Coherent Ion Transport in a Multi-electrode Trap Array 

by 

Ryan Bowler 

B.S., Physics, University of Washington, 2008 

B.A., Mathematics, University of Washington, 2008 

A thesis submitted to the 

Faculty of the Graduate School of the 

University of Colorado in partial fulfllment 

of the requirements for the degree of 

Doctor of Philosophy 

Department of Physics 

2015 
This thesis entitled: 

Coherent Ion Transport in a Multi-electrode Trap Array 

written by Ryan Bowler 

has been approved for the Department of Physics 

David J. Wineland 

Cindy Regal 

Date 

The fnal copy of this thesis has been examined by the signatories, and we fnd that both the 

content and the form meet acceptable presentation standards of scholarly work in the above 

mentioned discipline. 
iii 

Bowler, Ryan (Ph.D., Physics) 

Coherent Ion Transport in a Multi-electrode Trap Array 

Thesis directed by Dr. David J. Wineland 

Quantum information processors are predicted to enable a signifcant speedup in solving certain classes of problems compared to their classical counterparts. For one scheme of a processor 

implementation, required elements include qubits with long coherence times, laser-induced universal gates, information transport, and motional-state initialization using a second coolant ion species. 

Techniques described in this thesis include the use of a multiple electrode segmented trap, wherein 

information transport is achieved through the transport of qubit ions between spatially distinct locations. While experiment timescales had previously been dominated by ion transport and re-cooling 

of motional states, I have achieved transport and separation of qubit ions on timescales comparable to quantum logic gates. This work has been extended to achieve fast transport of qubit and 

coolant two-species ion chains. The developed techniques for ion transport result in low motional 

excitations, reducing the need for re-initialization of ion motional states. This can enable signifcant 

improvement in the operation time of an ion-based quantum information processor. 
Dedication 

This thesis is dedicated to my parents. 
v 

Acknowledgements 

The work described in this thesis is the culmination of years of work carried out with the 

support of many people. It is impossible to give a special thanks to each person who had played 

an important role during my time in graduate school, though I trust that many people know that 

they’ve played an important role even if not mentioned here. When I had frst joined the ion storage 

group, I had been given the great opportunity to work with John Jost, David Hanneke, and Jonathan 

Home, from whom I had learned a great deal about physics and operating complex experiments. 

Didi Leibfried has an enormous enthusiasm for the physics going on in the ion storage group, and 

I’ve greatly appreciated his enthusiasm for the experiments I had worked on during my time with 

the group. I have also had the privilege of working closely with John Gaebler, Ting Rei Tan, and 

Yiheng Lin, who have all had diverse and useful input over the years. I consider the support I’ve 

had from many friends invaluable during graduate school, and I’d like to thank Geoff White, Paul 

Pham, Adam Wagner, Nathan Morrison, Cecily Nicewicz, and Justin Kneip for sustained support 

over the years through the tough ups and downs. Most importantly, I’d like to thank my adviser 

Dave Wineland, who has a vast amount of knowledge and experience in research. His guidance is 

a driving force for the preeminent research I’ve had the privilege of being involved in during my time 

with the group. 
Contents 

Chapter 

1 Introduction 1 

2 9Be+ Qubit 7 

2.1 9Be+Atomic Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 

2.2 Hyperfne Structure and Magnetic Fields . . . . . . . . . . . . . . . . . . . . . . . . 8 

2.3 Doppler Cooling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 

2.4 State Initialization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 

2.5 Stimulated Raman Transitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 

2.6 State Detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 

3 Motional Dynamics of Trapped Ions 26 

3.1 Linear RF Paul Trap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 

3.1.1 Idealized System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 

3.1.2 Nulling Micromotion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 

3.1.3 Normal Modes of Multiple Trapped Ions . . . . . . . . . . . . . . . . . . . . 32 

3.1.4 Harmonic Oscillator States . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 

3.1.5 Re-ordering Ions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 

3.2 Non-classical States of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 

3.2.1 Coherent Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 

3.2.2 Thermal Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 
vii 

3.2.3 Squeezed Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 

3.3 Laser-Motion Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 

3.3.1 Motional-Sideband Laser Interactions . . . . . . . . . . . . . . . . . . . . . 43 

3.3.2 Interactions in the Presence of Multiple Normal Modes . . . . . . . . . . . . 46 

3.3.3 Doppler Cooling on Normal Modes . . . . . . . . . . . . . . . . . . . . . . . 48 

3.3.4 Sideband Cooling of Normal Modes . . . . . . . . . . . . . . . . . . . . . . 49 

3.4 Analyzing Motional States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 

3.5 Sympathetic Cooling with 24Mg+ and 25Mg+ . . . . . . . . . . . . . . . . . . . . . 54 

4 Apparatus 56 

4.1 Multiple Electrode Wafer Trap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 

4.2 Trap Geometries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 

4.2.1 Quantum Logic Trap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 

4.2.2 X-junction Trap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 

4.3 Containment System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 

4.3.1 Vacuum Components . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 

4.3.2 Loading Ions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 

4.3.3 Imaging System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 

4.3.4 Magnetic Field Coils . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 

4.4 Lasers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 

4.4.1 9Be+ Blue Doppler Beam Line . . . . . . . . . . . . . . . . . . . . . . . . . 74 

4.4.2 9Be+ Red Doppler Beam Line . . . . . . . . . . . . . . . . . . . . . . . . . 75 

4.4.3 9Be+ Raman Beam Line . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 

4.4.4 25Mg+ Doppler Beam Line . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 

5 Experiment Control 80 

5.1 Introduction to FPGAs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 

5.2 Experiment-Control FPGA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 
viii 

5.3 Arbitrary Waveform Generator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 

5.3.1 Hardware . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 

5.3.2 User Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 

5.3.3 Memory Storage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 

5.3.4 Possible Future Implementations . . . . . . . . . . . . . . . . . . . . . . . . 94 

5.3.5 Experimental Implementations . . . . . . . . . . . . . . . . . . . . . . . . . 97 

6 Coherent Transport Dynamics of Trapped Ions 102 

6.1 Adiabatic Ion Transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 

6.2 Coherently-driven Ion Transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 

6.3 Transport of 9Be+ Experiments and Results . . . . . . . . . . . . . . . . . . . . . . 108 

6.4 Ion Separation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 

6.5 Separation of Two 9Be+Experiments and Results . . . . . . . . . . . . . . . . . . . 115 

7 Optimization of Multi-species Ion Transport via Feedback on Quantum Motion 121 

7.1 Methods for Optimizing Two-Species Transport . . . . . . . . . . . . . . . . . . . . 123 

7.2 Transport of a Two-Species Ion Chain Experiments and Results . . . . . . . . . . . 125 

8 Conclusions and Outlook 132 

Bibliography 135 

Appendix 

A Transport Gate 141 

B AWG Spec Sheets 145 
Tables 

Table 

3.1 Normal modes for a 9Be+-

25Mg+ linear ion crystal . . . . . . . . . . . . . . . . . . 36 

3.2 Doppler-cooling limits n ̄limit for each axis for 9Be+ . . . . . . . . . . . . . . . . . . 49 

4.1 UV and visible wavelengths for the 9Be+ lasers . . . . . . . . . . . . . . . . . . . . 72 

5.1 Commands sent from the control computer to the AWG . . . . . . . . . . . . . . . . 92 

7.1 Normal modes for a 9Be+-

24Mg+ chain and the Doppler cooling limits n ̄limit . . . . . 126 

7.2 Normal modes for a 9Be+-

25Mg+ chain and the Doppler cooling limits n ̄limit . . . . . 129 
Figures 

Figure 

1.1 Drawing of a wafer ion trap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 

1.2 Pie chart for SU(4) time-distribution . . . . . . . . . . . . . . . . . . . . . . . . . . 4 

1.3 Pie chart for Randomized Benchmarking time-distribution . . . . . . . . . . . . . . . 5 

2.1 Diagram for atomic structure of 9Be+ . . . . . . . . . . . . . . . . . . . . . . . . . 9 

2.2 Zeeman shift for the S1/2 hyperfne states . . . . . . . . . . . . . . . . . . . . . . . 12 

2.3 Diagram for atomic structure of 9Be+ with state preparation beams . . . . . . . . . . 16 

2.4 Raman transition diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 

2.5 Detection histogram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 

3.1 An idealized linear RF Paul trap . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 

3.2 Detection in the presence of micromotion . . . . . . . . . . . . . . . . . . . . . . . 33 

3.3 Diagram for a coherent state of motion . . . . . . . . . . . . . . . . . . . . . . . . . 39 

3.4 Diagram for a thermal state of motion . . . . . . . . . . . . . . . . . . . . . . . . . 41 

3.5 Diagram for a squeezed state of motion . . . . . . . . . . . . . . . . . . . . . . . . 42 

3.6 Raman beam alignment for beams interacting with normal modes along the trap axis 44 

3.7 Types of Raman sideband transitions between two spin levels . . . . . . . . . . . . 47 

3.8 Diagram of the Raman sideband cooling process on a single mode of motion . . . . 50 

3.9 Plot of Rabi π-times and rates for different number states . . . . . . . . . . . . . . . 51 

3.10 Rabi fopping trace for a thermal state of motion for a single 9Be+ . . . . . . . . . . 52 
xi 

3.11 Rabi fopping trace for a thermal state of motion for a 9Be+-

25Mg+ ion pair . . . . . . 53 

4.1 Photograph of the quantum logic trap . . . . . . . . . . . . . . . . . . . . . . . . . 59 

4.2 A model of the electrodes for the quantum logic trap . . . . . . . . . . . . . . . . . 61 

4.3 A close-up model of the quantum logic trap illustrating where ions would be trapped . 61 

4.4 Schematic of the side view of the quantum logic trap . . . . . . . . . . . . . . . . . 62 

4.5 Photograph of the X-junction trap . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 

4.6 A model of the electrodes for the X-junction trap . . . . . . . . . . . . . . . . . . . . 66 

4.7 Diagram of the vacuum system . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 

4.8 Images collected from a CCD camera imaging fuorescence from 25Mg+ and 9Be+ . 70 

4.9 Diagram for the relative alignment of the ion imaging system, magnetic feld coils, 

incident laser light, and the ion trap . . . . . . . . . . . . . . . . . . . . . . . . . . 73 

4.10 9Be+ Blue Doppler beam line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 

4.11 9Be+ Red Doppler beam line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 

4.12 9Be+ Raman beam lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 

4.13 25Mg+ Doppler beam line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 

5.1 Flow diagram overview of the AWG . . . . . . . . . . . . . . . . . . . . . . . . . . 86 

5.2 Photograph of the AWG master board . . . . . . . . . . . . . . . . . . . . . . . . . 88 

5.3 Diagram of the Simple Dual-Port RAM input and output interface . . . . . . . . . . . 92 

5.4 Example waveform with flter compensation . . . . . . . . . . . . . . . . . . . . . . 98 

5.5 Pulse shaping of a microwave signal using the AWG to specify the profle . . . . . . 100 

6.1 Schematic for the diagonally-opposite DC electrodes of the quantum logic trap . . . . 108 

6.2 Rabi fopping trace on the MAS and MSS after transport from zone A to B . . . . . . 109 

6.3 Rabi fopping trace on the MAS following transport in which α(tT ) =6 0 . . . . . . . . 110 

6.4 Fixed MSS drive time t after round-trip transport for different wait times in zone B . . 111 

6.5 Ramsey-type interference experiment comparing contrast with and without transport 113 
xii 

6.6 Calculation of 1 dω during the separation waveform . . . . . . . . . . . . . . . . . 116 ω2 dt

6.7 Change in ion position and normal mode frequency during separation . . . . . . . . 117 

6.8 Ion fuorescence in zone A as a function of the offset potential during separation . . . 119 

6.9 Rabi fopping traces of the MAS of the ion in zone A and zone B after separation . . 120 

7.1 Schematic for the diagonally-opposite DC electrodes of the quantum logic trap . . . . 125 

7.2 Rabi fopping traces on the MAS for 9Be+-

24Mg+ following the optimization procedure 128 

7.3 Schematic for the diagonally-opposite DC electrodes of the X-junction trap . . . . . . 128 

7.4 Rabi fopping traces on the MAS for 9Be+-

25Mg+ following the optimization procedure 131 

B.1 AWG Spec Sheet 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146 

B.2 AWG Spec Sheet 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 

B.3 AWG Spec Sheet 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 

B.4 AWG Spec Sheet 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 

B.5 AWG Spec Sheet 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 
Chapter 1 

Introduction 

Quantum information processing (QIP) takes advantage of quantum mechanics to perform 

certain computations more effciently than currently possible on a classical computer. Some examples are to simulate other quantum systems of interest [1, 2], factoring large numbers [3], or 

searching unsorted databases [4]. Typically, the basic unit of quantum logic is the quantum bit, or 

“qubit,” which comprises two levels of any quantum system. Trapped atomic-ions are a promising 

basis for QIP; our ion-based qubit comprises two atomic hyperfne states of the ion, and “quantum 

logic” is performed by a sequence of laser or microwave pulses to drive transitions between the 

qubit states [5, 6]. The building blocks for QIP include a universal set of operations for quantum 

logic, transport of information about the processor, and qubits that can be prepared in an initial 

state, can be reliably measured, and have a long coherence time. 

For large-scale QIP based on trapped ions, information transport could be achieved by designing an ion trap with an array of interconnected trap zones, where quantum information is transported by confning the ions in potential wells that are moved between trap zones [6, 7, 8]. The 

ion trap can be designed with many electrodes, where potentials on the electrodes form threedimensional harmonic wells via a combination of static and oscillating electric felds, and an additional time-varying component to electrode potentials can transport the ions between zones. The 

necessary transport protocols include the separation of ions from a shared trap, combination of 

selected ions into a single trap, and transport of ions through junctions for re-ordering. Re-ordering 

ions and being able to combine them into a single trap is necessary for entangling any pair of qubits. 
2 

The qubit, with states labeled |0i and |1i, can exist in the states |0i, |1i, or any superposition 

of the two states due to the quantum nature of the qubit system. Using the notation that the state 

for two qubits Q1 and Q2 is written as |Q1Q2i, then any two-qubit pure state can be written as 

Ψ = c00|00i+c01|01i+c10|10i+c11|11i, (1.0.1) 

with the normalization constraint ∑|ci j|

2 = 1 on the four complex coeffcients that describe the state. 

This makes many-qubit systems far more interesting than classical-bit systems. A key element for 

quantum information that has no analog in the classical world is the entangled state. An example 

of such a state is given by 

Ψ = √

1 (|00i+|11i), (1.0.2) 2 

which has the key characteristic that one qubit state is not independent or "separable" from the 

other. 

The shared vibrational motion between multiple ions in a single trap zone can act as an “information bus” to entangle two ions [5]. By using laser beam confgurations that can drive the qubit 

states and the ions’ shared harmonic motion, it is possible to engineer qubit-motion entanglement 

and in turn use the shared motion to entangle two qubits [5, 9, 10, 11]. The success rate, or “fdelity,” 

of the entangling gate is typically highest when the ions start in their ground state of vibrational motion, necessitating laser cooling to the near-ground state of motion in the trap. Ambient heating, 

motional excitations due to fuctuating stray electric felds, is an important source of decoherence 

for gates that use the shared motion as an information bus. In our scalable QIP architecture, entanglement between any two qubit ions requires separating, transporting, and combining ions in 

shared trap zones, which can cause additional motional excitations. Laser cooling can return the 

ions’ motion to the ground state, but if performed on the qubit ions, cooling disrupts the qubit state. 

To get around this, another ion species with very different laser wavelengths for cooling is trapped 

along with the qubit ions. Laser cooling on this “coolant ion” can cool the shared motion for all the 
3 

Figure 1.1: Wafer trap geometry. Ions are confned at the center between the wafers, where the 

green circles denote confned ions (not to scale). The ion trap is formed by static potentials on 

electrodes (grey and blue portions) and oscillating potentials on “RF” electrodes (red portions) to 

form a three-dimensional harmonic oscillator potential. Slots in the wafers allow laser-access to the 

ions. By fabricating many similar electrodes, a large array of trap zones can be created. 

ions, a technique known as “sympathetic cooling,” without disrupting qubit states. 

There are several trap construction methods for this QIP scheme that have been used, and 

to-date there are two principal methods. The electrodes can be manufactured onto insulating wafers 

(as depicted in Fig. 1.1), where the electrodes are made from depositing conducting material onto 

a wafer made from a laser-machined insulator [12, 13, 14, 15]. The minimum size for electrode 

features is set by the laser beam’s minimum focus size, and typical sizes have electrode widths of 

∼ 100μm. Another approach is through a lithographic process to produce the electrodes on a single 

planar surface, which allows smaller feature sizes down to ∼ 30μm size scales [16, 17, 18]. For 

large-scale quantum information processing, development of larger trap architectures is necessary, 

such as those in Refs. [13, 18, 19, 15, 20, 21]. While the lithographic process has become more 

popular, many of the existing demonstrations of scalable trap architectures have been carried out in 

wafer traps. 

Transport and separation of ions in a wafer trap was frst demonstrated in Ref. [12], where it 

was shown that ion transport can be achieved with little motional excitation if performed adiabatically 

(that is, in a time much longer than the ion trap’s vibrational period) and that the separation of 
4 

Figure 1.2: Time-distribution between various processes in Ref. [27]. Computer control includes 

preparing electronics for the experiment, laser control includes preparing the sequencing of laser 

pulses, and the largest portions are due to transport and the initialization of the motional states 

through laser cooling. 

two ions from a single trap can also be performed. For work in Ref. [22], a trap was designed 

with electrode dimensions roughly half the size of those in Ref. [12], and greater control over ion 

separation was achieved, attributed to a smaller size-scale of the electrodes. Work reported in 

Ref. [23] showed the ability to transport ions on timescales comparable to quantum logic, but with 

the trade-off of large motional excitations. Transport experiments have also been carried out in 

a wafer trap to show reliable, adiabatic transport through a junction with low motional excitations 

[15], which is important for re-ordering ions. Excitations due to transport were largely attributed to 

electric feld noise from electrode surfaces [24, 25, 26] and imperfect transport waveforms due to 

slow electronics for the transport potentials. 

Demonstrations of our approach to scalable QIP method-sets with trapped ions were reported 

in Refs. [29, 27, 28] using 9Be+ as the qubit ion and 24Mg+ as the coolant ion. These demonstrations combined all of the building blocks of ion-based QIP, including long qubit coherence times, 

transporting ions between zones, ground-state cooling with another ion species, and laser-driven 

entanglement using the shared harmonic motion for universal logic gate sets. However, these experiments involved ion transport and laser cooling on long timescales. While each quantum logic 

operation typically took ∼ 10μs, ion transport required hundreds of microseconds to remain in the 

adiabatic regime to suppress unwanted excitations of the ions’ motion. The distribution for the durations of various processes in Ref. [27] is shown in Fig. 1.2 and the distribution for the durations 
5 

Figure 1.3: Time-distribution between various processes in Ref. [28]. The distribution is very similar 

to that for Ref. [27], where the largest portions are due to transport and cooling. 
6 

of various processes in Ref. [28] is shown in in Fig. 1.3. The timescales for QIP are dominated 

by laser-cooling the harmonic motion after transport, followed by the ion transport durations. A 

demonstration of an error correction algorithm, which is conditioned on the detection of errors, has 

also been carried out in this multi-electrode scheme with similar timescale discrepancies [30]. The 

timescale discrepancies between transport/cooling and quantum logic imposes a temporal bottleneck for large-scale quantum information processing. Therefore, to achieve large-scale QIP with 

trapped ions, transport must be performed on a timescale comparable to quantum logic with littleto-no motional excitations. 

The work described in this thesis achieved orders-of-magnitude improvements in ion transport, characterized both by transport speeds on timescales similar to quantum logic gates and 

highly-mitigated excitations in ion motion during transport. This thesis describes experiments transporting and separating 9Be+ qubit ions and transporting qubit and coolant ion pairs of 9Be+-

24Mg+

or 9Be+-

25Mg+. To begin with, Chapter 2 discusses the 9Be+ qubit and the ion-laser interactions 

with the qubit states. Then, Chapter 3 discusses the motional dynamics of trapped ions, including 

the laser interactions with the harmonic motion of the ions and relevant laser interactions with the 

coolant ions. Chapter 4 discusses the apparatus for the experiments, including the ion traps and the 

laser systems used during the experiments. Chapter 5 describes the electronics used to control the 

experimental sequences and the electronics for controlling ion transport. In Chapter 6, I describe 

the theory and experiments for coherent ion transport with qubit ions on timescales comparable to 

quantum logic. This chapter also discusses the theory and experiments for separating a pair of qubit 

ions into distinct trap zones. In Chapter 7, I describe the experiments for applying an optimization 

routine for transporting qubit and coolant ion pairs on quantum logic timescales. I close with some 

discussion for future directions to further improve transport during QIP operations. 
Chapter 2 

9Be+ Qubit 

A quantum information processor requires good quantum bits, or qubits, which implies that 

they have long coherence times for information storage over the duration of processor operations. 

A qubit also must be able to be prepared in a well-defned initial state, have a good method set for 

manipulating the qubit states and good measurement effciency. A scalable processor requires a 

universal logic gate set and the ability to transport information about the processor. A qubit comprises two levels of any quantum system, which for this work comprises two electronic states of the 

9Be+ ion. This chapter provides a description of the qubit states and how we manipulate them for 

our approach to quantum information processing. While 9Be+ ions have a diverse internal electronic structure, our use of lasers and microwaves allow us to isolate transitions between specifc 

two-level subsets to carry out our experimental operations for qubit manipulations. Additional laser 

control is used for state initialization and detection of the qubit state. In this chapter, I’ll discuss the 

energy levels in 9Be+ and motivate the choice of two specifc hyperfne sublevels of the 9Be+ ground 

state as our qubit manifold. Then I’ll describe the laser- and microwave-interactions to control and 

manipulate the internal states and laser-cooling of the 9Be+ qubit ion. 

2.1 9Be+Atomic Structure 

Beryllium is the lightest alkali-earth metal, and when ionized, it becomes an alkali-like atom 

with a single outer electron. The single outer-electron confguration is a relatively convenient structure, and the 9Be+ atomic structure is well-studied. The relevant atomic structure for the work in 
8 

this thesis is diagrammed in Fig. 2.1. The ground state level is the 2s2S1/2 state, which is split 

into two hyperfne levels F = 1 and F = 2 (which arises from the coupling of outer-electron angular 

momentum J to the nuclear spin I = 3/2 with total angular momentum F = I +J [31]), where the 

hyperfne states have ∼1.25 GHz splitting. For the work in this thesis, the principle number n = 2 

and electron spin S = 1/2 will not change and thus will be omitted in the discussion below. Our 

qubit comprises a pair of hyperfne states in the S1/2 manifold (the upper of which has a radiative 

decay time of about 1015 s [6]), which we describe in the total angular momentum |F,mFi basis. 

The other relevant states are the lowest excited orbital angular momentum states, two P

states, split by fne structure (the coupling of outer-electron spin to its orbit angular momenta [31]). 

The P1/2 and P3/2 are split by ∼ 197 GHz [32], and both states are coupled to the S states with 

∼313 nm laser light. The P states have a linewidth of Γ ' 2π × 19.4 MHz. The P1/2 state has a 

hyperfne splitting of 237 MHz [32], and the higher P3/2 state has a very small <1 MHz hyperfne 

splitting which we neglect in our experiments. 

2.2 Hyperfne Structure and Magnetic Fields 

In the experiments, we apply an external magnetic feld to our ions. This provides a quantization axis and splits the hyperfne structure into magnetic sublevels by the Zeeman effect. The 

Hamiltonian including both the hyperfne interaction and the magnetic feld interaction is 

H = hAI · J +μB(gJJ +gII)·B, (2.2.1) 

where h is Planck’s constant, A is a hyperfne constant in Hz (for 9Be+ splitting of the S1/2 state, A=-

625.008837048(10) MHz), μB is the Bohr magneton, gJ = 2.00226206(42) and gI/gJ = 2.134779853(2)×

10−4 give the respective Lande ́ g-factors for electron angular momentum J and nuclear spin I for 

9Be+, and B is the externally-applied magnetic feld (numbers from Ref. [33]). The direction of B, 

or Bˆ, sets the spin quantization axis, which for this analysis I choose as the zˆ-axis. 

The frst term in Eq. 2.2.1 can be rewritten as 
9 

9

9Be+ Level Diagram

2p 2P1/2

2p 2P3/2

2s 2S1/2

313 nm

F = 2

F = 1

F = 1

F = 2

mF = –2

mF = –1

mF = 0

mF = 1

mF = 2

mF = –2 mF

 = –1 mF

 = 0 mF

 = 1 mF

 = 2

mJ = 3/2

mJ = 1/2

mJ = –1/2

mJ = –3/2

Hyperfine < 1 MHz

197.2 GHz

237 MHz

1.25 GHz

Figure 2.1: Energy level diagram of 9Be+. The fine structure splitting of the excited P

orbital is 197.2 GHz. The hyperfine splitting is 237 MHz for the P1/2 manifold and less

than 1 MHz for the P3/2 manifold. The hyperfine splitting of the ground S1/2 states is

1.25 GHz. The S ↔ P transition is electric dipole allowed at 313 nm.

Figure 2.1: Atomic structure diagram of 9Be+. The hyperfne splitting of the S1/2 state is ∼1.25 GHz. 

The fne structure splitting of the P states is ∼197.2 GHz and the hyperfne splitting of the P1/2 state 

is 237 MHz while that of the P3/2 state is less than 1 MHz. The S to P dipole-allowed transitions 

are coupled by 313 nm light. Hyperfne sublevels are shown with split mF sublevels of each F state 

arising from an external magnetic feld. 
10 

1 1 hAI · J = hA(IzJz + I+J− + I−J+), (2.2.2) 2 2 

where I+/− = Ix±iLy and J+/− = Jx±iJy operators can be written by the usual raising and lowering 

p operators for angular momentum L+/−|L,mLi = L(L+1)−mL(mL ±1)|L,mL ± 1i [31]. The 

second term in Eq. 2.2.1 can be written as 

μBB(gJJz +gIIz). (2.2.3) 

Solving for the dependence on B can be done by constructing an operator matrix for each mI

and mJ level and solving for the eigenvalues. Grouping terms for the above-diagonals involving I+

the below-diagonals involving I− as I± J m constructs terms for a matrix such that I mJ

1

I

± J m = hA I mJ 2 

p I(I +1)−mI(mI ±1) p J(J +1)−mJ(mJ 1). (2.2.4) 

The terms for the matrix diagonal are 

Dmi,mj = hAmImJ +μBB(gjmJ +gImI). (2.2.5) 

These terms are put into the following matrix for the 9Be+ hyperfne sublevels of the S1/2 state. 

⎛ ⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝ 

D3 1 0 0 0 0 0 0 0 

2 , 2 

0 D1 1 0 0 I

+

1 J−1 0 0 0 

2 , 2 2 2 

0 0 D−1 1 0 0 I

+ J−

1 0 0 

2 , 2 −1

2 2 

0 0 0 D−3 1 0 0 I

+ J−

1 0

2 , 2 −3

2 2 

0 I

−

3 J+ 0 0 D3 0 0 0 

2 ,−1 −1

2 2 2 

0 0 I

−

1 J+ 0 0 D1 0 0 

2 ,−1 −1

2 2 2 

⎞ ⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ 

(2.2.6) 

0 0 0 I

− J+ 0 0 D−1 0 

2 ,−1 −1 −1

2 2 2 

0 0 0 0 0 0 0 D−3 

2 ,−1 2 
11 

For J = 1/2, the eigenstates can be solved analytically, where the analytic solution to the system 

is referred to as the Breit-Rabi formula. Solving for the eigenvalues of the above matrix yields the 

magnetic feld B-dependence of each hyperfne mF = mI +mJ sublevel. The results of diagonalizing 

the matrix are plotted in Fig. 2.2 for the B-feld dependence for each |F,mFi sublevel; for the rest of 

this thesis, states labeled |F,mFi are assumed to be a state in the S1/2 manifold. 

A key feature of the Zeeman shift is that for certain choices of B, the slope of the Zeeman shift 

vs. B is the same for certain pairs of |F,mFi states. This means that certain choices of feld B and 

a corresponding pair of two particular |F,mFi states can form a qubit that is frst-order insensitive to 

fuctuations from external felds (a “feld-insensitive” qubit). The trivial solution for a feld-insensitive 

qubit is the pair |2,0i ←→ |1,0i at B = 0, but since we need to spectrally resolve the mF levels, this 

choice is unsuitable. 

There are several pairs of states at non-zero B that form a feld-insensitive qubit with spectrallyresolvable mF levels; there are nine such pairs in all with calculations in Ref. [34], but we have 

chosen two for the convenient experimental-access in our lab in the intermediate-feld regime of 

the Zeeman shift. The transition |2,0i ←→ |1,1i is feld-insensitive at B = 11.945 mT and was 

used to demonstrate a qubit coherence time of ∼ 15 s [35]. At that magnetic feld, |2,0i ←→ |1,1i

has a qubit splitting of 1207.49584322 MHz. The choice of |2,1i ←→ |1,0i is feld-insensitive 

at B = 11.964 mT, with a splitting 1207.35280753 MHz, and was used to perform experiments 

involving the entanglement of two qubits [29, 27, 28] with similar coherence timescales. The 

|2,1i ←→ |1,0i qubit choice was used for the work described in this thesis. 

2.3 Doppler Cooling 

Working with an ion requires strong spatial localization. Part of this localization comes from 

Doppler cooling, a laser-light induced process to reduce the average kinetic energy of an ion held in 

an ion trap. Our ion trap uses a combination of static and time-varying electric felds to confne the 

ion in an approximately three-dimensional harmonic well (to be discussed in Chapter 3). A review 
12 

Figure 2.2: Zeeman shift as a function of magnetic feld of the S1/2 hyperfne states calculated 

via the Breit-Rabi formula. The states are labeled in the |F,mFi basis. Two of the frst-order 

feld-fuctuation insensitive pairs of hyperfne states that are available to our experiments are 

the |2,0i ←→ |1,1i states at B = 11.945 mT, which have a splitting of 1207.50 MHz, and the 

|2,1i ←→ |1,0i states at B = 11.964 mT, which have a splitting of 1207.35 MHz. 
13 

of laser-cooling can be found in Ref. [36], and Doppler cooling on an ion was demonstrated in Refs. 

[37, 38] and were the frst demonstrations of laser-cooling of atoms. 

The typical treatment for Doppler cooling is for a laser beam used for cooling, detuned from 

resonance for a closed two-level transition. A laser beam is applied to the spatial region of the ion 

trap with its frequency set red-detuned by Δ0 from the S1/2 to P3/2 resonance. The steady-state 

scattering rate is [31] 

Γ I/Isat

γ = (2.3.1) 2 1 +I/Isat + (2Δ/Γ)2 ,

where Γ is the excited state’s linewidth, Isat is the saturation intensity [31], and Δ = Δ0 −k · v is the 

detuning which varies with Doppler shift kv for light with wavenumber k and ion velocity v along the 

beam. For slowly-varying v, an ion with a velocity component opposite to the laser beam sees a 

blue detuning in the ion frame, bringing the light closer into resonance and resulting in a scattering 

force 

Fsc = ~kγ (2.3.2) 

and a corresponding “damping coeffcient” from Fdamping ' −∂F

∂ωkv = −μv [39, 31] such that 

8kΔ/Γ2 

μ = − Fsc (2.3.3) 1 +I/Isat + (2Δ/Γ)2 

This is a slowing force if Δ is negative (red-detuned) and heats the ion otherwise. 

When light scatters off of the ion, photons fy off in random directions imparting momentum 

kicks ~k. The emitted photons lead to diffusion in velocity space, but the ion’s velocity components 

along the beam experiences a slowing force. As the ion’s velocity along the beam direction slows, so 

too does the scattering rate as the Doppler shift relaxes the beam’s detuning away from resonance. 

The minimum temperature is reached at an equilibrium between absorption and re-emission, called 

the Doppler cooling limit. The Doppler cooling limit is reached with Δ0 = −Γ/2 and I/Isat  1, 

corresponding to a temperature [31] 
14 

~Γ kBTmin = , (2.3.4) 2 

where kB is Boltzmann constant. 

In an ion trap, there are three axes, and it is necessary to be able to consider Doppler cooling 

along each axis, which requires considering the alignment and polarization of the Doppler cooling 

beam with respect to our ion-trap geometry [40]. Our Doppler cooling light is σ+-polarized and 

is red-detuned from the closed two-level cycling transition |S1/2,F = 2,mF = 2i ←→ |P3/2,F =

3,mF = 3i which has a linewidth of Γ ' 2π×19.4 MHz. In the case of a Doppler cooling beam with 

wavevector k and considering cooling along an axis i, the Doppler cooling temperature is given by 

[40, 39] 

p ~Γ 1 +I/Isat fsi kBTmin = (1+ ) (2.3.5) 4 fi

 ́ where fi = k

ˆ2 and fsi = Ps(kˆ 

s)k

ˆ

si

2 dΩ for scattered light with wavevector ks and spatial distribution i

1 Ps. For isotropic scattering such that fsi = 1/3 for each axis i and k

ˆ = √ (xˆ+yˆ+zˆ) with a beam 3 

intensity I/Isat  1, then Eq. 2.3.5 agrees with the former cooling limit given by Eq. 2.3.4. 

The experiment uses a Doppler cooling beam aligned with the external magnetic feld axis Bˆ, 

which lies at 45◦ to the xˆ axis with respect to our ion trap. In spherical coordinates, the scattered 

light distribution for the σ−-polarized dipole scattering is given by [41] 

Ps(k

ˆ 

s) = (3/16π)[1 +cos2 θs] (2.3.6) 

where Bˆ · k

ˆ 

s = cosθs. For example, the projection of the scattered light onto the xˆ-axis gives k

ˆ

si

2 =

(sinθs cosφs +cosθs)2/2 in spherical coordinates. The integration evaluates to fsi = 7/20 and the 

incident Doppler cooling beam’s alignment projected onto the xˆ-axis gives fi = 1/2, and putting 

these into Eq. 2.3.5 with I/Isat  1 solves for the theoretical Doppler cooling limit of ∼0.4 mK. 
15 

2.4 State Initialization 

Preparing our qubit in a well-defned initial state in every experiment is a vital part of our qubit 

operations. We perform optical pumping on 9Be+ with laser light, and from there we can perform 

coherent transitions to work with our qubit manifold. The P states decay with 1/Γ=8.2 ns, providing 

rapid spontaneous decay back to the S1/2 manifold. The S ←→ P transition can be accessed with 

dipole radiation at 313 nm. By selecting σ+-polarized light propagating along the B direction, the 

light drives mF ←→ mF +1 transitions. The proper combination of laser light can then prepare 9Be+

in the |2,2i initial state. Once in our initial state, we can coherently excite transitions as needed 

about the hyperfne manifold for qubit manipulations. 

Several laser beams are used for state initialization and are shown in Fig. 2.3. We assume 

that the S1/2 states are initially randomly populated. The Blue Doppler (BD) laser beams play a 

role in state initialization (while this is not their principle purpose, the role in optical pumping is 

important). Blue Doppler beams include both near-resonant (BD) and far-detuned (BD detuned) 

lasers coupling the |S1/2,F = 2,mF = 2i ←→ |P3/2,F = 3,mF = 3i transition. The BD detuned 

beam is ∼ -400 MHz detuned from this transition and is used to pre-cool very hot ions. This beam 

has several times the saturation intensity, and couples all of the F = 2 states and F = 1 states 

to P3/2 with mF ←→ mF +1 transitions, and therefore pumps all S1/2 state-population towards the 

|2,2i state. The BD beam is near-detuned by −Γ/2 ' −2π × 9.7 MHz from the transition and at 

less than the saturation intensity, which is chosen to satisfy the conditions for reaching the Doppler 

cooling limit. It would be on the order of milliseconds to reliably pump to the |2,2i state from all of 

the F = 2 and F = 1 states with the BD beams due to the far off-resonance transitions, so after the 

BD beams are applied to transfer population to near the |2,2i state, dedicated repumping beams 

are used to more effciently prepare the initial state. 

The core repumping beams couple the S1/2 state to P1/2 state with σ+-polarized light. These 

are used following some amount of state preparation by the BD beams. A resonant repumping 

beam, or the “repumper” beam, resonantly drives the |S1/2,F = 2,mF = 1i ←→ |P1/2,F = 2,mF =
16 

F = 2

F = 1

mF = –2

mF = –1

102.8 86.7 76.4 69.1

86.5 76.2

1207.495,843

F = 2

F = 1

mF = –2

mF = –1

39.9 28.5 23.2 20.1

28.2 23.1

261.8

2

P3/2

197 GHz

2

P1/2

313.20 nm

2

S1/2

mJ= 3/2

mJ= 1/2

mJ= –1/2

mJ= –3/2

mF = –3

mF = +3

223.4

BD detuned

BD

RD

repumper

Figure 2.3: Atomic structure diagram of 9Be+ at B=11.94 mT showing the state-preparation laser 

couplings. The blue Doppler (BD) beams couple S1/2 to P3/2 and red Doppler (RD) beams couple 

to the P1/2 state with σ+-polarized light to optically pump into the |2,2i initial state. 
17 

2i transition. The |P1/2,F = 2,mF = 2i state dominantly decays by the dipole-allowed transitions 

to the |2,2i, |1,1i, and |2,1i states. Another beam, the Red Doppler (RD) repump beam, so-called 

historically due to being detuned to the red of the |S1/2,F = 1,mF = 1i ←→ |P1/2,F = 2,mF = 2i

transition, is used as well. The detuning avoids a “dark state” that can arise from a coherent process 

from interference between the two repumping beams when they are at the same detuning from the 

P1/2 state [42], and to avoid this, the RD beam is detuned by ∼ -20 MHz. As the incident repumping 

light is σ+-polarized, once population is entirely pumped to the initial |2,2i state, the beams will no 

longer be able to drive transitions to the P1/2 manifold due to selection rules. For any population in 

states of the S1/2 manifold other than the |2,2i, |1,1i, and |2,1i states, any off-resonant transitions 

from S1/2 to P1/2 or P3/2 can then allow decay to other states in the S1/2 manifold (for example, 

when the |1,−1i state is excited, it can decay back to states other than those listed). However, the 

BD detuned and RD beams off-resonantly pump F = 2 and F = 1 states towards the |2,2i state. 

The combination of these repumping beams therefore asymptotically prepares the ion in the |2,2i

state. 

It is best to keep both repumping beam intensities below the saturation intensity, or else imperfect polarization components including σ− (since light is along the B-axis, there is no π-polarization) 

can lead to signifcant depumping of |2,2i. Each RD beam can have its intensity independently varied for an optimal value. The state initialization sequence begins with BD detuned, BD, RD, and 

repumper beams applied for the order of a ∼2 ms duration. Following this, we then apply the BD, 

RD, and repumper beams for ∼500 μs. Lastly, the RD and repumper are applied for ∼10 μs in the 

fnal stage of state preparation. The combination of all of the beams prepare the |2,2i state with 

near-unity population (Ref. [34] estimates repumping with an error on the order of 10−5 , accounting for imperfect polarization). Whenever the state needs to be re-initialized in the middle of an 

experiment, the repumper beams are applied for ∼10-30 μs to prepare the |2,2i initial state. 
18 

2.5 Stimulated Raman Transitions 

After state initialization to the |2,2i state, we coherently transition to the |2,1i state and from 

there perform qubit rotations on the |2,1i ←→ |1,0i transition. Here, I use the notation for qubit 

states | ↓i ←→ | ↑i. However, the qubit splitting of 1.2 GHz and the Zeeman splitting between 

mF states of ∼100 MHz makes direct laser transitions impractical. A common technique is to use 

two-photon stimulated Raman transitions [43, 6] to control the S1/2 hyperfne structure with two 

beams separated in frequency by the transition splitting. For the following, I consider an idealized 

three-level system. 

Incident laser light from two beams have the electric felds 

Ei = εˆiEi cos(ki · x−ωLit +φi) (2.5.1) 

for i ∈ {1,2}, with L1 and L2 denoting the two beams with polarizations εˆi wavevectors ki and frequencies ωLi. Laser beam L1 performs coupling from an initial ground state | ↓i to an intermediate 

state |ii spaced from | ↓i by ωi (in our case, the S ←→ P transition splitting; while both P states 

contribute, this analysis will consider coupling to a single intermediate state) detuned by Δ from 

resonance. Laser beam L2 similarly couples the fnal excited state | ↑i to the intermediate state 

|ii with detuning Δ, and |ωL2 − ωL1| = ω0 is the splitting between the desired hyperfne transition 

(diagrammed in Fig. 2.4). Hence, ωL1 = ωi −Δ and ωL2 = ωi −Δ−ω0 in terms of the internal state 

frequencies. As of this thesis, our two Raman beams are ∼ -250 GHz red-detuned from the P1/2 

state. 

The wavefunction for the three-level system for the states s is 

Ψ = ∑ Cs(t)|si, (2.5.2) 

s=↓,↑,i 

where Cs is the normalized coeffcient for each state. The interaction Hamiltonian is 

HI = −μ ̄ ·(E1 +E2) (2.5.3) 
19 

Figure 2.4: Diagram of a stimulated Raman transition coupling the ground state | ↓i to excited 

state | ↑i. Laser L1 couples the ground state | ↓i to an intermediate state |ii detuned by Δ from 

resonance. Laser L2 likewise couples the excited state | ↑i to an intermediate state |ii with a 

detuning Δ. The laser frequencies are split by |ωL2 − ωL1| = ω0, each beam with respective kvectors k2 and k1. 
20 

for dipole coupling between the dipole moment μ ̄ and the incident lasers. It is convenient to transiH0t/~HIe−iH0t/~ form to the interaction picture by taking HI

0 = e , where H0 = Hinternal is the timeindependent Hamiltonian for the three internal atomic states (this is effectively like transforming to 

the frame of the atomic states for the interaction). Transforming to the interaction picture and taking 

the rotating wave approximation gives 

HI

0 = ~g1 exp(i(k1 · x+φ1)) exp(iΔt)| ↓ihi|+~g2 exp(i(k2 · x+φ2)) exp(iΔt)| ↑ihi|+h.c., (2.5.4) 

with multiplying factors g1 = −E1 h↓ |μ ̄ · εˆ

1|ii and g2 = −E2 h↑ |μ ̄ · εˆ2|ii as short-hand. Then, applying 2~ 2~

Schrodinger’s equation for the laser interactions, we have  ̈

C ̇

↓ = −ig1 exp(i(k1 · x+φ1)) exp(iΔt)Ci 

C ̇

↑ = −ig2 exp(i(k2 · x+φ2)) exp(iΔt)Ci 

∗ ∗ C ̇

i = −i(g1 exp(−i(k1 · x+φ1))C↓ +g2 exp(−i(k2 · x+φ2))C↑) exp(−iΔt). (2.5.5) 

If the detuning Δ is much larger than the coupling rates g1 and g2 and larger than the natural 

linewidth Γ of the state |ii, which is the case for the 9Be+ P states, then there is relatively little 

population in the |ii state, i.e., C ̇

i is much less than C ̇

↑,↓ and therefore can be neglected. However, 

Ci must still be integrated to include in the C ̇

↑,↓ terms. 

ˆ ∗ ∗ C ̇

i = − i(g1 exp(−i(k1 · x+φ1))C↓ +g2 exp(−i(k2 · x+φ2))C↑) exp(−iΔt)dt (2.5.6) 

Integrating each term separately by parts gives frst the term with C↓ as 

 ˆ  C↓-term C↓ exp(−iΔt) 1 = −ig∗

1 exp(−i(k1 · x+φ1)) − + C ̇

↓ exp(−iΔt)dt . (2.5.7) i iΔ iΔ

With another round of integration by parts, this becomes 
i 

21 

  ˆ  C↓-term C↓ exp(−iΔt) 1 C ̇

↓ exp(−iΔt) 1 = −  ̈ ig1 

∗ exp(−i(k1 ·x+φ1)) − + − + ( C↓ exp(−iΔt) iΔ iΔ iΔ iΔ

(2.5.8) 

with a similar expression for the term including C↑. The increasing factors of 1/Δ are increasingly 

vanishing, and so we only keep the terms frst-order in 1/Δ, giving the coeffcients 

C↓ exp(−iΔt) C↑ exp(−iΔt) C ̇

↓ = ig1 exp(iΔt)[ig∗ +ig∗ exp(−i(δk · x+δφ)] 1 2 iΔ iΔ

C↓ exp(−iΔt) C↑ exp(−iΔt) C ̇

↑ = ig2 exp(iΔt)[ig∗ +ig∗ exp(i(δk · x+δφ)]. (2.5.9) 1 2 iΔ iΔ

where δk · x + δφ denote the differences between the k-vectors k2 − k1 and phases φ2 − φ1 of the 

two beams. 

The coeffcients in Eq. 2.5.9 give rise to an effective two-state system, coupling | ↓i to | ↑i

with very limited population in |ii during the process. The corresponding effective Hamiltonian is 

then 

 |g1|

2 g1g∗ |g2|2 g2g∗  2 1 H = −~ | ↓ih↓ |+ | ↓ih↑ | exp(−i(δk · x+δφ))+ | ↑ih↑ |+ | ↑ih↓ |exp(i(δk · x+δφ)) Δ Δ Δ Δ

(2.5.10) 

This Hamiltonian is in the form of the familiar interaction with monochromatic radiation in a two-level 

system, which induces Rabi oscillations between the states | ↓i and | ↑i [31], but is characterized 

by the phase difference between the two laser beams δk · x+δφ and a Rabi rate given by 

g∗

1g2 ΩRaman = − . (2.5.11) Δ

The diagonal terms of Eq. 2.5.10 are the usual AC Stark Shifts [31], which can be accounted for 

with a suitable tuning of the laser frequencies. 

The Rabi rate in Eq. 2.5.11 depends both on the incident laser beam intensities (or, the 

electric feld amplitude) and depends inversely on the detuning Δ (which also dictates population 
22 

in |ii), offering tradeoffs between our available laser beam powers, possible detunings, and qubit 

operation speeds. The steady-state scattering rate off of the intermediate state γ (see Eq. 2.3.1) 

introduces a loss mechanism if |ii can scatter to a state outside of | ↓i and | ↑i, which is possible as 

the three-level system considered above is only an idealization of the 9Be+ atomic structure during 

a Raman transition. 

This analysis can be extended to include multiple intermediate states, which applies to our 

specifc case as the Raman transitions are driven in the presence of both P1/2 and P3/2 states. For 

a suitably large detuning from all intermediate states, such that every respective occupation Cj for 

the j

th intermediate state is very small, then the Raman transitions induce Rabi oscillations where 

the Rabi rate has contributions from Raman scattering off of each intermediate state. The general 

expression is then [44] 

E1E2 h↑ |μ ̄ · εˆ

2|iihi|μ ̄ · εˆ1| ↓i ΩRaman = ∑ (2.5.12) 4~2 

i Δi 

for intermediate states i with detuning Δi from the respective intermediate state. 

Our stimulated Raman transitions are driven with lasers composed of a combination of σ−-, 

π-, and σ+-polarized light. Beam L1 has π-polarization with k1 perpendicular to the magnetic feld 

B direction. Beam L2, with k2, is either co-propagating or 90◦relative to k1 and has σ+ and σ− polarization components. This enables ΔmF = ±1 transitions when implementing stimulated Raman 

transitions. For a detuning Δ from the P1/2 state and detuning Δ−Δf from the P3/2 state, where Δf

is the fne-structure splitting between the P states, then using appropriate Clebsch-Gordan coeffcients allows the Rabi rate to be rewritten as [44] 

g1g2 Δf ΩRaman = √ (ε1,πε2,σ+ +ε1,σ−ε2,π) , (2.5.13) 6 Δ(Δ−Δf)

where g1,2 are the matrix elements E1,2 |h↓ |μ ̄ ·σˆ +|P3/2i| and εi, j is the j

th-polarization component 2~

of the i

th laser beam. 

During experiments, after state initialization into the |2,2i state, we drive a Raman transition 
23 

to the |2,1i state, and from there perform coherent qubit rotations on the |2,1i ←→ |1,0i qubit 

states. Our incident laser beams typically have ∼1 mW of power, waist w0 ' 30μm, and drive Rabi 

oscillation π-times of ∼ 5μs. The states in the S1/2 hyperfne manifold have an effectively infnite 

lifetime, and to the degree that spontaneous scattering can be suppressed, enable near-perfect 

Rabi fopping between any choice of states, the linewidth only Fourier limited from pulse duration. 

2.6 State Detection 

Detecting the fnal measured qubit state is necessary to discern the result of a quantum 

information process, and this requires the ability to distinguish fnal qubit state populations of |2,1i

and |1,0i to a high degree. The approach with 9Be+, and ions in general, is to use state-dependent 

resonant fuorescence. In some cases, the process rapidly scatters photons on a resonant transition 

without any optical pumping out of the transition manifold. We refer to the state used for detection 

as the “bright” state. 

We use a laser beam at 313 nm with σ+ polarization to resonantly couple the |S1/2,F =

2,mF = 2i ←→ |P3/2,F = 3,mF = 3i states. Due to selection rules, the only possible decay path 

when the P3/2 state decays is back to the |2,2i state. Thus, this closed transition cycle rapidly 

scatters photons off of the 9Be+ ion from the bright |2,2i state. Before detection, we transfer 

population from |2,1i to |2,2i, therefore enabling effcient detection of the |2,1i state (we often 

refer to |1,0i as our “dark” qubit state). Our detection beam is derived from our BD beam shifted 

onto resonance. A portion of the solid angle of scattered light can be under observation to collect 

the light and distinguish whether or not the ion is in the bright state. Typically the beam intensity is 

less than Isat such that spontaneous emission dominates the return to the bright state from P3/2. 

Scattering off of a dark state, such as |1,1i, is suppressed because transitions from this state are 

∼1.2 GHz detuned from the bright transition with very little scattering from the detection beam. 

The observed scattering rate for the resonant fuorescence is limited by the P3/2 decay rate 

of 8.2 ns, the solid angle for photon collection, and quantum effciency of our detector. The steady-
24 

state scattering rate γ from the detection beam (see Eq. 2.3.1) with detuning Δ=0 for resonant light 

with I/Isat=1/2 gives γ=Γ/6 and we can expect that in a 250 μs detection, ∼5000 photons will be 

scattered off of the bright state. In practice, our photon detection system will observe ∼30 photons 

in that duration for an ion in the bright state. When the ion is in the dark state, no photons should be 

collected, but typically there is a background of ∼1-2 photons primarily from stray background light. 

One source of error in detection is due to off-resonant scattering off of the dark state. To 

further discriminate the bright and dark states, an “electron shelving” technique [45] transfers population in the dark qubit state to a state which is farther detuned from the detection beam. When 

detecting the qubit state, |2,1i population is transferred to |2,2i for detection and the |1,0i population is transferred to |1,−1i. 

Repeated detections over many trials builds up statistics that ideally obey a Poisson distribution, that is, a distribution that describes random results from a fxed average rate. The Poisson 

distribution obeys the probability density function 

μx −μ e , (2.6.1)

x! 

where μ here is the mean value and x is the number of counts in the distribution (see Fig. 2.5 for 

an example of collected data). When detecting multiple ions, there is a mean value for μ0 for zero 

ions in the bright state, μ1 for one ion in the bright state, μ2 for two ions bright, etc, for the number of 

ions under detection light, which follows a normalized sum of Poisson distributions. More in-depth 

discussion on detection can be found in Ref. [34]. 
25 

0 5 10 15 20 25 30 35 40 45

0

0.05

0.1

0.15

0.2

counts

Figure 2.5: Example of detection data (blue histograms) from state-dependent resonant fuorescence for a 250 μs detection duration from an ion repeatedly prepared in a superposition of bright 

and dark states for 100 detections. The counts have been ft to normalized Poisson distributions 

(red dashed line). In this case, the data is consistent with measuring the ion in the bright state 

with ∼50% probability and average 32.86 PMT counts, and the ion in the dark state with ∼50% 

probability and average 0.80 PMT counts. The distributions for the bright and dark states are highly 

distinguishable. 
Chapter 3 

Motional Dynamics of Trapped Ions 

To work with a trapped ion, we need to maintain a very strong spatial localization. By Earnshaw’s theorem, it is impossible to have a purely confning static electric feld with no local charge, 

and so to trap an ion, more complex felds are necessary. Two common types of ion traps are the 

RF Paul and the Penning trap. The Penning trap uses a combination of static electric and magnetic felds for confnement. The RF Paul trap, which uses a combination of time-varying and static 

electric felds, creates an effective confning potential, or pseudopotential. Our ion trap forms a 

three-dimensional harmonic oscillator, confning the ion to about ∼10 nm in every direction when 

they are cooled to near their ground state. We trap both 9Be+ and 24Mg+, and here I consider the 

motion of both trapped ion species. 

We apply Doppler and Raman-sideband laser-cooling to the trapped ions to cool them to near 

the ground state of motion. The application of Raman pulses detuned by the oscillator frequencies 

allows us to couple directly to motional modes, and a judicious use of pulses coupled with repumping can be used to cool the ions to near the zero-point energy of the trap. At cold temperatures, the 

motion of the trapped ion is best described in terms of the number states of the quantum harmonic 

oscillator. The motion is inherently non-classical, and there are a few specifc distributions of number states that are important in the dynamics of trapped ions. In this chapter I describe the quantum 

dynamics of for single and multiple trapped ions. I will begin with a description of the linear RF 

Paul trap and its harmonic confnement. Then I will describe some non-classical states of motion. 

Following this, I will describe the motional-state laser interactions, including motional state analysis 
27 

and state initialization through laser cooling. The last section is on the atomic structure of 24Mg+

and 25Mg+, which are used for “sympathetic” laser-cooling ions without disrupting the qubit-state of 

the 9Be+ ions. 

3.1 Linear RF Paul Trap 

3.1.1 Idealized System 

An ionic qubit may provide a promising basis for a QIP. However, it is not practical to work 

with ions without strict environmental control. For instance, coupling to a noisy environment, such 

as fuctuating electric or magnetic felds [6], can cause decoherence of the qubit state. In addition, it 

is diffcult to selectively work with individual qubit ions without control over the spatial position of an 

ion. Selected ions could be lost from collisions with background gas, and it is desirable to confne 

ions in a system with little coupling other external environmental factors. Fortunately, a reasonable 

degree of decoupling from the environment can be achieved with an ion trap held in a vacuum 

system [46, 6, 35, 47]. 

One advantage to trapping ions is that their net charge allows the use of electric potentials 

for strong confnement (ions typically remain trapped for several hours). On this topic, Earnshaw’s 

theorem states [31], “A charge acted on by electrostatic forces cannot rest in stable equilibrium in 

an electric feld.” The theorem can intuitively be understood from the fact that if there are no local 

static sources of an electric feld E ̄  ̄ , then ∇ · E = 0 in the region, i.e., there is no “sink” in three 

spatial dimensions. Fortunately, there are some ways around this problem. 

One solution is to use time-varying electric felds combined with static electric felds. This 

solution is known as a linear radio-frequency (RF) Paul trap [48]; it is used for the work in this thesis. 

The basic idea of the Paul trap is to supply a combination of electric felds which would create a 

quadrupole, or “saddle point”, potential. If this saddle point oscillates rapidly, it can generate stable 

confnement, i.e., it becomes an effective potential minimum in two or three dimensions at once. 

A linear Paul trap can be built using conducting electrode rods in a quadrupole confguration; 
28 

Figure 3.1: An idealized linear RF Paul trap with four parallel rod electrodes. A RF voltage is 

applied to diagonally opposite electrodes. The RF provides a harmonic pseudopotential along the 

radial directions, the x and y axes, wherein the ions are trapped at the potential null. The DC 

voltages, which are referenced to RF ground, are applied to another diagonally opposite pair of rod 

electrodes. The DC electrodes can be segmented and supplied with a combination of high outersegment voltages and low inner-segment voltages to provide confnement along the z axis (also 

referred to as the trap axis). 
29 

an idealized diagram of this design is shown in Fig. 3.1. Applying time-varying electric felds can 

lead to stable ion confnement. Two pairs of diagonally opposite conducting rods carry an RF voltage 

V = V0 cos(Ωt), where Ω is the applied frequency. The other pair of rods are referenced to the RF 

ground. In this setup, we consider the potential due to the RF in the x and y radial directions, and 

z axis, which is also referred to as the trap axis. To achieve confnement along the trap axis, we 

create a static potential difference U0 between the high DC electrode segments and the low DC 

electrode segments (as shown in Fig. 3.1). The electric potential φ must satisfy Laplace’s equation, 

∇2φ = 0. The potential satisfying the boundary conditions is given by [49, 6, 39] 

V0 cos(Ωt) 2 2 2 2 φ = [αz +βx +γy2] +κU0[α0

z +β0x +γ

0

y2], (3.1.1) 2R2 

where κ is an overall geometric factor and R is the distance from the trap axis to the surface of an 

electrode. Laplace’s equation demands that α+β+γ = 0 and that α0 +β0 +γ0 = 0. For the case 

α = 0, the oscillating potential is uniform along the z axis, or “trap axis,” for infnite length rods. This 

gives a generally good approximation in a linear Paul trap, as typical electrode scales are much 

larger than the confnement spatial extent. For our electrode geometry in Fig. 3.1, γ = −β gives the 

desired potential, where the terms in the potential in Eq. 3.1.1 proportional to cos(Ωt) are akin to 

rotating a saddle-point potential about the trap axis. Sometimes geometric factors are included to 

model different (and possibly imperfect) quadrupole confgurations [39], but they are omitted from 

this analysis. 

For the static potential, α0

, β0, and γ0 are determined by the electrode geometry. Laplace’s 

equation constrains the coeffcients in Eq. 3.1.1 to α0 + β0 + γ0 = 0, and for our symmetric confguration, β0 = γ0

, and for an ion of mass m and charge q = |e|, the potential can be rewritten in the 

more common harmonic oscillator notation to be 

ω2 2 − 2 φs = m [z

1 (x +y2)], (3.1.2) z 2q 2 

where ω2 = 2κqU0/m is the center-of-mass oscillation frequency along the trap axis. The potential z
30 

in Eqs. 3.1.1 is the lowest-order expression for the potential, which is reasonably valid for ions near 

the trap center provided that the spatial extent of the trapped ion’s motion is much less than the 

ion-electrode distance R. The total potential is then given by 

V0 cos(Ωt)

 x2 −y2  m 1 2 Φtot = + ωz

2[z

2 − (x +y2)]. (3.1.3) 2 R2 2q 2 

The classical equations of motion in the x and y directions are 

d2x +[ax +2qx cos(2ς)]x = 0, (3.1.4) dς2 

d2y

dς2 +[ay +2qy cos(2ς)]y = 0. (3.1.5) 

These equations use the following variable substitutions: ς = Ωt/2, ax = ay = −(4qκU0/mΩ2), 

and qx = −qy = (2qV0/Ω2mR2). The notation used in this derivation is the usual notation seen 

in ion trap literature, where qi is not an electric charge and ai is not a lowering operator of the 

quantum harmonic oscillator. Typically the relationship ai < qi

2  1 holds true. These equations 

are known as the Mathieu equations, which can be solved with known Floquet solutions [49, 6]. 

They are classical equations but are suffcient for describing the the potential wells The solutions of 

the Mathieu equations, to the lowest order in ai and qi, give us[6] 

h i qi qi

ui(t) = Ai cos(ωit)[1 + cos(Ωt)]+Bi sin(ωit)sin(Ωt) , (3.1.6) 2 2 

where i denotes the x or y directions, the amplitude Ai depends on initial conditions, and Bi '

2 (ai +qi /2)1/2 . Three motional frequencies arise, with two in the radial direction at Ω±ωi, and the 

2 last at frequency ωi = BiΩ/2. For ai < qi  1, ωi is much smaller than Ω. According to Eq. 3.1.6, 

oscillation terms at Ω ± ωi have a much lower amplitude by at least a factor of qi/2; these terms 

give rise to “micromotion.” If the micromotion is neglected, the potential can be expressed as 

m 2 φp = ω2(x +y2), (3.1.7) r 2q
31 

for radial confnement where the static potential is negligible. Equation 3.1.7 forms a “pseudopotential” harmonic confnement in the radial directions with frequency 

√ √ ωr ' qV0/( 2ΩmR2) = qxΩ/(2 2). (3.1.8) 

Hence, we have the result that the linear RF Paul trap forms three-dimensional harmonic confnement governed by the potentials in Eqs. 3.1.2 and 3.1.7 and with frequencies ωz and ωr. However, 

critically, the Eq. 3.1.7 has an overall 1/m dependence and thus the radial potential varies with 

mass. The linear RF Paul trap has strong confnement, and it’s typical to maintain a trapped ion for 

a mean lifetime of four hours, which is suffcient to collect experimental data. 

There are both stable and unstable solutions depending on ai and qi (which could be translated to design parameters such as R and Ω for real-trap construction). A detailed analysis of 

“stability regions” in the parameter space of ai and qi can be found in Refs. [49, 39], though the 

motion is always stable for ai < q2  1. Another issue can arise if the potential is not a perfect i

quadrupole. Geometric imperfections can be accounted for in geometric factors such as κ, but 

for more serious design considerations, an analysis of higher-order terms in the potential may be 

necessary; details of this approach can be found in Ref. [50]. 

The radial symmetry for the pseudopotential is broken by the static DC potentials along the 

trap axis, which gives rise to anti-confnement in the radial directions. The total potential can be 

written with the geometric factors z2 −αx2 −(1 −α)y2 at the position of the ions. This means that 

the radial frequencies are given by 

q ωx = ωr 1−αω2 

z/ω2 r, (3.1.9) 

q ωy = ωr 1 −(1 −α)ωz

2/ωr2 . (3.1.10) 

Or, if ωx and ωy can be independently evaluated such as through an experimental measurement, 

then the pseudopotential can be calculated with 
 

32 

! ω2 −ω2 1 x y α = 1 − , (3.1.11) 2 ω2 

z

q ωr = ω2 

x +αω2 z. (3.1.12) 

3.1.2 Nulling Micromotion 

Unexpected or non-uniform static electric felds from the trap electrodes can lead to the 

trapped ions being pushed off of the pseudopotential null. In this situation, the radial micromotion at 

the trap frequency Ω will frequency modulate the laser beam interactions [51]. Heating can occur 

when the laser frequency is tuned near, but above the center frequency of a micromotion sidebands. 

To compensate for the effects of micromotion, a “shim,” or differential voltage on diagonally-opposite 

pairs of electrodes, can be used to push the ion onto the trap axis. By observing fuorescence of 

an ion as a shim is varied, the frequency modulation can be measured, which will correspond to 

the Bessel function J0(η) with modulation index given by η = A × k, where A is the micromotion 

amplitude at the displacement off of the pseudopotential null and k is the wavenumber of the light 

along the axis of the micromotion. A result of scanning a shim is shown in Fig. 3.2. 

3.1.3 Normal Modes of Multiple Trapped Ions 

The above treatment was for a single trapped ion. However, we often work with multiple 

trapped ions with different masses. Typically, ωz < ωr to force multiple ions in a single trap to stay 

aligned in a one-dimensional crystal along the trap axis (as depicted in Fig. 3.1). However, as more 

ions enter the trap, the crystal may undergo a transition such that it is no long purely along the trap 

axis (such as a “zig-zag” pattern). It is required, then, that for two ions, (ωr/ωz) > 1 and that for 

three ions (ωr/ωz) > 1.55. For a number of ions Nions > 3, there are several estimates, including 

(ωr/ωz) > 0.73N0.86 [52], though there is no exactly-known expression for maintaining a linear ions

chain along the trap axis. 
33 

Figure 3.2: Varying a differential “shim” across two diagonally-opposite electrodes spaced apart by 

' 320μm followed by detection of fuorescence on the cycling transition for a 250 μs duration. The 

shim values are applied as +Shim/2 and -Shim/2 on the opposite electrodes. The shim which gives 

maximum photon counts corresponds to a minimum in modulation, which occurs on the pseudopotential null. The non-zero minimum counts are due to background light. 
 

 

34 

As more ions are trapped, more normal modes must be included along the trap and radial 

axes as Nnormal = 3Nions. The problem becomes one of solving the frequencies of oscillation for all 

these normal modes given the equilibrium positions of ions, the Coulomb interaction between ions, 

and the variation in pseudopotential for different masses. I will assume that the frequencies for a 

single trapped 9Be+ ion are known through measurement as ωx, ωy, and ωz. For Nions in the trap, 

there is a Coulomb coupling between the charges, 

Nions Nions 2 keq Ucc = ∑ ∑ ,i < j, (3.1.13) |r ̄i −r ̄j| j=1 i=1 

where ke = 1/4πε0 is the Coulomb coupling constant, q = |e| the ion charge, and r ̄i, r ̄j are the 

positions of the ions in three dimensions. The equilibrium positions along the trap axis can be 

solved from the real solutions to the system of equations from 

i−1 2 Nions 2 keq keq −kzzi − ∑ ∑ = 0 (3.1.14) (zi −zj)2 + (zi −zj)2 

j=1 j=i+1 

for each i

th ion. For two ions, the inter-ion distance is given by d = (2keq2/kz)1/3 . 

The external potential is the same for each ion, and the axial confnement has no mass 

dependence. Therefore, the external potential on ions along the trap axis is 

Nions

Uz = ∑ 1

kz(r ̄i ·zˆ)

2 (3.1.15) 2 i=1 

where kz = mBeω2 is the axial spring constant, which is the same for all masses. The radial external z

potentials, properly scaled by mass, are 

  ! Nions 2 1 mBe Ux = ∑ mi ωr −αkz (r ̄i · xˆ)

2 , (3.1.16) 2 mi i=1 

  ! Nions 2 1 mBe Uy = ∑ 2 

mi

mi

ωr −(1 −α)kz (r ̄i · yˆ)

2 , (3.1.17) 

i=1 

for mass mi of the ion at the i

th position and α given by Eq. 3.1.11. The total potential, then, is 

UTot = Ucc +Ux +Uy +Uz and a coupling matrix can be formed. 
35 

Taking the frequency of small oscillations at the equilibrium positions, 

∂2U ki, j = , (3.1.18) ∂i∂ j

we form a matrix of spring couplings k with elements ki, j, and arrive at the eigenvalue problem [53] 

ki, j −ω2mi, j = 0. (3.1.19) 

√ To transform to mass-weighted coordinates, we take a diagonal matrix m with elements 1/ mi and 

make the transformation 

 ̄

kmass = m· k ·m. (3.1.20) 

The eigenvalues of the matrix in Eq. 3.1.20 give the normal modes of motion and the eigenvectors 

give the normal mode amplitudes in mass-weighted coordinates. 

I take as an example solving the normal modes for a two-ion, two-species crystal of 9Be+ and 

25Mg+. The results of using the measured parameters 

ωx = 2π×13.33 ×106 

ωy = 2π×11.83 ×106 

ωz = 2π×3.59 ×106 (3.1.21) 

are given in Table 3.1. 

3.1.4 Harmonic Oscillator States 

The motion of trapped ions can be described in the energy basis of the harmonic oscillator 

[6, 39] to very good approximation. The harmonic oscillator Hamiltonian is 

2 2 pˆ mω2zˆ Hˆ = + , (3.1.22) 2m 2 
36 

ω/2π (MHz) xˆBe yˆBe zˆBe xˆMg yˆMg zˆMg 

13.091 0.9997 0 0 0.0261 0 0 

11.560 0 0.9995 0 0 0.0312 0 

5.364 0 0 -0.9299 0 0 0.3677 

4.846 -0.0261 0 0 0.9997 0 0 

3.142 0 -0.0312 0 0 0.9995 0 

2.492 0 0 0.3677 0 0 0.9299 

Table 3.1: The normal modes for a 9Be+-

25Mg+ linear ion crystal with mass-weighted coordinates. 

where the operators pˆ, zˆ are that of the usual momentum and position, respectively, for mass m

and oscillator with frequency ω. The frequency ω can be for any normal mode of harmonic motion 

for a collection of trapped ions. If the operators zˆ and pˆ are rewritten with a = 2

1 (zˆ/z0 + pˆ/p0) and 

a† = 2

1 (zˆ/z0 − pˆ/p0), the Hamiltonian can be rewritten as 

† Hˆ = ~ω(a a+

1 ), (3.1.23) 2 

with the usual lowering operator a|ni = √n|n − 1i, raising operator a†|ni = √n+1|n + 1i, z0 =

p p ~/2mω, and p0 = −i ~mω/2 (the position and momentum wavefunction packet sizes). The 

value z0 offers a natural size scale for a trapped ion to be compared to trap sizes R or laser beam 

waists. A laser incident on an ion interacting with spin and motion has a convenient measure of the 

overlap between the trapped ion and incident laser light. This measure is the Lamb-Dicke parameter 

η = kz0, (3.1.24) 

where k is the incident laser’s wave vector projected along the axis for the oscillator mode associated 

with z0. 

1 For each mode of frequency ωi, there are discrete energy levels En = ~ωi(n+ 2 ) where n is 

an integer. Ions can be initialized in the lowest n = 0 energy state [54, 6] with high probability, and 

felds such as oscillating electric felds on the DC electrodes at frequency ωi can cause excitations 

that populate higher n-states. 
37 

3.1.5 Re-ordering Ions 

A mixed-species linear crystal of ions confned in an ion trap may not necessarily be in the 

desired order along the trap axis. When entangling two qubits, the ion crystal is often made up of two 

Be+ ions as well as two Mg+ ions, such as in Refs. [55, 29], and it is desirable to maintain symmetry 

in the crystal between the two qubit ions. For a 9Be+-

24Mg+ ion crystal, any anharmonicity in the 

trap potential can shift the normal mode frequencies depending on ion order [56]. Therefore, it is 

desirable to be able to re-order an ion crystal. The approach used in Refs. [29, 55, 56] and the work 

in this thesis takes advantage of the lower pseudopotential confnement for higher-mass ions. 

To re-order a 9Be+-

24Mg+ ion crystal, or “BM,” to either a BM or MB orientation, the heavier 

24Mg+ ion must be pushed off-axis and torqued to the desired side of the 9Be+ ion along the trap 

axis. First, a shim is applied with suffcient strength such that the minimum-energy confguration for 

the ion crystal lies along the radial direction. Then, with the shim still applied, a torque is applied 

by adding a uniform bias feld along the trap axis to push the 24Mg+ ion towards the desired side 

of the 9Be+ ion. Then the shim is removed, relaxing the ion crystal back onto the trap axis in the 

desired orientation. 

When working with a four-ion crystal with two 9Be+ ions and two 24Mg+ ions, the BMMB 

confguration is the easiest to achieve. To re-order the ions into BMMB, the axial confnement 

strength is increased. The radial confnement is weakened as the axial confnement is increased. 

Beyond a critical point, the 24Mg+ ions will be forced off of the trap axis before 9Be+ due to the 

relatively weaker radial confnement. The lowest-energy crystal confguration at this point is for the 

two 24Mg+ to symmetrically lie along the radial direction perpendicular to the trap axis, and to be 

positioned between the two 9Be+ ions along the trap axis, in a “diamond” shape. This forces the two 

24Mg+ ions at the center of the crystal, and if the axial confnement is then weakened, the crystal 

will return to the trap axis in the BMMB confguration. 
38 

3.2 Non-classical States of Motion 

3.2.1 Coherent Distribution 

The most classical-like state for the harmonic oscillator is the coherent state. In positionspace, this is a state where the ground-state harmonic oscillator wavefunction is displaced from the 

trap minimum. For example, a coherent state can be excited by a sudden displacement of the trap 

minimum. The wavefunction is then displaced from the trap minimum with an amplitude number 

zmax |α| = 2z0 

where z0 is the wavefunction packet size for the ground-state wavefunction and zmax is 

the maximum displacement of the wavefunction center from the center of the trap. 

The coherent state can be pictured in a phase-space image with a phase φ = ωt as the phase 

evolves in a circle for a fxed amplitude |α|, depicted in Fig. 3.3a, which is analogous to the classical 

behavior for the harmonic oscillator. The coherent state is related to the classical solutions of the 

harmonic oscillator by 

1 z(t) p(t) |α|e−iωt = ( + ), (3.2.1) 2 z0 p0 

for the position z(t) and momentum p(t) in the well. 

The state |αi is an eigenstate of the lowering operator such that 

a|αi = α|αi. (3.2.2) 

It is easy to verify that this is satisfed by 

αn −|α|

2/2 |αi = ∑ √ e |ni. (3.2.3) 

n n! 

From this, the probability of each n-state is Pn = |hn|αi|2 , which gives the distribution of number 

states as 

n ̄n −n ̄ Pn = e , (3.2.4)

n! 
39 

Figure 3.3: Diagram for a coherent state of motion. (a) Phase-space depiction of a coherent state 

|αi. The state evolves in a circle where φ = ωt is the angle with respect to the coordinate zˆ-axis as 

given by Eq. 3.2.1. (b) Plot of the probability of each n-state for a coherent state with n ̄=10. 

which is a Poissonian distribution over number states with mean value 

n ̄ = |α|

2 . (3.2.5) 

A coherent distribution over number states with n ̄ = 10 is shown in Fig. 3.3b. 

The displacement operator which creates the coherent state Dˆ (α)|n = 0i = |αi is given by 

αa a Dˆ (α) = e

†−α∗

. (3.2.6) 

In Ref. [57], a coherent state was generated by applying a classical, resonant electric feld to a trap 

electrode. The Hamiltonian for an oscillating potential along the normal mode axis zˆ takes the form 

eE0z0 i(ωt−φ) −e−i(ωt−φ) V(t) = −eE0zˆsin(ωt −φ) = i (a+a ) †)(e (3.2.7) 2 

for ion charge e and feld amplitude E0. In the interaction picture for the well, in which a → ae−iωt, 

and neglecting rapidly-oscillating terms with frequency 2ω, this becomes 

eE0z0 −iφ −a† iφ V(t) = i (ae e ). (3.2.8) 2 

The time-propagation operator for the interaction is then 

eE0z0 † U(t) = exp[− t(ae−iφ −a eiφ)], (3.2.9) 2~
40 

which is the displacement operator of Eq. 3.2.6 with 

eE0z0

α(t) = teiφ. (3.2.10) 2~

This gives that the coherent state amplitude increases linearly with an applied resonant electric 

feld. 

3.2.2 Thermal Distribution 

The thermal state often appears in ion trap motion which is most conveniently described as 

an incoherent distribution of number states. This motional distribution appears when there is a 

collection of random coherent kicks to the ion. This is the case for the motional distribution following 

Doppler cooling [36]. It also arises from a collection of random trap displacements over many 

observations of the motion. A common issue arises from ambient heating with a heating rate n ̇ ̄ 

due to fuctuating electric feld noise on trap surfaces radiating and interacting with the trapped ion 

[24, 25, 26] (typical heating rates for our systems is n ̇ ̄ ∼ 0.5 quanta/ms). Measuring the ion motion 

from these random excitations over many measurements will give rise to observing the thermal 

distribution. 

Treating the ion as being in contact with a thermal reservoir at temperature T, the motion has 

a probability of a particular number state 

1 

e−βEn e−β~ω(n+2 )

Pn = = , (3.2.11) Z Z

where β = 1/kBT, kB being the Boltzmann constant, and Z is the partition function. The partition 

function is the sum over all states, 

∞ −β~ω(n+2 ) Z = ∑ e

1 

, (3.2.12) 

n=0 

and this geometric series sums up to 
41 

Figure 3.4: Plot of the probability for each n-state for a thermal state with n ̄=10. 

Z = e−β~ω/2 

1 −e−β~ω . (3.2.13) 

From this, we have an expression for Pn. The zero-point energy cancels and we arrive at 

Pn −β~ωn(1 −e−β~ω = e ). (3.2.14) 

The average number occupation n ̄ evaluates to 

n ̄ =

∞

∑ nPn

n=0 

= 1 . eβ~ω −1 (3.2.15) 

After some algebra, this can be used to arrive at the probability distribution for number states, which 

is 

n ̄n

Pn = (3.2.16) (n ̄ +1)n+1 ,

and is plotted for n ̄=10 in Fig. 3.4. 

It is not necessarily sensible to describe a single trapped ion as having a temperature T. 

However, we can reliably use Eq. 3.2.16 to approximate energy from a thermal expectation, such 

as for estimating the motional distribution following Doppler cooling on normal modes of motion. 
42 

Figure 3.5: Diagram for a squeezed state of motion. (a) Phase-space depiction of a squeezed 

state. The state evolves with φ = ωt as the angle with respect to the coordinate zˆ-axis. As depicted, 

the state along the zˆ-axis was squeezed with contraction by βs and the pˆ-axis expanded by βs at 

the start of state evolution. (b) Plot of the probability for each n-state for a squeezed state with 

βs=40. (c) The same probability distribution with a factor of ten smaller vertical scale to showcase 

the relatively long “tail” of the distribution. 

3.2.3 Squeezed Distribution 

The minimum-uncertainty state of the harmonic oscillator is the ground state. The variance 

in both position (Δz2 = (1/mω)~/2) and momentum (Δp2 = (mω)~/2) clamps the distribution of 

2 2 position and momentum to Δz ·Δp = ~2/4. A physical process may then “squeeze” the distribution 

in z, which will expand the distribution in p, and the state time-evolves with an out-of-phase breathing 

in the distributions of z and p in the harmonic well. The squeezed state can be characterized by a 

squeezing parameter βs such that Δzs = Δz/βs [58, 39]. A phase-space depiction is shown in Fig. 

3.5a. 

A squeezed state can be generated with a force which modulates the trapping well at 2ω

(in Ref. [57], this was done with a parametric drive by two Raman beams at 90◦with a frequency 

difference of 2ω incident on an ion initially in the state |n = 0i) and hence only even-n states will 

engage in the distribution. The probability distribution for even n is given by [58, 39] 

p  n 2 βs βs −1 n! 2−n Pn = (3.2.17) βs +1 βs +1 [(n/2)!]

2 .

This distribution has an exceptionally long “tail,” that is, a signifcant occupation of number states 

out to large n, which is depicted in Fig. 3.5 for βs=40. 
43 

Typically, squeezed states take parametric excitations to drive any relevant amount of population into higher n states. An instantaneous change in a trap, for example, from an initial 2 MHz 

confnement to a fnal 200 kHz confnement (which is an unrealistic situation) would lead to a 

p squeezing parameter of z0, f /z0,i = ωi/ωf = βs = 3.16, and this squeezing parameter corresponds to Pn=0 ' 0.85 ground-state occupation. Further analysis in Ref. [58] includes a description 

for a coherently displaced squeeze state, which adds more subtleties to the squeezed state. For 

example, typically, the state obtained from squeezing and then coherently displacing a state is not 

the same as that from coherently displacing and then squeezing the state. 

3.3 Laser-Motion Interactions 

3.3.1 Motional-Sideband Laser Interactions 

The stimulated Raman transitions, discussed in Chapter 2, have an extremely narrow linewidth 

for transitions between states of the 9Be+ S1/2 hyperfne manifold. The “carrier” transition, or the 

on-resonance transitions between hyperfne states, can be driven with either Raman beams that 

are co-propagating (co-carrier) or 90◦ from each other (90-carrier). Interactions with the ions’ motion is achieved when the Raman beams are in the 90-carrier confguration and have a difference 

k-vector δk along the motional axis. By scanning the frequency of one or both of the laser beams, 

the Raman transition can resolve motional sidebands to the blue or red of the 90-carrier transition. 

For linewidths much narrower than the kth normal mode frequency ωk, given by Rabi rates much 

less than ωk, then the carrier as well as the motional sideband resonances are spectrally resolved. 

By giving a relative detuning between two laser beams such that L1 has frequency ω1 = ωi−Δ and 

L2 has frequency ω2 = ωi − Δ − ω0 − δ where δ = nωk, n an integer, then the beams resonantly 

drive a Raman transition which entangles the motion with the qubit. 

To describe the Raman transitions between two levels, we start with time-independent H0 =

~ω0 † Hspin+Hmotion = 2 σz+~ωza a, with the usual Pauli spin operator σz. The interaction Hamiltonian 

is 
44 

Figure 3.6: Diagram of the Raman beam alignment for beams L1 and L2 with δk along the trap axis. 

The beams are ideally aligned to interact exclusively with normal modes along the trap axis. 
45 

HI = −μ ̄ ·B, (3.3.1) 

for the coupling between the dipole moment μ ̄ = μS ̄ for spin S and the applied feld B = εˆBcos(kz−

ωt + φ) with wavenumber k and frequency ω. We assume that the feld is propagating in the zˆ 

direction and polarized in the xˆ direction. Then Eq. 3.3.1 becomes 

HI = ~Ω(σ+ +σ−)[ei(kz−ωt+φ) +e−i(kz−ωt+φ)] (3.3.2) 2 

1 1 where ~Ω = −μB/4 and with spin operators σ+ = 2 (σx +iσy), σ− = 2 (σx −iσy), and the coordinate z = z0(a+a†). 

Transforming to the interaction picture, in which a → ae−iωzt and σ+ → σ+eiω0t

, we take the 

rotating-wave approximation to get 

H0 −iωzt † = ~Ωσ+ exp(i[η(ae +a eiωzt)−δt +φ])+h.c. (3.3.3) I 2 

where δ = ω − ω0 is the detuning from the carrier resonance and η = kz0 is the Lamb-Dicke parameter. 

The wavefunction can be written as 

∞

Ψ = ∑ ∑ Cs,n(t)|si|ni (3.3.4) 

s=↓,↑ n=0 

for spin states |si, motional states |ni, and normalized complex coeffcients Cs,n(t) for each state. 

For the case of on-resonance transitions δ = (n0

−n)ωz, where n0 and n are integers (off-resonance 

transitions included in Ref. [6]), Schrodinger’s equation gives the set of equations  ̈

−i

(1−|n0−n|) −iφΩn0

,n C ̇

↓,n = e C↑,n0, 2 

0

−n|) iφΩn0

,n C ̇

↑,n0 = −i

(1+|n e C↓,n, (3.3.5) 2 
46 

which corresponds to two-level Rabi oscillations with a Rabi rate dependent on the motional interaction. The Rabi rates for these transitions are given by [59, 60, 6] 

iη(a+a Ωn0

,n = Ωn,n0 = Ω|hn

0

|e

†)

|ni|

= Ωe−η2/2[n<!/n>!]

(1/2)

η|n0

−n|

Ln

|n

<

0

−n|

(η2), (3.3.6) 

for driving between number states |ni to |n0

i with n< the lesser and n> the greater, and Ω is the 

90-carrier Rabi rate. Lα(x) is the generalized Laguerre polynomial defned by n

n � n+α xm

Lα

n (x) ∑ (−1)

m n−m m!

. (3.3.7) 

m=0 

The effects on the Raman transition due to the motion must be taken into consideration whenever driving transitions in the 90-carrier beam confguration. The most common cases in our experiments are to set δ to either be tuned to the frst motion adding sideband, i.e., n0 = n+1 which adds 

a quanta of motion, to the frst motion subtracting sideband n0 = n−1 which removes a quanta of 

motion, or the carrier n0 = n; these cases are diagrammed in Fig. 3.7 to show the transitions. 

3.3.2 Interactions in the Presence of Multiple Normal Modes 

Driving a single-ion Raman transition in the presence of multiple ions adds extra terms to the 

Rabi rates due to the addition of more normal modes of motion affecting the transition. Assume 

that the lasers interact with the j

th ion in a linear chain of length Nions ions. The Hamiltonian now 

includes the sum of each normal mode, giving 

3Nions ~ω0 † H0 j = 2 σz j + ∑ ~ωlal al. (3.3.8) 

l=1 

with index k corresponding to each normal mode. The interaction Hamiltonian in Eq. 3.3.3 is now 

given by 
47 

Figure 3.7: Depiction of transitions between two spin levels for the frst blue-detuned, or motion 

adding, sideband (adding one quanta of motion), the frst red-detuned, or motion subtracting, sideband (removing one quanta of motion), and the carrier transition. 
48 

3Nions −iωlt † HI j

0 = ~Ω

σ+j exp(i[ ∑ ηl

j

(ale +al eiωlt)−δt +φ])+h.c., (3.3.9) 2 l=1 

where the interaction now contains a sum over terms with the Lamb-Dicke parameters and frequencies for each mode. The interaction is therefore affected by every normal mode, and this gives rise 

to a Rabi rate for resonant transitions between states | ↓,nki and | ↑,n0

ki of [6] 

3Nions † 0 iηj

Ωn0

k,nk = Ω|hnp6=k,nk| ∏ e l(al+al )

|np6=k,nki|. (3.3.10) 

l=1 

The “spectator” modes p 6= k contribute multiplying factors to the driven sideband (often referred-to 

as Debye-Waller factors) given by Eq. 3.3.6. In the case of our 90◦ Raman beam confguration, 

ideally there are only interactions along the zˆ-axis of the trap reducing the contributions from 3Nions

modes to only Nions modes. 

Typically our beam waist is ∼30 μm, too large to only address a single ion in a same-species 

chain (on the order of a 5 − 10μm-long chain). In the case of Rabi oscillations on multiple ions, 

the dynamics are more complicated due to the multiple spins interacting with individual modes as 

in Ref. [54]. In the case of two ions, and in the regime where η1,η2  1 (the Lamb-Dicke regime), 

the frst sideband transition is 

√ −(η2 

1+η2

Ωn10

,n1(n2) ' Ωη1 n1>e 2)/2(1−n2η2

2) (3.3.11) 

for a sideband transition on mode 1 with spectator mode 2. 

3.3.3 Doppler Cooling on Normal Modes 

In Chapter 2, I described Doppler cooling to reduce the net kinetic energy of ions. Our Doppler 

cooling beam is not aligned purely along any one trap axis, thereby allowing laser cooling in all trap 

directions. Once an ion is confned, the average energy achieved from Doppler cooling leaves the 

ion in a thermal distribution of number states set by the Doppler cooling limit. From Chapter 2 Eq. 

2.3.5, the Doppler cooling limit for the zˆ-direction was given by 
49 

ω/2π (MHz) n ̄limit Axis 

12.26 0.54 xˆ

11.19 0.62 yˆ 

2.69 2.59 zˆ

Table 3.2: Doppler-cooling limits n ̄limit for each axis when 9Be+ is Doppler cooled on a transition 

with a linewidth of Γ ' 2π×19.4 MHz, where the incident light is σ+-polarized and in the direction 

√ xˆ+yˆ+ 2zˆ ˆ

k = . 2 

p ~Γ 1 +I/Isat fsi kBTmin = (1+ ) (3.3.12) 4 fi

with fi = 1/2 and fsi = 7/20 for a σ+-polarized beam that lies along our external magnetic feld 

aligned 45◦ to the trap axis. 

Generally, the concept of temperature only applies to particles in contact with some thermal 

reservoir. However, Doppler cooling prepares the motion in a thermal state [40, 36], and Eq. 3.2.15 

gives us n ̄ for the normal mode with frequency ω given kBTmin after Doppler cooling. An example 

calculation for the average motional occupations along the axial and radial directions for a 9Be+

√ xˆ+yˆ+ 2zˆ ion are shown in Table 3.2, assuming rˆ breaks into xˆ and yˆ directions such that k

ˆ = and 2 

that fx,y = 1/4. In reality, as the radial direction’s xˆ and yˆ components are not always perfectly 

known with respect to the beam direction, the Doppler cooling limit may be different than calculated 

between the two directions. 

3.3.4 Sideband Cooling of Normal Modes 

When entangling two ions, we use the normal modes of motion as an information bus to 

entangle two ions in a shared trap. It is typically best to prepare the normal modes of motion near 

the |n = 0i ground state. After Doppler cooling, we accomplish this ground-state cooling through 

Raman sideband cooling [61, 54]. A cycle of sideband cooling begins with a motion subtracting 

sideband (MSS) to remove a quantum, followed by optical repumping, which for η  1 incoherently 

performs a carrier-like transition on the ion back to the initial state. For 9Be+, we drive the |F =
50 

Figure 3.8: Diagram of the Raman sideband cooling process on a single mode of motion. First a 

motion subtracting sideband pulse is applied to the initial state | ↓i, for us the 9Be+ |2,2i state, 

to drive the transition | ↓,ni → | ↑,n − 1i where | ↑i in our case is the 9Be+ |1,1i state. This is 

followed by applying our repumping beams for 10 −30μs to dissipate the ions’ internal energy and 

perform a carrier-like transition. Many repetitions of this process prepare the ion near the |n = 0i

state of motion. 

2,mF = 2,ni → |F = 1,mF = 1,n − 1i transition on the MSS. Then, our repumping beams are 

applied for 10 −30μs to return to the |2,2i initial state without a signifcant change in the motion, 

which is like an incoherent carrier transition on |1,1i → |2,2i. This dissipates the motional energy 

(diagrammed in Fig. 3.8). After repeated applications, when most of the motional occupation is 

prepared in the |n = 0i motional state, the motion subtracting sideband will no longer be able to 

drive the transition |F = 2,mF = 2,ni → |F = 1,mF = 1,n−1i as there is no motional state lower 

than the |n = 0i state. 

As each normal mode of motion for the trapped ions is in a thermal state following Doppler 

cooling, even a low n ̄k on mode k will have motional occupation of higher |ni states. Therefore, it 

requires many applications of the Raman sideband cooling cycle. Since there is a different Rabi 

rate Ωn−1,n for each number state given by Eq. 3.3.10, it is most effcient to vary the duration of the 

motion subtracting sideband pulses by τn−1,n = π/Ωn−1,n for each Raman cooling cycle. Plots of 

the cooling pulse durations and the Rabi rates are given in Fig. 3.9 to show how the pulse durations 

vary when considering a single normal mode. Photon recoil from repumping and heating rates 

prevent perfectly achieving the |n = 0i state. However, for example, motion in a starting thermal 
51 

Figure 3.9: Plot of (a) Rabi π-times and (b) Rabi rates for different number states n taking 

Ω=1. These plots assume 313 nm light for the Raman transitions on 9Be+ in a trap with 

ω = 2π×2.69 MHz for the axial normal mode frequency and η ' 0.41. The fastest π-time occurs 

for n = 5. 

state of n ̄ = 3 after Doppler cooling, applying ∼20 cooling cycles will be suffcient to ground-state 

cool a single ion to n ̄ ' 0.1 quanta. 

3.4 Analyzing Motional States 

Driving Rabi oscillations on a motional sideband, which entangles motion and qubit, and 

measuring the qubit state as a function of pulse duration will allow observation of the probability distribution the driven normal mode. For a single ion initialized in the state | ↓,ni, the Rabi oscillations 

take the form 

P↓(t) = |cos(Ωn0

,nt)|

2 = 1 [1 +cos(2Ωn0

,nt)] (3.4.1) 2 

with Ωn0

,n given by Eq. 3.3.6. In general, the motion will be in a distribution of motional states, in 

which case the Rabi oscillations take on a beat-note signal due to interference of Rabi rates from 

multiple n-states. The Rabi oscillations are then [57, 6] 

" # ∞

P↓(t) = 11 + ∑ Pn cos(2Ωn0

,nt) , (3.4.2) 2 n=0 

where Pn is the probability of the ion being in the state |ni. A Fourier analysis of this signal yields 

the probability distribution of the |ni states. An example of analyzing the Rabi oscillations for a 
52 

Figure 3.10: Rabi fopping trace for a thermal state of motion with n ̄ = 1.3 ±0.1 and η = 0.202 for 

a single 9Be+ ion driving on the frst motion adding sideband. (a) The Rabi oscillations are shown 

with distinct interference in the signal due to a distribution of number states. (b) The distribution of 

number states extracted from the Rabi oscillations. Figures reproduced from Ref. [57]. 

single ion in a thermal state of motion with n ̄ = 1.3 is shown in Fig. 3.10 from the work in Ref. 

[57] (in which coherent, thermal, and squeezed states were all observed through the sideband Rabi 

oscillations on a single 9Be+ ion). Typicall for small values of n ̄ . 0.1, it is diffcult to distinguish 

between a thermal and coherent distribution. 

When there are multiple modes of motion, the Rabi oscillations include terms from all normal 

modes. Then, the Rabi oscillation signal becomes 

" # 3Nions ∞ 3Nions 1 0 iηl(al+a† 

l ) P↓(t) = 1 + ∏ ∑ Pnl cos(2Ω|hnp=6 k,nk| ∏ e |np=6 k,nki|t) , (3.4.3) 2 l=1 nl=0 l=1 

where the terms in Eq. 3.4.2 for Ωn0

,n have been replaced by the multi-mode case in Eq. 3.3.10 and 

the sum over probability Pn has been replaced by a sum over all products for each probability of the 

ion being in the state |nli for each normal mode l. For a specifc situation, the Rabi oscillations for a 

Raman transition in the 90-carrier confguration on a 9Be+-

25Mg+ ion chain’s axial modes is given 

by 

" # ∞ ∞ 1 † † iηm(am+am) iηn(an+a P↓(t) = 2

1 + ∑ ∑ PnPm cos(2Ω|hm,n

0

|e e n)|m,ni|t) , (3.4.4) 

n=0 m=0 

where number states |ni and |mi are for the two normal modes of motion (the in-phase INPH

mode and out-of-phase OOPH mode) with respective populations Pn and Pm, and for a sideband 
53 

0 50 100 150 200

0

0.2

0.4

0.6

0.8

1

Duration of MAS drive on OOPH (μs)

P

↓(t)

Figure 3.11: Rabi fopping trace from driving on the out-of-phase normal mode of a 9Be+-

25Mg+ ion 

pair. Here the Lamb-Dicke parameters are [ηINPH, ηOOPH]=[0.16, 0.27] and the curve fts to a thermal distribution of [n ̄INPH, n ̄OOPH]'[0.34, 0.01]. Histograms were collected from ion fuorescence 

many times, which were used to construct the Rabi oscillation trace. 

0 0 0 transition on the mode with states |ni, with n = n+1 for the MAS, n = n−1 for the MSS, or n = n

for a carrier transition, while the mode with states |mi is a spectator mode. An example is given in 

Fig. 3.11 for a motion adding sideband oscillation curve performed on a 9Be+-

25Mg+ ion chain. 

A shortcut for analyzing a thermal state of motion takes advantage of the convenient form of 

Pn for a thermal state. If it is known that the ion is in a thermal state, such as after Doppler cooling, 

then observing both the motion adding and subtracting sidebands can be used to quickly extract n ̄ 

for the thermal state. For a drive time td on-resonance with the sideband, the state evolves to 

∞

PMSS

↓ = ∑ Pn cos2(2Ωn−1,ntd)

n=1 

∞

PMAS

↓ = ∑ Pn cos2(2Ωn+1,ntd), (3.4.5) 

n=0 

 ̄ with Pn = nn

from Eq. 3.2.16. Then, taking the ratio gives (n ̄+1)n+1 

n ̄n

∑∞

n=1 n+1)n+1 cos2(2Ωn−1,ntd) ( ̄ R = nn . (3.4.6)  ̄ ∑∞

n=0 (n ̄+1)n+1 cos2(2Ωn+1,ntd)

Factoring out n+

n ̄

1 from the numerator and shifting the sum, we then have  ̄ 
54 

∑∞ n ̄n

n ̄ n=0 (n ̄+1)n+1 cos2(2Ωn,n+1td) n ̄ 

R = n ̄n = , (3.4.7)

n ̄ +1 ∑∞

n=0 (n ̄+1)n+1 cos2(2Ωn+1,ntd) n ̄ +1 

and thus 

R

n ̄ = (3.4.8) 1 −R

for a thermal state. This only requires comparing the initial-state fuorescence to the fuorescence 

after the sideband drives on a single ion. This ratio method also works in the presence of other 

normal modes, as the Debye-Waller factors will cancel. 

3.5 Sympathetic Cooling with 24Mg+ and 25Mg+

Laser-cooling the 9Be+ qubit-ions disrupts the qubit state of 9Be+, and so during sequences 

of two-qubit entanglement, it is necessary to be able to cool the ions’ motion without disrupting 

the qubit states. One solution to this problem is through “sympathetic cooling,” where laser cooling 

is applied to “coolant ions” to cool the shared motion in a trap with both coolant and qubit ions 

[6, 62]. By using another ion species with resonant wavelengths far-detuned from those of 9Be+, 

laser-cooling can be applied to the ion chain without disrupting qubit states. 

We sympathetically cool with either 24Mg+ or 25Mg+ as the coolant ion, which have resonant transitions at 280 nm, far from the 9Be+ transitions at 313 nm. For the work in this thesis, 

Doppler cooling was applied to Mg+ coolant ions, though Raman sideband cooling with 24Mg+ was 

instrumental as part of the method sets for work in Refs. [55, 29, 27, 28] before each entanglement operation between qubits in a shared trap zone. Entanglement in the presence of dissipation, 

where the dissipation was partially engineered by sympathetic cooling on 24Mg+, was demonstrated in Ref. [63]. Sympathetic cooling using an electromagnetic-induced-transparency effect 

to suppress motion-adding interactions while driving motion-subtracting interactions was demonstrated in Ref. [63] and described theoretically in Ref. [64]. We demonstrated this electromagneticinduced-transparency cooling by sympathetic cooling on a 9Be+-

24Mg+ ion pair to n ̄INPH ' 0.15 
55 

and n ̄OOPH ' 0.08 in 85μs, and this method could potentially reduce the temporal overhead for 

ground-state cooling in future work with scalable QIP. 

The 24Mg+ atomic structure has no hyperfne splitting, and the S1/2 states mJ = 1/2 and mJ =

−1/2 are split by ∼334 MHz at an external magnetic feld of B = 11.964 mT. The next lowest orbital 

states are the P1/2 and P3/2 states, which have a fne structure splitting of ∼2.75 THz [65], which are 

coupled to the S states by 280 nm laser light. Doppler cooling is achieved with laser light at 280 nm 

red-detuned from the transitions |S1/2,mJ = −1/2i ←→ |P1/2,mJ = 1/2i with σ+-polarized light 

and |S1/2,mJ = 1/2i ←→ |P1/2,mJ = 1/2i with π-polarized light. Both beams are necessary, as 

there is no closed two-level cycling transition between S1/2 and P1/2 at non-zero external magnetic 

feld. These transitions have a linewidth of Γ ' 2π×41 MHz, and so optimal Doppler cooling occurs 

with a beam detuned from |P1/2,mJ = 1/2i on the order of −Γ/2 ' −2π×20.5 MHz. However, a 

“dark state” can arise due to a coherent process from interference between the two cooling beams 

when they are at the same detuning from the P1/2 state [42], so an offset of ∼ 10 MHz detuning is 

used between the beams. Doppler cooling on 24Mg+ is covered extensively in Ref. [14], including 

dark state analysis. 

The 25Mg+ atomic structure does have a hyperfne splitting due to the nuclear spin of I = 5/2, 

splitting the S1/2 manifold into two hyperfne levels F = 2 and F = 3. A beneft to 25Mg+ is that 

at the external magnetic felds for a feld-insensitive qubit in 9Be+ at B = 11.945 mT, the 25Mg+

hyperfne states |S1/2,F = 3,mF = 1i and |S1/2,F = 2,mF = 0i are split by 1763.18440809 MHz 

and form a near-feld-insensitive qubit that could possibly be used as a qubit in future experiments. 

The hyperfne levels at our typical external magnetic feld values have a closed two-level cycling 

transition for Doppler cooling, achieved by σ+-polarized light that drives the |S1/2,F = 3,mF =

3i ←→ |P3/2,F = 4,mF = 4i transition with 280 nm light. The linewidth for this transition is Γ '

2π×41 MHz, so optimal Doppler cooling is achieved with a detuning by −Γ/2 ' −2π×20.5 MHz 

from the transition. The beam for far-detuned Doppler cooling is ∼ -500 MHz red-detuned from the 

S1/2 to P3/2 transition. 
Chapter 4 

Apparatus 

This chapter describes the apparatus for running our experiments. The key apparatus components include multi-electrode ion traps, a vacuum-containment system for the traps, and optics 

for laser beam control. To carry out our scheme for scalable quantum information processing, the 

traps are designed with multiple electrodes that can be used to create many spatially distinct trap 

regions for ion confnement [8]. We have two wafer traps in which the ion transport experiments 

described in this thesis have been carried out: the “quantum logic trap” with up to 6 trap zones and 

the “X-junction trap” with up to 18 trap zones, each with similar electrode size-scales. To interact 

with the ions, up to six laser sources, many servo loops to stabilize laser intensity and position, and 

pulse-shaped microwaves are incident on one or more trap regions to drive atomic transitions. We 

have the capability to trap both 9Be+ and 24Mg+ (in the quantum logic trap) or 9Be+ and 25Mg+

(in the X-junction trap) in a shared trap zone for sympathetic cooling. This chapter will highlight the 

most important components and technical details which the apparatus comprises. Much of the work 

in this thesis was carried out in the quantum logic trap, though the relevant apparatus is extensively 

covered in Ref. [14]. As of this thesis, experiments are carried out with the X-junction trap, and this 

chapter focuses on the most current apparatus setup. First, I will describe the multi-electrode ion 

traps. Then I will describe the vacuum containment system, including hardware for loading ions. I 

then describe magnetic feld coils and imaging system, which are external to the vacuum system. 

Lastly, I describe and diagram our laser beam setup for the ion-laser interactions. 
57 

4.1 Multiple Electrode Wafer Trap 

To produce a multi-electrode trap, the idealized rod electrodes described in Chapter 3 can 

be replaced with wafers that have conducting material deposited on insulating surfaces. The traps 

described here were fabricated from wafers made out of laser-machined alumina sputter-coated 

with gold to form electrode surfaces [12]. Electrode details for both RF and DC electrodes were frst 

machined into the wafer surfaces with an excimer laser. The minimum size for electrode features 

is set by the excimer laser’s minimum focus size, which at the time of construction was 20 μm. A 

slot was also machined into each wafer, which allows optical access to the trapping region. Next, 

a titanium adhesion layer was laid down on the electrode features and then sputter-coated with 

gold on top of the adhesion layer. The deposition process produces a surface layer of conducting 

traces over the machined alumina, and this process is repeated on both sides of the wafer to 

produce the trap electrodes. Thorough descriptions of wafer trap fabrication method can be found 

in Refs. [12, 50, 14]. Here, I will focus on the geometry features that are important in ion transport 

experiments. 

Wafer traps have been used for many experiments to demonstrate the building blocks for 

scalable ion-based QIP. Early work with multi-electrode ion traps involving transport led the way 

to improved electrode geometries. For instance, transport and separation of ions in a wafer trap 

was frst demonstrated in Ref. [12], where it was shown that ion transport can be achieved with 

little motional excitation. However, the separation, or splitting, of two ions from a single trap led 

to signifcant motional excitation. For the work in Ref. [22], a trap was designed with a relatively 

smaller electrode structure, as compared to the electrode sizes and ion spacings in the trap in 

Ref. [12]. The new trap enabled greatly reduced motional excitation during separation, which was 

attributed to separating the ions over a smaller DC electrode that acted as a narrow wedge. Another 

improvement was based on a complication reported by Ref. [24], in which it was shown that when 

loading ions into the trap, stray 9Be+ could coat electrode surfaces and could contribute to ambient 

heating rates of the ion’s motion. A mask was put in place in the trap of Ref. [12] to shield the 
58 

trap electrodes of the experimental zone during ion loading and thus mitigated metal deposition 

which in turn led to a reduction in motional excitations. Other negative motional effects come from 

electric feld noise on the electrode surfaces [24]. Electrically fltering the potentials applied to DC 

electrodes reduces noise levels that contribute to ambient heating rates of the trapped ion’s motion. 

All of these improvements were incorporated into subsequent wafer designs in the traps used for 

the work described in this thesis. 

4.2 Trap Geometries 

4.2.1 Quantum Logic Trap 

A multizone two-wafer trap, designed by M. Barrett and fabricated by J. Jost [14], was designed to be used to demonstrate the basic building blocks for scalable QIP in a multizone trap 

array. This trap is designated the “quantum logic trap.” The quantum logic trap was used for several experiments to demonstrate scalable techniques for QIP, which include the demonstration of 

feld-independent qubits with 9Be+ [35], entangling two separate mechanical oscillators [55], and 

experiments combining ion transport with quantum logic to demonstrate method-sets for a two-qubit 

quantum processor [29, 27, 28]. A photo of the quantum logic trap is shown in Fig. 4.2.1. This is 

the trap used for transport work in Ref. [66]. 

The quantum logic trap is composed of wafers with eight pairs of diagonally-opposite, segmented DC electrodes and a pair of diagonally-opposite RF electrodes. In-depth schematics of 

the electrode dimensions are contained in Ref. [14]. The spacing between each DC electrode 

segment on a single wafer is set by the minimum machining width of 20 μm. Each wafer has a 

laser-machined slot cut out of it for laser access to the trapping region between the two wafers. The 

narrow section of the RF-to-DC slot is 200 μm wide and is designated as the experiment region. 

Experiments are principally carried out in two different “interaction” zones, zones with laser access 

to address qubit-ions, which are located over 200 μm wide electrodes surrounding a narrow 100 μm 

wide “separation” electrode. The two zones provide confnement 370 μm apart, symmetric about 
59 

1.5 cm

Trap Waffers

Filter Boards

RF Electrode

DC Electrodes

Figure 4.1: Multizone wafer linear RF Paul trap, the “quantum logic trap.” Two trap wafers, each with 

segmented electrodes, are mounted together to form parallel planes spaced 200 μm apart. The 

trap has gold electrodes deposited on laser-machined alumina wafers. The wafers are attached to 

a flter board that electrically flters the DC potentials applied to the DC electrodes. 
60 

the separation electrode. The relevant dimensions are shown in a model of the trap in Fig. 4.2. Ions 

transported between interaction zones traverse the separation electrode, which is so-named as it is 

used to apply a separation “wedge” potential between ions to separate ions apart into distinct trap 

zones. The width of the separation wedge was limited by laser machining capabilities at the time 

of construction. In the wider portion of the slot, designated as the load region, the RF-to-DC slot is 

400 μm wide. The slot is wider and displaced from the experiment region to help mitigate deposition 

of metal onto the experiment-electrode surfaces during loading. 

The trap wafers are attached to flter boards to reduce electric feld noise on the DC electrodes 

(one of the flter boards is visible in Fig. 4.2.1). Each DC electrode has a corresponding trace on 

the flter board with a low-pass RC flter, so that externally-applied potentials have been fltered 

to reduce noisy felds and thus reduce ambient motional heating effects. The flters comprise an 

820 pF capacitor to RF ground and a 240 Ω resistor. These flters have a corner frequency of 

1/2πRC ∼800 kHz to flter high frequency noise near the trap axial frequency. External to the 

vacuum system, there is additional fltering by two in-series RC flters with corner frequencies of 

∼ 200 kHz, each with a 1 nF capacitor to RF ground and an 820 Ω resistor. The DC potentials, in 

the range of -10 V to +10 V, can maintain confnement in the axial direction with ωz/2π ≈ 2.7 MHz for 

a single 9Be+ ion. In this trap, a typical axial heating rate for a single 9Be+ ion is n ̇ ̄ ∼ 0.5 quanta/ms. 

A model in Fig. 4.3 illustrates the relative positions of a two-ion crystal along the trap axis between 

the two trap wafers. 

The radial confnement is provided by RF applied to the RF electrodes. The RF is supplied by 

an external signal generator producing ∼1.5 W of RF power that is inductively coupled to a quarterwave coaxial resonator made of copper. The resonator has Q ∼ 300 and a resonant frequency of 

150 MHz. The RF electrode potential has ∼180 V peak amplitude. The RF parameters used in the 

lab generate a pseudopotential with ωr/2π ≈ 11 MHz radial frequency for a single 9Be+ ion. 

The side-view dimensions of the wafers are shown in Fig. 4.4, which includes additional alumina wafers. The trap wafers are both 125 μm thick and spaced apart by 200 μm with by alumina 

spacers. As the experiment region’s slot is also 200 μm, the trap electrodes form a symmetric ge-
61 

Figure 4.2: A model of the electrodes for the quantum logic trap. (a) Model of the trap showing the 

diagonally opposite segmented-DC electrodes, RF electrode (red), and biasboard (purple) in the 

quantum logic trap. The electrode wafers are spaced by 200 μm, and the biasboard is 400 μm from 

the bottom of the trap electrode wafers and is used to produce a uniform bias feld orthogonal to 

the trap axis. (b) Dimensions for the experiment region. The slot between the DC electrodes and 

the RF electrode on each wafer is 200 μm in the “experiment” region and 400 μm in a wider “load” 

region. The DC electrodes are spaced diagonally apart by ∼280 μm. The wafers have diagonally 

opposite DC and RF electrodes to trap ions with a similar geometry to the idealized rod trap design 

discussed in Chapter 3. 

Figure 4.3: A close-up model of the trap wafers in the quantum logic trap. It illustrates where ions 

would be trapped along the trap axis (not to scale) between the two trap wafers. 
62 

Figure 4.4: Schematic of the side view of the quantum logic trap. The spacer material is made from 

alumina. The bias wafer provides a source of global bias felds in the trap. A neutral atom source 

emits atoms towards the load region. The load region is spatially displaced from the experiment 

regions and the neutral atom source is partially blocked by a mask wafer to prevent deposition on 

experiment electrodes. Ionizing radiation is incident on the neutral atoms; once ionized, the atoms 

can be trapped in the load zone. 
63 

ometry around the trap axis. Another wafer, the biasboard wafer, is spaced 400 μm below the trap 

wafers. It has a large surface electrode designed to generate a uniform bias shim across the trap, 

largely orthogonal to the trap axis. When loading ions, a neutral atom source sends a fux of atoms 

through the gap in the wafers. There is a mask wafer 1.5 mm beneath the bias-board wafer, with 

a large electrode typically held at RF ground, primarily used to block incident neutral atoms or ions 

from depositing on the experiment electrodes during loading, but to still allow laser beams to pass 

through the load region. 

4.2.2 X-junction Trap 

The “X-junction trap” was designed to demonstrate transport reliably through a junction with 

low motional excitations [67, 15], which is important for re-ordering ions in a scalable QIP multielectrode geometry. This X-junction trap was constructed with the same methods described in the 

previous section. It was designed and built by R. B. Blakestad [50]. A photo of the X-junction trap is 

shown in in Fig. 4.5. As of this thesis, it is also being used in work towards demonstrating scalable 

QIP and fast transport experiments. 

The X-junction trap is formed from two wafers with a total of 46 DC electrodes (Fig. 4.6). 

In-depth details of the design and geometry are found in Ref. [50]. There is an X-shape machined 

into the wafers to form a junction. Inside the junction, there are three exits. The exits to the left 

and right of the junction lead to two experiment regions with similar electrode geometry to each 

other, each allowing two convenient spatially-distinct ion-laser interaction zones and a separation 

electrode. These regions are designated E (experiment) and H (horizontal). The fnal exit is the 

region designated V (vertical); this zone does not contain a separation electrode. In both experiment 

regions, there are pairs of 200 μm electrodes surrounding a 100 μm separation electrode as in 

the quantum logic trap. Additional relevant electrode geometry includes the vertical region, which 

has two 200 μm electrodes, and within the junction which is surrounded by two 100 μm-width DC 

electrodes per wafer. In the experiment and vertical regions, the wafer RF-to-DC slot is 200 μm. A 

load zone is connected to E through a 45◦ arm with a 300 μm slot width. 
64 

Figure 4.5: Multizone, balanced-geometry wafer trap, the “X-junction” trap. Two wafers, each with 

segmented electrodes, are mounted together to form parallel planes spaced 250 μm apart. The 

trap has gold electrodes deposited on laser-machined alumina wafers. The wafers are attached to 

a flter board that electrically flters the DC potentials applied to the DC electrodes. 
65 

A trapped 9Be+ ion has an axial frequency of ωz/2π ≈ 3.6 MHz. The two interaction trap 

zones in the E and H regions provide confnement 340 μm apart about the separation electrode. 

The DC electrodes, as in the quantum logic trap, are connected to external potentials through a 

flter board, which can be seen in Fig. 4.5. The flter board has the same RC components as in the 

quantum logic trap’s flter board, with an 820 pF capacitor to RF ground and a 240 Ω resistor giving 

a corner frequency of ∼800 kHz. An additional external RC flter applied to DC potentials has a 

1 nF capacitor and a 1 kΩ resistor, giving a corner frequency of ∼160 kHz. In this trap, a typical 

axial heating rate for a single 9Be+ ion is n ̇ ̄ ∼ 0.2 quanta/ms. 

The various wafers to form the X-junction trap are similar to that of the quantum logic trap, 

but the spacings differ somewhat. The trap wafers are 125 μm thick. The electrode wafers are 

displaced by a 250 μm alumina spacer. When higher order terms in trap potential were evaluated, 

R. B. Blakestad found it benefcial to use a larger wafer spacing relative to the DC-to-RF slot to 

more closely approach a perfect quadrupole. This wafer geometry is referred to as a balanced twowafer geometry. There is a biasboard wafer to produce shim bias felds, seen as the rectangular 

back-plane with a large slot comparable to the size of the trap wafers in Fig. 4.6. The biasboard 

wafer is spaced from the trap wafers by a 500 μm alumina spacer. The alumina spacer between 

the bottom trap wafer and the biasboard has a surface that extends beyond the trap wafer region 

to also serve as the flter board. Instead of an attached mask wafer like in the quantum logic trap, 

there is a stainless-steel shield (not shown) to shield the experiment regions from the neutral atom 

sources during loading. 

Each wafer has RF electrodes diagonally opposite from each other across the junction. The 

RF electrodes on a single wafer are connected to form a continuous conductor via a 70 μm wide 

bridge. This bridge is necessary for maintaining a confning pseudopotential inside the junction. 

A quarter-wave coaxial resonator made of copper delivers RF to both of the RF electrodes. The 

resonator has a loaded Q ∼ 42 and a resonant frequency of 83 MHz. For better spectral purity, an 

external half-wave helical resonator is in series with the coaxial resonator with a Q ∼ 145 for superior 

signal fltering on the applied RF to reduce motional excitations, particularly when traversing the 
66 

Figure 4.6: A model of the electrodes for the X-junction trap. (a) The X-junction trap can be used 

to transport ions to one of three possible regions from an X-shaped ensemble of electrodes. The 

electrode wafers are 125μm thick and spaced apart by 250μm. The X-junction trap has a total of 18 

trap zones that include two experiment regions E and H and a vertical region V. In the experiment 

and vertical regions, the laser-machined slot spacing is 200 μm. A loading region, L, is connected 

via a 45◦ arm far from the junction with a slot-width of 300 μm. The biasboard is 500 μm from the 

bottom of the trap electrode wafers and is used to produce a uniform bias feld orthogonal to the 

trap axis. The biasboard is machined with a large slot to allow laser access to all trap zones. (b) 

Dimensions for the experiment region E. It has similar geometry to the experiment region in the 

quantum logic trap, except that the diagonal spacing of the DC electrodes is 320μm. The RF bridge 

across the junction can be seen on the right. 
67 

junction [67, 15]. An external signal generator applies ∼1.6 W of RF that is inductively coupled to 

the resonator. The RF electrode potential has ∼200 V peak amplitude. These RF parameters lead 

to a pseudopotential with a ωr/2π ≈ 12 MHz radial frequency for a 9Be+ ion. 

4.3 Containment System 

4.3.1 Vacuum Components 

Collisions between trapped ions and background gas lead to knocking ions out of the ion trap. 

Therefore, the ion traps are isolated from the environment by a vacuum system. Our wafer traps are 

enclosed in a quartz cell with fused-silica windows for laser access, which is kept at low pressure. 

The vacuum pressure is kept at < 10−10 torr, which is typical in ion traps. The layout of our vacuum 

system is shown in Fig. 4.7. To maintain the low pressure, an ion pump continuously removes 

background gas. An ion gauge provides a measurement of the pressure in the system. A titaniumsublimation pump evaporates a titanium layer in the local vicinity to capture stray background gas 

(primarily H2) in the system. This pump is run infrequently, only at times when we suspect that there 

is an excess of background gas about the ion trap. 

There are electrical feedthroughs to bring various potentials from outside the vacuum into 

the system. There are 25-pin ribbon-cable connectors (one socket in the quantum logic trap, two 

in the X-junction trap) to carry potentials to the DC electrodes. Additional electrical feedthroughs 

supply currents to neutral-atom sources of 9Be and 24Mg (quantum logic trap) or 25Mg (X-junction 

trap) for the frst stage of loading ions. Currents, ∼1 A, heat metal structures holding bulk neutralatom material inside the vacuum system, and the heat causes neutral atoms to evaporate into the 

chamber. 

The ion trap and the RF resonator extend out from the side of the vacuum chamber. A quartz 

cell encloses the trap and resonator, keeping them under vacuum. In Fig. 4.7, the quartz cell 

surrounding the trap is shown as a nonuniform cylinder; the ion trap is surrounded by four windows 

that allow laser beams access to the trap from multiple directions. The windows are made from 
68 

Figure 4.7: Diagram of the vacuum system used to maintain low pressure around the ion traps. The 

trap is enclosed in a quartz cell. The vacuum is kept at less than 10−10 torr. The DC and RF to the 

trap electrodes are introduced from outside of the vacuum system. 
69 

UV-grade fused silica. The slots in the trap wafers allow beams to pass from the windows, through 

the trap, and then exit out the other side. The beams which enter through various viewports are 

for Doppler cooling, ion-internal-state preparation, and Raman transitions, which are sometimes 

required in multiple zones in a single experiment. 

4.3.2 Loading Ions 

To load ions into the trap, a neutral atom source emits atoms towards the loading zone. 

There are two neutral atom sources: one is a 9Be wire wrapped around a tungsten flament and 

the other is a stainless-steel tube loaded with isotope-enriched 24Mg (in the quantum logic trap) or 

25Mg (in the X-junction trap). Atoms are emitted by heating their supporting metal structure; the 

heating evaporates atoms into the vacuum. As neutral atoms are emitted, they are ionized by either 

electron-bombardment from an in-vacuum electron gun or by photo-ionization (PI) light from laser 

2 sources. PI light is tuned to resonantly excited neutral 9Be from 2s2 to 2s2p, or 25Mg from 3s

to 3s3p, after which a second photon excites the electron to continuum. We also shine Dopplercooling light on the load region to cool ions once ionized in the trap zone beneath the trap depth. A 

typical loading rate in our experiment is one ion in every ∼10 s. 

The 9Be atoms are ionized through PI laser light with a 235 nm wavelength. To generate the 

PI light, frst a laser-source at 532 nm with 5 W of power enters an optical cavity with a Titanium 

Sapphire (ti:sapph) crystal to generate a mode-locked pulsed laser at 705 nm with time-averaged 

power of ∼600 mW. This is then tripled through frst a doubling stage with second harmonic generation (SHG) crystal and then a single-pass summing stage with a SHG crystal to a 235 nm source 

with time-averaged power of ∼10 mW, aligned to be incident on the vacuum chamber towards the 

load region. PI light with a 285 nm wavelength is used to ionize the neutral 25Mg. This is generated 

starting with a near-infrared seed laser at 1140 nm which is amplifed and sent through single-pass 

SHG crystal to achieve 570 nm light. This is followed by a doubling cavity using a BBO crystal for 

the fnal 285 nm UV light with 100 μW incident on the load region. 
70 

Figure 4.8: Images collected from a CCD camera simultaneously imaging fuorescence from 25Mg+

on the left and 9Be+ on the right. (a) Two ion species trapped in a shared trap zone with an 

approximate ∼4 μm spacing. (b) After applying a bias-feld orthogonal to the trap access by a 

shim across diagonally-opposite DC electrodes, the ion crystal twists off of the trap axis due to the 

different pseudopotential confnement strength for the different-mass ions. 

4.3.3 Imaging System 

One of the windows in the quartz cell is used as a view port for the imaging system (shown 

in Fig. 4.9). A charge-coupled device (CCD) camera is typically used when loading ions so that 

we can see the ions as they are loaded into the trap, though during experiments we image the ion 

fuorescence on a photo-multiplier tube (PMT) for greater quantum effciency. When our Dopplercooling or resonant laser light scatters off of the ion, ion fuorescence is collected through several 

focusing optics. Our optics allow us to spatially resolve multiple ions in a single trap zone when 

imaging on the CCD camera. 

The current imaging system as of this thesis has an initial stage for focusing the fuorescence 

from trapped ions, which is a 10x objective lens capable of imaging two different wavelengths (see 

Ref. [68] for details). Both 9Be+ light at 313 nm and 25Mg+ light at 280 nm can be imaged simultaneously. The next stage of focusing comes from a curved mirror which gives 5x magnifcation, 

for a fnal 50x magnifcation. See Fig. 4.8 for CCD image captures of two ion species simultaneously imaged in a single trap zone in the X-junction trap. The PMT is used to collect experimental 

data through detecting photons scattered from either ion species (see Chapter 2). For the experiments in this thesis, we only detect 9Be+, however, the ability to image detection light from both 

ion species opens up the possibility of performing QIP experiments on qubit states from both ion 

species simultaneously. 
71 

4.3.4 Magnetic Field Coils 

We require a ∼ 120 Gauss magnetic feld to achieve the feld-insensitive qubit splitting we 

use for QIP as discussed in Chapter 2. The feld is generated by a pair of water-cooled coils 

of copper in a Helmholtz confguration, looped in a rectangular shape with dimensions 40 cm by 

20 cm, separated from each other by 8 cm and centered on the ion position, as diagrammed in 

Fig. 4.9. The coil is made of twenty turns of hollow, square copper tubing, with 0.157” width and a 

0.1” diameter round bore coated with kapton insulation. We apply 63 A of current to each coil to 

produce the desired feld strengths at the position of the ion confnement. This current can generate 

∼300 W of heat, necessitating water cooling. An additional pair of coils with 40 turns of 20 AWG 

copper wire are also used for adjustments to the feld. We periodically intersperse experiments with 

a measurement of the resonant transition frequency for a 9Be+ hyperfne transition, and based on 

the result alter the current through the smaller coils until the correct magnetic feld is applied to 

achieve our feld-insensitive qubit transition. 

4.4 Lasers 

Incident laser light interacts with ions by entering through fused silica windows in the quartz 

cell and passing through the slots etched into the trap wafers. The magnetic feld defnes the quantization axis for the ions’ atomic states. Thus, the incident direction of light relative to the magnetic 

feld Bˆ affects the allowed selection rules for internal-state transitions. Since the laser beam electric 

felds must be orthogonal to the beam direction, any beams traveling along the quantization axis 

do not have π-polarized light with respect to the ion, as discussed in Chapter 2. We need beams 

for carrier Raman transitions (those which directly drive transitions between hyperfne states) along 

with a confguration with a difference k-vector δk along the trap axis to interact with axial motion. 

Therefore, we need multiple laser-beam directions incident on our ions to perform all of the transitions that we need for our experiments. As shown in the diagram in Fig. 4.9, there are four windows 

that provide different beam directions. The diagram’s caption describes the various laser beams’ 
72 

Beam Visible Wavelength (nm) Visible Power (mW) UV Wavelength (nm) 

BD 626.266 330 313.13 

RD 626.395 230 313.2 

Raman 626.610 2050 313.31 

Table 4.1: Visible and corresponding UV wavelengths and visible powers for the 9Be+ lasers. 

alignments with respect to the trap axis and magnetic feld. 

To-date, there are no commercially available UV sources with the powers necessary for our 

experiments, so our UV laser beams are derived from longer-wavelength sources. The light necessary for ion-laser interactions with 9Be+ is at 313 nm for both Raman transitions and those used 

for Doppler cooling beams (the Blue Doppler beamline) and repumping beams (the Red Doppler 

beamline). These are generated from laser sources that begin with seed lasers in the IR and 

then sent through amplifers. The lasers are then single-pass summed through a temperaturecontrolled MgO:PPLN crystal held at ∼190 K temperature to produce visible wavelengths in the 

red at ∼626 nm [69]. The wavelengths needed after the frequency-summing is similar enough for 

all the beams such that power from the seed lasers at 1050 nm and 1550 nm can be re-used for 

different beams. 

After the visible light is generated, the frequencies for Blue Doppler and Red Doppler are 

stabilized to an I2 Doppler-free saturated absorption spectroscopy signal. The Raman beams, used 

for the Raman transitions between hyperfne states in 9Be+, are detuned by -250 GHz from the 

S1/2 to P1/2 transition (see Chapter 2), though the absolute frequency is not critical. Rather, the 

difference-frequency between two Raman beams is more critical, and therefore the beams are not 

stabilized to any frequency reference. Then, the visible light is sent into a bow-tie, or 4-mirror ring, 

cavity with a frequency-doubling BBO crystal at one of the foci [70, 71, 69]. The cavities consist of 

four mirrors, one of which is mounted on a PZT. The cavity length is stabilized to stay on-resonance 

through feedback on the PZT using a Hansch-Couillaud lock system [72]. The wavelengths and 

powers for the visible for 9Be+ beams are listed in Table 4.1 along with the corresponding UV 

wavelengths. 
73 

Figure 4.9: The relative alignment of the ion imaging system, magnetic feld coils, incident laser light, 

and the ion trap. The 9Be+ Raman beams, aligned 90◦ to each other labelled Be Co (so-called for 

sometimes having co-propagating beams with a relative detuning for carrier Raman transitions that 

are motion-insensitive) and Be 90 (so-called for the relative allignment to Be Co for motion-sensitive 

Raman transitions), are depicted entering upper and lower ports of the left side of the diagram. 

9Be+ blue Doppler and detection beams labelled Be BD, far-detuned Doppler cooling beam labelled Be Detuned, and repumping beams labelled Be Rd and Be Repumper (names respective 

to the transitions discussed in Chapter 2), enter on the upper right window and pass through the 

experiment region E of the X-junction trap. The 25Mg+ Doppler cooling beam labelled Mg BD and 

the far-detuned Doppler cooling beam labelled Mg Detuned co-propagate with the 9Be+ Doppler 

beams. Some of the Doppler cooling light is redirected towards the loading region L when loading 

ions into the trap. The coils generate a ∼120 Gauss magnetic feld at the position of the trapped 

ions. The imaging system is meant to focus light scattered off of the ions onto a CCD camera or a 

PMT to observe fuorescence from the trapped ions under incident detection light. 
74 

The 25Mg+ Doppler-cooling light is derived from a 560 nm dye laser with a bow-tie cavity.. 

The same dye laser setup was used for 24Mg+ for the quantum logic trap, with details described in 

Ref. [14]. The dye laser cavity is stabilized in long-term to a I2 Doppler-free saturated absorption 

spectroscopy signal. In short-term the laser is stabilized to an external reference Fabry-Pérot cavity 

with feedback on a PZT-mounted mirror and a PZT-adjustable etalon within the dye laser cavity. 

The visible light has a wavelength of 559.269 nm with ∼300 mW of power. This is sent through a 

frequency-doubling cavity similar to those used for 9Be+ light for a UV wavelength of 279.63 nm. 

All of the beams are position-stabilized before entering the trap apparatus. After the UV beam 

exits the doubling cavity, the position stabilization is achieved by frst placing two mirrors mounted on 

PZTs as part of the beam path, and then picking off a small amount of laser light from the UV beam 

in two places farther down-stream, closer to the trap apparatus. The picked-off light is incident on a 

position-sensing quadrant detector, and as the beam deviates from a desired position, feedback on 

the PZT-mounted mirrors realigns the beam position. All of these beams pass through double-pass 

acousto-optic modulators (AOMs) to shift their frequencies. To switch certain laser beams on or off 

during an experiment, a TTL signal from our experiment-control FPGA switches the RF fed to the 

AOMs on or off to turn on or extinguish the diffracted beam. Following the UV beamlines, the beams 

then are passed through UV-compatible fbers to deliver the beams to the fused-silica windows to 

then pass through the ion trap [73]. Then, before entering the trap, the beams pass through λ/4 

and λ/2 waveplates to select the necessary polarizations followed by a lens to focus the beams 

to ∼ 30μm waists at the position of the ions. This beam waist is suffciently large to illuminate our 

typical trapped ion chains, but is much smaller than the distance between trap zones. 

4.4.1 9Be+ Blue Doppler Beam Line 

The Blue Doppler UV beam line for Doppler cooling (discussed in Chapter 2) is diagrammed 

in Fig. 4.10. This beam line synthesizes three beams for the |S1/2,F = 2,mF = 2i ←→ |P3/2,F =

3,mF = 3i, denoted simply S1/2 to P3/2, transitions. AOMs are used to shift the input beam frequency to the desired values. The UV laser beam’s frequency is initially tuned so that shifting the 
75 

input beam by ∼ +600 MHz shifts the beam to near-resonance. This beam, which we call BD, is 

then tunable between optimal Doppler cooling with a red-detuning of −Γ/2 ' 2π×9.7 MHz from the 

S1/2 to P3/2 transition and on-resonance with S1/2 to P3/2 for detection. The beam for far-detuned 

Doppler cooling, or BD detuned, comes from shifting the input frequency by ∼ +200 MHz, which is 

∼ -400 MHz red-detuned from the S1/2 to P3/2 transition. The BD beam enters the trap with ∼ 5μW 

of power, and the BD detuned beam with ∼ 500μW of power. 

4.4.2 9Be+ Red Doppler Beam Line 

The Red Doppler UV beam line for repumping into the initial state for 9Be+ (discussed in 

Chapter 2) is diagrammed in Fig. 4.11. This beam line synthesizes two beams for the |S1/2,F =

2,mF = 1i ←→ |P1/2,F = 2,mF = 2i repumper and |S1/2,F = 1,mF = 1i ←→ |P1/2,F = 2,mF =

2i RD transitions used for optically pumping for state initialization. AOMs are used to shift the beam 

frequency to the desired values. The UV laser beam’s frequency is initially tuned so that it is inbetween the resonances for the |S1/2,F = 2,mF = 1i ←→ |P1/2,F = 2,mF = 2i and |S1/2,F =

1,mF = 1i ←→ |P1/2,F = 2,mF = 2i transitions, and this beam is then split into two beams to 

be tuned to resonance for the repumper beam by +800 MHz and RD beam by −310 MHz. This 

difference in frequency then spans the hyperfne splitting for the S1/2 states. The RD beam enters 

the trap with ∼ 10μW of power and the repumper beam with ∼ 5μW of power. 

The Blue Doppler beams are combined with the Red Doppler beams through a 50/50 beam 

splitter. When we desire to have these beams in an “off” state, to avoid leaked light incident on the 

ions due to imperfect extinguishing of the RF to the AOMs or pick-up in the AOM transducers, we 

use a “switch AOM” confguration. This comprises two additional 200 MHz AOMs (not diagrammed), 

through which the beams are aligned for the +1 order and then the -1 order through the pair of 

AOMs. This results in no net shift in the beam frequencies, but allows a strong extinguishing of the 

beam when an off state is desired. After the Blue Doppler and Red Doppler beams pass through 

the switch AOM confguration, they are coupled into a UV-compatible fber and delivered to the 

fused-silica input windows to the ion trap. 
76 

4.4.3 9Be+ Raman Beam Line 

The Raman beam lines for driving Raman transitions between hyperfne states of the S1/2 

hyperfne manifold in 9Be+ are diagrammed in Fig. 4.12. The beamline synthesizes two beams 

from the input UV beam. The Co beamline is so-named for being capable of having co-propagating 

Raman transition beams for carrier-transitions on the hyperfne states. There is a 600 MHz AOM 

which is in a double-pass confguration carefully aligned such that both the 0th order and +1 order 

beams leave the AOM overlapped with each other, and are co-propagating for the rest of the beamline. This enables carrier Raman transitions with a single beamline. The Co beamline then passes 

through a double-pass Co AOM to allow shifts in the beam such that the Co coupled with another 

Raman beam span the S1/2 hyperfne splittings. For motion-sensitive Raman transitions, there is 

also a 90 beamline so-called for being 90◦ to the Co beam when entering the ion trap. This beam 

passes through a double pass AOM, so that the Co is shifted by +620 MHz in the double-pass Co 

AOM and the 90 beam shifted by -420 MHz in the double-pass 90 AOM to span the S1/2 hyperfne 

splittings. The beams then pass through UV fbers which delivers the beams to the trap, and the Co 

and 90 beams each enter the ion trap with ∼40 mW of power. 

4.4.4 25Mg+ Doppler Beam Line 

The 25Mg+ UV beam line for Doppler cooling (discussed in Chapter 3) is diagrammed in Fig. 

4.13.. This beam line synthesizes two beams for the |S1/2,F = 3,mF = 3i ←→ |P3/2,F = 4,mF =

4i transition, denoted as S1/2 to P3/2 transition. The UV laser beam’s frequency is initially tuned so 

that shifting the input beam by ∼ +710 MHz with AOMs shifts the beam to near-resonance. This 

beam, which we call MG BD, is used for near-detuned cooling light with a detuning of −Γ/2 '

−2π×21 MHz from the S1/2 to P3/2 transition. The beam for far-detuned Doppler cooling, or MG 

detuned, is ∼ -500 MHz red-detuned from the S1/2 to P3/2 transition. The MG BD beam enters the 

trap with ∼ 30μW of power, and the MG detuned beam with ∼ 1 mW of power. 
77 

Figure 4.10: 9Be+ Blue Doppler beam line beginning with UV from the doubling cavity on the left. 

The boxes with frequency shifts denote AOMs in the beam path. The output path of the double-pass 

AOM confguration has a small vertical offset, which is then refected off of a mirror in a different 

direction from the input path. After the doubling cavity, the beam has a small amount of power 

diverted to quadrant position detectors to stabilize the beam position. The beam enters the BD 

detuned split AOM, so-called for splitting the Blue Doppler BD beam from the far-detuned cooling 

beam BD detuned, and when this AOM is switched on, the +1 diffracted order is sent to a +200 MHz 

double-pass AOM labeled BD SW (so-named as it is used as the switch for BD) to generate the 

near-detuned cooling light or resonant light. The BD detuned SW AOM is used to put +200 MHz 

on the far-detuned beam through the +1 diffracted order, and the BD beam is recombined with BD 

detuned by being alligned co-propagating with the +1 order BD detuned SW path. 

Figure 4.11: 9Be+ Red Doppler beam line beginning with UV from the doubling cavity on the left. 

The boxes with frequency shifts denote AOMs in the beam path. The output path of the doublepass AOM confgurations has a small vertical offset, which is then refected off of a mirror in a 

different direction from the input path. After the doubling cavity, the beam has a small amount of 

power diverted to quadrant position detectors to stabilize the beam position. The beam passes a 

λ/2 waveplate followed by a Glan-laser designed to split the beam into two distinct beams with 

orthogonal polarizations for the repumper and RD beams (same-polarization achieved again with 

a λ/2 waveplate in the repumper path). The repumper is double-passed through an AOM for a 

+800 MHz shift and the RD is double-passed for a -310 MHz shift. The Red Doppler beams are 

then recombined on a 50/50 beam splitter before then combining with the Blue Doppler beams on 

another 50/50 beam splitter. 
78 

Figure 4.12: 9Be+ Raman beam lines beginning with UV from the doubling cavity on the left. The 

boxes with frequency shifts denote AOMs in the beam path. The output path of the double-pass 

AOM confgurations has a small vertical offset, which is then refected off of a mirror in a different direction from the input path. After the doubling cavity, the beam has a small amount of power diverted 

to quadrant position detectors to stabilize the beam position. The beam passes a λ/2 waveplate 

followed by a Glan-laser designed to split the beam into two distinct beams with orthogonal polarizations for the Co and 90 Raman beams (same-polarization achieved again with a λ/2 waveplate 

in the 90 path). The Co beam passes through the AOM600 which is in a double-pass confguration carefully alligned such that both the 0th order and +1 order leave the AOM co-propagating to 

drive carrier Raman transitions. For motion-sensitive Raman transitions, we use the 90 beamline, 

so-called for being 90◦ to the Co beam when entering the ion trap. During Co-90 motion-sensitive 

transitions, the AOM600 is switched off and the Co is shifted by +620 MHz in the double-pass Co 

AOM and the 90 beam shifted by -420 MHz in the double-pass 90 AOM to span the S1/2 hyperfne 

splittings. The AOMs have bandwidths that allow scanning ranges of ∼ ±50 MHz, allowing us to 

achieve our desired hyperfne transitions. The beams then pass through UV fbers which delivers 

the beams to the trap. 
79 

Figure 4.13: 25Mg+ Doppler beam line beginning with UV from the doubling cavity on the left. The 

boxes with frequency shifts denote AOMs in the beam path. The output path of the double-pass 

AOM confguration has a small vertical offset, which is then refected off of a mirror in a different 

direction from the input path. After the doubling cavity, the beam has a small amount of power 

diverted to quadrant position detectors to stabilize the beam position. The beam enters the MG 

Split AOM, so-called for splitting the Blue Doppler MG BD beam from the far-detuned cooling beam 

MG detuned, and when this AOM is switched on, the +1 diffracted order is sent to a +200 MHz 

double-pass AOM labeled MG BD to generate the near-detuned cooling light. The MG detuned 

AOM is used to put +200 MHz on the far-detuned beam through the +1 diffracted order, and the MG 

BD beam is recombined with MG detuned by being alligned co-propagating with the +1 order. The 

beams then pass through UV fbers which delivers the beams to the trap. 
Chapter 5 

Experiment Control 

As the complexity grows in our experiments, so too does the need for adaptable and scalable 

experimental control. Our experimental needs include precise control over the timing, frequencies, 

and phases of laser or microwave pulses, controlling the timing for ion transport through timevarying trap potentials, and PMT count collection for state detection. Additionally, an experimental 

sequence may requiring branching (a change in the experimental sequence based on the result of 

state detection) which is useful for things such as for error correction during quantum information 

processing. We also need rapid adaptability from day to day and experiment to experiment, such as 

being able to rapidly measure and calibrate motional sideband frequencies, calibrate the external 

magnetic feld, and other regular calibrations interspersed with the main experimental work. Our 

experimental control comes in large part from a home-built system which comprises a graphical 

user interface (GUI) on a host computer, TTL and direct digital synthesis (DDS) interface sequenced 

by a feld-programmable gate array (FPGA), and peripheral software and hardware control. This 

experiment FPGA system forms the core of the interface between us and the quantum information 

processor. 

A key component to our approach to scalable QIP is to achieve information transport through 

the physical transport of ions between spatially-distinct trap regions. This transport requires fast and 

precise electronics to control the trap potentials if transport is to be performed in a way such that 

motional states of ions are preserved. To address this need, I developed a novel device dedicated 

to effcient and fast arbitrary voltage outputs in a ±10 V range for our experiments, which can be 
81 

used for trap confnement, ion transport, and for quantum logic pulse shaping. It is designed with 

a 50 MHz update rate, which is much faster than both the ion trap oscillation frequency and our 

Raman transition Rabi rates. The device is based around an FPGA as a controller that updates 

the inputs to digital-to-analog converters (DACs) that in turn output the voltages to a fast amplifer. 

It has the capability to functionalize multiple waveforms and switch between them based on TTL 

inputs during experiments. 

In this chapter, I frst give a brief introduction to FPGA technology. I then describe how it is 

used for both timing and frequency control of the atom-laser interactions as well as measurement of 

the ion states. For the bulk of the chapter, I describe the FPGA-based electronics that I developed 

for an arbitrary waveform generator; a summary of the hardware and applications is detailed in 

Ref. [74], with expanded information provided in this chapter and full schematics in Appendix B. Its 

primary use is for precision control of the trap potentials, described here with specifc attention to 

its features for scalable methods for ion transport. It is also used for laser- and microwave-pulse 

shaping, and I briefy describe this application. 

5.1 Introduction to FPGAs 

FPGAs are logic ICs (integrated circuits) that are reconfgurable for user-specifed logic operations. They were frst commercially available from Xilinx, though there are now a number of 

manufacturers. An FPGA chip largely comprises internal, on-chip arrays of logic gates, which are 

confgured to perform whatever logic sequences the user demands. There are input and output 

logic signals to the FPGA, which the FPGA interprets through its confgured internal logic gates. 

The FPGA thus is confgurable to functionalize its outputs based on its inputs. The FPGA also 

comprises logic blocks that may include registers, memory, buffer drivers, multiplexers, and mathematical operations. This is a technology that allows enormous amounts of digital adaptability. 

The FPGA logic-gate confguration, or gateware, is user-defned through computer communication. A common communication method is through a joint test action group (JTAG) programmer. 
82 

Specifc data that contains the confguration information for the on-chip logic is often derived from 

a hardware description language, typically VHDL or Verilog; the FPGAs in our lab are typically designed with VHDL. FPGA manufacturers often provide software tools to assist in synthesizing code 

into an FPGA confguration. The language is designed to be convenient for specifying logic sequences. The logic on the FPGA updates each clock cycle from an externally-supplied clock signal. 

This makes the logic timing of the FPGA deterministically known to the user. The language syntax 

defnes processes in parallel operations, thereby specifying simultaneous operations at each clock 

cycle in a deterministic way. 

Each physical input or output line to and from the FPGA chip can be described in VHDL; in 

code, it is treated as a signal, which is an object operated upon in a way similar to a variable. Signals 

are used either for logic gates (such as if statements) or to produce a derived logic output. An 

FPGA confguration, then, can be used as a sequencer of logic operations, potentially conditioned 

on external signals. This is used for controlling logic pulse sequencing by having logic output signals 

from the FPGA directly control switching between on and off states in devices such as RF switches, 

or more generally having inputs and outputs communicate with a variety of ICs. In addition to being 

an array of logic gates, FPGAs have on-chip blocks of static RAM. The RAM can hold data to be 

stored and dynamically accessed during FPGA operations. In this way, a host computer could send 

data that signals the FPGA to store information in block RAM, such that the FPGA later has access 

to data in real-time. Uploading information to RAM provides a way to load data before an experiment 

begins so that the FPGA can run an experimental procedure purely from within its own memory with 

no loss in run time due to computer interface. 

5.2 Experiment-Control FPGA 

In our typical experiments, such as in Refs. [27, 28, 66], we need many laser pulses with varying frequencies, durations, and phases, which are possibly interspersed with transport waveforms, 

and followed by detection of ion states through collection of photon counts from a photo-multiplier 
83 

tube (PMT). The experiments are controlled by a single “experiment” FPGA that is used to control 

and sequence all of these necessary operations. This experiment FPGA uses the Virtex-IV FPGA 

from Xilinx, interfaced through a commercial evaluation board, which is confgured to receive data 

from a host experiment-control computer (connected through a PCI slot) to specify the timing and 

sequencing of the experiment. 

The in-depth details for this pulse sequencer are in Ref. [34], so I will restrict the discussion 

to the key features when running experiments. The experiment FPGA is clocked by an external 

synthesizer at 62.5 MHz. The FPGA provides our experiments with 16 independent TTLs. It can 

control of up to 32 independent direct digital synthesizers (DDSs), which are principally used to set 

frequencies and phases of laser pulses by feeding the digitally synthesized RF signals from the 

DDSs into acousto-optic modulators (AOMs) and defectors (AODs). Most TTL channels control 

the on/off states of RF switches for DDS signals fed to the AOMs and AODs, thus controlling the 

timing of the laser pulses. One TTL is dedicated to triggering ion transport waveforms. Before an 

experimental sequence begins, the host computer downloads a specifc sequence of TTL and DDS 

settings to the experiment FPGA into its block RAM. The FPGA reads RAM in real time, controlling 

the experiment as the user had defned. 

A DDS produces an RF source from a digital phase accumulator and a sine-wave look-up table along with an input reference clock, with fnal analog output from a DAC. The phase accumulator 

divides the desired output frequency by the reference clock and uses this to calculate the step-size 

at which an address register steps through the sine-wave lookup table, where the register points to 

locations in the sine-wave look-up table. The phase accumulator typically has far more digital precision than typically available in the look-up table or the DAC output such that it maintains a much 

higher-precision calculation of the phase than available in the fnal output of the DAC. The DDSs 

in the experiment, the AD9858, have a 32-bit phase accumulator, 14-bit phase-offset adjustment, 

and an output voltage from 10-bit digital-to-analog conversion. To ensure accuracy and stability, 

the DDSs are synchronized to a 1 GHz signal derived from the 10 MHz Hydrogen maser source at 

NIST. 
84 

An experimental sequence ends with collecting PMT counts from ion state detection. The 

FPGA has a dedicated TTL input channel which reads photon counts from a PMT for the detection 

duration, typically ∼ 200μs. The count data is then received by the host computer over many 

repeats of an experiment to build up suitable statistics at the end of many rounds of experimental 

sequences. The FPGA is designed to allow the conditional branching off of the detection results if 

necessary, which has applications in error correction or in functionalizing certain experiments. 

We have a graphical user interface (GUI) to an experiment-control program that allows us to 

conveniently intersperse many user-defned experiments. Our own parser language, which serves 

as a high-level language for our digital control, had been designed for defning experiments through 

our FPGA system. This allows us to use a very simple language to specify relatively complex sequences and careful timing for our lasers, waveforms, and other experimental parameters. The 

experiment-control program translates the results of parsing our digital control language into a 

stream of data that the FPGA interprets, stores into RAM, and then runs to produce the physical 

realization of our experiment. 

5.3 Arbitrary Waveform Generator 

In demonstrations of our multi-electrode trap array scheme for QIP, ion transport required 

hundreds of microseconds to remain in the adiabatic regime, where the forces applied to the ion 

cause only small displacements from the center of the confning potential, while each quantum logic 

operation typically took ∼ 10μs [29, 55, 27, 28]. For larger scale quantum information processing, 

development of larger trap architectures is necessary; some examples of the work towards larger 

multi-electrode trap designs can be seen in Refs. [13, 18, 19, 15, 20, 21]. However, the time 

scale discrepancy between quantum logic and ion transport poses a bottleneck for rapid quantum 

information processing. In addition, slow transport leads to an increased motional excitation due to 

heating from electric feld noise originating from electrode surfaces [24], and therefore increases the 

laser cooling needed to initialize motion close to the quantum ground state via Doppler and Raman 
85 

sideband cooling, which in turn further increases the incurred time penalty for ion transport. To 

overcome these challenges, it is necessary to make ion transport times comparable to gate speeds 

which will necessarily require leaving the adiabatic limit where transport can not be considered slow 

compared the ion motional trap period. 

The ambient heating in ion traps is the dominant source of motional excitation when performing adiabatic ion transport. Work in Ref. [25] reported a 100-fold improvement in heating rates from 

electrode surface cleaning, but this method cannot easily be applied to our multiple-wafer traps and 

transport durations still need to be reduced. Experiments in Refs. [55, 67] used commercially available digital-to-analog converters (DACs) with update rates of 500 kHz for ion transport. However, 

update rates lower than the frequencies of ion motion in the confning potential well are unsuitable 

for precise control of the potential on time scales similar to or faster than the trapped ions’ motional 

period. In addition, harmonics of the 500 kHz update rate could cause signifcant heating of the 

ions’ motion if not heavily fltered [15]. A further requirement comes from large-scale error correction algorithms that may require the ability to quickly alter ion transport conditioned on the detection 

of errors. For example, a demonstration of an error correction algorithm, which is conditioned on 

the detection of errors, has been carried out in this multi-electrode scheme [30] had a similar slow 

transport. Hence, our electronic needs for this quantum information processor scheme include fast 

DAC update rates compared to confnement frequencies, which can be used to reduce the duration of ion transport with low motional excitations on a quantum logic time scales. We also require 

the ability to control many electrodes at once with low latency in accessing DAC output patterns 

(waveforms) that can be conditionally selected. 

To this end, I addressed our technical needs by creating a fast multi-channel arbitrary waveform generator (AWG) with an update rate of 50 MHz, well beyond currently relevant trapping frequencies (typically ∼1 MHz). The design is easily scalable to control a large number of electrodes. 

The AWG is based on an FPGA, which controls an ensemble of DACs that provide arbitrary analog 

outputs in a ±10 V range. The device is capable of simultaneously storing multiple “branches,” 

different sets of waveforms that can be selected by use of digital logic inputs on the device with a 
86 

Figure 5.1: A fow diagram overview of the AWG. The host computer transmits waveform data via 

USB to the FPGA which is stored in on-chip RAM. The FPGA transmits a 16-bit digital signal derived 

from its RAM to each DAC, whose output is then amplifed. 

40 ns latency. This section covers the hardware which comprises the AWG and the software control, 

followed by the underlying operation of the FPGA, and then gives examples of experimental uses. 

5.3.1 Hardware 

The core of the AWG is the Xilinx Spartan-3E XC3S500E PQ208 model FPGA, coded in 

VHDL using the Xilinx ISE Design Suite. The FPGA is directly connected to three 16-bit Analog 

Devices AD9726 DACs. The AD9726 DAC was chosen because of its high speed, its linearity, 

and its use of low voltage differential signal (LVDS), which is resistant to pick-up noise. The FPGA 

is clocked by a 50 MHz crystal oscillator. Each FPGA has 32 output signals (16-bit LVDS) and 

two derived 50 MHz clock-sources connected on-board to each DAC (three DAC channels per 

FPGA). Our experiment-control computer transmits waveform data via USB to the on-board FTDI 

FT245RL USB-to-parallel-FIFO (frst-in-frst-out), which provides an interface to upload waveform 

data a byte at a time (8 bits) from a host computer to the FPGA. When data is transmitted, the 

FPGA interprets the data stream and stores information for each DAC channel in three segments 

of on-chip, independent static random access memory (RAM) in the FPGA. An overview for the 

operation of the AWG is shown in Fig. 5.3.1. 

There are two types of AWGs, a “master” and a “slave”, with a small number of hardware 
87 

differences. The VHDL code used is the same for both masters and slaves, however the FT245RL 

and a 50 MHz crystal oscillator are only present on a master. Slaves are connected to the master 

through board-to-board interconnects, through which the waveform data and the clock signal are 

routed to be processed by their respective FPGAs, each identifed by an on-board hex switch. One 

master and one to fve slaves can be connected together to provide up to 54 synchronous channels 

on a single USB connection (limited by clock signal integrity). Separate masters are referenced 

to their own oscillators and hence have timing offsets of at most 20 ns. This caused negligible 

errors in ion experiments at NIST (though it is possible to bypass the on-board crystal oscillator to 

use a single external clock signal for all masters used in experiments to clock the AWG at different 

speeds or to achieve better synchronous operation). Each AWG master is identifed over the USB 

connection by a unique serial number programmed into the FT245RL, so in principle any number 

of channels can be controlled from a single computer (limited by the computer’s USB architecture). 

The DACs are clocked with 50 MHz signals derived by the FPGA from the crystal oscillator. Each DAC’s output is referenced to a precision 1.25 V source (Linear Technology LTC6652), 

specifed to have 10 ppm/◦C and 2.1 ppm noise, for stable and consistent potentials out of every 

channel. The differential signal outputs of each DAC are sent to the Analog Devices AD8250 highspeed amplifer with a gain of 10 to provide the fnal output potentials; the DAC and amplifer system 

is confgured for a ±10 V output with 16-bit linear resolution, stable down to roughly 30 μV and with 

∼ 1Ω output impedance. A 50 Ω resistor can be added in series with the output to prevent ringing 

in long cables. From observing after the resistor, we measure a white noise power spectral density 

√ of '120 nV/ Hz with an external 50 Ω load. The amplifer’s output has a measured slew rate of 

40 V/μs; although this limits the voltage swing at each clock update, it is suffcient for our experimental implementations. A photograph of a master board is shown in Fig. 5.2 with the discussed 

components highlighted. 
88 

17.8 cm

AD9726 DAC

AD8250 Amplifier

XC3S500E PQ208

USB Connector

FT245RL

Board-to-board Interconnect

Figure 5.2: A master board. The key components described in the text are highlighted. The host 

computer communicates with the master board via USB. The FT245RL handles the interface with 

the XC3S500E PQ208 FPGA on up to three boards (a master and two slaves) on a shared communication channel through a board-to-board interconnect. The FPGA drives three AD9726 DACs 

whose output is fed to AD8250 amplifers, which creates the fnal three output potentials. Reproduced from Ref. [74]. 
89 

5.3.2 User Control 

Before an experiment begins, a computer transmits waveform data via USB to be uploaded 

to the AWG’s RAM for each channel. The FT245RL manufacturer provides a support library of 

C++ commands provided by the manufacturer. The support library includes commands to open 

communication with the FT245RL, to send bytes of data to the FT245RL (which in turns flls a data 

buffer which is read by FPGA), and to read data such as each FT245RL’s serial number, which is 

used for identifying each master, or to possibly read bytes sent by the FPGA. This has been integrated into our experiment-control program, but computer control also exists as stand-alone control 

programs in C++ and Python. Before running an experiment, the experiment-control FPGA system 

is preconfgured to intersperse laser pulses and waveform triggers and the AWG is preconfgured 

with waveform data, so that an experiment runs without delays due to RAM upload times. When 

scanning an experimental variable involving waveforms, such as the waveform duration or a voltage offset, only the affected channels are redownload to minimize download times before running 

experiments. 

Using the gateware at the time of this thesis, the AWG is able to operate in one of four 

modes: (1) Output specifed by a single value for the output potential at each 20 ns clock cycle 

update, (2) Output given by values separated in time for specifed durations, (3) Output given in 

terms of quadratic polynomials specifed by a duration and coeffcients, and (4) Output specifed in 

terms of cubic polynomials. Options 2, 3, and 4 can provide increasing levels of data compression 

to overcome limited amounts of RAM and reduce waveform upload durations from the host computer. We generally use cubic spline interpolation of the desired waveform, which enforces 1st and 

2nd derivative continuity in the fnal waveform and each cubic polynomial can have a duration of 

anywhere from 240 ns to ∼655 μs. When running experiments, we typically constraint the initial 

and fnal derivatives to be zero, thereby smoothing end points of potentials. The interval between a 

TTL trigger and when a waveform begins to run depends on the time required to read the initial data 

from memory (longest is the 4th mode where 240 ns, or twelve clock cycles, are required before 
90 

the waveform begins), but after this initial delay the memory is read continuously so that voltage 

updates are continuous with no loss due to read-time on each clock cycle. 

Masters have input TTL ports routed to all connected slaves, and the port to signal that a 

waveform should begin running is labeled as F1. The rising edge of the TTL trigger signals the start 

of a waveform on all connected channels (a “run trigger”). Each waveform is assigned a branch 

number by the user and put into a corresponding segment of RAM in the FPGA. A branch is selected 

through three external TTL inputs labeled as F2, F3, F4 with F4 the most signifcant bit and F2 the 

least. The TTL inputs switch waveforms with 40 ns (two clock cycles) latency between branches 

and allow the user to switch between waveforms in response to events during the experiment. By 

tying more than one master to a single TTL, many masters (and their respective slaves) can be 

run synchronously. Each waveform branch may further be broken up into a sequence of waveform 

“steps”, where a step is a collection of waveform data run off of a single TTL trigger within the 

selected branch after which the AWG awaits the next TTL trigger. There is an additional line F5 

which can be confgured as an input or output and is independent on every master and slave board. 

The F5 line is currently only used for diagnostics but can potentially be used for additional branches, 

synchronous output triggers, or an additional input port for counting TTL pulses such as from a PMT. 

The AWG’s branching lets us functionalize our waveforms to concatenate a basis-set of transport 

operations that can be triggered on the fy, possibly dependent on experimental results, without the 

need for delays due to downloading new waveform data. 

The gateware is designed to accept command codes which the master FPGA interprets and 

routes accordingly to the proper master or slave FPGA when the experiment-control computer transmits data. The commands sent from the control computer are listed in Table 5.1. The master routes 

a line to every connected FPGA to signal that there is incoming data from the FT245RL 8-bit wide 

transmit buffer. When uploading waveform data to RAM, the user frst sets which FPGA to talk 

with (CMD_FPGAADDR) which is set by an on-board HEX switch from 0 to 15, then sets which 

specifc DAC connected to that FPGA out of three (CMD_SETDACX) that will receive the waveform data. After that, any new data only interacts with the specifc DAC channel’s respective RAM. 
91 

The starting communication address for writing waveform data can be set to begin at a predefned 

location (CMD_SETMADDR), such as at the start of a specifc branch. When sending waveform 

data, frst the length of bytes to be sent is specifed (the length set by CMD_BURST), followed by 

the stream of data (CMD_WRITEBURST). Branch data points are specifc memory addresses that 

defne a memory address register that tells the FPGA where to look for the beginning of waveform 

data when the respective branch is selected. When the branch input TTLs select a specifc branch 

followed by a run trigger, waveform data is read starting from the respective branch data point up to 

the next branch or to the maximum address (set by CMD_SETLEN). 

5.3.3 Memory Storage 

Each channel has on-chip block RAM allocated for its waveform specifcations. The interface 

to the block RAM was generated using the Xilinx Block Memory Generator. The AWG implements a 

Simple Dual-port RAM diagrammed in Fig. 5.3. This allows two independent ports to the RAM: port 

A has an input to specify an address at which to write to memory and port B has a separate input 

to specify an address for reading from memory. The independent addresses allow the option to 

maintain write and read locations independent of one another. Port A has data written at the RAM 

address labeled ADDRA, clocked at CLKA, and with data from the FT245RL at DINA so that data 

transmitted from the experiment-control computer is uploaded to the desired channel’s memory 

when the write-enable line goes high. Port B has data read from RAM at the address ADDRB, 

clocked at CLKB (for the current gateware, CLKB and CLKA are the same), and with data output 

at DOUTB, which is interpreted for waveform outputs. It takes two clock cycles to interact with the 

RAM at a specifc memory address for either port. On a given clock cycle, the address is read, and 

on the following clock cycle, the memory at that address location will be available for input or output. 

Therefore, care must be taken in the timing to a clock-cycle delay between specifying a memory 

address and having new data available. 

Each channel’s RAM can hold 110 kilobits, but this could be rapidly consumed for complex experiments with ion transport. This memory is set for 16-bit wide data that is 6144 memory locations 
92 

Command Hex Function 

CMD_FPGAADDR 00 Sets the address to fag which FPGA is to receive signals from 

the FT245RL amongst connected PDQ boards 

CMD_BURST 01 Sets burst length for how much waveform data will be sent 

CMD_SETMADDR 02 Sets the starting communication memory address for writing the 

waveform to memory registers 

CMD_WRITESINGLE 03 Write waveform data 

CMD_WRITEBURST 04 Write a stream of waveform data to the memory register with size 

’burst length’ 

CMD_SETLEN 06 Maximum waveform address in the memory register 

CMD_MODE 08 Running mode for all the DACs (00, 01, 02, or 03) 

CMD_SETDACX 1X Choices for which of X (0, 1, 2) of three channels for incoming 

waveform data 

CMD_SETBRANCHX 2X Setting branch memory addresses for each channel where X (0, 

..., 7) is the branch number for current DAC 

Table 5.1: Commands sent from the control computer to the AWG through the FT245RL USB 

interface. 

Figure 5.3: Diagram of the Simple Dual-Port RAM input and output interface generated by the Xilinx 

Block Memory Generator. Port A is used to write to the RAM and port B for reading out the RAM. 

The CLKA and CLKB clock inputs are both referenced to the same global clock as the rest of the 

FPGA logic. ADDRA specifes the writing address for data DINA when the write enable line WEA is 

high, and ADDRB specifes the address for read-out of the RAM data via DOUTB. 
93 

deep (each memory location has two parity bits for a total of 18-bit wide data, but this is unused 

in the current gateware and could potentially be used to hold additional information about a memory location). The 16-bits comprise a pair of 8-bit-wide data transmissions through the FT245RL 

USB connection. The memory depth corresponds to memory addresses specifed with 13-bits, and 

memory registers include addresses for upload locations in RAM, addresses for reading waveform 

data in RAM, and the branch-point address register. 

In the 4th mode, which uses cubic polynomials to specify waveform outputs, a single cubic 

polynomial uses ten memory locations. The initial voltage voltage for each channel uses 16 bits 

for the starting voltage: 0 to 32767 specifes 0 V to 10 V, and 32768 to 65536 specifes the 2’scomplement (taking the bit parity fip and adding one) for -10 V to 0 V. The frst linear coeffcient 

uses 32 bits with 16 bits of fractional precision. The quadratic and cubic coeffcients use 48 bits with 

32 bits of fractional precision. The duration for the polynomial is specifed in number of clock cycles 

with 16 bits (up to a 32767×20 ns duration). A 2’s-complement duration signifes that the AWG 

should wait for a TTL trigger before running the next polynomial in memory. Thus, when operating 

in the 4th mode, up to 614 polynomials can be stored in each channel’s RAM. 

To convert a real time polynomial, 

y(t) = a+bt +ct2 +dt3 , (5.3.1) 

into a form used by the AWG, the voltage output must be computed using a discrete sum over clock 

cycles. As such, at the nth clock cycle during waveform output, the coeffcients for waveform output 

are evaluated by the FPGA as 

γn = γn−1 +δ

βn = βn−1 +γn

αn = αn−1 +βn. (5.3.2) 

This corresponds to a voltage output given by 
94 

n(n+1) n(n+1)(n+2) Vout = α0 +nβ0 + γ0 + δ0 (5.3.3) 2 6 

at the nth clock cycle, given initial coeffcients α0, β0, γ0, and δ0. These values are computed from 

the desired continuous polynomial by the relationships 

δ0 = d

γ0 = (c−d)/2 

β0 = b−c/2 +d/6. (5.3.4) 

Before uploading these values to the RAM for an AWG channel, these values then must be scaled by 

the user to the necessary precision for the RAM storage and converted from the input units of volts. 

The maximum fnal output voltage of 10 V corresponds to the signed 16 bit number 215 −1 = 32767 

applied to the input of each DAC. For example, γ0 is converted as 232γ × (32767/10V), with the 

factor of 232 included for the 32 fractional bits. In the 4th mode, the FPGA expects to read data for 

a cubic polynomial of a total duration of ntot clock cycles in the sequence α0, ntot, β0, γ0, and δ0, 

after which it will interpret the next data as α0, ... for the next polynomial. For the 1st (, 2nd (, 3rd)) 

mode, data is likewise interpreted as the sequence α0 (, ntot (, β0, γ0)). 

5.3.4 Possible Future Implementations 

5.3.4.1 Faster DAC Output 

The AWG has built-in hardware and protocols for the computer-communication interface, 

which does not need to change for different output interfaces. Currently, the AD9726 DACs are 

clocked at 50 MHz, but this can be increased to 100 MHz by confguring the FPGA gateware to derive a 100 MHz internal clock from the input 50 MHz oscillator and correspondingly clock the DACs 

at 100 MHz. Additional speed can be attained from using an interpolating DAC. Such DACs, for 

example the Texas Instruments DAC5681Z, can further up-convert the clock frequency internally to 
95 

interpolate between the user-specifed voltage values faster than otherwise possible from the FPGA 

update rate. For such a feature, however, the PCB design for the AWG must be updated for the new 

DAC pin connections. 

The AD8250 amplifer is specifed to have approximately a factor of 5 slower speed than 

the fastest DAC response-time. To obtain a faster DAC signal, two options are available. One is 

to bypass the amplifer entirely, directly accessing the dual-ended DAC output (this approach has 

been implemented on some AWGs to feed the outputs directly to an external amplifer). The other 

is to choose a different on-board amplifer. The AD8250 was one of the fastest options at the time of 

the AWG’s creation. However, there are faster amplifers now available, such as the Analog Devices 

ADA4895, and this could be implemented to further speed up the overall AWG output speed. 

5.3.4.2 Use as a DDS Source 

The uploaded waveform data to the AWG could be chosen in such a way to behave as a sine 

look-up table in block RAM to be used to generate an RF signal, limited by acceptable sampling 

rates and the FPGA update rate (for example, a properly well-defned signal must be a divisor of 

the DAC update rate). This will generate a fxed-phase sine wave, and could possibly be used to 

inject RF on a trap electrode super-imposed on the confning voltage signal. However, a superior 

approach would be to alter the AWG to replace the three DAC channel outputs and instead use an 

array of DDSs. This would require reconfguration on both the hardware and VHDL level. 

The AWG could be re-confgured to generate a sequence of RF signals by storing sequences 

of phase and frequency settings in block RAM to be sent to dedicated DDS chips. The computercommunication methods already in the AWG gateware may remain fxed for transmitting data and 

uploading data to RAM. The AWG has 48 LVDS bits (currently to control the three DAC output 

channels); with suitable changes to the traces on the PCB, the fnal AWG output can replace outputs 

to DACs for arbitrary signals with outputs to DDSs for phase and frequency settings. The AWG can 

be used to control a sequence of RF signals by updating phase and frequency settings derived from 

block RAM. The available 48 LVDS bits can be distributed amongst DDSs as needed (offering the 
96 

choice of more bit precision vs. number of DDS channels per board; the current experiment-control 

system uses 8-bit precision DDSs in addition to phase settings which is typically suitable for our 

needs). The interpretation of block RAM for the fnal output from the FPGA to the DDS chips must 

be altered in VHDL, and an additional physical input line would be needed for a reference clock to 

the DDSs. This would allow an experimenter to control many DDSs per USB connection. 

5.3.4.3 Expanded Memory 

Future demonstrations of scalable QIP will possibly require very complex and lengthy transport sequences. If experiments require an expansion in available memory on the AWG beyond that 

already provided by data-compression via cubic splines, there are more effcient uses of the block 

RAM. The current way that polynomials are stored in memory is by specifying polynomial coeffcients with units of numbers of clock cycles, which are summed each clock update to calculate the 

proper output. This satisfes the current fractional precision necessary for most waveform uses. 

Memory storage could be made more effcient if it is known in advance if less fractional precision 

in the coeffcients is necessary for experiments. For example, very rapidly-changing polynomials 

could remove the lowest elements of fractional precision to free up memory. For suffciently small 

higher-order polynomial terms, memory can also be saved by only operating in 2nd or 3rd modes 

of operation. The memory usage could be made more effcient by replacing the coeffcients with 

voltage updates per some number of clock cycles. For example, the quadratic coeffcient γ can be a 

16 bit value added to the desired voltage output Vout every nγ clock cycles. This would eliminate the 

need for any fractional storage in each channel’s RAM, which could reduce the required memory to 

specify a cubic polynomial to eight memory locations deep rather than ten. 
97 

5.3.5 Experimental Implementations 

5.3.5.1 Driving Filtered Electrodes 

The high update rates of the AWG relative to confnement frequencies in our traps enables 

variation of electrode potentials for ion transport on timescales faster than the ion oscillation period. In ion transport experiments, the AWG produced the time-varying electric potentials in a 

multi-electrode linear Paul trap (diagrammed in Fig. 5.4a). These experiments demonstrated that 

transport and separation over distinct trap zones on quantum logic time scales can be performed 

with suppressed excitation in the trapped ions’ fnal motional state. 

A complication arises from a chain of resistor-capacitor (RC) low-pass flters that is applied 

to each electrode to reduce electric feld noise from external sources at the trap frequencies. To 

achieve the high speed of the fast ion transport waveforms, we must precompensate for the lowpass effects of the flters in order to achieve the desired time-dependence of the electrode potentials 

[75]. The flters for each electrode in the quantum logic trap, from AWG-side to trap-side, are a pair 

of 200 kHz low-pass RC flters followed by one 800 kHz low-pass RC flter for our quantum logic 

ion trap, which have resistors {R2, R1, R0} = {820, 820, 240} Ω and capacitors {C2, C1, C0} = {1, 1, 

0.82} nF. The input potential Vinput to a single stage of the RC flters in the flter chain are related to 

the desired output potential Vout put by 

dVout put Ri Vinput = Vout put +RiCi + (Vout put −Vprev) , (5.3.5) dt Rprev

where Ri and Ci are the RC component values of the i

th layer of the flter chain and Vprev and 

Rprev are from the solution to the previous layer of fltering trap-side. One example of numerically 

solving for the effects of the flter system is shown in Fig. 5.4b. However, this approach is ultimately 

limited by the maximum voltage output and amplifer speed, so it cannot be done arbitrarily fast. 

The transport waveforms altogether in Ref. [66], including that shown in Fig. 5.4b, were specifed 

by cubic splines that occupied ∼ 6 % of each channel’s memory. 
98 

0 2 4 6 8 10 10

5

0

5

10

Time (μs)

Potential (V)

Filter input

Desired waveform

(a)

(b)

Figure 5.4: (a) Schematic of the ion trap electrode structure showing the two diagonally opposite 

segmented DC electrodes (not to scale). Outer electrodes O1 and O2 are used to fne tune the 

potential while ions move over electrodes A, X, and B during the experiments in Ref.[66]. (b) Example flter compensation for electrode X during transport of an ion from over electrode A to over 

electrode B. The desired waveform is generated from simulation and the potentials generated by 

the AWG must account for flter distortion. The waveform from the AWG must overshoot in voltage 

to achieve the desired 8 μs transport potentials. Reproduced from Ref. [74]. 
99 

5.3.5.2 Waveform branching 

The capability to quickly switch between waveform branches allows complex sequences of 

waveforms to be built up from a series of simpler component waveforms. For example, the experiments in Ref.[28] required multiple instances of two 9Be+ qubit ions undergoing operations with 

quantum logic in a shared trap zone for two-qubit operations and the separation of the ions to 

distinct zones for one-qubit operations. The full sequence required as many as 16 separation and 

recombination waveforms in a single experiment. We used three waveform branches preloaded into 

AWG memory and selected with TTL inputs during the experiments: (1) the static trap potential, (2) 

the separation waveform, and (3) the recombination waveform. Because the three waveforms are 

reused and need to be combined in various sequences, we can defne the corresponding branch 

for each waveform and the experimental sequences of these three waveform branches were concatenated on the fy without the necessity to upload a new set of waveforms for each sequence. 

5.3.5.3 Shaped Microwave Pulses 

Microwaves can be used to drive transitions between hyperfne states. In one implementation, the AWG was used for pulse shaping of a microwave signal at ∼ 1.69 GHz that drives the 

|S1/2,F = 3,mF = 1i ←→ |S1/2,F = 2,mF = 1i hyperfne transition in 25Mg+. For the experiments 

in Ref. [76], the AWG was used to prevent ringing of high power (∼45 dBm) pulsed microwave 

signals during turn on/off by shaping the pulse with a smooth rise and fall. In this case, the microwaves were ramped on and off in 10 μs. A multiplier combines the AWG shaping waveform 

and the microwave signal into a shaped microwave pulse, which is then amplifed. Resonances for 

the hyperfne transition, shown in Fig. 5.5, taken by varying the frequency of the microwave signal 

followed by a state-dependent fuorescence measurement, are demonstrated for two cases: a rectangular pulse shape (Fig. 5.5a), and a Gaussian pulse shape (Fig. 5.5b). The transition profles 

are related to the Fourier transform of the pulse shape, a sinc2 and a Gaussian, respectively, with 

pulse durations chosen for approximately equal full width at half max for each profle. 
100 

pulse shape: 

pulse shape: 

-

0

(kHz)

fluorescence (a.u.)

1.0

0.6

0.4

0.8

-50 0 50

a)

b)

0.2

1.0

0.6

0.4

0.8

0.2

-100 100

(

(

Figure 5.5: Pulse shaping of a microwave signal using the AWG to specify the profle. The microwave drives the hyperfne transition in 25Mg+. Shown is the ion fuorescence in arbitrary units 

vs. microwave frequency. Here ν0 ∼ 1.69 GHz denotes the frequency splitting of the two hyperfne 

levels. (a) Observed sinc2 lineshape from a rectangular pulse shape for a 20 μs duration. (b) Observed Gaussian lineshape from a Gaussian pulse shape. The FWHM pulse duration was 45 μs, 

with the Gaussian tail truncated to a total duration of four times the FWHM. Reproduced from Ref. 

[74]. 
101 

5.3.5.4 Laser Pulse Shaping and Stabilizing 

As the ion-laser interactions are typically pulsed, our pulse sequencer switches laser beams 

on and off as needed. For some applications, greater control over the optical pulse amplitude is 

desired. Laser intensity noise can be a source of error both singe-qubit Rabi oscillations and in 

laser-driven two-qubit entanglement. We use a feedback loop to measure the laser intensity and 

control it. However the feedback loop doesn’t behave well when we just switch it on instantaneously 

to do a square pulse. One way to deal with this problem is to ramp the laser on and off smoothly so 

the feedback loop can follow and stay locked. 

To this end, we use the Analog Devices AD5390 vector multiplier which can be used as a 

quadrature amplitude modulator, or sometimes known as an I/Q modulator. These modulators have 

a local oscillator RF signal with two RF input ports and two DC control “I” and “Q” input ports, 

where I and Q are components which give an amplitude and phase offset in the complex plane. 

The ADL5390 was chosen for its frequency range and linear amplitude response to the inputs. The 

signal sent to the modulator’s I port is determined by the output of an analog proportional-integral 

(PI) servo with an integrator hold capability. The PI is confgured to minimize |V+ −V-|, where V+ is 

a signal from a photodiode derived from the laser beam and V− is an output channel of the AWG. 

Laser beam attenuation is controlled by utilizing the -1 diffracted order of the AOM for the laser 

in a double-pass confguration. When a laser pulse is desired, the servo’s response to a pulse 

shape from the AWG into the V− port drives the I port of the I/Q modulator, thereby increasing the 

amplitude of the RF signal fed into the AOM. Noise within the bandwidth of the servo is attenuated 

as the laser beam increases intensity to its maximum value. The RF signal is switched off when the 

beam is not in use. 
Chapter 6 

Coherent Transport Dynamics of Trapped Ions 

The large timescale discrepancy between the time it takes to perform quantum logic (∼ 10μs 

per operation) and to perform transport and recooling (∼1-2 ms per operation) in previous demonstrations of our QIP architecture [29, 27, 28] are unacceptable in large-scale QIP. Previous ion transport experiments, such as those in Refs. [12, 15], have demonstrated cold transport with adiabatic 

transport durations, that is, much longer than the period of oscillation for the trapped ion. Excitations, due to non-ideal transport waveforms and accumulated thermal heating in our traps, made 

more signifcant for longer transport durations [24, 25, 26], degrade our two-qubit entanglement 

through the shared motion. Finite qubit coherence times also contribute to imperfect entanglinggate fdelity that can surpass acceptable error levels accrued during long operation times (practical 

fault-tolerant entangling gate errors typically are cited as having a threshold of 10−4 [77, 78]). It is 

critical to reduce QIP operation time well-beneath qubit coherence times. Therefore, it is a requisite that ion transport durations are at least on timescales comparable to quantum logic, and that 

transport is performed with mitigated motional excitations to reduce overhead from recooling. 

Transport is performed by applying time-varying potentials, or “waveforms,” to the ion trap 

electrodes. For signifcantly short transport durations, diabatic excitation effects due to accelerations can become signifcant. Additionally, former limitations in commercially available DACs that 

were used to supply waveforms could cause unwanted excitations. Harmonics of the update rate, 

with a fundamental frequency of ' 500 kHz, could have Fourier components at the ions’ axial 

frequencies and thus cause motional excitations during transport [15]. During ion separation or 
103 

recombination, the axial frequencies could go quite low and cross through a resonance of the fundamental or harmonics of the DAC update rate, which can also lead to undesirable excitations. 

With suffciently fast and precise DACs for transport, the steps in the potentials can be approximately continuous with respect to the ions’ motion. The electronics to generate the waveforms in 

the experiments described here were supplied by the arbitrary waveform generator with a 50 MHz 

update rate described in Chapter 5. This allows waveforms that approximately match any position 

profle for transport, and the motion can be coherently excited and de-excited with timing precision 

on timescales faster than ion trap periods. 

This chapter discusses the experimental results towards improvements on ion transport characterized by both orders-of-magnitude faster timescales over former implementations as well as 

highly-mitigated motional excitations. I begin with an overview of some experiments with ion transport prior to my research, which was generally in the adiabatic regime, followed by a description 

some of the theoretical treatments for speeding up ion transport in ways that consider limited motional excitations on fast timescales. The theoretical discussions include that for separating two ions 

into distinct trap zones, which necessarily includes changing the axial trap frequencies. I will then 

describe my experimental results for coherent diabatic transport of one and two 9Be+ qubit ions 

between two trap zones and the separation of two 9Be+ ions into distinct trap zones on timescales 

comparable to quantum logic durations and with mitigated motional excitations. I also discuss the 

diabatic transport of two-species ion chains. These results pave the way for ion-based scalable 

QIP, combining all the building blocks in a way that can potentially have information processing 

timescales well beneath acceptable coherence-time limitations. 

6.1 Adiabatic Ion Transport 

The faster that ions are transported, as durations approach timescales comparable to the axial 

trap periods, the more that motion can be coherently excited due to fast accelerations. If transport 

durations are much slower than the trap periods, referred-to as “adiabatic transport”, then excitations 
104 

will be mitigated. A multi-electrode wafer trap was the earliest platform for ion transport [12]. A 9Be+

ion was transported with a position profle of z(t) = sin2(πt/2T) × 1.2 mm, T the total duration, 

traversing an intermediate electrode, with varied velocities in a 2.9 MHz well. When T ∼ 50μs, 

for example, the timescales were much longer than the motional period and thus the transport was 

adiabatic, and in turn motional excitations along the axis were ∼0.01 quanta. However, as durations 

were decreased, as low as T ∼ 20μs, large motional excitation was observed to the point of being 

diffcult to measure the number-state occupation. The trap from these experiments also had a 

“separation wedge” electrode which was 800 μm wide. Two ions were brought over this electrode 

and split into two separate trap regions with a 95 % success rate. The fnal motion incurred a great 

amount of motional excitation and separation was not perfectly reliable, though it had promise for 

splitting a two-ion chain into spatially distinct wells. 

Transport through a T-junction in an ion trap was reported in Ref. [13] with 112Cd+ and transport through the junction in the X-junction Trap with 9Be+ was reported in Ref. [67], demonstrating 

that ions can traverse a junction in the electrode architecture to allow a which-way transport for 

ion re-ordering. Low-excitation ion transport through the X-trap’s junction was achieved in work 

reported in Ref. [67] and improved upon in Ref. [15]. Here, ions entering the junction and exited 

at either 0◦ or 90◦ degree angles from the incident trajectory and transported back to the starting 

position with average velocities of 2 m/s over a ∼1 ms duration and only incurred ∼0.2 quanta 

excitation, which is promising for future scalable QIP implementations. 

In Ref. [22], a similar trap design to the quantum logic trap was used in a demonstration of 

quantum teleportation between qubits which involved the separation of ions. A crystal of three 9Be+

qubit ions underwent various stages of separation and combination along with several entanglement 

steps. The success of the teleportation experiment depended heavily on two-qubit entanglement 

operations not degrading due to accumulated motional excitations in the shared motion. In those 

experiments, only 0.1 quanta of excitation was seen in the stretch mode of motion in a two-ion 

crystal after a separating three ions into groups of one and two in distinct trap zones across a 1 ms 

separation duration. The separation wedge was 100 μm wide, and the improved separation result 
105 

compared to that in [12] was attributed to the relatively narrower separation wedge electrode. 

Several theoretical considerations have been made towards ideal ion transport. Optimal adiabatic transport was investigated in Ref. [79] by looking at the quantum dynamics of the forced 

harmonic oscillator. A number of time-varying transport profles were analyzed, with one of the 

2  ́

0 

t

e−t

02

most interesting results due to the error function transport, where erf(t) = √πdt

0 is the standard error function. This was calculated to be a highly effective adiabatic profle for a suitably precise 

waveform for timescales down to small numbers of trap periods. However, in Ref. [23], transport at 

high speeds was demonstrated with an error function profle on timescales under 4 trap periods for 

40Ca+ ions, and resulted in fnal excitations outside the adiabatic regime with a 1% ion loss rate, 

necessitating the development of rapid transport with a higher success rate. Simulated analysis 

of trap designs that are optimal for transport in conjunction with optimal transport waveforms have 

been carried out as well [80]. 

Even under perfect adiabatic transport, it may not be possible for the ions’ motion to remain 

ground-state cooled across QIP operations due to ambient heating [24, 25, 26] (our systems have 

typical heating rates of n ̇ ̄ ∼ 0.5 quanta/ms). Our solution to the acquisition of motional excitations 

during transport and heating effects arose out of sympathetic cooling. Our qubit ions, 9Be+, can 

share a trap zone with a 24Mg+ or 25Mg+ coolant ions during transport. In [81], work with the quantum logic trap brought together all the building blocks for scalable QIP which involved separation, 

recombination, and sympathetic cooling in traps which held both Be+ and Mg+ ions. However, an 

additional latency arises due to sympathetic cooling, as the ground-state cooling method also required roughly 1 - 2 ms of cooling time. Having both qubit and coolant ions to sympathetically cool 

motional excitations acquired due to imperfect transport and ambient heating rates in the trap is an 

added consideration for rapid transport with multiple ion species. 
106 

6.2 Coherently-driven Ion Transport 

One approach to rapid ion transport is to fnd transport profles which stay adiabatic on 

timescales down to a few trap oscillation periods at high speeds such as discussed in Ref. [79], 

which includes an error function profle. There have been several theoretical investigations on 

diabatic transport with minimally excited fnal states, including optimized electrode and transport 

potential analysis [80, 82]. An approach to perform diabatic transport is to coherently excite and 

de-excite the motion across the waveform by engineering transport waveforms with suitable accelerations with no net excitation at the transport destination. As long as the harmonic trap frequency 

ω is held constant, excitations will always be coherent. Precise transport requires varying the confnement potential in a way that approximately matches the theoretical transport profle with respect 

to the trapped ions, which is accomplished with our arbitrary waveform generator with a 50 MHz 

update rate (see Chapter 5), and this also allows us to control the transport timing with precision 

much faster than the trap oscillation period. For an ion with mass m, trap frequency ω, and with a 

transport profle z0(t

0

), then the coherent state amplitude α is given by [83] 

r  ˆ t  mω −iωt iωt

0

α(t) = −e z ̇0(t

0

)e dt0 . (6.2.1) 2~ 0 

That is, as long as the Fourier component of z ̇0(t

0

) at the trap frequency vanishes when integrated 

over the transport duration, the ion will end up in its initial state at the destination z0(t). The simplest 

case is for the trapped ion to suddenly move with a constant velocity v and then stop after a total 

transport duration tT (z ̇0 = 0 for t < 0 and t > tT , and z ̇0 = v for t ∈ [0,tT ]). This gives us 

  r  mω v α −iωtT (tT ) = i 1−e . (6.2.2) 2~ ω

For an ion initially in the ground state of motion, then for the condition that ωtT = 2πN, N an 

integer, the ion returns the ground state at the destination. This is similar to an approach using 

step-wise accelerations of neutral atoms in [84] performing transport on the 1 s timescale. For noninteger N, the ion will be left in a coherent state different from the ground state. Due to the fnite 
107 

response times of our trap electrode flters and drive electronics, sudden starts and stops are not 

practical. Nevertheless, Eq.(6.2.1) gives a criterion for transport functions with low excitations at the 

destination. In the case of multiple same-species ions, there will be no differential force, so only the 

common mode of motion needs to be considered so that in principle this approach is effective for 

many ions. 

Considering transport for multi-species ion chains requires more careful consideration due 

to the potential for excitations of all normal modes of motion. The work in Ref. [85] produced 

a theoretical treatment for multi-species ion transport with no residual excitations, with specifc 

attention to the case of 9Be+ and 24Mg+. Under transport which coherently drives the motion, the 

out-of-phase normal mode of motion will be excited, unlike in the same-species case. One approach 

decomposes the transport profle into a fnite-length Fourier series, such that the coeffcients are 

used as degrees of freedom to solve for ideal transport with the constraints that z(0) = zi, z(tT ) = zf , 

z ̇(0) = 0, z ̇(tT ) = 0, and that the integral for coherent excitations for each mode ωj proportional to 

c(ωj) =  ́ tT z ̇(t)eiωjtdt vanishes. The position profle, 0 

Nions+2 1 π(2l −1)t

z(t) = (zf −zi)×( + ∑ al cos[ ]) +zi, (6.2.3) 2 l=1 tT

is used to satisfy Nions + 2 conditions for Nions normal modes, and already has the property that 

z ̇(0) = z ̇(tT ) = 0. The constraint to eliminate coherent excitations is imposed by having each coeffcient al proportional to the term 

ˆ tT π(2l −1) π(2l −1)t

cl(ωj) = − sin[ ]eiωjt

dt, (6.2.4) tT 0 tT

and that ∑N

l=

ions

1 

+2 

cl(ωJ)al = 0. The linear system of equations to solve for the coeffcients becomes 
108 

Figure 6.1: Schematic of the electrode structure for the diagonally-opposite DC electrodes of the 

quantum logic trap. During experiments, the ions are separated and recombined in zone X, and 

transported between zones A and B. The waveforms consist of time-varying potentials, with common potentials for each diagonally-opposite pair of electrodes. 

⎡ ⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎣ 

1 1 1 ... 1 

1 −1 −1 ... −1 

0 c1(ω1) c2(ω1) ... cNions+2(ω1)

. . . . . 

. . . . . 

. . . . . 

⎡ ⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎣ 

⎤ ⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎦ 

1/2 

a1 

a2 

. 

. 

. 

⎤ ⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎦ 

=

⎡ ⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎣ 

0 

1 

0 

. 

. 

. 

⎤ ⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎦ 

. (6.2.5) 

0 c1(ωN ... 0 ions) c2(ωNions) cNions+2(ωNions) aNions+2 

This method assumes that the excitations during transport are small enough such that the ion 

motion remains in the harmonic regime, and does not lead to signifcant higher-order terms due to 

the Coulomb coupling. 

6.3 Transport of 9Be+ Experiments and Results 

Single-species ion transport experiments were carried out in the quantum logic trap (Fig.6.1). 

We initialized a single 9Be+ ion in zone A in a trap with an axial frequency of ω/(2π) ' 2MHz with 

trap potentials {VO1,VA,VX,VB,VO2} = {1.289, 0.327, 2.173, 0.310, 1.311} V). The axial motion 

was Raman sideband laser-cooled to initialize the motion to n ̄ ' 0.1 and optically pumped and 

initialized to the qubit state |F = 2,mF = 1i = | ↓i of the 9Be+ S1/2 manifold with magnetic feld 

B ' 11.9 mT. It was then transported 370 μm in a total transport duration tT ≈ 8μs to zone B (A 

→ B), with a velocity v ' 46 m/s, with a waveform that approximated the conditions for Eq. 6.2.2. 
109 

0 20 40 60 80

0

0.2

0.4

0.6

0.8

1

Duration of sideband drive in zone B (μs)

P

↓(t)

MSS Fit

MAS Fit

Figure 6.2: Rabi fopping trace on the MAS and MSS after transport from zone A to B. Here we ft to 

thermal distributions that give n ̄ = 0.19 ±0.02 from the MAS ft and n ̄ = 0.17 ±0.01 from the MSS 

ft. The axial frequency was ω/(2π) = 1.972(1) MHz with a Lamb-Dicke parameter η = 0.479. 

Reproduced from Ref.[66]. 

We varied ω by applying a single, universal scaling factor to our electrode voltages to optimize the 

transport and found ωOpt/(2π) = 1.972(1) MHz to minimize coherent excitations. 

Following A → B transport, qubit sideband transitions were driven in zone B on the |2,1i ←→

|1,0i transition. The ion was then transported back to zone A (B → A) to determine P↓(t) through 

state-dependent resonance fuorescence [57]. As an approximation, we ft to the function 

" # ∞ −γt P↓(t) = 1

1+e ∑ Pn cos(2Ωn±1,nt) , (6.3.1) 2 n=0 

which is the function from Chapter 3 Eq. 3.4.2 including a phenomenological decay rate γ. The fnal 

excitations were consistent with the trap’s ambient heating rate, with a minimal fnal state measured 

as having n ̄ = 0.19 ± 0.02 from the MAS ft and n ̄ = 0.17 ± 0.01 from the MSS ft, corresponding 

approximately to N = 16 (Fig.6.2). Excitation was also minimized for values of ω/2π differing from 

ωOpt by integer multiples of t

−1 . T

We also transported two 9Be+ ions (A → B) in the same well. If the potential well is maintained at constant frequency throughout the waveform, there is no differential force on samemass ions, and the center-of-mass (COM) mode should be excited as for a single ion while the 
110 

0 20 40 60 80 100 120

0

0.2

0.4

0.6

0.8

1

Duration of MAS drive in zone B (μs)

P

↓(t)

Figure 6.3: Rabi fopping trace on the MAS following transport in which α(tT ) 6= 0. The fgure shows 

a coherent state ft with n ̄ = 6.4 ± 0.2, corresponding to |α| = 2.53 ± 0.04. The axial frequency 

was ω/(2π) = 1.919(2)MHz and the Lamb-Dicke parameter was η = 0.486. Reproduced from 

Ref.[66]. 

√ “stretch” mode of motion (the normal mode for out-of-phase amplitudes with frequency ωstr = 3ω) 

should not be excited. We initialized both modes to n ̄ ' 0.1 and after transport, we observed 

n ̄COM = 0.35±0.1 with negligible excitation of the stretch mode. 

For non-optimal values of ω/(2π), P↓(t) measurements showed that the populations of the 

fnal motional states were consistent with a coherent state of motion; however, these measurements 

do not verify the coherences between number states. To verify that the transport excites a coherent 

state, we implemented A → B transport in a well of frequency ω/(2π) = 1.919(2) MHz to try to 

maximize α(tT ) in zone B, corresponding to approximately N = 15.5 and a state with n ̄ = 6.4±0.2 

from a MAS ft (Fig.6.3). We then applied a uniform electric feld Ez = E0 cos(ωtE +φE) for a phasespace displacement αE to remove the coherent excitation from diabatic transport as an RF deexcitation. This was followed by a MAS drive of duration t, and then B → A transport to determine 

P↓(t). By adjusting E0,tE, and φE (relative to the time when transport started), we could make 

αE = −α(tT ). Under these conditions, we measured a fnal state at n ̄ = 0.19 ± 0.02, consistent 

with our heating rate and our best optimized cold transport. 

When α(tT ) 6= 0 in zone B, by waiting an appropriate delay time Td ±N2π/ω in B, and then 
111 

Figure 6.4: Fixed MSS drive time t after A → B → A “cold” transport for different wait times in 

zone B. The points at which P↓(t) changes the least indicates a higher n = 0 component of the fnal 

motion. The difference in wait time between low n ̄ features is consistent with ω for our maximal 

coherent drive. 
112 

reversing transport B → A, the ion would be returned to its ground state in A (A → B → A “cold” 

transport). By varying the wait time Td and applying a motion-subtracting sideband drive of fxed 

duration tMSS set to the π/2 time for an ion with n ̄ = 0, we could observe the interactions of the 

coherent states driven between A → B transport and B → A transport. The variation in P↓(tMSS) as 

Td is varied indicates a change in the fnal motional state in zone A, where the smallest change in 

P↓ corresponds to the ion returning to near its ground state, shown in 6.4. 

It is necessary that qubit coherences are reliably maintained during transport, a feature 

demonstrated in Refs.[12, 15]. Under transport with minimal fnal excitation we can expect that 

since any number state receives the same phase-space displacement across the transport, initial motional states should maintain the same relative coherences before and after transport. To 

demonstrate this, we performed a Ramsey-type interference experiment in which the qubit and motion are entangled. After state initialization, a MAS π/2 pulse was applied to transform the initial 

state | ↓,n = 0i to the qubit-motion entangled state √

1 (| ↓,n = 0i + | ↑,n = 1i). To establish a 2 

baseline reference, we performed a Ramsey experiment with free precession time approximately 

equal to the transport duration, but with the ion held stationary in zone A, with a measured a fringe 

contrast of 85(2)%, limited primarily by imperfect initial ground-state cooling and our relatively large 

Lamb-Dicke parameter. The Ramsey experiment with A → B → A transport between Ramsey 

pulses, both transport steps under optimized transport, resulted in a measured 86(2)% contrast, 

which was consistent with no loss in coherence of the qubit-motion entangled state (Fig. 6.5). 

6.4 Ion Separation 

The theoretical treatment to our approach to ion separation, which is an inherently adiabatic 

approach, was described in Ref. [86], including analysis of optimal potentials and ion trap geometries for separation. Some of the theoretical basis for separation was expanded upon in Ref. [87]. 

When separating multiple ions confned in a single zone into two separate zones, it is impossible 

to preserve the motional mode frequencies. To lowest order, the required external potential can be 
113 

0 1 2 3 4 5

0

0.2

0.4

0.6

0.8

1

Phase

P

↓(t)

With Transport

Without Transport

Figure 6.5: Ramsey-type interference experiment with MAS π/2 pulses in zone A. The contrasts 

without transport (contrast of 85(2)%) and with transport (contrast of 86(2)%) are consistent with 

no loss in coherence. The phase offset is due to imperfect matching of the delays between the two 

MAS pulses for the case with transport and the case without. 
114 

described as the sum of a quadratic and a quartic term for 0 ≤ t ≤ ts for total separation duration of 

ts, given by 

U(z) = a(t)z

2 +b(t)z4 , (6.4.1) 

with a(0) > 0, b(0) = 0 and a(ts) < 0, b(ts) > 0. Due to the Coulomb repulsion between the ions, 

the normal modes remain harmonic throughout the separation for small excursions about each ion’s 

local minimum. The inter-ion equilibrium distance d(t) during the separation process is given by the 

real solution to [86] 

b(t)d(t)

5 +2a(t)d(t)3 = 2πε

q

0 

. (6.4.2) 

Two same-species ions will have two axial frequencies, the COM and stretch modes, respectively 

given by 

ω2

1 = (2a(t) +3b(t)d(t)

2)q/m (6.4.3) 

ω2

2 = ω21(1 +ε) (6.4.4) 

where ε = q2/(πε0mω2

1d(t)3) [86]. The COM and stretch normal mode frequencies go through 

their minima near the point when a vanishes, at which point the external potential is mostly quartic. 

For very large inter-ion distances, the two modes are equal; that is, the ions are in reasonably 

independent wells with frequencies ω1 ' ω2. 

The separation waveform comprises two main segments. The ions began in a shared trap at 

the center of electrode X with respect to Fig. 6.1. For the frst segment, a(t) was decreased and 

b(t) increased by lowering the potentials on electrodes A and B while simultaneously increasing 

the potentials on outer electrodes O1 and O2 until the mode frequencies approximately reach their 

minimum values. The potential for electrode X was also lowered to a value which was simulated to 

minimize the trap’s potential curvature. In the second segment, the a(t) term passes through a zerocrossing through the application of an increasing potential on electrode X to produce a separating 
115 

wedge potential (or simply a “wedge”). The fnal value of electrode X for producing the wedge is the 

result of simulating a step-wise increase on electrode X, iterated until suitably-separated wells are 

found. The spatial extent and precision of the wedge is governed by the ion-to-electrode distances; 

therefore, smaller trap electrode sizes and ion-to-electrode distances can maintain stronger confnement and produce a narrower wedge, which could lead to reduced excitations. Higher maximum 

voltages on electrodes could also maintain stronger confnement. 

During separation, diabatic changes in both the curvature and equilibrium ion positions of the 

axial potential can lead to coherent displacement of the motional states. Both the inter-ion distance 

and the frequencies have an inter-dependency, so controlling the rate of change in frequency also 

controls the relative velocities between the ions. The largest excitation during separation coincides 

with roughly the time at which the value for a(t) changes sign. Initially, the waveform voltages during 

the frst segment were ramped with sin2(πt/2T) for duration T between initial and fnal values. 

The second segment linearly ramped the potential on electrode X to its fnal value. To suppress 

excitations, we approximated the adiabatic condition for ramping a harmonic well, 1 dω  1 [88], ω2 dt

where ω here corresponds to ω1 of Eq. 6.4.3. We numerically solved for 1 dω across the waveform 

ω2 dt

and imposed a maximum value of 1 dω = 0.025 for the COM mode during the ramp-down of the ω2 dt

dω harmonic well and 1 = 0.015 around the sign change of a(t) in Eq. 6.4.1 to shape the waveform ω2 dt

to a more adiabatic form, with the results shown in Fig. 6.6. Simulations for the ion positions and 

frequencies as a function of time are shown in Fig. 6.7. 

6.5 Separation of Two 9Be+Experiments and Results 

We ran a separation waveform (duration ' 340μs) to partition a linear chain of nine 9Be+

ions into two chains with all possible number combinations. We frst Doppler cooled the chain in 

zone A and optically pumped to | ↓1,↓2,··· ↓9i. It was then transported to zone X for separation 

into two groups in zones A and B. A variable offset potential VO2 applied to electrode O2 imposed 

a bias electric feld in zone X that shifted the center of the ion chain relative to the separation 
116 

0 0.5 1 1.5 2 2.5 3 3.5 4

x 10−5

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

time (s)

ω

−2

d

ω/dt

Original separation

Adjusted separation

Figure 6.6: Calculations of 1 dω during the separation waveform. The original separation waveform ω2 dt

corresponds to simulations producing a waveform where the potentials for the frst segment were 

ramped with sin2(πt/2T) for duration T between initial and fnal values, and the second segment 

linearly ramped the potential on electrode X to the fnal value. We imposed a maximum value of 

1 dω 1 dω = 0.025 for the COM mode during the ramp-down of the harmonic well and = 0.015

ω2 dt ω2 dt

around the sign change of a(t) to produced the adjusted separation waveform. The local minimum 

dω in 1 during the waveform corresponds to a hand-off between the two waveform segments, which ω2 dt

in practice takes several microseconds. 
117 

0 0.5 1 1.5 2 2.5 3 3.5 4

x 10−5

−1

−0.5

0

0.5

1

x 10−4

Time (s)

Ion Position (

μm)

0 0.5 1 1.5 2 2.5 3 3.5 4

x 10−5

0

1

2

3

4

5

Time (s)

Axial Frequency (MHz)

Common Mode

Stretch Mode

Figure 6.7: Change in ion position and normal mode frequency during separation for the two ions 

and the two normal modes. The normal mode frequencies converge as the ions are split into 

separate wells. 
118 

wedge to control the number of ions in each group. Ions in zone A were detected with resonance 

fuorescence, with the average photon counts approximately proportional to ion number. Following 

detection, all ions were recombined and transported to zone A where the total number of ions was 

rechecked to ascertain that no ions were lost during the waveform. As shown in Fig.6.8, increasing 

VO2 led to increased ion numbers in zone A. By adjusting VO2 to the center of a certain step, we 

could reliably partition the ions into two groups of predetermined number. 

In a second experiment, two 9Be+ ions were initialized over zone X, whose center is defned 

as z = 0 relative to Eq.(6.4.1), which were initialized to n ̄ ' 0.1 on both the COM and stretch modes 

with an initial well frequency of ω/(2π) ' 2.6 MHz ({VO1,VA,VX,VB,VO2} = {2.433, -0.3763, 

-1.7089, -0.3831, 2.473} V) and optically pumped to | ↓1↓2i. Near the point of ion separation, simulation predicted that the minimum COM mode frequency was '700 kHz, the minimum stretch 

mode frequency was '880 kHz, and the inter-ion distance was ∼ 17μm. The ions were separated 

into one ion in zone A and one ion in zone B, separated by ∼ 130μm. The fnal motional frequencies were 2.85 MHz and 2.77 MHz respectively, for {VO1,...VO2} = {4.441, -5.252, -0.649, -5.411, 

5.952} V, with the total waveform duration of 55 μs. After separation, qubit MAS transitions of varying duration t were driven on either the ion in zone A or B followed by recombination in zone A where 

P↓(t) was determined through state-dependent resonance fuorescence for each ion (Eq. 6.3.1). 

The frst separation segment took 17μs, during which it was necessary to ramp VO2 from 2.473 

to 5.952 V to compensate for trap-geometry asymmetries and to keep the chain centered over the 

separation electrode, with the fnal value set by minimizing n ̄ for both ions. 

In practice, the potential in Eq. 6.4.1 may have odd-order terms due to trap imperfections, 

leading to asymmetry during the separation process. During the second segment, we applied a 

constant relative difference between electrodes A and B to compensate for a “tilt” in the potential and 

to make the potential well as symmetric as possible about electrode X. We also obtained best results 

by applying a variable offset to the waveform potential to electrode X to tune the quartic component 

b(t) in the potential. These parameters enabled a fne-tune balancing of the fnal excitation between 

the two ions. Experimentally, we found that near-equal excitations to both ions in their respective 
119 

−0.2 −0.1 0 0.1 0.2

0

20

40

60

80

Relative potential offset on electrode O2 (V)

Average PMT counts from Zone A

Figure 6.8: Ion fuorescence in zone A as a function of the offset potential on electrode O2 during 

separation relative to a central value. Each step increase of the fuorescence corresponds to the 

presence of an additional ion in zone A after separation. Fluorescence of a single ion led to approximately 10 average photomultiplier tube (PMT) counts in a 200 μs detection time, but as more ions 

were added in zone A the crystal was no longer uniformly illuminated by the detection beam and 

the total fuorescence dropped below 10 counts per ion. Reproduced from Ref.[66]. 
120 

Figure 6.9: Rabi fopping trace of the MAS of the ion in zone A and zone B after separation. The 

ft to the MAS drive in zone B is to a coherent state with n ̄ = 1.9 ±0.1, and |α| = 1.38 ±0.04 (η =

0.404). The trace for the ion in zone A looks very similar giving n ̄ = 2.1 ±0.1, or |α| = 1.45 ±0.03 

(η = 0.399). 

fnal zones gave the lowest total excitation to both ions. 

After separation, fts to MAS transitions were consistent with coherent states of n ̄ = 2.1±0.1 

in zone A and n ̄ = 1.9±0.1 in zone B (Fig.6.9). With RF de-excitation of each ion, we could reduce 

n ̄ to 1.4±0.1 and 1.6±0.1 in zones A and B respectively, consistent with thermal distributions. The 

reductions indicated partial coherence in the fnal states, but the residual excitations could not be 

explained by ambient heating. Simulations of heating using measured values of heating vs. ω for a 

single ion only accounts for Δn ̄ ≈ 0.2. The fact that we couldn’t reduce n ̄ further could be explained if 

the phase of the coherent states changed signifcantly from experiment to experiment. For example, 

for a change in VO2 of + 3 mV to shift the ions towards zone A, the motion of the ion in zone A had 

n ̄ ' 1.14 ± 0.07 while the ion in zone B had n ̄ > 15, which after an RF de-excitation left the ion 

in zone B in a motional state consistent with a thermal distribution having n ̄ = 3.8 ±0.5. With the 

additional bias feld, the ion in zone B is closer to the maximum of the wedge during separation and 

so it would be expected to gain more kinetic energy as it falls towards the center of its respective 

well. Following the results in Ref. [66], it was reported in Ref. [89] that separation was achieved in 

80 μs with n ̄ ∼ 4 quanta on each ion with similar procedures to that described here. 
Chapter 7 

Optimization of Multi-species Ion Transport via Feedback on Quantum Motion 

For quantum information processing (QIP) based on trapped ions [5, 6], a large-scale processor might be achieved with an array of inter-connected trap zones where quantum information 

is transported by moving trapped ions between zones [7, 62]. Ion transport can be achieved by 

applying time-varying potentials, or “waveforms”, to segmented trap electrodes. This multizone ion 

transport approach to QIP was demonstrated with a mixed-species linear chain of ions along the 

trap axis, where the ions were separated into two 9Be+-

24Mg+ pairs (a 9Be+ “qubit” ion and a 

24Mg+ “coolant” ion) accompanied by motional excitation that could be removed with sympathetic 

laser cooling to cool all of the normal modes of the axial motion [29, 27]. However, separation, 

transport, and laser cooling accounted for durations ∼100 times longer than for quantum logic 

gates. For harmonic traps, it’s only necessary to cool the center-of-mass (COM) normal mode 

of motion for same-species chains of ions. Rapid ion transport with mitigated motional excitation 

was demonstrated in Refs. [90, 66] for same-species ions, but due to motional excitations from 

stray or fuctuating electric felds [24], it may still be necessary to implement sympathetic laser cooling. Thus, we must consider rapid ion transport for a qubit- and coolant-ion pair which mitigates 

excitations in all normal modes of motion. 

In the work here, ions are confned in a multizone linear Paul trap, with the confnement along 

the trap axis characterized by an angular frequency ω. The transport waveforms are determined by 

simulations of the potentials from individual electrodes that form a segmented trap array, the results 

of which are linearly superposed. The waveforms are designed to transport ions along the trap axis 
122 

while holding ω constant, and they are generated by electronics with an update rate of 50 MHz, 

which is much higher than the confning frequencies [74]. Unlike the case for rapid transport of a 

same-species ion chain as in Refs. [90, 66], different-species ions in a linear chain will experience 

different accelerations along the trap axis due to the unequal mass and thus non-COM modes of 

motion may be excited. Additionally, the transverse confnement depends inversely on the ion mass 

such that stray felds, those which can be diffcult or impractical to simulate due to effects which 

may include electrode charging or electrode-geometry imperfections, can “twist” a two-species ion 

chain along directions which contain components of both transverse and axial motion and in turn 

can cause coupling between axial and transverse motion. Excitations in the transverse motion also 

have nonlinear couplings to the axial motion [91]. To compensate for excitations during transport, 

we produced a gradient-method algorithm to adjust compensating felds in our ion trap electrodes 

to experimentally feedback on the quantum motion of the qubit ion. We have observed that during 

rapid transport, axial motion can be highly excited beyond what we theoretically predict, and that this 

is due primarily to traversing stray felds that cause signifcant diabatic excitations of the transverse 

motion. 

We investigated the dynamics of multiple ion species during transport between spatially distinct locations in multizone linear Paul traps. To account for imperfections in the transport, we 

implemented a gradient-method algorithm with feedback on the quantum motion of the ions using 

control parameters comprising trap-electrode potentials. This technique was implemented in two 

separate traps. A 9Be+ and 24Mg+ ion pair in a harmonic well was transported 370 μm within 32 μs 

with a peak velocity of ∼ 26 m/s with an average gain of 0.2 motional quanta. A 9Be+ and 25Mg+

ion pair in a harmonic well was transported 340 μm within 18 μs with a peak velocity of ∼ 42 m/s 

with an average gain of 0.1 motional quanta. This optimization method facilitates fast transport 

of multiple ion species that can signifcantly reduce the time overhead in certain architectures for 

scalable quantum information processing with trapped ions. 
123 

7.1 Methods for Optimizing Two-Species Transport 

The qubit is formed from two 2s2S1/2 electronic ground-state hyperfne levels of 9Be+ |F =

2,mF = 1i ≡ | ↓i and |F = 1,mF = 0i ≡ | ↑i, separated in frequency by ω0/(2π) ' 1.2 GHz, where 

F and mF are the ion’s total angular momentum and its projection along the quantization magnetic 

feld axis (B ' 11.9 mT). Ion motional quantum states can be described in the number basis |ni

1 with energy ~ω(n+ 2 ). To characterize the motional states experimentally, we use two laser beams 

which can be detuned from each other by ω0 +ω or ω0 −ω to drive motion adding sideband (MAS) 

spin-fip transitions | ↓,ni → | ↑,n+1i or motion subtracting sideband (MSS) transitions | ↓,ni →

| ↑,n − 1i on the axial normal mode of frequency ω via stimulated-Raman transitions. Since the 

corresponding Rabi rates are n-dependent, the populations of number states can be determined 

from the probability P↓(t) of the state | ↓i as a function of drive time t, derived from state-dependent 

resonance fuorescence [57]. Due to the presence of more than one mode of motion for a 9Be+-

24Mg+ ion chain, the probability P↓(t) depends on both motional modes. We ft to the function 

∞ ∞ 1 † † −γt iηm(am+am) iηn(an+a P↓(t) = (1 +e ∑ ∑ PnPm cos[2Ωt|hm,n

0

|e e n)|m,ni|]) (7.1.1) 2 n=0 m=0 

where number states |ni and |mi are for the two normal modes of motion with respective populations Pn and Pm, and γ is a phenomenological decay rate. This is the same as the equation in 

Chapter 3 Eq. 3.4.4, but with a phenomenological decay rate γ. In this case the mode with states 

|ni is driven as n0 = n+1 for the MAS, n0 = n−1 for the MSS, or n0 = n for a carrier transition, while 

iη(a+a the mode with states |mi is a “spectator” mode. The matrix elements |hn0

|e

†)

|ni| are given by 

iη(a+a −η2/2(n<!/n>!)1/2η|n0−n|

L|n |hn

0

|e

†)

|ni| = e 0

−n|

(η2 n ) (7.1.2) <

for each normal mode [6]. The populations of the number states for both normal modes can be 

determined by P↓(t). For a two-ion chain, the two axial modes are the in-phase (INPH) and out-ofphase (OOPH) modes. 
124 

2

Our initial transport ansatz for the position profle is the error function, defned as erf(t) =

 ́ t

√ 0 e−t

0

dt

0

, to perform rapid transport that remains adiabatic with the respect to trap motion [79], π

though other possible functions which allow potentially faster adiabatic transport were investigated 

in Ref. [85] for a multi-species ion chain. The functional form we used for the position along the trap 

axis is given by 

1 +erf(4[t −tT /2]/tT )

z(t) = (zf −zi) +zi, (7.1.3) 2 

where zi is the trap’s initial position, zf is the trap’s fnal position, and tT the total duration of the 

transport. The function is truncated at t = 0 and at t = tT , though the trap potentials for the transport 

waveform with the position profle in Eq. 7.1.3 were constrained to match the trap position and 

velocity at the initial and fnal positions with cubic spline interpolation. As a baseline, a single 

9Be+ had been transported with a waveform that transports the ion according to Eq. 7.1.3 with 

Δn ̄ ' 0.1 for transport durations as low as tT ' 11μs. However, when the waveform transported a 

two-species chain of ions according to Eq. 7.1.3, n ̄OOPH and n ̄INPH could be observed to incur as 

much as Δn ̄ ∼ 10 of excitation after transport. 

To optimize the transport, the transport waveform was designed to move the ions with the 

position profle in Eq. 7.1.3, and the waveform was divided into discrete segments of evenly-spaced 

waveform durations for a total of Nseg segments, and each segment was sequentially optimized. The 

ions were transported across the distance up to the current segment’s fnal position in a one-way 

transport, and then the transport was reversed to return the ions to the initial position in a round-trip 

transport. A carrier transition was then driven for a fxed drive duration set to the carrier π-time from 

Eq. 7.1.1 for the case when n ̄INPH = 0 and n ̄OOPH = 0, followed by resonance fuorescence to 

make a measurement that indicates the |nINPHi = 0 and |nOOPHi = 0 occupations. A basis-set of 

feld-control parameters are independently varied for the current segment to feedback on the axial 

excitations, and after the results that minimize axial excitations for each parameter were found, they 

were then all applied to the current waveform segment. This procedure is iterated as needed for Nit
125 

Figure 7.1: Schematic of the quantum logic trap structure showing the two diagonally opposite 

segmented DC electrodes (not to scale). The ions are initialized in zone A, and the transport 

waveform carries ions across 370 μm from zone A to zone B. Trap radio-frequency electrodes (not 

shown) are referenced to a common ground potential. 

iterations for each segment, after which the next segment is concatenated and then optimized in 

the same fashion. This produces time-varying compensation felds across the entire transport that 

minimize axial excitations. 

7.2 Transport of a Two-Species Ion Chain Experiments and Results 

Our frst implementation of the algorithm transported the ions in the quantum logic trap across 

a distance of 370 μm, the distance from zone A to zone B shown in Fig. 7.1, and tT =32 μs with a 

peak velocity of ∼ 26 m/s. Simulations calculate that during transport, each mode has a numerically estimated maximum coherent state displacement |αINPH| ∼ 0.53 and |αOOPH| ∼ 0.04 before 

returning to the ground state at the transport destination. Initially, a 9Be+-

24Mg+ ion chain was 

prepared in zone A with electrode voltages {VO1, VA, VX, VB, VO2}={2.433, 0.593, 4.124, 0.584, 

2.473} V and Doppler cooled simultaneously with both Doppler cooling on 9Be+ and Doppler cooling on 24Mg+ for a 500 μs duration, which prepares transverse and axial motion in a thermal state 

[40, 36]; see Table 7.1 for the normal modes and Doppler-cooling limits. Modes with a larger amplitude for 24Mg+ are more effciently cooled with the 24Mg+ Doppler cooling light, while modes 

with a larger amplitude for 9Be+ are more effciently cooled with the 9Be+ Doppler cooling light. 

After Doppler cooling, the axial motion was Raman sideband laser-cooled to n ̄INPH ' 0.1 and 

n ̄OOPH ' 0.1 and the 9Be+ qubit ion was optically pumped to | ↓i before transport. 
126 

ω/2π (MHz) xˆBe yˆBe zˆBe xˆMg yˆMg zˆMg n ̄limit

12.113 0.9998 0 0 0.0177 0 0 0.55 

11.029 0 0.9997 0 0 0.0203 0 0.63 

4.673 -0.0177 0 0 0.9998 0 0 4.78 

4.035 0 0 -0.9258 0 0 0.3780 1.58 

3.524 0 -0.0203 0 0 0.9997 0 6.49 

1.902 0 0 0.3780 0 0 0.9258 8.67 

Table 7.1: The normal modes for a 9Be+-

24Mg+ chain with mass-weighted coordinates and the 

Doppler cooling limits n ̄limit for each mode. The x and y coordinates are those for the transverse 

motion and the z coordinate is that for the axial motion. Doppler cooling light lies along the direction 

√ xˆ+yˆ+ 2zˆ k

ˆ = . The cooling limits are from Doppler-cooling light with both 24Mg+, which cools on 2 

a transition with linewidth Γ ' 2π×41 MHz and is used to calculate n ̄limit for modes with a larger 

amplitude for 24Mg+, and Doppler cooling on 9Be+ which cools on a transition with linewidth Γ '

2π×19 MHz and is used to calculate n ̄limit for modes with a larger ampitude for 9Be+. 
127 

The waveform is designed with time-varying common voltages to diagonally-opposite electrodes. To optimize the transport, we used four feld-control parameters, three of which were potentials generated by applying a differential “shim” voltage onto the diagonally-opposite electrode pairs 

A, X, and B in Fig. 7.1, given by +V and −V where V was varied over the range [-50,50] mV; the 

last parameter was from a bias voltage applied to a planar “biasboard” electrode 400 μm beneath 

the linear array electrode structure, varied over the range [-200,200] mV. After the optimization had 

completed, the ions were transported to zone B and a MAS sideband transition was driven, followed by transport back to zone A to determine P↓(t) to ascertain the success of the algorithm. 

Acousto-optic defectors (AODs) directed the laser beams to either zone A or B. The optimization algorithm was ran with Nseg = 6 and Nit = 3, which optimized to minimal fnal excitations of 

n ̄INPH = 0.27 ±0.03 and n ̄OOPH = 0.31±0.04 consistent with thermal distributions from fts to the 

MAS on the INPH and OOPH modes, respectively (Fig. 7.2). For comparison, a ft to a coherent 

state for either mode gives n ̄INPH = 0.22 ±0.02 and n ̄OOPH = 0.23 ±0.02, which are roughly the 

same for the results to the thermal fts. 

Our second implementation of the algorithm transported the ions in the X-junction trap across 

a distance of 340 μm, the distance from zone A to zone B with respect to Fig. 7.3, and tT =18 μs 

with a peak velocity of ∼ 42 m/s. Simulations calculate that during transport, each mode has a 

maximum coherent state displacement |αINPH| ∼ 1.03 and |αOOPH| ∼ 0.08 before returning to the 

ground state at the transport destination. Initially, a 9Be+-

25Mg+ ion chain was prepared in zone A 

with electrode voltages {VO1, VA, VX, VB, VO2}={1.379, -2.867, 0.639, 0.678, 0.124} V and the 

chain was Doppler cooled on 9Be+ and 25Mg+ for a 500 μs duration; see Table 7.2 for the normal 

modes and Doppler-cooling limits. Modes with a larger amplitude for 25Mg+ are more effciently 

cooled with the 25Mg+ Doppler cooling light, while modes with a larger amplitude for 9Be+ are 

more effciently cooled with the 9Be+ Doppler cooling light. After Doppler cooling, the axial motion 

was Raman sideband laser-cooled to n ̄INPH ' 0.1 and n ̄OOPH ' 0.01 and the 9Be+ qubit ion was 

optically pumped to | ↓i before transport. 

Four feld-control parameters were used: shims applied to the diagonally-opposite electrode 
128 

Figure 7.2: Rabi fopping traces on the MAS for each normal mode along the transport axis for 

transport from zone A to zone B following the optimization procedure. We ft to thermal distributions 

that are consistent with n ̄INPH = 0.27 ±0.03 and n ̄OOPH = 0.31 ±0.04. (a) Rabi fopping trace on 

the MAS for the INPH mode with Lamb-Dicke parameter ηINPH = 0.184. (b) Rabi fopping trace 

on the MAS for the OOPH mode with Lamb-Dicke parameter ηOOPH = 0.310. 

Figure 7.3: Schematic of the X-junction trap structure showing the two diagonally opposite segmented DC electrodes (not to scale). The ions are initialized in zone A, and the transport waveform 

carries ions across 340 μm from zone A to zone B. Trap radio-frequency electrodes (not shown) are 

referenced to a common ground potential. 
129 

ω/2π (MHz) xˆBe yˆBe zˆBe xˆMg yˆMg zˆMg n ̄limit

13.091 0.9997 0 0 0.0261 0 0 0.48 

11.560 0 0.9995 0 0 0.0312 0 0.59 

5.364 0 0 -0.9299 0 0 0.3677 1.09 

4.846 -0.0261 0 0 0.9997 0 0 4.59 

3.142 0 -0.0312 0 0 0.9995 0 7.34 

2.492 0 0 0.3677 0 0 0.9299 6.50 

Table 7.2: The normal modes for a 9Be+-

25Mg+ chain with mass-weighted coordinates and the 

Doppler cooling limits n ̄limit for each mode. The x and y coordinates are those for the transverse 

motion and the z coordinate is that for the axial motion. Doppler cooling light lies along the direction 

√ xˆ+yˆ+ 2zˆ k

ˆ = . The cooling limits are from Doppler-cooling light with both 25Mg+, which cools on 2 

a transition with linewidth Γ ' 2π×41 MHz and is used to calculate n ̄limit for modes with a larger 

amplitude for 25Mg+, and Doppler cooling on 9Be+ which cools on a transition with linewidth Γ '

2π×19 MHz and is used to calculate n ̄limit for modes with a larger ampitude for 9Be+. 
130 

pairs A, X, and B in Fig. 7.3, given by +V and −V, where V was varied over the range [-30,30] mV, 

and a bias voltage applied to a planar electrode 500 μm beneath the linear array electrode structure 

varied over the range [-1,1] V. In this case, optical access to zone B was not possible, so following 

one-way transport, the transport was reversed by a highly-adiabatic return to zone A with tT =60 μs 

with a peak velocity of ∼ 13 m/s, where simulations calculate that during transport, the maximum 

coherent displacements are |αINPH| ∼ 0.09 and |αOOPH| ∼ 0.007. After round-trip transport, a 

MAS sideband transition was driven followed by detection to determine P↓(t). In this experiment, 

two 2s2S1/2 electronic ground-state hyperfne levels of 9Be+ |F = 2,mF = 2i ≡ | ↓i and |F =

1,mF = 1i ≡ | ↑i were used. The optimization algorithm was ran with Nseg = 5 and Nit = 2 to 

converge on suitable results, which optimized to minimal fnal excitations of n ̄INPH = 0.19 ± 0.02 

and n ̄OOPH = 0.06 ± 0.01 consistent with thermal states from fts to the MAS on the INPH and 

OOPH modes, respectively (Fig. 7.4). For comparison, a ft to a coherent state for either mode 

gives n ̄INPH = 0.17±0.02 and n ̄OOPH = 0.12±0.01, which are roughly the same for the results to 

the thermal fts. 

In summary, we have demonstrated adiabatic transport of a qubit- and coolant-ion chain on 

∼20-30 μs timescales, which approach those of quantum logic gate operations. When transporting 

two-species ions, we observed high motional excitations that could be prevented by experimentally optimizing the transport through feedback on the quantum motion. Following the optimized 

transport, the motion was comparable to the initial laser-cooled state. The methods described here 

should be straight-forward to extend to longer chains of multi-species ions. The optimization accounts for multiple normal modes of motion, and in principle, such a transport optimization should 

extend to very large collections of ions or atoms. 
131 

Figure 7.4: Rabi fopping traces on the MAS for each normal mode along the transport axis for 

transport from zone A to zone B with a highly adiabatic return to zone A following the optimization 

procedure. We ft to thermal distributions that are consistent with n ̄INPH = 0.19±0.02 and n ̄OOPH =

0.06 ± 0.01. (a) Rabi fopping trace on the MAS for the INPH mode with Lamb-Dicke parameter 

ηINPH = 0.157. (b) Rabi fopping trace on the MAS for the OOPH mode with Lamb-Dicke parameter 

ηOOPH = 0.270. 
Chapter 8 

Conclusions and Outlook 

Enormous amounts of work remain before scalable QIP with trapped ions (or any physical implementation) can be achieved. The improvements in ion transport described in this thesis should 

help progress towards a realization of of a multi-electrode ion-based quantum information processor. However, there are still many improvements on ion transport that remain. 

Potentially, one of the biggest challenges to improve transport is to achieve ion separation 

with no residual excitation of ion motion. As of now, the state-of-the-art in separation has produced 

residual excitations on the order of 1 quantum that have proven diffcult to suppress [66, 89]. A possible way to improve separation over former experiments would be to employ a feedback algorithm 

similar to that used for two-species 9Be+-

24Mg and 9Be+-25Mg transport by using a set of electrode potentials to optimize the separation potential. An additional complication for separating the 

multi-species 9Be+-

24Mg+-24Mg+-9Be+ chain is that separation may require a feld-compensation 

feedback algorithm to avoid transverse mode excitations as in the case for transport. A judicious 

selection of electrode parameters may be able to achieve near-ideal waveforms to mitigate excitations of axial and transverse normal modes throughout the separation process. There is a potential 

alternative that can avoid the need for combination and separation for each two-qubit entanglement 

step. By bringing ions in two separate wells suffciently close to each other without combining into 

the same trap, the Coulomb interaction between ions can supply an information bus between separate wells [92, 93]. This method may circumvent the need to fully combine and separate ions in 

some cases. 
133 

When transporting a two-species ion crystal, we observed increasingly strong transverse 

excitations with increasing speeds, and this in turn made it more diffcult to observe minima in 

excitations while optimizing the compensation felds for the waveform. Future experiments could 

perhaps frst optimize the compensation felds and then apply the results of the feld compensations 

to a faster waveform. There are alternative theoretical ideas for ways to move multi-species ions 

very rapidly and return to the ground state [85], which may lend to transport of a two-species 

crystal with similar low duration and excitation as those obtained in single-species transport of 

Refs. [66, 90]. By ground-state cooling the transverse motion before transport, there should be 

less overall excitation during transport. A possible way to cool the transverse motion could be to 

use the techniques for sympathetic electromagnetic-induced-transparency cooling with a twisted 

9Be+-25Mg+ ion chain before transport [94]. Experiments for reducing the heating rates through 

ion-cleaning of surface traps has achieved a hundred-fold improvement in heating rates which also 

may reduce overall excitations during transport [25, 95]. 

Transporting a multi-species ion chain reliably through a junction remains an open problem. 

The pseudopotential near the junction differ for the two species due to different mass, which may 

lead to additional complications, such as requiring different waveforms for reliable transport. A 

possible solution to circumvent loss in the junction is to transport each ion species separately and 

then recombine qubit and coolant ions after transport through the junction. However, reducing the 

number of necessary coolant ions would alleviate some possible complications that might arise with 

diffculties for transporting 9Be+-

25Mg+ through the RF junction. One solution to possible diffculties 

in transporting multiple species in the junction could be to restrict a single coolant ion to a dedicated 

“refrigerant” zone. When sympathetic cooling is required before an entangling gate, a coolant ion 

and two qubit ions would be combined to a 24Mg+-

9Be+-9Be+ chain in a refrigerant zone, transporting the qubit 9Be+ ions through the junction as necessary. Asymmetry in such a crystal makes 

entanglement with the shared motion more desirable on a 9Be+-

9Be+ chain, so following sympathetic cooling, the crystal could be separated into 24Mg+ and 9Be+-

9Be+ to entangle the two qubit 

ions. In Ref. [66], it was demonstrated that it is possible to asymmetrize separation such that the 
134 

motion is selectively excited in one well with less excitations in the other. Work in Ref. [22] had 

reliable low-excitations of the stretch mode of 9Be+-

9Be+ after separation of 9Be+-9Be+-9Be+ into 

9Be+ and 9Be+-9Be+, which was the mode used for subsequent entanglement on the 9Be+-9Be+

chain. Therefore, following separation into 24Mg+ and 9Be+-

9Be, the stretch mode could be kept 

near the ground-state cooled motion and used for entanglement. 

The ion transport experiments described in this thesis have demonstrated a high-degree of 

control over the motion for trapped ions during diabatic ion transport in multi-electrode ion traps. 

Ion transport has been demonstrated with signifcant speed-up in the operation times for transport, 

which can help reduce operation times for scalable ion-based QIP well-beneath qubit coherence 

times. The mitigated motional excitations alleviate the timescales for recooling before entangling 

gates as well. Our recent scalable QIP demonstrations had a distribution of operations which were 

dominated by transport and recooling, but by applying the methods described in this thesis in future 

QIP experiments, it is possible that the temporal distribution of tasks during an experiment could 

become dominated by quantum logic operations. 
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Appendix A 

Transport Gate 

A possible approach to reduce the resources required for QIP experiments with a multielectrode trap architecture is to have dedicated regions where beams are always turned on, which 

alleviates issues due to duty cycle by having continuous beam position and intensity stabilization. 

Ions then can be transported through the beam to perform qubit rotations or entanglement without 

the need for switching on or off beams or redirecting beams to different zones [96]. The precise 

control achieved over ion transport described in this thesis allows the ion velocity to be easily set to 

convenient values for our laser beam parameters. This section assumes that ions travel at approximately constant velocity. 

The proposal in Ref. [96] includes using the Doppler shift in the frame of the ions between 

two Raman beams as a detuning from a shared normal mode of motion to drive an entangling gate 

during adiabatic ion transport. Our desirable velocities are on the order of v & 10 m/s to reduce 

gate durations to acceptable values. For v = 30 m/s and for a laser beam at an angle γ with respect 

to the transport axis, the Doppler shift is Δω = |k||v| cos(γ) ' 2π×68 MHz, which is too large for 

a practical gate detuning from a ∼ 5 MHz normal mode. Instead, we can achieve rapid transport 

with Doppler shifts that can be canceled by shifting the laser beam frequencies with AOMs and thus 

allow us the freedom to set an ideal detuning δ0 in the beams and maintain a fxed angle γ = π/4. 

Typical AOMs in our lab have a bandwidth of ∼ 100 MHz, allowing a suffcient range of tuning to 

account for relevant Doppler shifts. 

For a Gaussian laser beam feld, traversing a beam at constant velocity would be equivalent 
142 

to a Gaussian-shaped temporal laser pulse. For a beam propagating along zˆ and with transverse 

radial direction rˆ, the electric feld is given by 

    E0 w0 −r2 kr2 

E(r,t) = exp exp −i(kz−ωt +φ−arctan(z/zr) + +h.c., (A.0.1) 2 w(z) w(z)2 2R(z)

p where the beam waist is w0, w(z) = w0 1 + (z/zr)2, R(z) = z[1 + (zr/z)2], and zr = kw2

0/2. For 

ions near the beam’s focus, such that |z/zr|  1, we can make the approximations that w(z) ' w0, 

arctan(z/zr) ' 0, and kr2/2R(z) ' 0. Then, the feld simplifes to 

  E(r,t) = E0 

exp −r

2

2 

exp[−i(kz−ωt +φ)]+h.c. (A.0.2) 2 w0 

For the case of single-qubit rotations, we assume that a Raman transition is driven by copropagating laser beams (Chapter 2). The Rabi rate is proportional to the square of the electric 

feld, and so for a maximum Rabi rate Ωm at the center of the beam with the substitution τ =

√ w0/ 2vsin(γ) in Eq. A.0.2 for an ion moving with velocity v across a beam of waist w0, the Rabi 

rate varies as 

Ω(t) = Ωm exp −(t/τ)

2 . (A.0.3) 

The qubit rotation angle is then given by 

ˆ tT

√ θ = Ω(t)dt = Ωmτ πerf[tT /τ] (A.0.4) −tT

for a transport time 2tT . As an example for a π-rotation, assuming that an ion has v = 30 m/s 

and traverses a beam with w0 = 30μm at 45◦ to the transport axis, then τ = 1μs. Taking the 

approximation tT → ∞ gives that we need a beam with Ωm ' 2π×282 kHz, which is typical for our 

systems. 

Entangling two qubit ions can be achieved when traversing beams in a 90◦-degree angle from 

each other, each 45◦ to the transport axis, to drive motion-sensitive transitions. This requires that 
143 

the inter-ion distance is much less than twice the beam waist so that both ions have approximately 

the same interaction while traversing the beams. For two 9Be+ ions in a well with ωz = 2π ×

3.6 MHz, the ion distance is d = (2keq2/kz)1/3 ' 3.92μm (from Chapter 3), approximately an order 

of magnitude less than w0. One gate scheme requires the ability to simultaneously drive a transition 

detuned from the red- and blue-sideband transitions on the qubit ions [97, 98]. This can be achieved, 

for example, by using an electro-optic modulator (EOM) such that the modulation due to the EOM 

suppresses the carrier and has two sidebands detuned from the stretch mode of motion for two 

9Be+ ions. For the work in Ref. [28], we used two AOMs to shift the frequencies of two beams 

which were then combined to have co-propagating sidebands. The entangling-gate Hamiltonian is 

given by 

h i † iδ0t HI = Ω(t) (σ+1 +σ+2)⊗ae−iδ0t + (σ+1 +σ+2)⊗a e +h.c. (A.0.5) 

where σ+i is the usual Pauli spin operator for qubit i = 1,2 and δ0 is the detuning from a normal 

√ mode. Consider the stretch mode ωstretch = 3ωz ' 2π × 6.2 MHz, which would not be excited 

during ion transport. Nominally, there is no residual motion following the entangling gate. A statedependent coherent state is driven by the gate, which is proportional to 

ˆ tT

α(t) ∝ exp[−(t/τ)

2]eiδ0t

dt, (A.0.6) −tT

which gives that if the Fourier transform vanishes at δ0, then the motion and spins are disentangled 

after the transport. Taking the approximation that tT → ∞, and that α(−∞) = 0, and with the 

substitution p = δ0τ/2, the coherent state evolution during the entangling gate is given by [96] 

√π(erf[t/τ−ip] +1) α(t) = r , (A.0.7)

h√ i 2 erf 2p

where erf[z] = −ierf[iz]. This expression vanishes for large p. For example, if p = 4, then 

δ0 ' 2π×1.273 MHz. With that detuning, simulations show that for Ωm = 2π×250 kHz, the fnal entanglement error is ε ' 3.13×10−7 , well beneath the fault-tolerant threshold of ∼ 10−4 , and 
144 

that α(∞) ' 3.15 × 10−7 . It may also be convenient to consider performing an entangling gate 

with the same considerations using Gaussian-shaped laser pulses incident on stationary ions, as 

precision timing for starting and stopping the gate interaction is replaced with asymptotic behavior 

for disentangling the qubit and motion states at the end of the entagling gate. 
Appendix B 

AWG Spec Sheets 
146 

1 2 3 4

DCBA

4 3 2 1

D

CBA

Title

Number Revision Size

B

Date: 7-Feb-2011 Sheet of File: E:\engineering jobs\..\DAC_CARD_1OF5.schDrawn By:

C1 C2

0.01uF

C3

DGND

GND 10GND 17GND 27GND 37GND 52GND 53GND 70GND 79GND 85GND 95GND 105GND 121GND 131GND 141GND 156GND 173GND 182GND 188GND 198GND 208

VCCAUX 7VCCAUX 44VCCAUX 66VCCAUX 92VCCAUX 111VCCAUX 149VCCAUX 166VCCAUX 195

VCCINT 13VCCINT 67VCCINT 117VCCINT 170

POWER

JTAG/CONTROL

DONE 104 PROG_B 1TCK 158TDI 207TDO 157TMS 155

U1EXC3S250/500E PQ208

IP 54IO_L01P_2/CSO_B 55IO_L01N_2/INIT_B 56IP_L02P_2 57IP_L02N_2 58VCCO_2 59IO_L03P_2/DOUT/BUSY 60IO_L03N_2/MOSI/CSI_B 61IO_L04P_2 62IO_L04N_2 63IO_L05P_2 64IO_L05N_2 65IO_L06P_2 68IO_L06N_2 69IP_L07P_2 71IP_L07N_2/VREF_2 72VCCO_2 73IO_L08P_2/D7/GCLK12 74IO_L08N_2/D6/GCLK13 75IO/D5 76IO_L09P_2/D4/GCLK14 77IO_L09N_2/D3/GCLK15 78 IP_L10P_2/RDWR_B/GCLK0 80 IP_L10N_2/M2/GCLK1 81 IO_L11P_2/D2/GCLK2 82 IO_L11N_2/D1/GCLK3 83 IO/M1 84

IO_L12P_2/M0 86 IO_L12N_2/DIN/D0 87 VCCO_2 88 IO_L13P_2 89 IO_L13N_2 90 IP 91

IO_L14P_2/A23 93 IO_L14N_2/A22 94IO_L15P_2/A21 96 IO_L15N_2/A20 97 IO/VREF_2 98 IO_L16P_2/VS2/A19 99 IO_L16N_2/VS1/A18 100 IP 101 IO_L17P_2/VS0/A17 102 IO_L17N_2/CCLK 103

U1BBANK2XC3S250/500E PQ208

C4

C5

C6

C7 C8

C9 C10

0.01uF

C11 C12

VCCINT12

D3_CLK+D3_CLKD3_CLK_IN+D3_CLK_IND2_CLK+D2_CLKD2_CLK_IN+D2_CLK_INRESET

D1_CLKD1_CLK+

D3_CLK+D3_CLKD3_CLK_IN+D3_CLK_INRESET

BANK3 CLKDAC_CARD_2OF5.sch

D3_CLK+D3_CLKD3_CLK_IN+

D3_CLK_INRESET

D2_CLK+D2_CLKD2_CLK_IN+

D2_CLK_INRESET

D1_CLK+D1_CLKD1_CLK_IN+D1_CLK_INRESETF3F2F1

BANK1 CLKDAC_CARD_4OF5.sch

D1_CLK+D1_CLKD1_CLK_IN+

D1_CLK_INRESET

1OE 11Y1 21Y2 3GND 41Y3 51Y4 6VCC 72Y1 82Y2 9GND 102Y3 112Y4 123Y1 133Y2 14GND 153Y3 163Y4 17VCC 184Y1 194Y2 20GND 214Y3 224Y4 234OE 24 3OE 25 4A4 26 4A3 27 GND 28 4A2 29 4A1 30 VCC 31 3A4 32 3A3 33 GND 34 3A2 35 3A1 36 2A4 37 2A3 38 GND 39 2A2 40 2A1 41 VCC 42 1A4 43 1A3 44 GND 45 1A2 46 1A1 47 2OE 48

U274LVC162244

D0 1D4 2D2 3

VCCIO 4

D1 5D7 6GND 7

NC 8

D5 9D6 10D3 11PWREN 12RD 13WR 14

USBDP 15

USBDM 16

3V3OUT 17

GND 18

RESET 19

VCC 20

GND 21TXE 22 RXF 23

NC 24

AGND 25

TEST 26

OSCI 27

OSCO 28

U3FT245R

DBVDD33VCCAUX25VCCINT12

DBVDD33VCCAUX25VCCINT12

POWERDAC_CARD_5OF5.sch

C13 C14

0.01uF

C15 DGND

VCCAUX25

C16

DGND

VDD 4 OUT 3

GND 2 TRIS 1

X1XTAL OSC DGND

CLOCKF1_AF2_AF3_A

VCCAUX25

C170.01uF

C180.01uF

+5D

VCCAUX25

C971uF

DGND

M0M1M2

D0D1D2D3D4D5D6D7

SCKD1

D7D6D5D4D3D2

RD

RD

NC 4D+ 3D- 2VCC 1GND 5

J1USB MINIB

DGND

C190.01uF

DGND

DGND

F1F2F3

DGND

C984.7uF

DGND

R14.7kR610k

SW1RESET

VCCAUX25

R24.7k DGND

Q1MMBTA06

R8470

R9470

D1 VCCAUX25 R10

330

R11 68R12 68R13 68

DBVDD33

DGND

1 23 45 67 89 10

JP1HEADER 5X2 SM

TMS_INTDI_INTDOTCK_INDBVDD33

1 23 45 6

JP3 DGND

M0M1M2

DIN

SI 1SCK 2RESET 3CS 4 WP 5

VCC 6

GND 7

SO 8

U4AT45DB

DINMOSISCKCS

R34.7k DBVDD33 R4

R54.7k

DGND

C200.01uF DGND

DGND

C100

VCCAUX25

C21 C22

0.01uF

C23

DGND

VCCAUX25

C10110uF

VCCAUX25VCCAUX25

C10210uF

GCLK

D0

CSINIT_BMOSI

868481

R14 R15

1k

R16

F1_AF2_AF3_A

GCLK

VCCAUX25

C12810uF

C129

10uF

F1TTL input F2in

F3in

DGND

G2

G1G2F4_A

GO_1GO_2F4

F5

D0D1D2D3D4D5D6D7

GCLKGO_1GO_2F4

1 23 45 67 89 10

11 1213 1415 1617 1819 2021 2223 2425 2627 2829 3031 3233 3435 3637 3839 4041 4243 4445 4647 4849 5051 5253 5455 5657 5859 60

JP2A99242-ND

D0DGNDD1D2D3D4D5D6D7GCLKGO_1GO_2F4

DGND

DGND DGNDDGND DGNDDGND DGNDDGND DGNDDGND DGNDDGND DGNDDGND DGNDDGND DGNDDGND DGNDDGND DGNDDGND DGNDDGND DGNDDGND DGND

DGND

D0D1D2D3D4D5D6D7GCLKGO_1GO_2F4F3F2F1

DATA BUSDAC_CARD_5OF5.sch

G1

RXF

GO_1GO_2

D1_CLK_IND1_CLK_IN+

D2_08ND2_08P

D2_05PD2_05N

RXF

VCCAUX25

F3F2F1

D2_08ND2_08PD2_05ND2_05P

TP12

INIT_B

Vss 3

RST 1

Vdd 2

U18MCP121T

DGND

VCCAUX25

D2_CLK+D2_CLKD2_CLK_IN+D2_CLK_INRESETGO_1GO_2D2_08ND2_08PD2_05ND2_05P

BANK0 CLKDAC_CARD_3OF5.sch

F4in

F5out

DGND

F4_A

R23 R38

VCCAUX25

Trigger

R51

D22.5V

F3F2F1

F3F2F1

F5F4

DGND

21

JP13RESET
147 

6 I L /

DCpM

/ L I 6

D

CpM Title

Number Revision Size

p

DateQ KyFebyI866 Sheet ofFileQ EQ\engineering jobs\PP\DMC_CMRD_IOFVPschDrawn pyQ

DL_6VNDL_6VPDL_6/NDL_6/PDL_6LNDL_6LPDL_6INDL_6IPDL_66NDL_66PDL_68NDL_68PDL_8XNDL_8XPDL_8UNDL_8UP

DL_88PDL_88NDL_86PDL_86NDL_8IPDL_8INDL_8LPDL_8LNDL_8/PDL_8/NDL_8VPDL_8VNDL_8HPDL_8HNDL_8KPDL_8KN

DL_6VNDL_6VPDL_6/NDL_6/PDL_6LNDL_6LPDL_6INDL_6IPDL_66NDL_66PDL_68NDL_68PDL_8XNDL_8XPDL_8UNDL_8UPDL_8KNDL_8KPDL_8HNDL_8HPDL_8VNDL_8VP

DL_88PDL_88NDL_86PDL_86NDL_8IPDL_8INDL_8LPDL_8LNDL_8/PDL_8/N

IO_L86P_L IIO_L86N_L LIO_L8IP_L /IO_L8IN_L9VREFL VIP HIO_L8LP_L UIO_L8LN_L XIO_L8/P_L 66IO_L8/N_L 6IIP 6/IO_L8VP_L 6VIO_L8VN_L 6HIO_L8HP_L 6UIO_L8HN_L 6XIP9VREF_L I8VCCO_L I6IO_L8KP_L9LHCLK8 IIIO_L8KN_L9LHCLK6 ILIO_L8UP_L9LHCLKI I/IO_L8UN_L9LHCLKL IVIP IH IO_L8XP_L9LHCLK/ IU IO_L8XN_L9LHCLKV IX IO_L68P_L9LHCLKH L8 IO_L68N_L9LHCLKK L6 IP LI IO_L66P_L LL IO_L66N_L L/ IO_L6IP_L LV IO_L6IN_L LH

VCCO_L LU IO_L6LP_L LX IO_L6LN_L /8 IO_L6/P_L /6 IO_L6/N_L /I IP /L

IO9VREF_L /V VCCO_L /H IO_L6VP_L /K IO_L6VN_L /U IO_L6HP_L /X IO_L6HN_L V8 IP V6

U6MpMNKLXCLSIV89V88E PQI8U

CLKVDD 6REXT ICLKVDD LCLKGND /CLKc VCLKy HCLKGND KDGND UDVDD XDp6Vc 68Dp6Vy 66Dp6/c 6IDp6/y 6LDp6Lc 6/Dp6Ly 6VDp6Ic 6HDp6Iy 6KDp66c 6UDp66y 6XDpVDD I8DpGND I6Dp68c IIDp68y ILDpXc I/DpXy IVDpUc IHDpUy IKDCLK_OUTc IUDCLK_OUTy IXDpVDD L8DpGND L6DCLK_INc LIDCLK_INy LLDpKc L/DpKy LVDpHc LHDpHy LKDpVc LUDpVy LXDpVDD /8 DpGND /6 Dp/c /I Dp/y /L DpLc // DpLy /V DpIc /H DpIy /K Dp6c /U Dp6y /X Dp8c V8 Dp8y V6 DVDD VI DGND VL SDO V/ SDIO VV SCLK VH CSp VK RESET VU REFIO VX FSMDJ H8 SDR_EN H6 MDVDD HI MDGND HL MCVDD H/ MCGND HV MVDDI HH MGNDI HK MVDD6 HU MGND6 HX IOUTp K8 IOUTM K6 MGND6 KI MVDD6 KL MGNDI K/ MVDDI KV MCGND KH MCVDD KK MDGND KU MDVDD KX SPI_DIS U8

UVMDXKIH

INy 6

DGND I

yVs L

M8 / M6 V WR H

OUT K

cVs U

REF X

INc 68

LOGIC

UUMDUIV8

R/8/XPX 8P6gR/6/XPX 8P6g MGND

CI/8P86uF

R/HIP8k 8P6g

CIH8P86uF

MGND

MGND

c6V

y6V

y6V

c6L

MGND

MGND

DGND

DVDDIV

DGND

DVDDIV

CIK8P86uF

DGND

CIU8P86uF

DpVDDLL

CIX8P86uF

DpVDDLL

CL88P86uF

CL68P86uF

CLI

8P86uF

CLL

8P86uF

CL/

8P86uF

CLV

8P86uF

CLH

8P86uF

CLK

8P86uF

CLU

8P86uF

CLX

8P86uF

cLPLVMGNDcIPVV

cIPVVcLPLV

MGNDMGND

DpVDDLL

DGNDDGND

C/88P86uF

C/68P86uFDL_CLKc

DL_CLKy

DL_CLK_INc

DL_CLK_INy

MGND

C/I C/L

8P86uF

C// DGND

VCCMUXIV

VCCMUXIV

VCCMUXIVVCCMUXIV

DGND

MGND

C/V8P86uF

C68L68uF

6

C

/

UCI

8 I6

L

/

U H

V KX p

M

D

C

EF

SWICRD6HRM8Sp

R6U6k

R6X6k

RI86k

RI66k DVDDIV

DGND

SUS6S/SI

SU

SI

S6

S/

/LLI6/H

DpVDDLLCLKVDDIVDVDDIVVCCMUXIV

DpVDDLLCLKVDDIVDVDDIVVCCMUXIV

POWERDMC_CMRD_VOFVPsch

RESET

DGND

DVDDIV

JIOUTPUT

CIV

LX88pF

L66

6IuH

R6K/XPX

L6/

LPXuH

MGND

RVU

DNC 6

VIN I

SHDN L

GND / GND V VOUT H GND K GND U

U6XLTCHHVI

cLPLV

MGND
148 

I L / V

DCBp

V / L I

D

CBp Title

Number Revision Size

B

DateU KyFebyL6II Sheet ofFileU EU\engineering jobs\PP\DpC_CpRD_/OF8PschDrawn ByU

DL_I8NDL_I8PDL_IVNDL_IVPDL_I/NDL_I/PDL_ILNDL_ILPDL_IINDL_IIPDL_I6NDL_I6PDL_6WNDL_6WP

DL_66PDL_66NDL_6IPDL_6INDL_6LPDL_6LNDL_6/PDL_6/NDL_6VPDL_6VNDL_6GPDL_6GNDL_6KPDL_6KN

DL_I8NDL_I8PDL_IVNDL_IVPDL_I/NDL_I/PDL_ILNDL_ILPDL_IINDL_IIPDL_I6NDL_I6PDL_6WNDL_6WPDL_6KNDL_6KPDL_6GNDL_6GP

DL_66PDL_66NDL_6IPDL_6INDL_6LPDL_6LNDL_6/PDL_6/NDL_6VPDL_6VN

IP I8WIO_L6IP_6 IG6IO_L6IN_6 IGIIO_L6LP_6 IGLIO_L6LN_69VREF_6 IG/IO_L6/P_6 IGVIO_L6/N_6 IG8IO_L6VP_6 IGKIO_L6VN_69VREF_6 IGHIP IGWIO_L68P_6 IKIIO_L68N_6 IKLIP_L6GP_6 IKVIP_L6GN_6 IK8VCCO_6 IKGIO_L6KP_69GCLKV IKKIO_L6GN_69GCLK8 IKHIO9VREF_6 IKWIO_L6HP_69GCLKG IH6IO_L6HN_69GCLKK IHI

IP_L6WP_69GCLKH IH/ IP_L6WN_69GCLKW IHV IO_LI6P_69GCLKI6 IH8 IO_LI6N_69GCLKII IHG IO IHK

IO_LIIP_6 IHW IO_LIIN_6 IW6 VCCO_6 IWI IO_LILP_6 IWL IO_LILN_69VREF_6 IW/ IP IWVIO_LI/P_6 IWG IO_LI/N_6 IWKIO_LIVP_6 IWW IO_LIVN_69VREF_6 L66 VCCO_6 L6I IO_LI8P_6 L6L IO_LI8N_6 L6/ IP L6V IO_LIGP_6 L68 IO_LIGN_69HSWpP L6G

UIDBpNK6XC/SL869866E PQL6H

CLKVDD IREXT LCLKVDD /CLKGND VCLKc 8CLKy GCLKGND KDGND HDVDD WDBI8c I6DBI8y IIDBIVc ILDBIVy I/DBI/c IVDBI/y I8DBILc IGDBILy IKDBIIc IHDBIIy IWDBVDD L6DBGND LIDBI6c LLDBI6y L/DBWc LVDBWy L8DBHc LGDBHy LKDCLK_OUTc LHDCLK_OUTy LWDBVDD /6DBGND /IDCLK_INc /LDCLK_INy //DBKc /VDBKy /8DBGc /GDBGy /KDB8c /HDB8y /WDBVDD V6 DBGND VI DBVc VL DBVy V/ DB/c VV DB/y V8 DBLc VG DBLy VK DBIc VH DBIy VW DB6c 86 DB6y 8I DVDD 8L DGND 8/ SDO 8V SDIO 88 SCLK 8G CSB 8K RESET 8H REFIO 8W FSpDJ G6 SDR_EN GI pDVDD GL pDGND G/ pCVDD GV pCGND G8 pVDDL GG pGNDL GK pVDDI GH pGNDI GW IOUTB K6 IOUTp KI pGNDI KL pVDDI K/ pGNDL KV pVDDL K8 pCGND KG pCVDD KK pDGND KH pDVDD KW SPI_DIS H6

UGpDWKLG

DL_CLKc

DL_CLKy

DL_CLK_INc

DL_CLK_INy

INy I

DGND L

yVs /

p6 V pI 8 WR G

OUT K

cVs H

REF W

INc I6

LOGIC

UWpDHL86

RVLVWPW 6PIg

RV/VWPW 6PIg

pGND

CVG6P6IuF

RVKLP6k 6PIg

CVH6P6IuF

J/OUTPUT

pGND

pGND

cI8

yI8

yI8

cI/

pGND

pGND

DGND

DVDDL8

DGND

DVDDL8

CVW6P6IuF

DGND

C866P6IuF

DBVDD//

C8I6P6IuF

DBVDD//

C8L6P6IuF

C8/6P6IuF

C8V

6P6IuF

C88

6P6IuF

C8G

6P6IuF

C8K

6P6IuF

C8H

6P6IuF

C8W

6P6IuF

CG6

6P6IuF

CGI

6P6IuF

c/P/VpGNDcLP8V

cLP8Vc/P/V

pGNDpGND

DBVDD//

DGNDDGND

CGL6P6IuF

CG/6P6IuF

pGND

CGV CG8

6P6IuF

CGG DGND

VCCpUXL8

VCCpUXL8

VCCpUXL8VCCpUXL8

pGND

CGK6P6IuF

DGND

CI6VI6uF

DBVDD//CLKVDDL8DVDDL8VCCpUXL8

DBVDD//CLKVDDL8DVDDL8VCCpUXL8

POWERDpC_CpRD_8OF8Psch

RESET

DVDDL8

DGND

GO_IGO_L

DL_68P

DL_68N

DL_6HP

DL_6HN

RLLIk

DGND

CGW

/W66pF

LI/

ILuH

R8KVWPW

LIG

/PWuH

pGND

RG6

c/P/V

DNC I

VIN L

SHDN /

GND V GND 8 VOUT G GND K GND H

ULILTCGG8L

pGND
149 

6 I L /

DCBp

/ L I 6

D

CBp Title

Number Revision Size

B

DateQ KyFebyI866 Sheet ofFileQ EQ\engineering jobs\PP\DpC_CpRD_/OFVPschDrawn ByQ

D6_6VND6_6VPD6_6/ND6_6/PD6_6LND6_6LPD6_6IND6_6IPD6_66ND6_66PD6_68ND6_68PD6_8XND6_8XPD6_8UND6_8UP

D6_88PD6_88ND6_86PD6_86ND6_8IPD6_8IND6_8LPD6_8LND6_8/PD6_8/ND6_8VPD6_8VND6_8HPD6_8HND6_8KPD6_8KN

D6_6VND6_6VPD6_6/ND6_6/PD6_6LND6_6LPD6_6IND6_6IPD6_66ND6_66PD6_68ND6_68PD6_8XND6_8XPD6_8UND6_8UPD6_8KND6_8KPD6_8HND6_8HPD6_8VND6_8VP

D6_88PD6_88ND6_86PD6_86ND6_8IPD6_8IND6_8LPD6_8LND6_8/PD6_8/N

IO_L86P_69p6H 68HIO_L86N_69p6V 68KIO_L8IP_69p6/ 68UIO_L8IN_69p6L 68XIP 668IO_L8LP_6 66IIO_L8LN_69VREF_6 66LVCCO_6 66/IO_L8/P_6 66VIO_L8/N_6 66HIP 66UIO_L8VP_69p6I 66XIO_L8VN_69p66 6I8IO_L8HP_6 6IIIO_L8HN_69VREF_6 6ILIP 6I/VCCO_6 6IVIO_L8KP_69RHCLK 6IHIO_L8KN_69RHCLK6 6IKIO_L8UP_69RHCLKI 6IUIO_L8UN_69RHCLKL 6IXIP 6L8 IO_L8XP_69RHCLK/ 6LI IO_L8XN_69RHCLKV 6LL IO_L68P_69RHCLKH 6L/ IO_L68N_69RHCLKK 6LV IP9VREF_6 6LH IO_L66P_69pI 6LK IO_L66N_69p6 6LU IO_L6IP_6 6LX IO_L6IN_69p8 6/8

IP 6/I VCCO_6 6/L IO_L6LP_6 6// IO_L6LN_6 6/V IO_L6/P_6 6/H IO_L6/N_6 6/K IP 6/U

IO_L6VP_69HDC 6V8 IO_L6VN_69LDC8 6V6 IO_L6HP_69LDC6 6VI IO_L6HN_9LDCI 6VL IP 6V/

U6CBpNK6XCLSIV89V88E PQI8U

CLKVDD 6REXT ICLKVDD LCLKGND /CLKc VCLKy HCLKGND KDGND UDVDD XDB6Vc 68DB6Vy 66DB6/c 6IDB6/y 6LDB6Lc 6/DB6Ly 6VDB6Ic 6HDB6Iy 6KDB66c 6UDB66y 6XDBVDD I8DBGND I6DB68c IIDB68y ILDBXc I/DBXy IVDBUc IHDBUy IKDCLK_OUTc IUDCLK_OUTy IXDBVDD L8DBGND L6DCLK_INc LIDCLK_INy LLDBKc L/DBKy LVDBHc LHDBHy LKDBVc LUDBVy LXDBVDD /8 DBGND /6 DB/c /I DB/y /L DBLc // DBLy /V DBIc /H DBIy /K DB6c /U DB6y /X DB8c V8 DB8y V6 DVDD VI DGND VL SDO V/ SDIO VV SCLK VH CSB VK RESET VU REFIO VX FSpDJ H8 SDR_EN H6 pDVDD HI pDGND HL pCVDD H/ pCGND HV pVDDI HH pGNDI HK pVDD6 HU pGND6 HX IOUTB K8 IOUTp K6 pGND6 KI pVDD6 KL pGNDI K/ pVDDI KV pCGND KH pCVDD KK pDGND KU pDVDD KX SPI_DIS U8

UKpDXKIH

D6_CLKc

D6_CLKy

D6_CLK_INc

D6_CLK_INy

INy 6

DGND I

yVs L

p8 / p6 V WR H

OUT K

cVs U

REF X

INc 68

LOGIC

U68pDUIV8

R///XPX 8P6g

R/V/XPX 8P6g

pGND

CHU8P86uF

R/U

IP8k 8P6g

CK88P86uF

pGND

pGND

c6V

y6V

y6V

c6L

pGND

pGND

DGND

DVDDIV

DVDDIV

CK68P86uF

DGND

CKI8P86uF

DBVDDLL

CKL8P86uF

DBVDDLL

CK/8P86uF

CKV8P86uF

pGND

CKH

8P86uF

CKK

8P86uF

CKU

8P86uF

CKX

8P86uF

CU8

8P86uF

CU6

8P86uF

CUI

8P86uF

CUL

8P86uF

cLPLVpGNDcIPVV

cIPVVcLPLV

pGNDpGND

DBVDDLL

DGNDDGND

CU/8P86uF

CUV8P86uF

pGND

CUH CUK

8P86uF

CUU DGND

VCCpUXIV

VCCpUXIVVCCpUXIV

VCCpUXIV

DGND

CUX

8P86uF

DGND

C68V68uF

DBVDDLLCLKVDDIVDVDDIVVCCpUXIV

DBVDDLLCLKVDDIVDVDDIVVCCpUXIV

POWERDpC_CpRD_VOFVPsch

RESET

DVDDIV

DGND

F6FIFL

J/OUTPUT

C/K

LX88pF

L6I

6IuH

RVH/XPX

L6V

LPXuH

pGND

RVX

cLPLV

DNC 6

VIN I

SHDN L

GND / GND V VOUT H GND K GND U

UI8LTCHHVI

pGND
150 

1 2 3 4

DCBA

4 3 2 1

D

CBA

Title

Number Revision Size

B

Date: 7-Feb-2011 Sheet of File: E:\engineering jobs\..\DAC_CARD_5OF5.schDrawn By:

EN 1IN 2GND 3

OUT 4

ADJ 5

U1129302 DGND

C10610uF

EN 1IN 2GND 3

OUT 4

ADJ 5

U1229302 DGND

C10710uF

21

JP5+5 D

DGND

R491.6k

R24 1k

C90

0.01uF

C10810uF

R251k

DGND

R261k

R27 1k

C91

0.01uF

C10910uF

R281k

DGND

DVDD25_INVCCINT12_IN

DBVDD33_IN

12345

JP6AUX D

DGND

+5D

21

JP7+5 A

AGND

EN 1IN 2GND 3

OUT 4

ADJ 5

U1329302

C11010uF

EN 1IN 2GND 3

OUT 4

ADJ 5

U1429302

C11110uF

R501.6kR291k

C92

0.01uF

C11210uF

R301k

R311kR321k

C93

0.01uF

C11310uF

R331k

AGNDAGND

AGNDAGND

+3.3V_IN+2.5V_IN

+3.3V+2.5V 1

23

JP8AMP

EN 1IN 2GND 3

OUT 4

ADJ 5

U1529302

C11410uF

R341k

R35 1k

C94

0.01uF

C11510uF

R361k

DGND

DGND

VCCAUX25_IN

EN 1IN 2GND 3

OUT 4

ADJ 5

U1629302 DGND

C11610uF

C95

0.01uF

C11710uF

R371k

DGND

L1

L2

C11910uF

C11810uF

1 23 45 67 8

JP9DAC D

1 23 45 67 8

JP10FPGA

DBVDD33_IN_AUXDVDD25_IN_AUX

DVDD25_IN_AUXDBVDD33_IN_AUX

VCCINT12_IN_AUXVCCAUX25_IN_AUX

VCCAUX25_IN_AUXVCCINT12_IN_AUX C120

10uF

C12110uF

DGND

L3

L4

AGND

L5

1 23 45 67 8

JP11DAC A

+3.3V_IN_AUX+2.5V_IN_AUX

123

JP12AUX A

+2.5V_IN_AUX+3.3V_IN_AUX

C12210uF

C12310uF

L6L7

C124

10uF

C125

10uF

AGND

L8

L9

+15-15

DBVDD33

DVDD25

CLKVDD25

TP2

VCCAUX25

VCCINT12

TP4TP3

TP5TP6TP7TP8

TP10TP11

1 23 45 67 89 10

11 1213 1415 1617 1819 2021 2223 2425 2627 2829 3031 3233 3435 3637 3839 4041 4243 4445 4647 4849 5051 5253 5455 5657 5859 60

JP4A99203-ND

D0D1D2D3D4D5D6D7

GCLKGO_1GO_2

D0DGNDD1D2D3D4D5D6D7GCLKGO_1GO_2

DGND

DGND DGNDDGND DGNDDGND DGNDDGND DGNDDGND DGNDDGND DGNDDGND DGNDDGND DGNDDGND DGNDDGND DGNDDGND DGNDDGND DGNDDGND DGND

bottom

D0D1D2D3D4D5D6D7

GCLK

GO_1GO_2

F4

+5A

F4

D313V

R5510k

+13

AGND

F3F2F1

F3F2F1

TP13

Alternative to +13 regulator circuit