. For example, in order to have optical access to the ion, one might design
 a trap so that the gap between the middle and top segments is at the same height
 as the focus. Fig. A.3 also shows that the trapping region can be placed above the
 185
-0.5 0.0 0.5 1.0 1.5 2.0
 0.0
 0.5
 1.0
 1.5
 z b02
 b 0
 2 s
 HzL
 Figure A.3: The dimensionless function  20s(z) is plotted as well as the horizontal line
 used to  nd values of z1; z2 that give rise to an extremum in the trapping potential
 at the focus of the paraboloid. The particular line drawn in the  gure gives a value
 of z2 that is equal to the focal length, meaning that one of the cuts in the paraboloid
 will be at the same height as the trapping region. Having z2 be at the same height
 as the trapping region would allow one to gain laser access to the ion without having
 to drill additional holes in the paraboloid.
 middle segment or within the middle segment, but never below the middle segment.
 186
Appendix B
 Probability amplitude for Raman transitions in a three-level atom
 using an optical frequency comb
 Assume the dipole operator  (r) couples j1i to j3i and j2i to j3i. If the solution
 to the time-dependent Schr odinger equation is expanded in the eigen-basis of H0 the
 probability amplitudes are governed by the following equations of motion,
  =
 X
 n
 cn(t)jni (B.1)
 _c1 + i!1c1 =
 i
 2
 X
 k
 ~Ek
  
ei!kt + e i!kt
  
 1;3c3 (B.2)
 _c2 + i!2c2 =
 i
 2
 X
 k
 ~Ek
  
ei!kt + e i!kt
  
 2;3c3 (B.3)
 _c3 + i!3c3 =
 i
 2
 X
 k
 ~Ek
  
ei!kt + e i!kt
  
( 3;1c1 +  3;2c2) ; (B.4)
 where ~Ek = Ek=~. Now make the following transformations:
 ~c1 = c1 (B.5)
 ~c2 = c2e
 i!hf t (B.6)
 ~c3 = c3e
 i!ct; (B.7)
 187
where !hf = !2  !1. The new equations of motion are,
 _~c1 =
 i
 2
 X
 k
 ~Ek
  
ei!kt + e i!kt
  
 1;3~c3e
  i!ct  i!1~c1 (B.8)
 _~c2 =
 i
 2
 X
 k
 ~Ek
  
ei!kt + e i!kt
  
 2;3~c3e
  i(!c !hf )t  i(!2  !HF )~c2 (B.9)
 _~c3 =
 i
 2
 X
 k
 ~Ek
  
ei!kt + e i!kt
   
 3;1~c1e
 i!ct +  3;2~c2e
 i(!c !hf )t
  
 i(!3  !c)~c3:
 (B.10)
 These transformations get rid of the fast oscillations in the probability amplitude
 for the excited state. If the laser is far from resonance so that the population of
 the excited is small, this transformation allows the adiabatic approximation _~c3  0.
 Using this approximation and de ning !1  0 \sea-level," the equations become,
 _~c1 =
 i
 4 
X
 k;k0
 ~Ek ~Ek0
  
ei!kt + e i!kt
   
ei!k0 t + e i!k0 t
  
j 1;3j
 2
  
~c1 + ~c2e
  i!hf t
  
(B.11)
 _~c2 =
 i
 4 
X
 k;k0
 ~Ek ~Ek0
  
ei!kt + e i!kt
   
ei!k0 t + e i!k0 t
  
j 2;3j
 2
  
~c1e
 i!hf t + ~c2
  
;
 (B.12)
 where  = !3 !c. In order to proceed, we want to make a rotating-wave approxi-
 mation, but is this allowed with a frequency comb? In the equations of motion for
 ~c1 and ~c2 there are terms that look like ei(2!c+(k+k
 0)!R)t. In our case !c is hundreds of
 THz, while !R is around 100 MHz. In general, this term can not be thrown out, but
 in our case we can use the shape of the pulse to argue for the validity of the RWA. In
 188
our case f(t) has a width of around 1 picosecond. This means that ~f(!) has a width
 of about 1 THz. The term ei(2!c+(k+k
 0)!R)t oscillates rapidly unless !R(k k0)   2!c,
 but the coe cients ~Ek ~Ek0 die o exponentially. ~Ek / ~f(k!R). The stationary terms
 are around k  !c=!R which have coe cients ~Ek / ~f(!c)  ~f(800THz)  0, allow-
 ing these terms to safely be ignored. In this approximation, the equations of motion
 become,
 _~c1 =
 i
 4 
X
 k;k0
 ~Ek ~Ek0
 h
 ei(k k
 0)!Rt + e i(k k
 0)!Rt
 i
 j 1;3j
 2
  
~c1 + ~c2e
  i!HF t
  
; (B.13)
 and the equation for ~c2 is similar. By picking out the stationary terms one arrives
 at the following di erential equation,
 _~c1 =
 ij e;gj2
 4 
 
2
 X
 k
 j ~Ekj
 2~c1 +
  
X
 k
 ~Ek ~Ek q +
 X
 k
 ~Ek ~Ek+q
 !
 ~c2
 !
 ; (B.14)
 where q  !HF=!R is assumed be an integer. In the case of symmetric pulses, ~f is
 an even function, which implies that ~Ek = ~E k and the  nal result becomes,
 _~c1 =
 ij e;gj2
 2 
 
X
 k
 j ~Ekj
 2~c1 +
 X
 k
 ~Ek ~Ek q~c2
 !
 (B.15)
 _~c2 =
 ij e;gj2
 2 
 
X
 k
 j ~Ekj
 2~c2 +
 X
 k
 ~Ek ~Ek q~c1
 !
 : (B.16)
 189
Now we can de ne the Rabi frequency  and a coe cient for the Stark shift  ,
   
j j2
 P
 k j ~Ekj
 2
 2 
(B.17)
  
2
  
j j2
 P
 k
 ~Ek ~Ek q
 2 
: (B.18)
 The expressions in Eq. (B.18) involve a sum over the di erent spectral components
 in the frequency comb which should be able to be approximated by an integral in the
 case where the repetition rate is small compared to the bandwidth of a single pulse.
 In this approximation, Eq. (B.18) states that the Rabi frequency is proportional to
 the autocorrelation of the  eld envelope evaluated at the hyper ne frequency.
 190
Appendix C
 Walsh function Identity
 This appendix provides an inductive proof of the following identity,
 Z 1
 0
 dxWAL(2n  1; x)ei2
 n+1 x
 nX
 l=0
 alx
 l = 0: (C.1)
 The construction of the Walsh functions is simple in terms of the elementary se-
 quences known as Rademacher functions R (n; t) = sign [sin (2n t)]. The dyadic or-
 dering of the Walsh functions allow them to be de ned in terms of the Rademacher
 functions as WAL (n; t) =
 Qm+1
 i=1 R (i; t)
 bi 1 when n is expressed as a binary number
 n = bm2m + :::: + b020 and bi = 0 or 1. With this de nition, it is easy to see that
 choosing the index 2n  1 for the Walsh function means that all the bi coe cients
 are 1. We now prove the base case, n = 1.
 Z 1
 0
 dxWAL(1; x)ei4 x
 1X
 l=0
 alx
 l (C.2)
 =
 Z 1
 0
 dxR(1; x)ei4 x
 1X
 l=0
 alx
 l (C.3)
 = a1
 Z 1
 0
 dxR(1; x)ei4 xx (C.4)
 = a1
  Z 1=2
 0
 dxei4 xx 
Z 1
 1=2
 dxei4 xx
 !
 (C.5)
 = a1
 Z 1=2
 0
 dxei4 x (x (x+ 1=2)) = 0: (C.6)
 191
The inductive step is taken by assuming that Eq. (C.1) is true and looking at the
 2n+1  1 case:
 Z 1
 0
 dxWAL(2n+1  1; x)ei2
 n+2 x
 n+1X
 l=0
 alx
 l
 =
 Z 1
 0
 dxR[1; x]::::R(n+ 1; x)ei2
 n+2 x
 n+1X
 l=0
 alx
 l (C.7)
 =
 1
 2
 Z 2
 0
 dx
 n+1Y
 i=1
 R(i; x=2)ei2
 n+1 x
 n+1X
 l=0
 al
  x
 2
  l
 (C.8)
 =
 1
 2
 Z 1
 0
 dx
 n+1Y
 i=1
 R(i; x=2)ei2
 n+1 x
 n+1X
 l=0
 al
  x
 2
  l
 +
 1
 2
 Z 2
 1
 dx
 n+1Y
 i=1
 R(i; x=2)ei2
 n+1 x
 n+1X
 l=0
 al
  x
 2
  l
 : (C.9)
 The next step is taken by notin that for an integer n  0, R(n+ 1; x=2) = R(n; x),
 which allows the expression to be written as,
 =
 1
 2
 Z 1
 0
 dx
 nY
 i=0
 R(i; x)ei2
 n+1 x
 n+1X
 l=0
 al
  x
 2
  l
 +
 1
 2
 Z 2
 1
 dx
 nY
 i=0
 R(i; x)ei2
 n+1 x
 n+1X
 l=0
 al
  x
 2
  l
 (C.10)
 =
 1
 2
 Z 1
 0
 dx
 nY
 i=1
 R(i; x)ei2
 n+1 x
 n+1X
 l=0
 al
  x
 2
  l
  
1
 2
 Z 2
 1
 dx
 nY
 i=1
 R(i; x)ei2
 n+1 x
 n+1X
 l=0
 al
  x
 2
  l
 : (C.11)
 192
We now make the substitution x0 = x  1 and take advantage of the fact that
 R(n; x+ 1) = R(n; x) for n  1.
 =
 1
 2
 Z 1
 0
 dxWAL(2n  1; x)ei2
 n+1 x
 n+1X
 l=0
 al
 2j
 xl
  
1
 2
 Z 1
 0
 dxWAL(2n  1; x)ei2
 n+1 x
 n+1X
 l=0
 al
 2l
 lX
 k=0
  
m
 k
  
xk (C.12)
 =
 1
 2
 Z 1
 0
 dxWAL(2n  1; x)ei2
 n+1 x
 nX
 l=0
 blx
 l; (C.13)
 which is zero by assumption since bl is a constant, thus concluding the proof.
 193
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