ABSTRACT
Title of dissertation: MULTI-SPECIES TRAPPED ATOMIC ION
MODULES FOR QUANTUM NETWORKS
Ismail Volkan Inlek, Doctor of Philosophy, 2016
Dissertation directed by: Professor Christopher Monroe
Joint Quantum Institute
University of Maryland Department of Physics
and
National Institute of Standards and Technology
Trapped atomic ions are among leading platforms in quantum information
processing with their long coherence times and high fidelity quantum operations.
Scaling up to larger numbers of qubits is a remaining major challenge. A network of
trapped ion modules offers a promising solution by keeping a manageable number of
qubits within a module while photonic interfaces connect separate modules together
to increase the number of controlled memory qubits. Since the generation of entan-
glement between qubits in different modules is probabilistic, an excessive number of
connection trials might result in decoherence on the memory qubits through absorp-
tion of stray photons. This crosstalk issue could be circumvented by introducing a
different atomic species as photonic qubits. Compared to a system that only uti-
lizes single species of atoms, there are also additional advantages in a multi-species
apparatus where attractive features of each atom can be employed for certain tasks.
In this thesis, I present experimental demonstrations of necessary ingredients of a
multi-species module for quantum networking. In these experiments, 138Ba+ ions
are intended to be used as photonic communication qubits with visible photon emis-
sion lines that are more convenient for current fiber optics and detector technologies
while 171Yb+ ions are used for storing and processing quantum information where
long coherence times available in hyperfine clock states make them suitable memory
qubits. The key experiments include demonstration of atom–photon entanglement
using the 138Ba+ qubit and utilizing the Coulomb interaction between 171Yb+ and
138Ba+ with spin-dependent forces for transfer of information from communication
to memory qubits.
MULTI-SPECIES TRAPPED ATOMIC ION
MODULES FOR QUANTUM NETWORKS
by
Ismail Volkan Inlek
Dissertation submitted to the Faculty of the Graduate School of the
University of Maryland, College Park in partial fulfillment
of the requirements for the degree of
Doctor of Philosophy
2016
Advisory Committee:
Professor Christopher Monroe, Chair/Advisor
Professor Alexey Gorshkov
Professor Mohammad Hafezi
Professor Alan Migdall
Professor Edo Waks
c© Copyright by
Ismail Volkan Inlek
2016
Dedication
To Erin
ii
Acknowledgments
Graduate school has been a challenging yet at the same time rewarding, ex-
citing and fun experience for me. First of all, I want to thank my advisor Chris
Monroe for making this possible. I am very fortunate to have had the chance to
benefit from Chris’ expertise and intuition. Lessons I learned through my errors
and successes in conjunction with his guidance has been an invaluable part of my
graduate school years.
In the Monroe group, I had the opportunity to work closely with very talented
people who were instrumental on achieving the results presented on this thesis.
I want to thank Dave Hayes, Kenny Lee, Susan Clark, Shantanu Debnath, Dave
Hucul, Grahame Vittorini, Clay Crocker, Ksenia Sosnova and Marty Lichtman for
their tireless work.
Even though I only briefly overlapped with Hayes, I was grateful to benefit
from his deep understanding of physics in that short period of time, especially during
the late night data acquisition sessions when we did not have improved entanglement
rates yet. Similarly, Kenny leaved shortly after I joined the group but it was always
fun listening to Coldplay with him in the lab and he was the person I learned how
to properly build electronics. Although Shantanu has been reassigned to another
project by the time I have joined the group, I inherited his ground work on FPGAs
and built up on it, which has been a critical part on achieving the results published
over the years. He has been the person I went when my code was not working
properly for one reason or other. Susan did a tremendous job as the postdoc in
iii
our project on re-setting up remote entanglement apparatus. Hucul was the senior
grad student in the ion-photon project for the most part, and I’m incredibly happy
he was there to teach me how to conduct an experiment and how to troubleshoot
problems when things do not work as expected. He has truly been a role model
for me. Grahame was by far the most interesting person I have interacted with,
in a good way. The lab environment has been a very enjoyable place to work
with him and through his expertise in ion trapping, we tackled many problems.
I also thank him for founding the “National Turkish-American Friendship Portrait
Gallery” and hopefully he will continue commissioning my art projects in the future.
Clay has been very helpful as the junior grad student on the experiments, especially
on setting things up for the barium atoms. His contribution to results presented
on this thesis has been significant. After Hucul and Grahame’s departure, Ksenia
and Marty become part of the project. Ksenia’s theoretical work has illuminated us
on characterizing barium ions and she has been improving rapidly on experimental
techniques since. Marty joined us in my last year and helped us getting these
results as well. I believe his expertise in optimization processes and automation will
be crucial in taking research efforts to next level. I also want to thank all the other
past and present group members, who were always there to lend an advice when I
needed it.
I want to thank my family who were very supportive of my Ph.D. pursuit even
when it meant I’ll be on the other side of the world. And finally, without my wife
Erin’s support and love, this work would not have been possible.
iv
Table of Contents
List of Tables vii
List of Figures viii
List of Abbreviations x
1 Introduction 1
1.1 Modular Quantum Networks . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Mullti-Species Modules . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2.1 Entanglement of Remote Qubits . . . . . . . . . . . . . . . . . 6
1.2.2 Entanglement Generation Probability . . . . . . . . . . . . . . 12
1.2.3 Crosstalk Considerations . . . . . . . . . . . . . . . . . . . . . 13
1.2.4 Additional Advantages of a Multi-Species Setup . . . . . . . . 18
1.2.5 Multi-Species Network Architecture . . . . . . . . . . . . . . . 18
1.3 Previous Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2 Experimental Setup 21
2.1 RF Paul Traps and Vacuum Chamber . . . . . . . . . . . . . . . . . . 21
2.2 Ytterbium and Barium Atomic Ions . . . . . . . . . . . . . . . . . . . 22
2.2.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.2.2 Neutral Sources and Photo-ionization . . . . . . . . . . . . . . 25
2.2.3 Methods for Getting Fluorescence Photons from Barium Ions . 28
2.2.3.1 Co-trapping for Verification of 138Ba+ Loading . . . . 28
2.2.3.2 138Ba+ Transition Wavelength Measurements with
HCL . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.2.4 The Optical Bloch Equations . . . . . . . . . . . . . . . . . . 30
2.2.4.1 Three level system . . . . . . . . . . . . . . . . . . . 33
2.2.4.2 Eight level system . . . . . . . . . . . . . . . . . . . 39
2.3 Double Imaging Setup . . . . . . . . . . . . . . . . . . . . . . . . . . 40
2.4 High Numerical Aperture Setup . . . . . . . . . . . . . . . . . . . . . 41
2.5 Re-ordering of the Ions . . . . . . . . . . . . . . . . . . . . . . . . . . 44
v
3 Ytterbium and Barium Atomic Ion Qubits 46
3.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
3.2 Ytterbium Hyperfine Qubit . . . . . . . . . . . . . . . . . . . . . . . 47
3.3 Barium Zeeman Qubit . . . . . . . . . . . . . . . . . . . . . . . . . . 50
3.3.1 Doppler Cooling . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3.3.2 State Initialization . . . . . . . . . . . . . . . . . . . . . . . . 52
3.3.3 State Detection without Shelving . . . . . . . . . . . . . . . . 54
3.3.4 Laser Stabilization . . . . . . . . . . . . . . . . . . . . . . . . 59
3.3.5 Magnetic Field Stabilization . . . . . . . . . . . . . . . . . . . 59
3.3.6 Atom-Photon Entanglement . . . . . . . . . . . . . . . . . . . 63
3.3.6.1 Electric Dipole Transition with Fast Pulses . . . . . 66
3.4 Coherent Rotations with 355-532 nm Pulses . . . . . . . . . . . . . . 68
3.4.1 The Rabi Frequencies . . . . . . . . . . . . . . . . . . . . . . . 73
3.4.2 Differential Stark Shifts . . . . . . . . . . . . . . . . . . . . . 75
3.4.3 Re-pumping from 2D5/2 Level with an Orange LED . . . . . . 76
3.4.4 Crosstalk Between the Qubits and Raman Beams . . . . . . . 80
4 Normal Mode Structure of Ytterbium-Barium Ion Chain 82
4.1 Normal Mode Eigenfrequencies and Eigenvectors . . . . . . . . . . . . 82
4.1.1 Longitudinal Modes . . . . . . . . . . . . . . . . . . . . . . . . 82
4.1.2 Transverse Modes . . . . . . . . . . . . . . . . . . . . . . . . . 86
4.2 Spectroscopy Measurements . . . . . . . . . . . . . . . . . . . . . . . 88
4.3 Sub-Doppler Cooling with Electromagnetically Induced Transparency 90
5 Phonon Mediated Interactions between Ytterbium and Barium Ions 100
5.1 Cirac-Zoller Interactions . . . . . . . . . . . . . . . . . . . . . . . . . 100
5.1.1 Coherent Information Transfer Operation . . . . . . . . . . . . 101
5.1.2 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . 102
5.2 Mølmer-Sørensen Interactions . . . . . . . . . . . . . . . . . . . . . . 105
5.2.1 Spin-Dependent Forces . . . . . . . . . . . . . . . . . . . . . . 106
5.2.2 Multi-Species Entanglement . . . . . . . . . . . . . . . . . . . 110
5.2.2.1 Experimental Considerations and Results . . . . . . 112
5.3 The Swap Gate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
6 Outlook 120
6.1 Entanglement of Remote Barium Qubits . . . . . . . . . . . . . . . . 120
6.2 Maintaining Phase Stability in a Multi-Species Network . . . . . . . . 124
Bibliography 126
vi
List of Tables
1.1 Overhead considerations in modular architectures . . . . . . . . . . . 5
3.1 Atomic properties of excited states . . . . . . . . . . . . . . . . . . . 74
vii
List of Figures
1.1 Modular Trapped Atomic Ion Network . . . . . . . . . . . . . . . . . 4
1.2 Photons Incident on a 50/50 Beam Splitter and Atom-Photon Entan-
glement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.3 Photonic Bell State Measurement . . . . . . . . . . . . . . . . . . . . 11
1.4 Crosstalk Between Memory and Communication Quibts . . . . . . . . 14
1.5 Distance Scaling of Crosstalk Mechanisms . . . . . . . . . . . . . . . 17
1.6 Multi-Species Network Architecture . . . . . . . . . . . . . . . . . . . 19
2.1 RF Paul Traps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.2 Photoionization of Neutral Barium . . . . . . . . . . . . . . . . . . . 27
2.3 Optogalvanic Effect and Transition Wavelength Measurements . . . . 31
2.4 138Ba+ Energy Levels . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.5 Simplified Three Level Λ System of 138Ba+ ion . . . . . . . . . . . . . 34
2.6 Three Level Λ System Time Dynamics . . . . . . . . . . . . . . . . . 38
2.7 Excitation Spectrum of Three Level Λ System . . . . . . . . . . . . . 39
2.8 Eight Level Λ System Time Dynamics . . . . . . . . . . . . . . . . . 40
2.9 Excitation Spectrum of Eight Level Λ System . . . . . . . . . . . . . 41
2.10 Double Imaging Setup . . . . . . . . . . . . . . . . . . . . . . . . . . 42
2.11 High Numerical Aperture Setup . . . . . . . . . . . . . . . . . . . . . 43
3.1 171Yb+ Clebsch-Gordan Coefficients . . . . . . . . . . . . . . . . . . . 47
3.2 171Yb+ Qubit Operations . . . . . . . . . . . . . . . . . . . . . . . . . 48
3.3 138Ba+ Clebsch-Gordan Coefficients . . . . . . . . . . . . . . . . . . . 50
3.4 Doppler Cooling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3.5 The 138Ba+ Qubit State Initialization . . . . . . . . . . . . . . . . . . 53
3.6 The 138Ba+ Qubit State Detection . . . . . . . . . . . . . . . . . . . . 56
3.7 State Detection/Initialization Optical Setup . . . . . . . . . . . . . . 57
3.8 60 Hz Magnetic Field Noise . . . . . . . . . . . . . . . . . . . . . . . 61
3.9 Atom-Photon Entanglement . . . . . . . . . . . . . . . . . . . . . . . 64
3.10 Excitation with Short Pulses . . . . . . . . . . . . . . . . . . . . . . . 67
3.11 Magnetic Dipole Transitions . . . . . . . . . . . . . . . . . . . . . . . 68
3.12 Two Photon Stimulated Raman Transitions . . . . . . . . . . . . . . 69
3.13 Stimulated Raman Transitions on 138Ba+ and 171Yb+ . . . . . . . . . 74
viii
3.14 Differential Stark Shift on 138Ba+ . . . . . . . . . . . . . . . . . . . . 77
3.15 Deshelving From 2D5/2 Levels with an Orange LED . . . . . . . . . . 79
4.1 Two Ion Longitudinal Mode Motional Amplitudes . . . . . . . . . . . 85
4.2 Two Ion Longitudinal Mode Frequencies . . . . . . . . . . . . . . . . 85
4.3 Two Ion Transverse Mode Motional Amplitudes . . . . . . . . . . . . 87
4.4 Two Ion Transverse Mode Frequencies . . . . . . . . . . . . . . . . . 87
4.5 Two Ion Transverse Mode Structure Dependence on Trap Parameters 89
4.6 Non-Copropagating Beam Geometry . . . . . . . . . . . . . . . . . . 90
4.7 Motional Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
4.8 EIT Cooling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
4.9 EIT Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
4.10 Motional Spectrum with EIT Cooling . . . . . . . . . . . . . . . . . . 98
5.1 State Transfer with CZ interactions . . . . . . . . . . . . . . . . . . . 103
5.2 Coherence Measurement of CZ interactions . . . . . . . . . . . . . . . 104
5.3 Single Qubit Mølmer-Sørensen Interaction . . . . . . . . . . . . . . . 109
5.4 Two Qubit Mølmer-Sørensen Interaction . . . . . . . . . . . . . . . . 112
5.5 The Raman Beam Configuration for 171Yb+ – 138Ba+ MS Interaction 113
5.6 Schematic of Control Electronics . . . . . . . . . . . . . . . . . . . . . 115
5.7 Multi-Species MS Interaction . . . . . . . . . . . . . . . . . . . . . . 118
6.1 Optical Pumping with a Beam Along Photon Collection Axis . . . . . 122
6.2 Entanglement of Remote 138Ba+ Qubits . . . . . . . . . . . . . . . . . 123
6.3 Phase Stability on a Multi-Species Module . . . . . . . . . . . . . . . 125
ix
List of Abbreviations
AOM Acousto-Optic Modulator
APD Avalanche Photo Diode
CZ Cirac-Zoller
EIT Electromagnetically Induced Transparency
EOM Electro-Optic Modulator
HWP Half Waveplate
HCL Hollow Cathode Lamp
IP In-Phase
MS Mølmer-Sørensen
NA Numerical Aperture
NV Nitrogen-Vacancy
OP Out-of-Phase
PBS Polarizing Beam Splitter
PMF Polarization Maintaining Fiber
PMT Photo Multiplier Tube
QWP Quarter waveplate
RF Radio Frequency
SPAD Single Photon Avalanche Diode
SNSPD Superconducting Nanowire Single Photon Detector
UV Ultraviolet
x
Chapter 1: Introduction
1.1 Modular Quantum Networks
Scaling up the number of qubits while maintaining quantum coherence in the
system is the current challenge in quantum information processing platforms [1–4].
Trapped ions are currently the leading candidate in this front for two major reasons.
First, since they are nature’s qubit, they are identical with each other. Using qubits
that only weakly interact with the environment, one can make use of frequencies
that are very close to each other, which is helpful in building a large scale system.
For instance, the hyperfine clock states of 171Yb+ are separated by [5]:
ω0/2pi = 12 642 812 118.5 + (310.8)B
2 Hz (1.1)
where B is the magnetic field in Gauss. Around 1 G, this corresponds to a sensitivity
of 0.6216 Hz/mG. Using integrated sensors in proximity to the ions, magnetic fields
can be stabilized to much better than 1 mG [6]. Consequently, it is practical to make
use of qubit frequencies that are within less than 1 Hz, an otherwise challenging task.
For fabricated qubits, such as superconducting circuits, NV centers or quantum
dots, the fact that current fabrication methods do not produce the exact same qubit
introduces additional complexity [7, 8]. While this is not a fundamental limitation,
1
relative phase evolution of the qubits needs to be kept track of in these platforms
which becomes increasingly difficult as the system size grows. For instance, the
two-qubit entangled state
|ψ〉en =
1√
2
(|0〉a|1〉b + ei∆abt|1〉a|0〉b) (1.2)
evolves with the frequency difference between the participating qubits “a” and “b”,
∆ab = ωa − ωb. As such, for a system of size N , there are O(N2) phases that need
to be monitored. More importantly, if the qubit frequencies are subject to fluctua-
tions due to environmental effects, these phase calibration requirements might even
prevent maintaining overall coherence properties of the whole system. Hence, the
indistinguishability and high levels of isolation of trapped ions from the environment
gives them an edge on this aspect of scalability.
Second, a straightforward approach of putting more qubits in close proximity
to each other is likely to fail [1] which necessitates a more modular approach for scal-
ing, similar to classical computing architectures, and trapped ions have shown more
promising results so far. Each platform has some limitation on the number of con-
trollable qubits achievable within a module. In NV centers, each electron spin has
limited number of neighbor nuclear spins that can be used as qubits [4]. Production
and calibration of small size superconducting circuit modules that eventually con-
nect to each other might be an easier scaling approach than manufacturing a single
chip with many qubits and the required connections. And with trapped atomic ions,
simply adding more qubits to the crystal chain is hampered by the increased number
of interactions between the qubits that need to be reliably controlled [9]. Also, this
2
size increase results in slower multi-qubit entangling operations, making them more
susceptible to decoherence mechanisms [10]. Increased likelihood of background gas
collisions and the resulting destabilization of the ion chain currently limits man-
ageable chains sizes to about 20 [11], and cryogenic traps are being considered to
minimize these collisions to push the number close to 100. For these reasons, a
modular approach is the way forward [12,13]; a small programmable quantum com-
puter has recently been demonstrated with trapped atomic ions [14], and such small
units can be connected together with re-configurable photonic interfaces to create a
network architecture (see Fig. 1.1). In this approach, quantum information is pro-
cessed at memory qubits while modules are connected by probabilistic entanglement
of communication qubits via photon interference [15].
There have been various experiments on different platforms that utilize optical
or microwave photons to entangle qubits at a distance [7, 13, 16, 17]. One impor-
tant parameter in scaling and entanglement propagation in the modular network
architecture is the ratio between the average entanglement generation time and the
entangled state coherence time. Trapped atomic ions is the only platform to date
that has achieved a ratio less than one, and it is three orders of magnitude smaller
than the closest platform (see Table. 1.1). As this ratio gets bigger, having multiple
modules connected at the same time in the network gets exponentially harder to
achieve and might result in unrealistic overhead in the resources required.
These two aspects, the indistinguishability of qubits and the fast generation
of entanglement between them compared to the coherence time, make trapped ions
attractive for use in a modular network architecture.
3
Figure 1.1: In this network architecture, each module holds a manageable number
of trapped ions. Entanglement within a module is generated using local Coulomb
interactions. Specific ions are assigned as communication qubits within a module;
scattered photons from these ions are collected into a N × N optical cross-connect
switch to link arbitrary modules with each other. Entanglement between commu-
nication qubits is heralded by a Bell state measurement on the collected photons
using beam splitters and photon detectors.
4
Platform entanglement coherence Ratio Overhead
generation (sec) time (sec) (x) (xx)
Trapped Ions [13] 0.22 1.12 0.2 0.71
Neutral Atoms [17] 0.1 2× 10−4 500 101350
Superconducting Circuits [7] 0.005 6× 10−6 800 102300
NV centers [16] 250 0.01 25000 10100000
Table 1.1: Entanglement generation time between qubits that are linked together
with photonic interfaces and the coherence time of this entangled state. The ratio
between these two numbers (x) is a big indication of how scalable a platform is
since the probability of successfully connecting multiple modules at the same time
gets lower as x grows. In a modular network, the overhead in resources scale as a
super-exponential (xx) for realizing fault tolerant operations [12].
Another promising approach to scaling is through quantum charge-coupled
devices where trapped ions are shuttled between trapping zones to bring together
short chains of ions for quantum algorithms [18]. Still, complexity of controlling the
hardware and optical requirements might limit the size of the system [12].
1.2 Mullti-Species Modules
Entanglement of remote qubits is the key ingredient in the construction of a
modular network. This operation is mediated through the quantum interference of
two photons on a beam splitter, and in this section I revisit the underlying theory.
The understanding of this mechanism is helpful in characterizing the crosstalk issues
related to these entanglement operations in a network architecture, which can be
5
eliminated by going to a multi-species apparatus.
1.2.1 Entanglement of Remote Qubits
The first demonstration of interference between two identical photons imping-
ing on a beam splitter was done by Hong, Ou and Mandel [19]. In this experiment,
identical photons incident on two input ports of a 50/50 non-polarizing beam splitter
were observed to bunch in either of the output ports. Since photon number resolu-
tion is difficult even with current detectors [20], this behavior was usually verified
by the absence of coincident detection of photons on separate output ports of the
beam splitter.
The operation of a 50/50 beam splitter can be expressed in terms of creation
operators [21]. First, a single photon in the nth port of the beamsplitter (see Fig.
1.2 (a)) can be obtained by applying operator a†n to the vacuum state; |1n〉 = a†n|0〉.
And, the effect of beam splitter on the photons is described by:
a†3 =
1√
2
(a†1 − a†2)
a†4 =
1√
2
(a†1 + a
†
2) (1.3)
where the sign difference is due to a low to high index of refraction change for light
going from port 2 to 3 [22]. Rewriting these equations:
a†1 =
1√
2
(a†3 + a
†
4)
a†2 =
1√
2
(a†4 − a†3) (1.4)
we can theoretically recreate the first interference experiment. For instance, when
6
two identical photons are incident on the input ports 1 and 2:
|1112〉 = a†1a†2|0〉
=
1
2
(a†3 + a
†
4)(a
†
4 − a†3)|0〉
=
1
2
(a†4a
†
4 − a†3a†3)|0〉
=
1√
2
(|0324〉 − |2304〉). (1.5)
With this simple analysis, it is evident that the photons bunch at either of the two
output ports of the beam splitter, and detectors placed at output ports 3 and 4
would not register at the same time assuming the incident photons are identical.
Another note of importance is that, here the photons are assumed to be in a single
state, or a single mode; hence only one set of creation operators is used. Identical
photons that are in a superposition of states do not necessarily exhibit this bunching
behavior as will be shown.
In principle, states that carry extra information can be distinguished from each
other in the lab. For example, perpendicular polarized photons can be separated
using a polarizing beam splitter, or cavities can be used to differentiate between
different frequency photons. Hence, introduction of another set of operators is nec-
essary if photons incident on a 50/50 beam splitter are in such a superposition.
We utilize these extra degrees of freedom for entanglement of remote qubits.
In this technique, a photon is first entangled with its parent atom [23]. Orthogonal
sets of horizontal and vertical polarization states {|H〉 = a†|0〉, |V 〉 = b†|0〉} are used
for transferring the information, resulting in a joint atom-photon state:
|ψ〉 = |↓ H〉+ |↑ V 〉 (1.6)
7
Port 2 Port 1
Port 4Port 3
50/50 
BS
2S1/2
2P1/2
(a) (b)
H
V
Figure 1.2: (a) A 50/50 non-polarizing beam splitter, with two input (1 and 2) and
two output (3 and 4) ports. The reflection happens inside the material from port
1 to 4 and in the air from port 2 to 3, introducing a pi phase shift to the field.
(b) Basic principle of generation a photon entangled with its parent atom. The
atoms are initialized to a pure state with optical pumping beams and another field
drives a dipole transition to excited level afterwards. Atom decays back to ground
state levels emitting a photon where the timescale is determined by the lifetime of
the excited level. The polarization state of this photon and the spin state of the
atom get entangled in the process. Capture and detection of these photons from
two separate qubits at the same trial cycle is required for entanglement generation
between them.
8
where I ignore normalization for simplicity in the rest of these calculations and the
procedure of creating this atom-photon entangled state is shown in Fig. 1.2 (b).
Then, the joint state of two such pairs is:
|ψ〉1 ⊗ |ψ〉2 = (|↓ H〉+ |↑ V 〉)1 ⊗ (|↓ H〉+ |↑ V 〉)2
= |↓↓ HH〉+ |↓↑ HV 〉+ |↑↓ V H〉+ |↑↑ V V 〉. (1.7)
Without loss of generality, we can rewrite this joint state in terms of entangled atom
and photon Bell states:
∣∣ψ+〉 = |HV 〉+ |V H〉 ∣∣ψ−〉 = |HV 〉 − |V H〉∣∣φ+〉 = |HH〉+ |V V 〉 ∣∣φ−〉 = |HH〉 − |V V 〉 (1.8)
for photons and similarly:
∣∣ψ+〉 = |↓↑〉+ |↑↓〉 ∣∣ψ−〉 = |↓↑〉 − |↑↓〉∣∣φ+〉 = |↓↓〉+ |↑↑〉 ∣∣φ−〉 = |↓↓〉 − |↑↑〉 (1.9)
for atoms. Then, the joint state in Eqn. 1.7 expressed with these Bell states is:
|ψ〉1 ⊗ |ψ〉2 =
∣∣ψ+〉
a
∣∣ψ+〉
p
+
∣∣ψ−〉
a
∣∣ψ−〉
p
+
∣∣φ+〉
a
∣∣φ+〉
p
+
∣∣φ−〉
a
∣∣φ−〉
p
(1.10)
where “a” and “p” subscripts denote atom and photon respectively. The effect of
beam splitter on photonic Bell states impinging from ports 1 and 2 can be seen
using Eqn. 1.4:
∣∣ψ+〉 = (a†1b†2 + b†1a†2)|0〉
= (a†4b
†
4 − a†3b†3)|0〉
= |03(HV )4〉 − |(HV )304〉 (1.11)
9
where we observe photon bunching of orthogonal polarization states in either output
ports with equal probability. In contrast, |ψ−〉 anti-bunches:
∣∣ψ−〉 = (a†1b†2 − b†1a†2)|0〉
= (a†3b
†
4 − b†3a†4)|0〉
= |H3V4〉 − |V3H4〉 (1.12)
and photons get separated to different ports. To complete the set of equations for
all four Bell states:
∣∣φ+〉 = (a†1a†2 + b†1b†2)|0〉
= (a†4a
†
4 − a†3a†3 + b†4b†4 − b†3b†3)|0〉
= |03(HH)4〉 − |(HH)304〉+ |03(V V )4〉 − |(V V )304〉∣∣φ−〉 = (a†1a†2 − b†1b†2)|0〉
= (a†4a
†
4 − a†3a†3 − b†4b†4 + b†3b†3)|0〉
= |03(HH)4〉 − |(HH)304〉 − |03(V V )4〉+ |(V V )304〉 (1.13)
resulting another bunching outcome, in which same polarization states emerge from
same output port.
As shown in Fig. 1.3, measurement on these photons can be carried out
using polarizing beam splitters (PBS) in conjunction with detectors, such as photo
multiplier tubes (PMT’s). Importantly, the outcomes for distinct impinging Bell
states give different coincident detection result, which allows us to do a Bell state
measurement on the joint photonic wave function. This in turn, projects the atomic
state of two qubits into an entangled state (see Eqn. 1.10). Only two out of the total
10
PBS
50/50 BS
PMT (H)(a)
(b)
PMT (V)
Figure 1.3: Polarizing beam splitters (PBS) can be used at the output ports of a
50/50 beam splitter to distinguish |H〉 and |V 〉 photons, where PMT(H)[PMT(V)]
detects |H〉[|V 〉]. (a) An incoming |ψ+〉 polarization state has two orthogonal polar-
ization states leaving the same output port of the beam splitter as can be seen from
Eqn. 1.11, which is later split apart by the PBS. Hence, coincident detection events
on the same output port side projects the atomic qubits to |ψ+〉. (b) Coincident
detection events involving different output ports with different polarization states
result in projection of atoms to |ψ−〉 (see. Eqn. 1.12). Incoming photonic |φ±〉
states leave from same output port with same polarization state (see. Eqn. 1.13),
hence these states are not misinterpreted as |ψ±〉
11
four Bell states can be measured with this setup, since even with photon number
resolving detectors, |φ+〉 and |φ−〉 cannot be distinguished from each other as they
only differ by a phase factor. However, it is still possible to distinguish all four Bell
states if photons are entangled in more than one degree of freedom (hyperentangled
state) [24]. The probabilistic nature does not affect the purity of entangled state,
as we post-select events where only two photons are detected, giving the commonly
used “heralded entanglement” name to this procedure [15].
1.2.2 Entanglement Generation Probability
In this heralded scheme, the successful generation of entanglement between
remote qubits depends on coincident collection and detection of photons from the
atoms. Hence, a large number of trials might be necessary until a successful event
happens. Considering 171Yb+ qubits [13], we can characterize this entanglement
probability looking at several factors that are involved:
PEnt =
1
2
(PbrPCG(
Ω
4pi
)PfcPqePloss)
2 (1.14)
where branching ratio (Pbr = 0.995) and Clebsch-Gordon coefficients (PCG =
2
3
)
amount to the generation probability of the single photon of interest after atom
excitations while collection solid angle ( Ω
4pi
= 0.1), fiber-coupling fraction (Pfc = 0.2),
photon detector quantum efficiency (Pqe = 0.35) and losses in the system (Ploss =
0.95) accounts for detection of these photons. The overall 1
2
factor comes from the
number of detectable Bell states, and the probability numbers are squared since
two photons from each atom needs to be collected and detected during the same
12
experimental trial. The largest number for PEnt ever achieved with trapped atomic
ions was the 10−5 [13] measured in our lab. Along with the ∼500 kHz experimental
repetition rate, an average entanglement generation rate of ∼5 s−1 was achieved.
Optimistically, this rate can be pushed to ∼500–5000 s−1 range with improvements
on fiber coupling and detector efficiencies.
1.2.3 Crosstalk Considerations
After looking at the basic principles of entanglement generation between re-
mote qubits and factors that determine number of trials required to create this
entangled state, we can now better understand the problems that might stem from
these operations in a modular network architecture.
In this setup (see Fig. 1.4), processing of quantum information and protection
of coherence on memory qubits while trying to entangle photonic qubits is required
for maximum utilization of resources. Hence, it is important to eliminate crosstalk
processes between them; excitation and optical pumping beams used in heralded
entanglement schemes or photons emitted by the communication qubits might be
absorbed by the memory atoms and result in corruption of the stored information.
In this section, the specific case of 171Yb+ qubits will be considered with hyperfine
clock states utilized for storing/processing information and a heralded entanglement
scheme relying on polarization encoding [13,25].
At a distance of r between the photonic qubit and the nearest memory qubit,
13
High Numerical Aperture Lens
Tr
ap
pe
d 
A
to
m
ic
 Io
n 
M
od
ul
e
Photonic Communication Qubit
Memory Qubits
To Fiber
σ
r
θ2
Figure 1.4: A trapped atomic ion module; the unit cell of network architecture.
Photons emitted from communication qubits are collected into a single-mode fiber
to be used in heralded entanglement of qubits in different modules. However, these
photons get emitted towards the memory qubits as well, and might be absorbed,
resulting in loss of information. Addressing only the photonic atoms with excitation
and optical pumping beams for heralded entanglement operation while not affecting
the memory qubits might be challenging as well. These crosstalk processes between
memory and communication qubits might compromise the quantum information
processing if they are close to each other.
14
the apex angle of the cone that covers the resonant absorption cross section is:
2θ(r) = 2Arctan(
√
3λ2/2pi2
r
) (1.15)
and, the ratio of the resulting solid angle to the total area is:
Ω
4pi
=
2pi
∫ θ
0
sin θ
′
dθ
′
4pi
=
1− cos[θ(r)]
2
. (1.16)
Probability of photon absorption by the memory qubit is proportional to this ratio.
However, there are two other factors that need to be included as well. First, the
polarization state and the frequency of the emitted photons affect how strongly they
can disturb the memory qubits, which amounts to a factor of ∼ 1
3
. Second, photons
are emitted from communication atoms with an exponential decaying temporal en-
velope [26], which could only result in a maximum excitation probability of 0.54 on
memory atoms from these photon wave packets [27]. Including these factors, the
crosstalk probability from these stray photons scale as:
Pct stray(r) ≈ 1
6
1− cos[θ(r)]
2
. (1.17)
Another crosstalk source is the individual beams needed to address the com-
munication qubit; the optical pumping beam for state initialization and the excita-
tion light to create single photons. To reduce these effects, the waist of the beam
addressing the communication qubit needs to be minimized. The intensity of the
light present at a distance r from photonic atom site can be found by assuming a
Gaussian beam shape of center intensity I0 and waist w0:
I(r) = I0 e
−2r2
w20 . (1.18)
15
The ratio of the photon scattering rate at the origin and at a distance r can be used
to calculate the crosstalk probability:
Pct beams(r) =
Γ(r)
Γ(0)
=
s(r)/(1 + s(r))
s(0)/(1 + s(0))
(1.19)
where s(r) = I(r)
Isat
and the beam is assumed to be on resonance. Setting I0 = Isat,
the excitation probability of an atom at distance r, while the atom at beam center
scatters a single photon is:
Pct beams(r) =
2e
−2r2
w20
1 + e
−2r2
w20
. (1.20)
Fig. 1.5 shows plots of two crosstalk mechanisms calculated in Eqn. 1.17 and 1.20.
Even with two orders of magnitude improvement on entanglement generation
probability, it is evident from Fig. 1.5(b) that, crosstalk might introduce consider-
able errors with typical atom separations of a few microns in trapped ion modules.
Introduction of a different atomic species as the communication qubit can elimi-
nate the crosstalk issues if the excitation and emitted light is far from any optical
resonance of the memory qubits [28]. This claim is verified in our multi-species
apparatus by showing that the coherence time of 171Yb+ qubit is not affected by a
nearby single photon source at a different resonance frequency or the laser field itself
that drives the dipole transition of the nearby atom. This experiment incorporated
a Ramsey measurement on 171Yb+ while Doppler cooling beams were incident on
the 138Ba+ atom to scatter many photons.
16
(a) (b) Average # photons absorbed 
after 10
5 trials
Pct_beams(r) with 2 µm waist
Pct_beams(r) with 5 µm waist
Pct_stray(r) 
2 µm waist
5 µm waist
Figure 1.5: (a) Probability of photon absorption versus atom spacing. For these
calculations, 171Yb+ atoms are considered. At close distances, the addressing beams
dominate the crosstalk effects. However, intensity at the tails of these Gaussian
beams quickly diminish and the stray photon contribution becomes the significant
factor. Curves for beams with 2 and 5 µm waists are shown. (b) Total probability
of photon absorption from these two mechanisms, PTot = Pct stray + Pct beams, versus
average number of photons absorbed at a distance r after 105 trials, p¯(r) = PTot(r)×
105 × 10. The factor of 105 is used here to match the number of trials it took to
entangle the remote qubits in our experiment [13]. The factor of 10 accounts for the
approximate number of photons that needed to be scattered form the communication
qubit for each trial. While we use single photons for entanglement generation, optical
pumping to a pure state introduces these extra photons. Single photon absorption
would result in total loss of information, hence the right axis can be considered as
crosstalk errors as well. As shown, to reduce the errors caused by crosstalk below
1% levels, the photonic atom needs to be separated from memory atoms by more
than 300 µm.
17
1.2.4 Additional Advantages of a Multi-Species Setup
In addition to solving the crosstalk problem, introducing a different species of
atom might enable extra features to a trapped ion module. The second species can
be used as refrigerant atoms to cool memory qubits [29] allowing for higher fidelity
quantum operations. Importantly, the coherence properties stored in the memory
qubits would not be affected by this operation, making it possible to execute more
quantum logic gates which might be necessary for realizing specific algorithms [14].
Attractive properties of each atom can be taken advantage of for certain tasks
as well. For example, atoms that are less susceptible to external noise sources might
be better suited as memory qubits with their long coherence times [30]. At the
same time, atoms that have strong photon emission lines in the visible or infra-
red bands would be more advantageous for use as photonic communication qubits,
benefiting from less attenuation on optical fibers to extend the networks to larger
distances [31] and higher quantum efficiency photon detectors [32] for increased
entanglement generation rates on modular architectures.
1.2.5 Multi-Species Network Architecture
The scaling approach with multi-species modules is shown in Fig. 1.6. After
entanglement of photonic link 138Ba+ qubits, the information is transfered to nearby
memory 171Yb+ qubits using the Coulomb interaction between them. This effectively
transfers the entanglement to memory qubits and increases the number of connected
processor atoms to scale up the system size. The operations required for realizing
18
Photonic Link: Ba+
  Memory qubits: Yb+
Module A Module B
Coulomb Int.
Photonic Bell 
State Analyzer
Figure 1.6: Different modules are connected by first entangling the 138Ba+ qubits
using flying photon qubits. After successful connection, the Coulomb interaction
between 171Yb+ and 138Ba+ atoms can be utilized for transferring information from
the photonic link to memory qubits.
this goal has been experimentally demonstrated and the results are shown later in
this thesis.
1.3 Previous Experiments
In the first three and a half years in the Monroe group, I have been a part of var-
ious experiments. The first experiment involved combining photonic and phononic
entanglement operations on a unit cell of modular architecture [13]. The aim on this
experiment was showing the validity of the modular scaling approach by combining
these different entanglement operations. Another important aspect was addressing
the requirement of faster timescale on creating entanglement than phase coherence
time of the entangled state.
Afterwards, we showed that phase coherence on this network architecture can
19
be maintained over space and time by using phase locking techniques [33], and this
is a critical requirement for scaling. Finally, we have shown how distinguishable
quantum memories can be entangled [34]. Even though trapped atomic ions can
be made identical with little effort, the techniques shown here can be beneficial on
other platforms where manufactured qubits might have non-identical frequencies.
Details on these experiments can be found on Refs. [13,33–35] and rather than
repeating the results and experimental considerations here, the focus of this thesis
will be on the multi-species apparatus.
20
Chapter 2: Experimental Setup
This thesis work has been conducted on an optical table with a long history
which produced many doctorate degrees. Hence, the details about certain aspects
of experimental setup are covered throughly in accompanying theses [35–37], where
only 171Yb+ qubits were employed. However, a major renovation took place in the
last two years to accommodate another species of atom. As such, in this chapter
I will briefly describe the setup and mainly focus on the details that came into
existence with inclusion of the second atomic species.
2.1 RF Paul Traps and Vacuum Chamber
We use RF Paul traps to hold onto positively charged atomic ions [38]. The ex-
perimental demonstrations done in this thesis work was realized in four-rod [36] and
segmented blade [35] traps (see Fig. 2.1). Even though these hand assembled traps
are primitive compared to fabricated traps with many electrical connections [39,40],
which are more suitable for complex shuttling procedures, allocating different zones
for various operations or shaping trapping potentials for equal spacing of ions, such
complexity was not required as experiments done so far only involved a maximum
number of two co-trapped ions. On the contrary, the hand assembled traps used
21
in this thesis work were better suited for the experiments conducted here. They
utilize two separate large optical access view-ports, which is convenient as one port
is mainly used for photon collection during heralded entanglement procedures, while
the other port is used for state detection of qubits and trapped ion imaging. Re-
cently, high optical access requirements started to be considered in fabricated trap
technologies, for example the Sandia HOA 2.0 trap [41], and will likely make hand
assembled traps obsolete in the near future.
To minimize the background gas collisions with the trapped ions, a high vac-
uum environment is needed. The procedures for reaching 10−11 Torr levels involve
baking the vacuum chamber enclosure and the RF Paul trap installed for several
days after initial cleaning procedures. These steps are investigated in greater detail
in Ref. [35,36].
2.2 Ytterbium and Barium Atomic Ions
2.2.1 Overview
As mentioned before, the past experiments conducted in my project used only
171Yb+ atomic ions as qubits. Other research projects in the Monroe group are
also exclusively utilizing 171Yb+ in their experiments [14, 42, 43]. 171Yb+ ions are
attractive in quantum information processing with long coherence times [5], conve-
nient high fidelity state detection schemes [44] and easy manipulation with external
fields [45]. However, UV photons emitted from the ions are not well suited for remote
entanglement schemes with current optical fiber and detector technologies. Addi-
22
RF
RF
DC
DC
Figure 2.1: The RF Paul traps used in this thesis work are constructed with four rod
(left) and segmented blade (right) structures. Four tungsten rods are held with an
alumina spacer with holes and two needles in the middle section provide confinement
in longitudinal axis with DC voltages. In the segmented blade trap, alumina plates
are laser machined and coated with gold. Segments allow application of different
DC voltages to shape the longitudinal confinement potential for changing the ion
spacings. In these traps, the typical applied RF field is ∼100-1000 V in the 10-40
MHz range where helical resonators are used to create these high voltage and high
frequency fields. Due to larger separations between the needles, the applied DC
voltages are considerably higher in four rod trap compared to segmented blade trap
to achieve similar axial confinement. Application of 200 V on both needles results
in a longitudinal mode frequency of ∼1 MHz in four rod trap while it only requires
∼10 V on the outermost segments of the blade trap to achieve this mode frequency.
23
tionally, due to issues and in view of advantages indicated in the previous chapter,
we have introduced another atomic species.
We decided on barium atomic ions as photonic communication qubits for sev-
eral reasons. First, 138Ba+ atoms have strong emission lines at 493 nm and 650 nm
wavelengths, deeper in the visible range compared to other available atomic species.
These photons can be sent longer distances with less attenuation in fibers for a
large scale distributed quantum networks. Single photon avalanche diodes (SPAD)
and superconducting nanowire single-photon detectors (SNSPD) are commercially
available and can reach up to 90% detection efficiencies at these wavelengths, in-
creasing the entanglement generation rate between remote qubits. Second, since
138Ba+ is close to 171Yb+ in mass, we can co-trap these ions with same RF field,
without resorting to application of multiple RF fields which might be necessary for
widely different massed species [46]. Lastly, the simple atomic structure of 138Ba+
would make photon generation schemes easier for entanglement operations between
different modules and sympathetic cooling of memory qubits for achieving longer
quantum information processing times. One disadvantage of these qubits is their
short coherence times as qubit states are encoded on the electron spin which is more
susceptible to magnetic field noise compared to hyperfine states. However, this does
not pose a big problem since entanglement between 138Ba+ qubits will be readily
transfered to memory qubits where this entanglement can be preserved for longer
periods. I will revisit these points more in detail later.
24
2.2.2 Neutral Sources and Photo-ionization
We load the ions into the trap by first heating a source of neutral atoms to
create a flux in the trapping region, and then photo-ionize them on the spot. For
ytterbium sources, a few milligrams of material is filled into a hypodermic needle
oven which is pointed towards the trapping region and a 3-4 A of current is passed
through to heat it up [36]. In contrast to ytterbium, barium rapidly oxidizes when
it is exposed to air, which makes putting a pure source into the vacuum chamber a
challenging task. A few different procedures were tried to minimize this oxidization.
Initially, a stainless steel tube filled with barium and closed with an indium seal
in an argon environment was tried as a source (purchased from Alvatec). The first
time we have tried this in a vacuum chamber, the indium seal was simply melted off
by running a small current (2 A), benefiting from a low melting point of 156 ◦C, and
barium source became available. However, following the same procedure resulted in
mixed results in subsequent trials for constructing another ion trap, ranging from
splattering of melted indium to shooting it off as a chunk. We attributed this to
pressure gradient across the seal due to argon gas on one side and vacuum on the
other, which means the initial source oven installed in the ion trap might have had
a defective seal. This would also explain the long period of unsuccessful 138Ba+
trapping trials we have experienced, where a thick barium oxide layer might have
been present and needed to be depleted first.
We have also tried another way of removing the indium seal by snipping it
off in an argon rich environment. While eventually we were able to trap using this
25
source, we had a hard time trapping initially. Eventually getting the oven red hot
to melt the source in the hopes of dissipating any oxide layer that might be blocking
the elemental source resulted in successful loading on this second ion trap.
Compared to these Alvatec ovens, a more traditional approach of quickly
putting barium inside a previously aligned oven tube in an argon rich environment,
and keeping the source at elevated temperatures during vacuum chamber bake-out
might work better. Nevertheless, the search for a reliable pure barium source instal-
lation is still in progress. One particular solution might use getter materials, where
barium is emitted from the material (BaAl4) at high temperatures while the alloy
prevents the oxidization of barium in air. Maintaining a directional neutral flux is
the main hurdle in this setup. Another approach might be using platinum filaments
implanted with barium [47]. Due to work function and ionization energy differences,
barium directly gets released as ions when a filament is heated. Directing ions to
the trap might pose a problem here as well, but recently this idea has been revisited
and different isotopes of barium ions were successfully loaded in Wes Campbell’s
group at UCLA.
After a flux of neutral atoms in the trapping region is maintained, these atoms
are photo-ionized with lasers. A 399 nm laser tuned to the 1S0 →1 P1 transition of
neutral Yb excites the atom and another photon from the 369 nm laser finalizes the
photo-ionization process [36].
For loading 138Ba+ , we have looked into commonly used schemes (see Fig.
2.2) and ultimately decided to use a 413 nm laser for loading in our apparatus. This
method only required the addition of one more readily available laser to the setup.
26
6s2    1S0
5d6p  3D1
6s6p  3P1
6s6p   1P1
79
1.
3 
55
3.
7
41
3.
3
Continuum
41
7.
1
56
0.
6
34
0.
1
23
7.
9
Γ/2π 
18.9 MHz
Γ/2π  
238 kHz
Γ/2π 
50 kHz
nm
(a) (b) (c)
Figure 2.2: Energy level diagram of neutral barium drawn to scale. Wavelengths and
natural linewidths are cited from Ref. [48, 49] and dashed lines show the maximum
wavelength that can be used for the second part of the photo-ionization process (a)
A strong dipole allowed transition is excited with a 553 nm laser and another with
less than 417 nm wavelength, which is already available for 171Yb+ qubits, can be
used for excitation to the continuum [50]. However, the lack of laser diodes around
553 nm make this scheme costly and impractical, as dye lasers or doubling from
1106 nm would be necessary. (b) A weak intercombination line is utilized in this
method [51], and another 413 nm photon from the same laser has enough energy to
complete the photo-ionization process. (c) Another method [52] involving a weaker
intercombination line which requires an extra UV source as well, making it less
appealing compared to (b).
27
In the ionization process, ∼2 mW of 413 nm laser beam is focused at the
center of the trap with a waist of ∼10 µm. We have noticed that, to trap 138Ba+ ,
the required flux from the barium oven is much more than is necessary for trapping
171Yb+ , as evidenced by the pressure reading in the vacuum chamber. With current
running through the barium oven, we see pressures climbing from 10−11 up to 10−9
Torr where we start to see non negligible photo-ionization rates of a few per minute.
Whereas, while trapping 171Yb+ , the pressure jump is less than 10−11 Torr. This
difference might be due to natural linewidth differences in the first excitation step
of photo-ionization process, or there might be a considerable amount of oxide com-
ponents in the barium source, reducing the percentage that can be photo-ionized
and necessitate a higher flux.
2.2.3 Methods for Getting Fluorescence Photons from Barium Ions
Rather than looking directly at the 138Ba+ fluorescence for the initial veri-
fication of loading these ions, we followed an indirect path by first verifying the
presence of these ions and afterwards trying to find the correct beam alignment and
wavelengths for addressing the transitions.
2.2.3.1 Co-trapping for Verification of 138Ba+ Loading
After loading an 171Yb+ ion into the trap, we first tried to co-trap a 138Ba+
by heating the neutral source and aligning the 413 nm photo-ionization beam along
the middle of the trap and directly looked for a shift on the 171Yb+ ion position to
28
verify presence of an extra ion in the trap. The 413 nm wavelength starting point
was obtained from past works [48, 51] and also corrections were made to account
for the Doppler shift due to the angle between the neutral flux direction and the
photo-ionization beam. Finding the absolute transition frequency is not a critical
requirement in this step, as the dispersion of the neutral flux broadens the transition.
One particular anecdote is worthwhile to point out here. Initially we tried this
approach while the 171Yb+ was fluorescing. Collisions due to flux from the barium
source when the 171Yb+ ions are in the excited state has been observed to get them
to the long lived 2F7/2 state. To bring the ions back to fluorescing levels, a 355 nm
laser was also incident on the 171Yb+ . However, the first ion trapped with this
method was actually verified from motional spectroscopy to be a BaO+ molecule.
We attributed this to the presence of a 355 nm laser which might have been stripping
electrons from the trap electrodes that collide with neutral source to ionize them,
or directly driving optical transitions on BaO for photo-ionization. To prevent this
issue, during 138Ba+ trapping trials, we blocked the 369 nm beams to make sure
171Yb+ is not excited and consequently do not end in the 2F7/2 state. After pressure
levels go back to normal values of around 10−11 Torr, the 369 nm beam is unblocked
to check the presence of any extra ion that might have been added in the process.
With trials around the guess value of the 413 nm transition including a ∼1
GHz of Doppler shift, the optimal loading point was found at 725.25980 THz, which
is consistent with reference point of 725.25890 THz [48].
29
2.2.3.2 138Ba+ Transition Wavelength Measurements with HCL
After the co-trapping verification, fluorescence measurements are the next step.
Even though transition wavelengths for 138Ba+ can be found from previous works
[48,51], calibration differences might introduce significant variations. For this reason,
we have directly measured these wavelengths using the optogalvanic effect in a hollow
cathode lamp (HCL) following techniques demonstrated in Ref. [53]. Fig. 2.3 shows
the setup for this measurement and the results.
Using the transition frequencies obtained with this method as a starting point,
fluorescence counts were able to be seen from a single 138Ba+ ion by scanning around
these frequency regions to find them more precisely.
A more complete level structure of the 138Ba+ ion is shown in Fig. 2.4. Since
the 2S1/2,
2P1/2 and
2D3/2 form a closed 3 level Λ configuration, two lasers addressing
the transitions between them is sufficient for fluorescence, Doppler cooling and some
of the qubit operations that will be covered in following chapters.
2.2.4 The Optical Bloch Equations
For the fluorescence considerations, the 171Yb+ atom can be treated as a two
level system since branching from 2P1/2 to
2D3/2 is negligible and the deshelving
laser does not directly couple back to the 2P1/2 level to bring the population back
to the cycling transition of 2S1/2 ↔2 P1/2 .
In contrast, a Λ configuration prevents carrying out a similar approach in the
138Ba+ . For a full characterization of interaction between a single 138Ba+ ion and
30
Ref.
Signal
High
Voltage
Lock-in Amplier
(a)
(b) (c)
Chopper Hollow Cathode Lamp
Laser Field
Figure 2.3: (a) The conductivity of the discharge gas changes when an optical
transition is excited [53], resulting in a voltage change which can be measured to
find these transitions. Using this method, we have measured (b) 2S1/2 → 2P1/2
and (c) 2D3/2 → 2P1/2 transition wavelengths of the 138Ba+ ion, finding the center
of transition at 607.42634 THz (493.54536 nm) and 461.31189 THz (649.86935 nm),
respectively. Even though we were not able to do a Doppler free measurement,
since HCL was not transparent, the net effect is a broadening rather than a shift
in the transition frequency, making this method suitable to find the approximate
wavelength of transitions.
31
6 2S1/2
6 2P1/2
6 2P3/2
5 2D3/2
5 2D5/2
60
7.
42
62
70
 T
H
z
461.311890 THz
487.9900 THz
γ/
2π
 =
 1
5.
16
 M
H
z
γ/2π = 5.30 M
Hz
65
8.
11
64
  T
H
z
γ/
2π
 =
 1
7.
66
 M
H
z
γ/2π = 6.55 M
Hz
γ/2π = 0.95 M
Hz
512.0020 THz
72
.9
%
27.1%
75
.6
%
21.5% 2.9%
τ = 31 s
τ = 80 s
δZeeman
6 2S1/2
6 2P1/2
6 2P3/2
5 2D3/2
5 2D5/2
µ02
µ0 =1.4 MHz/G
µ02/3
µ04/3
µ04/5
µ06/5
Figure 2.4: Relevant levels of the 138Ba+ drawn to scale. Branching ratios are given
as percentages and quoted from Ref. [54,55]. Transition frequencies (in THz) given
for 2S1/2 → 2P1/2 and 2D3/2 → 2P1/2 are from readings on a calibrated HighFinesse
WSU10 wavemeter in our lab and agrees with Ref. [48] which is quoted for the rest
of the transitions and the natural linewidths. Low lying D state lifetimes are quoted
from Ref. [56, 57]. Frequency splitting per Gauss between Zeeman sub-levels with
∆mJ = 1 (omitted in this figure) are given at upper right corner for each manifold.
32
two near resonant laser light fields, it is necessary to include all eight Zeeman sub-
levels in the participating levels. However, for simplicity, I start with a three level
system (see Fig. 2.5) and extend this formalism to include eight levels afterwards.
These interactions for the specific case of 138Ba+ has been investigated with greater
detail in Ref. [51,58] and calculations carried out here follow these works.
2.2.4.1 Three level system
The total Hamiltonian of the system includes the atom, the light fields and
the interaction between them:
H = Hatom +Hlight +Hint (2.1)
and the atomic Hamiltonian is described by:
Hatom =
3∑
n=1
h¯ωn|n〉〈n| (2.2)
where |n〉 are the atomic eigenvectors as shown in Fig. 2.5 (n = 1, 2, 3) with the
Bohr frequencies of ωn. Then, the matrix representation of this atomic Hamiltonian
is:
Hatom = h¯

ω1 − ω2 0 0
0 0 0
0 0 ω3 − ω2
 (2.3)
where the basis is defined by:
|1〉 =

1
0
0
 , |2〉 =

0
1
0
 , |3〉 =

0
0
1
 (2.4)
33
6 2S1/2
6 2P1/2
5 2D3/2 τ = 80 s
γg
γr
1
2
3
Ωg
Ωr
∆g
∆r
Figure 2.5: The parameters used in Bloch equations are shown here and describe
a simplified three level 138Ba+ ion interacting with two laser fields. Ω, γ and ∆
indicate the Rabi frequencies, natural linewidths and detunings of the laser fields,
respectively.
34
and zero energy level is chosen to be h¯ω2.
The atomic levels are coupled with the laser fields; 493 nm light (green) drives
the dipole transition between the states |1〉 and |2〉 while 650 nm light (red) couples
states |2〉 and |3〉. This interaction between the light field and the atom is described
by:
Hint = − ~D · ~E (2.5)
where ~D is the atomic dipole operator. Through this interaction, the light field is
only negligibly effected by losing a single photon, hence the electric field operator
~E can be treated classically with amplitude E0, polarization vector ~ and angular
frequency ω. Therefore, in the basis defined earlier, this interaction Hamiltonian
corresponds to:
Hint = h¯

0 Ωg
2
e+iωgt 0
Ωg
2
e−iωgt 0 Ωr
2
e−iωrt
0 Ωr
2
e+iωrt 0
 (2.6)
where the Rabi frequency is defined as:
h¯Ωg ≡ ~g · ~D12 E0g
h¯Ωr ≡ ~r · ~D23 E0r (2.7)
Consequently, the complete Hamiltonian is:
H = h¯

ω1 − ω2 Ωg2 e+iωgt 0
Ωg
2
e−iωgt 0 Ωr
2
e−iωrt
0 Ωr
2
e+iωrt ω3 − ω2
 (2.8)
with Hlight being eliminated by the classical approximation.
35
In order to include damping terms such as spontaneous emission from state |2〉
and the finite linewidth of the lasers, a density operator ρˆ is introduced to describe
the system:
ρˆ =
∑
n,m=1,2,3
ρnm|n〉〈m|. (2.9)
Here, the diagonal elements ρnn are the probability of the atom being in the |n〉
state, and the off-diagonals are called coherences. The Liouville equation reveals
the time dynamics of the density matrix elements:
dρ
dt
= − i
h¯
[H, ρ] + L(ρ) (2.10)
where L(ρ) is:
L(ρ) = −1
2
∑
k
(Cˆ†kCˆkρ+ ρCˆ
†
kCˆk − 2CˆkρCˆ†k). (2.11)
Decay to states |1〉 and |3〉 from state |2〉 via spontaneous emission are included by:
Cˆ1 =
√
γg|1〉〈2| and Cˆ2 = √γr|3〉〈2| , respectively. (2.12)
And similarly, the linewidths of green (γgl) and red (γrl) lasers are accounted by:
Cˆgl =
√
2γgl|1〉〈1| , Cˆrl =
√
2γrl|3〉〈3|. (2.13)
Transforming into a frame rotating at the laser frequencies, using the unitary
operator:
U =

e−iωgt 0 0
0 1 0
0 0 e−iωrt
 (2.14)
36
gives the Hamiltonian of the system:
H
′
=

∆g
Ωg
2
0
Ωg
2
0 Ωr
2
0 Ωr
2
∆r
 (2.15)
where detunings are defined as:
∆g = ωg − (ω2 − ω2)
∆r = ωr − (ω2 − ω3) (2.16)
and the density operator is:
ρ
′
= UˆρUˆ † . (2.17)
Under this transformation the damping terms do not change.
The optical Bloch equations are acquired by inserting H
′
, ρ
′
and L(ρ) into
Eqn. 2.10 :
~ρ ≡ (ρ11, ρ12, ..., ρ33)
~˙ρm =
∑
n
Mmn~ρn (2.18)
with the solution of:
~ρ(t) = eMt~ρ(0) (2.19)
for initial condition given by ~ρ(0).
Finally, the steady state solution, along with the
∑
n ρnn = 1 normalization
constraint, can be used to find the excitation spectrum:
~˙ρ = 0 =
∑
n
Mmn~ρn . (2.20)
37
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
ρ11
ρ33
ρ22
ρ11
ρ33
ρ22
|ρ13|
t (µs) t (µs)
(a) (b)
|ρ|
Figure 2.6: (a) The three level atom is optically pumped to state |3〉 when only a
resonant green laser is incident without a re-pumping red laser. The parameters
used are sg = 1, sr = 0, ∆g = 0, ∆r = 0, γgl = 0, γrl = 0 (b) Two laser fields drive
the atom into a dark state, a coherent superpositon of states |1〉 and |3〉, when green
and red lasers have the same detuning. Time evolution is shown for sg = 1, sr = 4,
∆g/2pi = −10 MHz, ∆r/2pi = −10 MHz, γgl = 0, γrl = 0.
In the following examples, the Rabi frequencies of transitions with green (i→
g) and red (i → r) laser interactions are expressed with corresponding saturation
parameters:
Ωi =
√
si
2
γi (2.21)
and the decay rates are that of a 138Ba+ atom, γg/2pi = 15.1 MHz, γr/2pi = 5.3
MHz. Time dynamics of this three level atom when subject to laser fields is shown
in Fig. 2.6.
Since in photon counting experiments the typical integration time (0.1-1 s) is
much longer than time scale of settling to the steady state (< 1 µs), time dynam-
ics can be ignored and the excitation spectrum is evaluated at steady state. For
38
-80 -60 -40 -20 0 20 40 60 80
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
-80 -60 -40 -20 0 20 40 60 80
0
0.02
0.04
0.06
0.08
0.1
0.12
ρ11
ρ33ρ22
∆r / 2π (MHz) ∆r / 2π (MHz)
|ρ||ρ|
Figure 2.7: Density matrix diagonal elements of three level system at steady state
with parameters sg = 1, sr = 4, ∆g/2pi = −10 MHz, γgl = 0, γrl = 0 and the
red laser detuning ∆r/2pi is scanned. Dashed line shows the coherent dark state
(see Fig. 2.6 (b)) where ρ22 vanishes and the atom does not emit photons when
∆g = ∆r. Fluorescence measurement on an atom reveals ρ22. Compared to a two
level system where ρ22 = 0.5 for a saturated transition, introduction of an extra
level results in significant reduction in excited state population and consequently
measured brightness of the atom.
example, emitted green photons are:
# Green Photons = γg ρ22(tss). (2.22)
The excitation spectrum is shown in Fig. 2.7.
2.2.4.2 Eight level system
With Zeeman sublevels included, 138Ba+ forms an eight level system where
each manifold contributes 2J + 1 states that are lifted from zero field degeneracy by
the magnetic field, ∆ω/2pi ≈ 1
2
gJµ0B where gJ is the Lande´ g-factor (see Fig. 2.4).
Optical Bloch equations can be derived following the same procedure described
in the previous section, and the addition of a magnetic field introduces laser polar-
39
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
0
0.1
0.2
0.3
0.4
0.5
0.6
t (µs) t (µs)
ρ11
ρ22
ρ33
ρ44
ρ55
ρ66
ρ77
ρ88
1 2
3 4
5 6
7 8
2S1/2
2P1/2
2D3/2
(a)
(b)
|ρ| |ρ|
Figure 2.8: (a) Indexing of the 138Ba+ states. (b) The eight level 138Ba+ atom is
optically pumped to the states |5〉 and |8〉 when only a pi polarized laser field is
incident as there are no levels from these states that pi light can couple to. The
parameters used are sg = 1, sr = 2, ∆g/2pi = −10 MHz, ∆r/2pi = 10 MHz, γgl = 0,
γrl = 0 and B = 4 Gauss. Starting condition of |ψ(t = 0)〉 = |1〉 results in higher
occupation in |5〉 compared to |8〉 in the steady state.
ization effects and the field amplitude into the dynamics.
Fig. 2.8 and 2.9 show simulated time dynamics and the steady state excitation
spectrum of the 138Ba+ ion.
2.3 Double Imaging Setup
The atoms are imaged by collecting the spontaneously emitted photons on
2P1/2 → 2S1/2 decays to a camera. Since the wavelengths of these photons are
vastly different for 171Yb+ and 138Ba+ , a double imaging setup (see Fig. 2.10) is
constructed to have the same focus for both point sources at the detector surface.
40
-80 -60 -40 -20 0 20 40 60 80
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
-80 -60 -40 -20 0 20 40 60 80
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
∆r / 2π (MHz) ∆r / 2π (MHz)
ρ33+ ρ44
ρ55+ ρ66+ ρ77+ ρ88
ρ11+ ρ22
|ρ| |ρ|
Figure 2.9: For simplicity in presentation populations in each manifold are summed
up. Extra Zeeman levels result in multiple dark resonances compared to the single
coherent dark state of the three level system. The parameters are set to sg = 2,
sr = 4, ∆g/2pi = −25 MHz, γgl = 0, γrl = 0, B = 4 Gauss with the laser fields
polarized perpendicular to the quantization axis.
2.4 High Numerical Aperture Setup
In addition to a double imaging setup, a second view-port allows collecting
photons from the opposite side of the vacuum chamber. The re-entrant window on
this side is designed to accommodate a NA-0.6 PhotonGear lens with a working
distance of about ∼17 mm. This objective collects ∼10 % of the emitted light and
will be used to couple photons to a single mode fiber for eventual entanglement of
remote qubits. However, so far it is only utilized in a free space setup and mainly
used for collecting 493 nm photons (see Fig. 2.11).
41
NA=0.36Short Pass Filters
Flip Mirror
Cam
era
PMT
Achromatic
Lens
Pinhole
Adj.
Lens
Ions
High
NA
Setup
BPF
BPF
Figure 2.10: The photons from both 171Yb+ and 138Ba+ are collected with an
NA=0.36 lens stack. Short pass filters are used to separate 369 nm and 493 nm
light and an adjustable lens corrects for chromatic aberration. Band pass filters
(BPF) are used to block the room light and other laser fields while letting through
369 and 493 nm fields and additionally a pinhole of diameter 100-200 µm spatially
filters the laser background from the ion signals. Finally an achromatic lens provides
extra magnification of the image formed at pinhole. A flip mirror is used to direct
light between a PMT, used in photon resolving for state detection of 171Yb+ and
fluorescence measurements, and a camera. Since the camera is only used for loading
verification and ion number confirmation, an inexpensive CMOS camera (PointGrey
Blackfly 2.3 MP Mono GigE PoE) is sufficient in this setup.
42
NA=0.6 Pinhole HWP PBS
PMT(H)
PMT(V)
Flip Mirror 
APD
Double
Imaging
Setup
Naked
Eye
Atom-Photon
Entanglement Setup
BPFIons
Figure 2.11: For correcting aberrations [59] and maximizing the photon collection,
NA=0.6 lens is placed on a five axis pitch, yaw and translation stage. A 50 µm
diameter pinhole spatially filters the 138Ba+ photons from background. Flip mirrors
are used for switching between different applications; atom-photon entanglement
setup incorporates polarization analyzing optics and photo multiplier tubes (PMT’s)
with low dark counts to minimize errors while avalanche photo diode (APD) with
higher quantum efficiency at 493 nm region allows getting shorter integration times.
High numerical aperture and the sensitivity of the human eye to visible photons make
it possible to see a single 138Ba+ with naked eye, which is not possible with 171Yb+
and other ions with strong dipole transitions around UV wavelengths. In future
applications of entanglement of remote 138Ba+ qubits, atom-photon entanglement
setup can be replaced with a single mode fiber coupling optics.
43
2.5 Re-ordering of the Ions
In a multi-species system, different atoms might be required to occupy specific
positions. For example, communication qubits need to be coupled into a single mode
fiber for generating entanglement between them, and a change in the position will
considerably reduce the coupling efficiency with optics aligned to a certain spot.
Additionally, laser beams addressing a particular species at a certain position likely
do not have the same intensity at different locations which will change the interaction
dynamics.
The experiments presented in this thesis are carried out with only one of each
species co-trapped in the same chain, and the ions are observed to swap positions on
minute timescales possibly caused by the background gas collisions. Two different
hardware implementations are used to keep the ions in a desired configuration.
The first approach relies on changing the applied DC voltages to push the ions
to a spot where the micromotion [38] is not compensated. These voltages are applied
for ∼200 ms and then brought back to the original micromotion compensated values.
This results in brief heating of the ions by the Doppler cooling beams to melt the
crystal, and re-crystallizing in a random configuration when the optimal cooling
conditions are satisfied again by applying the initial voltages.
Another method is realized by RF voltage modulation. At low RF powers,
the ions align along the transverse axis due to weaker confinement, and they get
re-ordered in the longitudinal direction when switched back to high RF. This is
realized by amplitude modulation of the RF voltage with control DC voltages.
44
Both methods probabilistically re-order the 171Yb+ -138Ba+ crystal in the cor-
rect configuration and for only two ions these approaches have been satisfactory
since it takes only a few trials on average to get the right formation. However, with
inclusion of more memory and communication qubits to the modules, a determinis-
tic re-ordering method might be necessary such as has been realized in Paul traps
with sophisticated electrode structures [60].
45
Chapter 3: Ytterbium and Barium Atomic Ion Qubits
3.1 Overview
In our multi-species apparatus, we encode the memory qubits in the 2S1/2
manifold hyperfine clock states of 171Yb+ atomic ions, |F = 0,mF = 0〉 ≡ |⇓〉 and
|F = 1,mF = 0〉 ≡ |⇑〉. They are well suited to be used as memory qubits with their
long coherence times and easy state initialization/detection schemes [5]. The qubit
frequency separation of 12 642 812 118.5 + ∆ Hz can be conveniently addressed
with microwave sources and pulsed lasers [45] for quantum operations, and a small
second order dependence to external magnetic field (∆ = 310.8 B2 Hz with field
amplitude B in G) is the source of long coherence time.
As communication qubits, we use the electron spin of 138Ba+ atomic ions in
the 2S1/2 level, |J = 1/2,mJ = −1/2〉 ≡ |↓〉 and |J = 1/2,mJ = +1/2〉 ≡ |↑〉 [61].
The frequency splitting between qubit states depend on the external magnetic field
and it is given by 2µ0B ≈ 2.8 B MHz where µ0 is the Bohr magneton (1.4 MHz/G)
and B is magnetic field in G. Current optical fiber and photon detector technologies
make 138Ba+ an attractive communication qubit compared to other common trapped
atomic ion species owing to its visible 493 and 650 nm photon emission lines.
For the eventual construction of a multi-species network, there are several
46
2S1/2
2P1/2
22 2
22
-2-2
-2 -2 -2
2
2
2S1/2
2P3/2
6
6
3 3
334 11
4
-1
4
4
1
1
1
-1
-1
⇓
⇓
⇓
⇓
Figure 3.1: For ease of representation, Clebsch-Gordan coefficients are shown as
6 × C2 in here. The transitions with a negative sign indicate a minus sign on the
corresponding coefficient.
essential single qubit quantum operations that must be performed in the modules,
such as qubit initialization, state detection, coherent rotations and atom–photon
entanglement. In this chapter, I present the experimental techniques used for these
operations.
3.2 Ytterbium Hyperfine Qubit
The excited states of the atoms can be utilized for coherent and dissipative
qubit operations. The relative coupling strength of dipole allowed transitions and
decay probabilities from an excited level to different ground states are characterized
with Clebsch-Gordan coefficients, which are shown on Fig. 3.1 for 171Yb+ atoms.
Fig. 3.2 summarizes the Doppler cooling, state detection and initialization proce-
dures for the 171Yb+ atom.
Doppler cooling is realized by driving the F = 1↔ F = 0 transition between
47
2S1/2
2P1/2
2D3/2
3[3/2]1/2
F=0
F=1
F=0
F=1
F=1
F=2
F=1
F=0
2S1/2
F=0
F=1
2S1/2
F=0
F=1
2P1/2 F=0
F=1 2P1/2 F=0
F=1
x x
(a) (b) (c)
2/3
1/3
∆ (GHz)
2S1/2
2P1/2
2D3/2
3[3/2]1/2
12.6
2.1
0.9
2.2
(d)
Figure 3.2: (a) Doppler cooling beam is detuned from the transition to achieve an
optimal temperature. Only a small amount of optical intensity on the sideband is
sufficient as the off-resonant pumping probability is low. (b) State detection of the
qubit relies on selective photon scattering from F = 1 levels. Branching needs to
be considered here, as any mixing between F = 0 and F = 1 population would
result in reduced fidelity. This is prevented by forbidden decays from the F = 0
level of 3[3/2]1/2 to F = 0 of
2S1/2. (c) The optical pumping beam is produced by
adding 2.1 GHz sidebands to the detection beam with an EOM. After each photon
scattering event, the atom decays to the F = 0 level with a probability of 1/3.
Even though it is not a limiting factor in most applications, the carrier needs to
be eliminated for faster optical pumping as scattering with this component would
result in photon scattering without decays to F = 0. This can be realized with using
an AOM rather than EOM to add the necessary 2.1 GHz. (d) Table of hyperfine
splittings in different manifolds.
48
2S1/2 and
2P1/2 levels with a detuning δ ≈ −γ/2 where γ/2pi ≈ 19.6 MHz [26].
Off-resonant coupling to the F = 1 level result in pumping to the F = 0 level of
the 2S1/2 manifold. Hence, a second beam is necessary to put the population back
to main cooling cycle and it is produced by introducing a 12.6+2.1 GHz sideband
with an EOM.
A resonant beam is turned on for the state detection, and population in F = 1
scatters many photons while F = 0 scattering rate is greatly reduced since the
laser is detuned by 14.7 GHz from the closest allowed transition. By looking at the
number of collected photons in a certain time window, a single shot measurement
is done on the 171Yb+ qubit state. The background light from the environment and
the scattered laser light from the trap electrodes that might find a way to the photon
detectors are minimized by spectral and spatial filters to prevent these photons from
being registered as atom signal. The main limitation with this method is off-resonant
pumping to the F = 0 level [5,62] which abruptly stops the fluorescence of the qubit
state |⇑〉. In our setup, with a 0.36 NA lens and 30 % efficient photon detectors,
we can detect the qubit state with a fidelity of F ≈ 99.2 % in around 200 µs. With
the employment of faster optics (NA=0.6), a record 171Yb+ state detection fidelity
of 99.9% is demonstrated in Kim group at Duke University [44].
The state of the 171Yb+ qubit can be initialized to |⇓〉 by turning on a laser
resonant with F = 1 of 2S1/2 and
2P1/2. After each scattering event, the atom decays
to F = 0 with 1/3 probability. The low probability of pumping back from F = 0
with this laser beam results in near perfect state initialization with a timescale of
less than 1µs.
49
2S1/2
2P1/2
2S1/2
1/3
-    1/3
2/3
-    2/3
1
1 2/3
1/3
1/3
2/3
2P3/2
Figure 3.3: Clebsch-Gordan coefficients of allowed dipole transitions on 138Ba+ atom.
Even though 2P3/2 states are not directly populated in the experiments conducted
here, they play an important role on the stimulated Raman transition that will be
analyzed later in this chapter.
3.3 Barium Zeeman Qubit
The Clebsch-Gordan coefficients for 138Ba+ atom are given in Fig. 3.3. While
decays to levels 2D3/2 and
2D5/2 change the time dynamics of the operations, the
individual matrix elements related to the coupling between 2P1/2 ↔ 2D3/2 and
2P3/2 ↔ 2D5/2 level do not play an important role on most of the operations
described here for 138Ba+ , therefore they are not shown.
50
2S1/2
2P1/2
2D3/2
3[3/2]1/2
2S1/2
2P1/2
2D3/2
171Yb+ 138Ba+
20
0
1
50 1
3
1
369 nm
650 nm
493 nm
93
5 
nm
Figure 3.4: The numbers on dashed lines indicate approximate relative branching
ratios; the small decay probability to the 2D3/2 level (1/200), and bow-tie con-
figuration of the 171Yb+ excitation scheme result in a simplified Doppler cooling
mechanism where it is sufficient to consider 2S1/2 and
2P1/2 manifolds. In contrast,
a much higher branching ratio (3:1) and dark states that might be introduced in Λ
system of 138Ba+ necessitates all three levels and two laser fields to be considered in
the cooling layout.
3.3.1 Doppler Cooling
The 171Yb+ atom can be treated as a two level closed system in Doppler cooling
applications, as the small branching ratio introduces a negligible divergence from
the assumed two level structure and additionally, re-pumping from the metastable
2D3/2 level can be achieved without a transition back to the
2P1/2 level (see Fig.
3.4(a)). In contrast, since the levels 2S1/2,
2P1/2 and
2D3/2 form a Λ configuration
in 138Ba+ as shown in Fig. 3.4(b), these manifolds and two laser fields that couple
them need to be considered in the Doppler cooling process of this atom.
Interestingly, a numerical analysis shows that the Doppler cooling process in
this configuration is mainly dominated by the 2S1/2 to
2P1/2 transition and the fre-
51
quency of the 2D3/2 to
2P1/2 laser has almost no effect [63]. Still, for optimal cooling,
the 650 nm laser should have enough optical intensity to saturate the transition to
prevent population trapping in the 2D3/2 levels, and the frequency of this laser needs
to be adjusted to prevent driving the atom to a coherent dark state in conjunction
with the 493 nm laser [64].
Additionally, in a non-zero magnetic field, the transitions between 2S1/2 and
2P1/2 Zeeman sublevels have different frequencies, hence adjustment in the detuning
of the 493 nm laser is necessary for efficient cooling. This shift from the nominal
optimal detuning of δ = −γ/2 approximately amounts to −4
3
µ0B (−23µ0B) for
light polarized perpendicular (parallel) to the quantization axis where the magnetic
field B is in units of G and µ0 = 1.4 MHz/G [63]. Finally, optical pumping to a
particular Zeeman sublevel must be avoided by adjusting the polarization of both
lasers. For instance, a pure pi polarized 650 nm light would lead to population
trapping at 2D3/2, mJ = ±3/2 levels, which can be prevented by introducing circular
polarized components. Even with these adjustments, there are inevitable dark states
introduced with the 2D3/2 ↔ 2P1/2 transition [65] which the atom evolves out of
at the Larmor frequency. This results in an overall drop in fluorescence rate and
Doppler cooling speed.
3.3.2 State Initialization
We initialize the 138Ba+ qubit with polarization selectivity of the dipole tran-
sitions between the states |↓〉, |↑〉 and the levels at the excited 2P1/2 manifold (see
52
2S1/2
2P1/2
x
1/3
2/3
Figure 3.5: Levels involved in optical pumping of the 138Ba+ qubit, and decay
probabilities to different qubit states. After scattering N photons, the probability
of the qubit being initialized to the |↓〉 state is: P (|↓〉) = 1− (2/3)N . Decays from
2P1/2 to
2D3/2 manifold is not included for simplicity, as these events effect overall
speed and not the efficiency.
Fig. 3.5). When a 493 nm σ− polarized laser light field is incident on the atom,
the state |↑〉 gets excited to the 2P1/2 |mJ = −1/2〉 level and decays to the |↓〉 state
with 1/3 probability. Since the |↓〉 state does not have any allowed dipole transitions
with this σ− light which might put the population back into the state |↑〉, the atom
gets optically pumped to the |↓〉 state after several scattering events. The qubit can
be initialized to the |↑〉 state as well following same procedure with a σ+ light.
Since the efficiency of this operation depends on the polarization purity of
the light, a polarization maintaining fiber is used to send in this light to prevent
long term drifts and eliminate occasional calibration requirements. Even though a
single quarter waveplate (QWP) is enough to turn the linear polarization output of
the fiber into the circular light necessary for the optical pumping, we use a set of
53
three waveplates, quarter-half-quarter, to have arbitrary control on the polarization
state in order to account for polarization rotations that might be introduced during
reflections on mirrors and birefringence of the view-ports.
To achieve the maximum optical pumping efficiency, the fluorescence counts
from the atom are monitored while the laser beam is incident, and waveplates are
adjusted to minimize these counts. Since realizing the perfect polarization also de-
pends on alignment between the quantization axis and the beam wave vector ~k, fine
adjustment on the perpendicular magnetic field components is generally necessary
to tilt the field along the ~k. We obtain a typical optical pumping time of less than 1
µs with > 99.5 % efficiency, which is ultimately limited by the polarization optics.
3.3.3 State Detection without Shelving
In this network architecture, entanglement generated between 138Ba+ qubits
in different modules is immediately transferred to neighboring 171Yb+ qubits. Con-
sequently, 138Ba+ qubits only act as a link between memory qubits in the system
and quantum information processing ultimately does not rely on state detection of
these communication qubits. However, qubit state detection for 138Ba+ atoms is still
useful for troubleshooting of operations until the state can be reliably transferred
to and read out on memory qubits. Since 138Ba+ atoms lack an isolated cycling
transition for qubit states, a common detection method on 138Ba+ relies on shelv-
ing one of the qubit states to a long lived state and looking at the fluorescence of
the remaining population [66]. This method requires two additional lasers, one for
54
shelving and another for depopulating the long lived state. As this detection method
is not necessary in the long run, we have implemented a different state detection
technique which does not require additional lasers.
In our 138Ba+ state detection technique (see Fig. 3.6(a) and (b)), after con-
ducting an experiment, we first excite the 2S1/2 to
2P1/2 transition with 493 nm σ
+
polarized light. If the ion is in the |↓〉 state, it scatters three photons in average
before getting optically pumped to the |↑〉 state. In contrast, the |↑〉 state does not
scatter any photons due to absence of excited state lines this it can couple to. In
the next experimental cycle, we send σ− polarized light and this time only the |↑〉
state scatters photons. We cycle between σ+ and σ−, and look at the distribution
of photon detection events corresponding to the polarization of the excitation beam
to infer the 138Ba+ qubit state.
As it is evident from the description, one of these detection beams is identical
to the optical pumping beam. Hence, nominally only one extra beam addition to the
setup is necessary for this detection scheme. To create this second beam, we send
another orthogonally polarized beam through the same polarization maintaining
fiber (see. Fig. 3.7), and rotations through the optics turn it into the opposite
circular polarization state compared to the optical pumping beam.
Even though some information can be extracted from the qubit with the use
of a single beam, we reduce statistical effects on state detection accuracy stemming
from the probabilistic nature of this method by switching between the orthogonally
polarized beams. For instance, the number of photon detection events in a certain
time window (T) from 138Ba+ qubit (α|↓〉+ β|↑〉) subject to only σ− beam on each
55
Init.
Exp.
σ+
σ−
2S1/2
2P1/2
(b)
σ+σ−
(a)
(c)
PMT
APD
time
P(
   
  )
Figure 3.6: (a) 138Ba+ detection timing sequence. (b) Relevant energy levels on
138Ba+ and coupling of different levels with σ+ and σ− light. (c) State detection on
a single qubit Rabi flopping experiment. In this state detection technique, we utilize
already existing infrastructure that is required for achieving rapid entanglement
generation between modules; an 80% quantum efficient avalanche photodiode (APD)
and a lens that collects around 10% of the emitted light (NA=0.6). Hence, overall
data acquisition rate of our technique compared to a deterministic single shot state
detection method amounts to 8%. Using a photo multiplier tube (PMT) which has
lower dark counts compared to the APD, we get higher detection accuracy. However,
this switch also results in a lower data acquisition rate due to lower PMT quantum
efficiency of 30%.
56
HWP1
AOM(V)
AOM(H)
AOM(EIT) HWP2
HWP
QWP
QWP
50/50 
BS
PMF
PMF
PBS PBS
+1
+1
0
+1
To TrapFrom 493 nm Laser
Figure 3.7: 493 nm light is sent to initialization/detection optical setup through a
polarization maintaining fiber (PMF). HWP1 is used for optical power adjustments
between two arms, while HWP2 is necessary for aligning the polarization to the axis
of second PMF. AOM(H) and AOM(V) are used to switch between orthogonally
polarized detection beams; they couple into the same fiber and waveplate stack turns
H → σ+ and V → σ−. For optimal addressing of the dipole transitions, the RF
drive frequency difference between these two AOM’s should match the corresponding
transition frequency difference of 138Ba+ . This amounts to ∆(RF )/2pi = 8
3
×1.4×B
MHz where magnetic field amplitude B is in G. An extra beam is also incorporated
in this setup for electromagnetically induced transparency (EIT), whose function
will be explained in the following chapter. Numbers on the beams following AOM’s
indicate which diffraction order is being used.
57
trial would yield a Poisson distribution around the mean:
λ ∝ βΓ(I, δ)
ΓMax
PeffN(T ) (3.1)
where N(T ) is number of trials in this time window, Peff is the overall photon detec-
tion efficiency of the system, and finally Γ(I, δ) is the scattering rate as a function
of the intensity, I, and detuning δ. Consequently, different superposition states
produce distributions that overlap each other, leading to inaccuracy in inferring the
state of 138Ba+ from the number of events where a photon is detected. Introduction
of the second beam results in a higher state detection accuracy by producing an-
other Poisson distribution that can be used, similar to the 171Yb+ case where both
qubit states have a measurable outcome. Overall, with two different circular po-
larized beams utilized, this scheme resembles a modified 171Yb+ state measurement
technique where only some random subset of all the measurements are taken into
account.
Additionally, the accuracy of this method is not susceptible to slow intensity
and frequency noise of the laser to a certain extent; as the polarization is swapped
for each trial, noise sources slower than the experimental repetition rate, typically
< 1 – 10 kHz, become common mode since they shift the mean value for both
distributions. We obtain a detection accuracy of about 98% with a PMT and 94%
with an APD as measured in Fig. 3.6(c), which is limited by polarization purity of
the beams and detector dark counts.
58
3.3.4 Laser Stabilization
For the operations described above, laser frequency stabilization requirements
are not rigorous. The final Doppler temperature is first order insensitive to frequency
fluctuations around the optimal detuning, optical pumping transitions can be over
saturated and kept on longer to eliminate effects of laser frequency changes and
swapping between σ+ and σ− polarized light on state detection cancels out slow laser
drift issues. Therefore, we found locking the 493 and 650 nm laser frequencies with a
calibrated wavemeter utilizing a slow software based feed forward PID loop sufficient.
The wavemeter (HighFinesse WSU10) measures up to 16 different wavelengths and
after each measurement the voltage output that is dependent on the difference from
set point is applied to the laser grating piezo actuator to correct for drifts. Estimated
long term stability of this passive locking technique is around 1 MHz, which is
satisfactory for the 138Ba+ operations.
Typical fluctuations of 493 and 650 nm laser optical power outputs or fiber
coupling efficiencies do not have significant effect on the 138Ba+ operations either,
hence a noise eater setup was not utilized here.
3.3.5 Magnetic Field Stabilization
In the multi-species approach, communication qubits do not require long co-
herence times as photonic entanglement generated between them gets quickly trans-
ferred to nearby memory qubits where this entanglement can be stored for longer
and used later. However, the short coherence time of the Zeeman 138Ba+ qubits,
59
which is a result of high magnetic field sensitivity of about 2.8 kHz/mG, might result
in errors during transfer operations.
The effect of magnetic field noise can be characterized by looking at the time
evolving density matrix of a single qubit prepared in |ψ〉 = 1√
2
(|↓〉+ |↑〉):
ρ(t) =
1
2
(|↓〉〈↓|+ |↑〉〈↑|+ eiφ(t)|↑〉〈↓|+ e−iφ(t)|↓〉〈↑|) (3.2)
where φ(t) =
∫ t
0
∆(t
′
)dt
′
in the frame rotating with qubit frequency and ∆(t) is the
noise term. The unknown evolution introduced to the density matrix due to this
extra phase results in decoherence.
Rather than directly measuring the coherence time of 138Ba+ , magnetic field
sensitive transition between the |⇓〉 and |F = 1,mF = −1〉 states of the 171Yb+ is
utilized in a Ramsey experiment for faster data acquisition. In our typical lab envi-
ronment, we have measured a coherence time T ∗2 of around 200 µs, which indicates
100 µs for the 138Ba+ qubits due to approximately twice the magnetic field sensi-
tivity. Since this coherence time is comparable to the time required for transferring
information [67], we have implemented a setup for canceling background magnetic
field noise to extend the coherence time.
A major contribution to magnetic field noise comes from lab equipment and
power lines at 60 Hz and higher harmonics. We have verified this effect by looking at
the correlation between the frequency separation of the |⇓〉 and |F = 1,mF = −1〉
states on 171Yb+ and the phase of the 60 Hz signal (see Fig. 3.8 (a)). Since the
experiments are typically conducted on a much faster time scale than 1/(60 Hz),
the result is sampling of the line signal at different magnetic field amplitudes rather
60
60 Hz Compensated
No Compensation
(a) (b)
Figure 3.8: (a) The effect of 60 Hz noise is evident with shown dependence of
frequency splitting between |⇓〉 and |F = 1,mF = −1〉 states on 171Yb+ (ω) to delay
from the line trigger. Blue circles are the experimental measurements on 171Yb+
and the line is a fit with 60 Hz and higher harmonics sinusoids. (b) A Ramsey
experiment conducted on 171Yb+, driving transition between |⇓〉 state and magnetic
field sensitive |F = 1,mF = −1〉 state. Coherence time is measured without (red
squares) and with (blue circles) 60 Hz compensation fields created using a SRS
DS345 waveform generator which is line triggered. The factor of two higher magnetic
field sensitivity of 138Ba+ qubit results in half of the coherence time measured here.
than dephasing on individual experiments.
A common way of solving this issue involves conducting experiments at a given
point of the 60 Hz cycle, such that the magnetic field is same for each experiment
[68,69]. Since photonic entanglement generation is probabilistic, this method is not
feasible for us as it will greatly reduce the entanglement generation rate between
two modules. Instead, initially we have tried an active magnetic field stabilization
technique where the fluctuations are measured with a detector placed close to the
trapped ions and corrections are fed forward to a coil to cancel them. However,
due to shielding effects and spatial inhomogeneity of magnetic field amplitude, it is
difficult to use a detector outside the vacuum chamber to actively stabilize the field
at the ion position [6].
61
As an alternative, we employed a passive approach where we use an arbitrary
waveform generator to apply a magnetic field at 60 Hz and higher harmonics with full
phase and amplitude control to minimize the background noise the ions experience.
Looking at the coherence time of the qubit directly as we adjust the parameters, we
were able to push this coherence time to about 8 ms as shown in Fig. 3.8 (b). As the
background field is subject to changes with addition of experimental apparatus or
alterations in the orientation of the existing electrical equipment, occasional recali-
bration of the parameters is generally necessary once in a few days timescale. Even
though it has not been realized in our setup yet, this process can be automatized
with existing experimental control electronics to make it more robust against such
disturbances.
Active stabilization of DC bias magnetic field is challenging for similar reasons,
and unlike 60 Hz there is no periodicity that can be utilized in a passive technique
since these effects are mostly caused by operation of nearby elevator, movements
of magnetized materials around the lab etc. Therefore, experiments where small
detunings can introduce large errors are conducted at after work hours to circumvent
this problem. A feed forward stabilization technique might be a valid solution to
this problem, where the qubit splitting is measured at regular intervals interrupting
other experimental sequences, and a controlled current source counters the changes
in DC field.
62
3.3.6 Atom-Photon Entanglement
As the first step on using 138Ba+ as our communication qubits we have shown
entanglement between the atom and the emitted photon following a procedure sim-
ilar to Ref. [23,70]. On this experiment, we initialize the qubit to the |↓〉 state, and
weakly excite it to the 2P1/2 |mJ = +1/2〉 level with σ+ laser light (see Fig. 1.2).
This weak excitation beam is generated by applying a 5 ns RF pulse to the same
AOM used in the qubit detection setup, and the corresponding transition probabil-
ity was observed to be approximately 10%. The ion decays back to the |↓〉 state
emitting a σ+ polarized photon, or to the |↑〉 state with a pi photon. We collect
the photons perpendicular to the quantization axis (see Fig. 3.9(a)); therefore pi
photons are registered as vertical polarized in the (|V 〉) basis while the σ+ photons
are registered as horizontally polarized (|H〉). Along this photon collection axis,
approximately a factor of two radiation intensity difference between pi and σ+ pho-
tons is counterbalanced with decay probabilities from Clebsch-Gordon coefficients,
resulting in a joint atom-photon state:
|ψ〉 = 1√
2
(|↓ H〉+ |↑ V 〉) . (3.3)
Following a photon detection event, the qubit state is measured to look at the
correlation between atom and photon states in the original basis. Next, the same
measurement is done by rotating polarization state, without changing the atomic
qubit:
Rp(4θ) ≡
cos2θ sin2θ
sin2θ −cos2θ
 , |H〉 =
1
0
 , |V 〉 =
0
1
 ,
63
0 π/4
0
0.2
0.4
0.6
0.8
1
PMT (V)
PMT (H)
π/2 3π/2
0
0.2
0.4
0.6
0.8
1
PMT (V)
PMT (H)
P(
   
  )
P(
   
  )
HWP  angle Qubit π/2 rotation angle
PMT(H)
PMT(V)PBS
B
HWP
(a)
(b) (c)
NA=0.6
138Ba+
Figure 3.9: (a) Single photon collection setup. (b) Photon polarization qubit is
rotated with a half wave plate (HWP) and we look at atom-photon qubit state
correlations after photon detection on PMT’s. (c) Photon polarization is rotated to
diagonal basis by fixing HWP at pi/8, |H〉 → |H〉−|V 〉 and |V 〉 → |H〉+|V 〉. Photon
detection on PMT’s project the atom to a superposition (PMT(H) → |↑〉 + |↓〉
and PMT(V) → |↑〉 + |↓〉). Following detection of a photon, we coherently rotate
these superposition states to |↓〉 and |↑〉 with a pi/2 rotation to show atom-photon
entanglement.
64
Rp(pi)|ψ〉 = 1√
2
(|↓ V 〉+ |↑ H〉) (3.4)
where a pi rotation on photon polarization qubit, Rp(pi), is realized by changing the
HWP angle by pi/4 (see Fig. 3.9 (b)).
Entanglement is verified by repeating the correlation measurement in the ro-
tated basis of both atom and photon qubits. First, the polarization state is rotated
to the diagonal basis by applying a pi/2 rotation:
Rp(pi/2)|ψ〉 = 1
2
(|↓ (V +H)〉+ |↑ (V −H)〉)
=
1
2
(|(↓ + ↑)V 〉+ |(↓ − ↑)H〉) (3.5)
and following a photon detection event, the atomic qubit is rotated by pi/2 around
the equatorial axis defined by rotation phase φ:
Ra(pi/2, φ = pi/2)Rp(pi/2)|ψ〉 = 1
2
(|↓ H〉+ |↑ V 〉)
Ra(pi/2, φ = 3pi/2)Rp(pi/2)|ψ〉 = 1
2
(|↓ V 〉+ |↑ H〉) . (3.6)
Correlation measurements between atom and photon qubit states in these rotated
bases are shown in Fig. 3.9 (c). We calculate entanglement fidelity to be F ≥ 0.86
from these measurements. Errors are attributed to polarization mixing due to non-
zero solid angle (10%) [36], multiple photon scattering from the atom in excitation
step (3%) and imperfect state initialization/detection (1%). These error sources can
be eliminated in future implementations by collecting photons along the quantization
axis [13, 71] and using pulsed lasers for fast excitation of atoms [15].
65
3.3.6.1 Electric Dipole Transition with Fast Pulses
In addition to preventing multiple excitation events, usage of pulsed lasers for
fast dipole transitions allow achieving a better temporal overlap on a beamsplitter
between the photon wave-packets coming from separate atoms. The speed require-
ments of these transitions depend on the lifetime of the atom in the excited state;
errors due to temporal mismatch scale as 1 − e−∆/τ where ∆ is the arrival time
difference and τ is the lifetime of the excited state.
Previous experiments on 171Yb+ have been conducted using a Spectra Physics
Tsunami Ti:Sapphire laser [15, 26, 72] that produces 1 ps pulses to realize ∆  τ
, and the same laser can be tuned to 986 nm region to be frequency doubled to
produce 493 nm light for 138Ba+ . However, this laser does not produce as much
optical power at this wavelength compared to 740 nm and the operation might be
compromised with air absorption lines which requires purging of the laser cavity.
As a more robust alternative, we have tried a Toptica FemtoFiber pro TVIS fiber
based laser to attain short laser pulses.
Fig. 3.10 (a) shows the spectrum of these laser’s pulses, as measured with a
optical spectrum analyzer. This laser produces 500 ps pulses with a repetition rate
of approximately 80 MHz and a pulse picker is used for extinction of subsequent
pulses. The electric field produced by the laser drives the optical dipole transition
between 2S1/2 and
2P1/2 level in which the angle of this rotation on Bloch sphere is
given by:
θ =
µ
h¯
∫ T
0
E(t)dt =
µ
h¯
∫ T
0
√
I(t)dt (3.7)
66
P(
   
  )
(a) (b)
Figure 3.10: (a) Optical spectrum is measured and fit to a hyperbolic secant func-
tion. The width is consistent with transform limited 500 ps pulses in the time
domain. (b) Excitation probability to 2P1/2|mJ = −1/2〉 with a single pi polarized
pulse is measured by looking at the qubit state. From the excited level, decay prob-
abilities are 1/3 and 2/3 to the states |↓〉 and |↑〉, respectively. Hence, the maximum
probability of finding the qubit in the |↑〉 state approaches 66 % following the ex-
citation and spontaneous emission. Available optical power from the laser was not
sufficient to drive a complete pi pulse even with a considerably well focused beam of
5 µm waist. (Data is presented with permission from Toptica Inc.)
where T is the total pulse duration and I(t) is the optical intensity. For measuring
this transition probability, 138Ba+ initialized in |↓〉 is excited with a pi polarized
pulse to 2P1/2|mJ = −1/2〉. Then, probability of this excitation and creating a
single photon that is entangled with the atom is:
P (2P1/2|mJ = −1/2〉) = sin2(θ/2)
= sin2(α
√
I) . (3.8)
For this calculation, the intensity of light field is assumed to be constant through the
pulse duration, however, this assumption does not affect the fitting to experimental
results (see Fig. 3.10 (b)) as the overall factors on the intensity can be taken out of
the integral given in Eq. 3.7.
67
⇑
P(
   
  )
T (ms)
Figure 3.11: The Rabi flopping between the |⇓〉 and |⇑〉 states in 171Yb+ with
microwave radiation. Oscillating field is applied for a total time of T to the qubit
initialized in the |⇓〉 state.
3.4 Coherent Rotations with 355-532 nm Pulses
The 171Yb+ qubit states can be rotated by applying microwaves resonant with
the hyperfine splitting. Similarly, RF radiation is suitable for manipulation of 138Ba+
Zeeman qubits. These oscillating fields drive magnetic dipole transitions and Rabi
flopping between the qubit states. Hence, single qubit gates can be easily imple-
mented by directing a microwave horn or a inductive coil towards the trap and
applying a few watts of resonant radiation (see Fig. 3.11).
The long wavelengths of the microwave and RF fields result in negligible mo-
mentum transfer to atoms during interaction, therefore, they do not resolve the col-
lective motion of the ions and are not suitable for two qubit entanglement operations
through utilization of spin dependent forces [73]. Large magnetic field gradients can
be applied to increase the effective coupling to the motion for realization of multi
qubit entangling gates [74, 75] but this approach is beyond the scope of this thesis
68
0
1
e
∆
ω0
δ
ωe0 ωe1
ωα,kα ωβ,kβ
Figure 3.12: Diagram of two photon stimulated Raman transition between ground
states |0〉 and |1〉 via coupling to the excited state |e〉. Two fields that couple the
qubit states are detuned from the excited state by ∆.
work.
In contrast to microwave and RF fields, optical beams can impart considerable
momentum with two photon stimulated Raman transitions where the ground states
are coupled through a virtual level. Fig. 3.12 shows the general theme of this
operation and specific considerations for 171Yb+ and 138Ba+ qubits will be carried
out later in this section. Even though frequency combs of a pulsed laser are used
in the experiments conducted here, these optical fields can be treated as continuous
waves (CW) without loss of generality [37,45].
Two optical fields that drive electric dipole transitions between the states
69
|0〉 ↔ |e〉 and |1〉 ↔ |e〉 are given by:
~En(r) =
En
2
~n(e
i(kn·r−ωnt−φn) + e−i(kn·r−ωnt−φn)) (3.9)
with n = α, β and polarization ~n. This interaction Hamiltonian in a frame rotating
with ωβ and under the rotating wave approximation is [73]:
HI = −D · E
=
h¯
2
[(g0,αe
i(kα·r−δt−φα)|e〉〈0|+ h.c.) + (g1,βei(kβ ·r−φβ)|e〉〈1|+ h.c.) + ∆|e〉〈e|]
(3.10)
and single photon Rabi frequencies are gi,n = −Di · nEn/2h¯. With detuning from
the excited level much larger than the atomic linewidth and coupling terms (∆ 
γ, |gi,n|2) excited state population is adiabatically eliminated [76]. Under another
rotating wave approximation:
HI =
h¯
2
[(Ωe−i(∆k·r−δt−∆φ)|1〉〈0|+ h.c.) + χ0|0〉〈0|+ χ1|1〉〈1|] (3.11)
where ∆k = ~kα − ~kβ and ∆φ = φα − φβ. The two photon Rabi frequency is given
by Ω = g∗1,αg0,β/2∆ and AC Stark shifts are χi = [|gi,α|2 + |gi,β|2]/4∆.
The motion in the harmonic trap can be considered at this stage. By assuming
∆k is along the x direction, we are only interested in motion along this axis. The
position operators for two atoms are expressed as:
ri = xi + qi,IP(a+ a
†) + qi,OP(b+ b†) (3.12)
where xi is the equilibrium position, a, a
† and b, b† are the usual creation and
annihilation operators, for the in-phase and out-of-phase modes, respectively. When
70
two atoms are of the same mass:
q1,IP = q2,IP =
1√
2
√
h¯
2MωIP
q1,OP = −q2,OP = 1√
2
√
h¯
2MωOP
(3.13)
where M is the ion mass and ωIP , ωOP are the normal mode frequencies. These
coefficients are analyzed in detail for the unequal mass case in the next chapter. In
the vibrational mode interaction frame, the Hamiltonian for two ions (m=1,2) is:
HI =
h¯
2
∑
m=1,2
[
(Ωme
−i[ηm,IP(ae−iωIPt+a†eiωIPt)+ηm,OP(be−iωOPt+b†eiωOPt)]
×eiδte−i(∆kxm−∆φ)|1m〉〈0m|+ h.c.) + χ0,m|0,m〉〈0,m|+ χ1,m|1,m〉〈1,m|
]
(3.14)
where the Lamb-Dicke parameters are defined as η = ∆kq. The optical phases
are introduced on this interaction through e−i(∆kxm−∆φ). Since the two Raman
beams are originating from the same laser source, fluctuations in the optical phase
of the laser does not cause shot to shot variations. However, if the beams are
not propagating through the same optical path, for instance when it is required for
addressing motion to introduce a ∆k vector, relative optical path length fluctuations
might be a source of decoherence on long timescales. The optical paths are passively
stable on the majority of experiments presented here, which are generally around
100-200 µs. Even though these non-copropagating qubit rotations are sensitive
to the optical phases as analyzed here, multi qubit entangling gates can be made
insensitive by introducing special geometries [33,73].
In the resolved sideband limit (Ω  ωIP, ωOP), frequency difference between
the Raman beams can be tuned to drive transitions between different qubit states
71
and eigenstates of the harmonic motion [38]. With δ adjustments, a stationary term
emerges from the interaction Hamiltonian and other terms can be eliminated with
a rotating wave approximation.
With δ set to 0, including the AC Stark shift effects, the transitions between
the internal qubit states are driven while the motional quanta is preserved. This
carrier interaction is given by [38]:
HcarI =
h¯
2
∑
m=1,2
∑
nIP,nOP
(ΩmDm,nIP,nOPei(∆kxm−∆φ)σ(m)+ + h.c.)|nIP, nOP〉〈nIP, nOP|
(3.15)
where spin operators are σ+ = |1〉〈0|, σ− = |0〉〈1|, and the Debye-Waller factor
Dm,nIP,nOP = e−
1
2
(η2m,IP+η
2
m,OP)LnIP(η2m,IP)LnOP(η2m,IP) suppresses the carrier transition
for large phonon occupation states through Laguarre polynomial of order nIP or
nOP [38].
A collective motional state can be decreased or increased by one quanta accom-
panying a spin flip. These interactions are called red and blue sideband transitions,
respectively. Setting δ = −ωIP gives the red sideband interaction:
HRSBI =
h¯
2
∑
m=1,2
(ηm,IPΩme
i(∆kxm−∆φ)D′n,n−1σ(m)+ a+ h.c.) (3.16)
and with δ = ωIP, blue sideband interaction is:
HBSBI =
h¯
2
∑
m=1,2
(ηm,IPΩme
i(∆kxm−∆φ)D′n,n+1σ(m)+ a† + h.c.) (3.17)
where D′ is the sideband Debye-Waller factors [38, 73]. These interactions between
fields and the atoms respectively drive |0, n〉 ↔ |1, n− 1〉 and |0, n〉 ↔ |1, n+ 1〉
transitions.
72
3.4.1 The Rabi Frequencies
The analysis above only considered the single excited level. In reality, there
are multiple levels involved in these stimulated Raman transitions on 171Yb+ and
138Ba+ atoms. These paths and their detunings from the excited level are shown in
Fig. 3.13 and the Rabi frequency depends on the coupling strength to these levels
and the detuning from them:
Ω =
∑
i
g∗0,ig1,i
2∆i
. (3.18)
The single photon Rabi frequencies, gij, show the coupling strength and are given
by [76]:
gij = Cγ
√
I
2Isat
(3.19)
where C is the Clebsch-Gordan coefficient, Isat is the saturation intensity and γ is
the natural linewidth of the transition.
Using the coupling coefficients given in Figs. 3.3 and 3.1 for the paths given
in Fig. 3.13 result in:
ΩYb =
I
12
[
− γ
2
1
∆1Isat,1
+
γ22
∆2Isat,2
]
(3.20)
ΩBa =
I
√
2
12
[
− γ
2
1
∆1Isat,1
+
γ22
∆2Isat,2
]
(3.21)
where subscript 1 (2) is used to describe parameters corresponding to 2P1/2 (
2P3/2)
level of the corresponding atom.
Plugging in the atomic parameters given in Table 3.1, the Rabi frequencies
73
2S1/2
2P1/2
2P3/2
532 nm
355 nm
44 THz
94 THz
66 THz
34 THz
138Ba+ 171Yb+
⇑     
⇑
     
↓     
↓
     
Figure 3.13: 532 and 355 nm photons off-resonantly couple to both 2P1/2 and
2P3/2
levels for stimulated Raman transitions. Detuning from the excited levels are shown
for these wavelengths.
171Yb+ 138Ba+
Excited Level 2P1/2
2P3/2
2P1/2
2P3/2
Natural Linewidth γ/2pi (MHz) 19.7 25.8 15.1 17.7
Saturation Intensity Isat (mW/cm
2) 51.0 95.1 16.4 35.7
Detuning ∆/2pi (THz) 34 -66 -44 -94
Table 3.1: Natural linewidth, saturation intensity and detuning from excited levels
for 2P3/2 and
2P1/2 levels. Saturation intensity is calculated using Isat =
h¯γω3e
12pic2
where
ωe is the resonant transition frequency from the ground state.
74
are:
ΩYb = I
(
− 2.75× 10−2 Hz
mW/cm2
)
(3.22)
ΩBa = I
(
2.62× 10−2 Hz
mW/cm2
)
. (3.23)
Even though the destructive interference between the two paths in 138Ba+ decreases
the Rabi frequency, lower saturation intensities and stronger coupling due to larger
dipole matrix elements result in a similar stimulated Raman transition rate for
171Yb+ and 138Ba+ with an equal 355 and 532 nm optical intensity.
3.4.2 Differential Stark Shifts
In addition to driving transitions, the optical fields cause AC Stark shifts of
the qubit states. Frequencies can be adjusted to account for this change in the
qubit splitting. However, intensity fluctuations might introduce detuning errors in
this case, therefore these Stark shift effects should be avoided if possible to prevent
errors in the qubit operations.
In 171Yb+ , these shifts are [76]:
χ⇓ =
I
24
[ γ21
∆1Isat,1
+
2γ22
∆2Isat,2
]
(3.24)
χ⇑ =
I
24
[ γ21
(∆1 − ωhf )Isat,1 +
γ22
2(∆2 − ωhf )Isat,2 +
3γ22
2(∆2 + ω
′
hf − ωhf )Isat,2
]
(3.25)
where ω
′
hf is the hyperfine splittings in
2P3/2 manifold. For
138Ba+ , with the Raman
75
beams polarized as shown in Fig. 3.13 they are:
χ↓ =
I
24
[ γ21
∆1Isat,1
+
2γ22
∆2Isat,2
+
3γ22
∆2Isat,2
]
(3.26)
χ↑ =
I
24
[ γ21
∆1Isat,1
+
2γ22
∆2Isat,2
+
2γ21
∆1Isat,1
+
γ22
∆2Isat,2
]
(3.27)
where first two terms in each expression are contributions of the pi polarized compo-
nent and the rest are from σ− light. Small Zeeman shifts on the levels are ignored
for simplicity as ∆ δZeeman. From symmetry, pi light shifts the levels by the same
amount and the differential shift comes mostly due to the circular polarized field.
Comparing these shifts to the effective coupling rate of the qubit states gives:
χ⇓ − χ⇑
ΩYb
= 1.87× 10−4 (3.28)
χ↓ − χ↑
ΩBa
= 0.707 . (3.29)
This significant differential Stark shift on 138Ba+ can be minimized by introducing
σ+ light to equalize the shifts on the |↑〉 and |↓〉 states as shown in Fig. 3.14. Even
though this results in a lower Rabi frequency, detuning errors that might stem from
long term intensity drifts can be minimized with this approach.
3.4.3 Re-pumping from 2D5/2 Level with an Orange LED
The spontaneous emission rate is very small during stimulated Raman inter-
actions as excited states are negligibly populated with large detuning from these
levels. These rates are given by [77]:
ΓY b =
2piI
24
[ γ31
∆21Isat,1
+
2γ32
∆22Isat,2
]
(3.30)
ΓBa =
2piI
8
[ γ31
∆21Isat,1
+
γ32
∆22Isat,2
]
(3.31)
76
σ−
π
Figure 3.14: The Rabi frequency (Ω) and corresponding shift in the qubit splitting
(∆χ = χ↓ − χ↓) is measured on the 138Ba+ atom where 532 nm pulses are used for
the rotation. Blue line shows a fit to measurement where only pi and σ− field is
incident for transition. This result in a considerable differential shift with a slope
of ∆χ/Ω = 0.69, which agrees with the calculated value of 0.707. Rotation of the
σ− polarized light to linear results in an equal σ+ contribution and reduces the
differential shift. Red squares shows the measurements with this adjustment, and a
fit gives ∆χ/Ω = 0.02. With more sensitive measurements, it might be possible to
get even lower differential shifts on 138Ba+ qubit.
77
where the configuration that minimizes the differential Stark shift for the 138Ba+
atom is considered. Consequently, the probability of spontaneous emission during a
pi pulse calculated using Eqns. 3.21 and 3.31 gives:
PYb = 3.2× 10−6 , PBa = 5.9× 10−6. (3.32)
Even though these probabilities are negligible for each individual rotation,
the atomic level structure of the 138Ba+ atom introduces another complexity to
the system. The atom decays to the 2S1/2 or
2D3/2 manifolds after spontaneous
emission from 2P1/2. These events can be considered as a quantum gate error and
operations restart in the next experimental cycle. However, decays from 2P3/2 might
end up in 2D5/2, and the
138Ba+ qubit will stay in this long lived metastable level
for τ = 31 s, resulting in down time on the system if a re-initialization mechanism
is not available. Only considering spontaneous emission from the 2P3/2 level and
accounting for branching ratio of 0.215 (see Fig. 2.4), the probability of populating
2D5/2 during a pi pulse is 1.8 × 10−7. Experimentally, the 138Ba+ atom is observed
to go dark at a rate of a few per minute with the 532 nm laser in use.
To solve this issue, we have utilized a high intensity orange LED (Luxeon Star
617nm) on the optical setup which deshelves the atom from 2D5/2 levels through
2P3/2 with absorption of a 614 nm photon (see Fig. 3.15). The available optical
power that can address this dipole transition is estimated to be ∼100 nW, only
∼ 10−7 of the total 1W due to the large spectral distribution of the emitted light
from the LED. With a measured ∼30 ms average deshelving time, this layout is
sufficiently fast compared to the timescale over which the population gets trapped
78
6 2S1/2
6 2P1/2
6 2D3/2
6 2P3/2
5 2D5/2
49
3 
nm
39
9 
nm
416 nm
614 nm
(a) (b)
∆ = 19 THz
τ ≈ 30 ms
Figure 3.15: (a) Instead of relying on a 532 nm laser for trapping population at the
2D5/2 level, a 399 nm Yb photo-ionization laser is used for faster characterization of a
617 nm LED. With both 493 nm and 399 nm laser beams incident on the 138Ba+ , the
ion is observed to go dark with an average rate of 0.5 s−1. Suspected mechanisms that
result in this behavior are shown here. (b) The fluorescence photons are monitored
to measure average deshelving speed with the orange LED. While the cooling and
Yb photo-ionization lasers are incident, a drop in the number of detected photons in
a certain time window signals the occupation of the 2D5/2 level. During this process,
the orange LED is kept on and the time the 138Ba+ atom stays dark is measured
which corresponds the deshelving timescale.
79
and therefore provides a cheap and easy alternative to a laser that might otherwise
be used for the same purpose.
3.4.4 Crosstalk Between the Qubits and Raman Beams
Crosstalk between 355 nm (532 nm) beams and the 138Ba+ (171Yb+) qubit
needs to be considered in our system as they couple to the 2P1/2 and
2P3/2 levels
with a detuning of 238 THz (248 THz) and 187 THz (347 THz). This results in
relative two photon Rabi frequencies:
ΩBa,355
ΩBa,532
=
1
187
− 1
238
1
44
− 1
94
= 0.109 ,
ΩY b,532
ΩY b,355
=
1
248
− 1
347
1
34
+ 1
66
= 0.026 (3.33)
on 138Ba+ and 171Yb+, respectively, for equal 355 nm and 532 nm optical intensities
on the Raman beams.
Since the qubit splitting frequencies are different, when set for driving tran-
sitions on their corresponding qubit, the 355 (532) nm Raman beams are unlikely
to address the transitions on 138Ba+ (171Yb+ ) qubit. However, due to usage of the
pulsed laser, there might be unwanted comb teeth combinations that drive crosstalk
transitions. This issue can be circumvented by adjusting the magnetic field ampli-
tude to change the 138Ba+ qubit frequency or the repetition rate of the pulsed laser
if the cavity is accessible.
Additionally, the number of possible cross transitions might be partially re-
duced by setting the polarization specifically for each of the individual qubits. For
instance, elimination of the pi polarized component on the 355 nm beams effectively
suppresses any transition they can drive on 138Ba+ qubit. In contrast, the same
80
approach does not apply to 532 nm beams as they need to have both pi and σ−
components for rotations on the 138Ba+ qubit.
81
Chapter 4: Normal Mode Structure of Ytterbium-Barium Ion Chain
In this chapter, the collective motion of the 171Yb+ – 138Ba+ Coulomb chain
in a linear Paul RF trap is reviewed. The modes of motion play an essential part in
entanglement of nearby qubits [14, 28, 78, 79], hence it is important to understand
how confinement dynamics are affected with unequal mass of the ions in a multi-
species system.
4.1 Normal Mode Eigenfrequencies and Eigenvectors
These calculations follow the methods described in Ref. [80–82] where the
atoms are considered to be well localized around their equilibrium position and
subject to small vibrations compared to separations in between them.
4.1.1 Longitudinal Modes
First, we look at the longitudinal direction in which the confinement is the
weakest and consequently the atoms line up on this axis. The total kinetic energy
is:
T =
m1x˙1
2
2
+
m2x˙2
2
2
(4.1)
82
where atom 1 (2) position is defined as x1 < 0 (x2 > 0). Potential energy is given
by the expression:
U =
kx21
2
+
kx22
2
+
1
x2 − x1 (4.2)
with trap spring constant k = m1ω
2
1 = m2ω
2
2. Here, the angular frequencies ω1,2
correspond to motional frequency of single confined atom. Equilibrium positions
are determined by:
∂U
∂x1
= 0⇒ x(0)1 = −(1/2k2)2/3
∂U
∂x2
= 0⇒ x(0)2 = (1/2k2)2/3 (4.3)
which are independent of the mass since the trapping potential only interacts with
the charges. Assuming a harmonic collective motion around the equilibrium posi-
tions, potential energy can be approximated with a Taylor expansion truncated to
second order:
U ≈ 1
2
∂2U
∂x21
∣∣∣
0
x21 +
1
2
∂2U
∂x22
∣∣∣
0
x22 +
∂2U
∂x1∂x2
∣∣∣
0
x1x2 (4.4)
where the first derivative terms vanish at equilibrium and positions are replaced with
displacements xi−x(0)i → xi. Dynamics are described by the Lagrangian L = T −V
and equations of motion are found from:
− ∂
∂t
∂T
∂x˙i
=
∂U
∂xi
. (4.5)
and putting Eqns. 4.1, 4.4 here result in:
−m1x¨1 = ∂
2U
∂x21
∣∣∣
0
x1 +
∂2U
∂x1∂x2
∣∣∣
0
x2
−m2x¨2 = ∂
2U
∂x1∂x2
∣∣∣
0
x1 +
∂2U
∂x22
∣∣∣
0
x2 (4.6)
83
These equations are represented in the following matrix form:
ω21
 2 −1
−1/α 2/α

x1
x2
 = ω2
x1
x2
 (4.7)
where ansatz xi(t) = xie
iωt is used and α = m2/m1. For equal mass ions, α = 1, this
eigenvalue problem has the well-known center of mass and breathing mode solutions
with the values of:
XCM =
1/
√
2
1/
√
2
 , ωCM = ω1
XBR =
 1/
√
2
−1/√2
 , ωBR = √3ω1 (4.8)
In the general case, the eigenvalues are:
ωOP = ω1
√√
α2 − α + 1 + α + 1
α
ωIP = ω1
√
−√α2 − α + 1 + α + 1
α
(4.9)
with corresponding normalized eigenvectors:
XOP = (− α +
√
(α− 1)α + 1− 1√
(α +
√
(α− 1)α + 1− 1)2 + 1
,
1√
(α +
√
(α− 1)α + 1− 1)2 + 1
)
XIP = (
−α +√(α− 1)α + 1 + 1√
(−α +√(α− 1)α + 1 + 1)2 + 1 ,
1√
(−α +√(α− 1)α + 1 + 1)2 + 1)
(4.10)
Instead of center of mass and breathing labels, a more appropriate in-phase (IP)
and out-of-phase (OP) description is used here. Mass ratio dependence of mode
motional amplitudes and frequencies are shown in Fig. 4.1 and 4.2.
84
x1
x2 x1
x2
Figure 4.1: Dependence of motional amplitudes x1,2 to mass ratio for in-phase (IP)
and out-of-phase (OP) modes. Dashed line shows 138Ba+ -171Yb+ mass ratio of
≈ 0.81 where XIP = (0.743, 0.668) and XOP = (−0.587, 0.809)
ωIP
ωOP
Figure 4.2: In-phase and out-of-phase mode frequencies normalized to single ion
frequency of ω1. For
138Ba+ -171Yb+ corresponding frequencies are ωIP = 1.05 ω1
and ωOP = 1.84 ω1.
85
4.1.2 Transverse Modes
Stronger confinement in the transverse direction compared to longitudinal axis
results in reduced sensitivity to heating and thermal motion in entanglement oper-
ations [83]. Hence, it might be more advantageous to use these modes in a multi-
species module.
Kinetic energy along this direction is given by:
T =
m1z˙1
2
2
+
m2z˙2
2
2
(4.11)
and potential energy is:
U =
kT z
2
1
2
+
kT z
2
2
2
+
1
|r2 − r1| (4.12)
where kT = m1ω
2
T and ωT is the transverse frequency of single ion of mass m1.
The Coulomb component can be approximated with a Taylor expansion assuming
displacement in the transverse direction is small compared to the separation between
the ions:
1
|r2 − r1| =
1√
(x1 − x2)2 + (z1 − z2)2
≈ 1
∆x
(
1− (z1 − z2)
2
2∆x2
)
=
1
∆x
− 1
2∆x3
(
z21 + z
2
2 − 2z1z2
)
(4.13)
The equilibrium position is again found by ∂U
∂zi
= 0 giving z
(0)
1 = 0 and z
(0)
2 = 0.
Using equilibrium positions found in Eqn. 4.3 for ∆x3, the Lagrange equation
of motion is: ω2T −
ω21
2
ω21
2
ω21
2α
ω2T
α
− ω21
2α

z1
z2
 = ω2
z1
z2
 (4.14)
86
z1
z2
z2
z1
Figure 4.3: Dependence of motional amplitudes z1,2 to mass ratio for in-phase (IP)
and out-of-phase (OP) modes. Dashed line shows 138Ba+ -171Yb+ mass ratio of
≈ 0.81 where ZIP = (0.256, 0.966) and ZOP = (0.950,−0.312)
ωIP
ωOP
Figure 4.4: In-phase and out-of-phase mode frequencies in transverse direction.
With single ion 171Yb+ frequencies of ωT = 2.5 MHz and ω1 = 0.9 MHz, the
138Ba+
-171Yb+ chain frequencies are ωIP = 2.715 MHz and ωOP = 2.39 MHz.
For equal mass, α = 1, this reproduces ZIP = (1/
√
2 , 1/
√
2) with ωIP = ωT
and ZOP = (1/
√
2 , −1/√2) with ωOP =
√
ω2T − ω21. Motional amplitudes and
mode frequencies for two ions with mass ratio m2/m1 = α and typical single ion
confinement potentials of ωT = 2.5 MHz and ω1 = 0.9 MHz are shown in Fig. 4.3
and 4.4.
Comparison of Fig. 4.1 and 4.3 reveal that, even with relatively close mass
of 171Yb+ and 138Ba+ , there is a big mismatch in motional amplitudes for a given
87
transverse mode, whereas this mismatch is not as significant for longitudinal modes.
Since the coupling of external fields to motional modes scales as η = ∆~k · Z, the
mismatch in mode amplitudes results in reduced entangling gate speeds as TGate ∝
1
ηΩ
[84, 85] where Ω is the two photon Rabi frequency proportional to intensity I
of the fields used and ∆~k = ~k1 − ~k2. Limitations in available optical power in our
setup resulted in switching to longitudinal modes to benefit from smaller mismatch
for faster gate operations.
Before getting to experimental measurements, I present how the motional am-
plitudes are affected by confining potentials to complete this analysis. From Eqn.
4.10, it is clear that relative motion of the ions are not affected by the confining
potentials and only depend on the mass ratio. The trapping potential determines
the spread of the wavefunction,
√
h¯
2mω
, which is a common factor for both ions.
In contrast, transverse mode structure depends on both the mass and confinement
ratios, ω1
ωT
, and Fig. 4.5 shows this dependence for the particular mass ratio of
α = mBa
mY b
≈ 0.81.
As an additional note, a similar analysis shows that as the number of ions in
the crystal chain increase, the mismatch on motional modes tend to decrease and
utilization of transverse modes might become more practical.
4.2 Spectroscopy Measurements
The analysis of previous section is experimentally verified by resolving the
motion of the ions with Raman spectroscopy. When two laser beams have a ∆~k
88
z2
z1
z1
z2
Figure 4.5: Weak longitudinal confinement results in bigger mismatch for the mo-
tional amplitudes in transverse modes. Therefore, increase in DC voltages to create
a tighter trap in longitudinal direction indirectly helps in motional amplitude match-
ing of 171Yb+ and 138Ba+ in transverse directions. However, as ω1 approaches ωT ,
this analysis is likely to fail since the ions might realign along a weaker axis.
vector along a motional axis, transitions between different Fock states can be driven
and the resonant Rabi frequencies are given by [38,73]:
Ωn,n ≈ Ω0(1− η2n)
Ωn,n−1 ≈ Ω0η
√
n
Ωn,n+1 ≈ Ω0η
√
n+ 1 (4.15)
where Ω0 is the base Rabi frequency, η = ∆~k · ~R is the Lamb-Dicke parameter
representing the coupling strength between the fields and the motion and n is the nth
eigenstate of the harmonic potential. The Rabi frequencies in Eqn. 4.15 correspond
to carrier, red and blue sideband transitions in the given order.
Fig. 4.6 presents the beam geometry for addressing longitudinal and transverse
modes. The 355 and 532 nm Raman beams coming from each direction are combined
on the same optical path using suitable dichroic filters. Spatial adjustments are done
on individual beams before combining to correct for the chromatic aberration of the
89
Yb+ Ba+ Yb+ Ba+
(a) (b)
x
z
Figure 4.6: 355 nm (purple) and 532 nm (green) beams can address the (a) trans-
verse, z, and (b) longitudinal, x, modes when there is a ∆~k vector along the motional
axis. By changing the direction of only one set of beams, switching between the two
axes is possible.
final lens to have both beams focus at the same spot.
The frequency of these Raman beams are scanned, keeping the time and in-
tensity constant, to reveal the motional spectrum (see Fig. 4.7). Even though
traditionally transverse modes have been used in the previous works on this setup
[13, 33, 45, 86], and among the Monroe group in general, a switch to longitudinal
mode took place to benefit from better coupling of the ions to external fields.
4.3 Sub-Doppler Cooling with Electromagnetically Induced Trans-
parency
Uncontrolled thermal motion of the ions can introduce decoherence into the
system during interaction with external fields if the motional amplitude is compara-
90
Out-of-phase Mode
In-phase Mode
Yb+ Ba+
Yb+ Ba+
Mismatch 0.25 : 0.96
Yb+ Ba+
Yb+ Ba+
Out-of-phase Mode
In-phase Mode
Mismatch 0.74 : 0.67
Mismatch 0.59 : -0.81
(a)
(b)
Frequency (MHz)
Frequency (MHz)
IPz
IPOP IP OP
Mismatch 0.95 : -0.31
IPzIPy IPyOPy OPy
OPz OPz
P t
ra
ns
.
P t
ra
ns
.
Figure 4.7: Motional spectrum centered around the carrier and negative (positive)
frequencies correspond to red (blue) sideband transitions. Purple (Blue) line shows
transition probability on 171Yb+ (138Ba+ ) with 355 (532) nm Raman beams. (a)
Both y and z transverse directions are resolved since ∆~k had projection to both
axes. In transverse directions, big mismatch between 171Yb+ and 138Ba+ motional
amplitudes is evident from the width difference of spectral lines corresponding to IP
and OP modes. (b) Closer motional amplitudes in the longitudinal direction results
in similar width lines for both the IP and OP modes.
91
ble to the field wavelength. These errors are suppressed in the Lamb-Dicke regime
(η2(n¯ + 1)  1) [87], and achieving high fidelity quantum logic gates generally re-
quires cooling motional modes nearly to the ground state (n¯ ≈ 0) [88]. Heating
mechanisms, such as electric field noise [38], result in divergence from these limits
and might be a limiting factor in quantum information processing in which long
algorithms are implemented.
In a multi-species apparatus, 138Ba+ atoms can be utilized to be refrigerants as
well to sympathetically cool the memory qubits in between quantum logic operations
[89]. With Doppler cooling, the steady state temperature is limited to [90]:
kBT =
h¯γ
2
. (4.16)
At this temperature, average phonon occupation numbers approximate to:
h¯ωn¯ ≈ h¯γ
2
⇒ n¯ ≈ γ
2ω
(4.17)
in which a typical motional mode frequency of ω/2pi = 2 MHz results in n¯ ≈ 4 with
γ/2pi = 15.1 MHz for the 138Ba+ atomic ion.
The heat can be further dissipated from Doppler cooled atoms by using nar-
rower transitions [91], Raman sideband cooling [92] or electromagnetically induced
transparency (EIT) techniques [93]. Compared to others, the EIT method can
address a wider frequency range to simultaneously cool multiple modes and also
provides faster cooling rates [94].
The EIT cooling mechanism is illustrated in Fig. 4.8. In a Λ system with one
92
excited and two ground states, a strong field creates atom-laser dressed states [95]:
|+〉 = cosθ|e〉+ sinθ|g1〉 (4.18)
|−〉 = sinθ|e〉 − cosθ|g1〉 (4.19)
(4.20)
where
tanθ =
√
∆2 + Ω2 −∆
Ω
(4.21)
and the frequencies are shifted from bare states by:
δ =
1
2
(
√
∆2 + Ω2 −∆). (4.22)
A weak field (Ω
′
< Ω) couples the state |g2〉 to dressed states with a Fano-like
resonance [96,97]. The coherent dark state at ∆ = ∆
′
is utilized in suppressing the
carrier transitions while a sharp resonance selectively addresses the red sideband
|g2, n〉 → |+, n− 1〉 when the induced light shift is set to the motional frequency,
δ = ω. Therefore, each excitation event reduces the phonon number by one quanta
and decays from |+〉 back to ground states and restarts the process. The minimum
average steady-state phonon number is given by [95]:
n¯ =
( γ
4∆
)2
(4.23)
with a maximum cooling rate of:
Wmax ≈ γω
4
√
∆2 + Ω2
. (4.24)
Depending on the requirements in the system, ∆ can be adjusted to find a compro-
mise between the cooling rate and the minimum temperature achievable.
93
∆+
+, n
+, n-1
g1
g2 g2
g2,n
g2,n-1
g2,n+1
(a) (b) (c)
δ
e _
_
_
, n
, n+1δ
ω
Ω Ω’
∆’
Figure 4.8: (a) Setting of the EIT cooling scheme on a Λ system, for instance utilizing
one of the 2P1/2 and both
2S1/2 levels. (b) A strong beam introduces dressed states
that are shifted by δ from the bare state frequencies, and a weaker probe beam
couples |g2〉 to |−〉 and |+〉 with a Lorentzian and Fano-like excitation spectrum,
respectively. At ∆ = ∆
′
, the absorption goes to zero for an ion at rest due to dark
resonance, which results in suppression of carrier transitions. (c) The harmonic
motion in the trap results in frequency modulation of the EIT beams if they have a
∆~k along the motion axis, and red sideband transitions are driven when δ = ω. Or
equivalently, missing δ from the narrow resonance is supplied by the motion of the
ion.
94
While the 2S1/2 ↔ 2P1/2 ↔ 2D3/2 configuration of 138Ba+ can possibly be used
as the Λ system for the EIT cooling, the laser frequency stabilization requirements
for realizing ∆ = ∆
′
makes it impractical since Ω and Ω
′
are provided with two
different lasers in this case. Instead, we use the 2S1/2 and
2P1/2 levels similar to
Ref. [29], in which the correspondence to the general picture of Fig. 4.8(a) is:
|g1〉 → |↓〉 , |g2〉 → |↑〉 , 2P1/2 |mJ = +1/2〉 → |e〉 . (4.25)
To prevent population trapping in 2D3/2 states, the 650 nm laser is kept on during
the EIT process but it does not participate in dynamics as the frequency difference
between the red and green lasers is approximately ∆ and does not introduce extra
coherent states.
In normal operating conditions, the frequency of the 493 nm laser is locked to:
(ωge − ωlaser)/2pi ≈ 80 MHz (4.26)
and the AOMs are used to shift this set frequency close to resonance for operations
described in the previous chapter. EIT beams are created by using AOMs with
a center frequency around 200 MHz, which results in ∆/2pi ≈ 120 MHz. The
strong σ+ beam that dresses the states is sent through the same fiber which is used
for resonant circular polarized beams of state detection/initialization operations as
shown in Fig. 3.7. For the weak probe field, the pi polarized beam is sent through
another fiber such that ~kσ −~kpi has a component along the motional axis of interest
(set for cooling longitudinal motion in this thesis work). Additionally, the AOM
drive frequency difference needs to match the splitting between |↓〉 ↔ |↑〉 to satisfy
∆ = ∆
′
.
95
Even though the 493 nm laser is subject to frequency fluctuations since it is
only passively stabilized, these drifts have negligible effect on EIT cooling perfor-
mance. First, the Stark shift, δ, on the dressed state is first order insensitive to
changes in detuning, ∆, as evident from Eqn. 4.22 with typical ∆  δ = ω and
∆ > Ω conditions. In addition, as both (Ω, σ+,∆) and (Ω
′
, pi,∆
′
) fields are pro-
duced from the same laser, detuning shifts are common mode. Hence, the ∆ = ∆
′
optimum condition is satisfied for EIT cooling even with a noisy laser source. It
should be noted that, due to magnetic field noise, there can be deviations from this
ideal situation. However, with typical noise amplitude of the order 1 mG, 138Ba+
qubit splitting frequency only changes a few kHz which is not resolved with wide
EIT features.
Fig. 4.9 shows the absorption spectrum and time dynamics of the EIT cool-
ing scheme employed on a co-trapped 171Yb+ -138Ba+ crystal. The atoms are first
Doppler cooled and then EIT cooling is applied to 138Ba+ while 171Yb+ is kept
on dark. The temperature of the crystal is extracted from the measured asymme-
try on the longitudinal sidebands [38] with non-copropagating Raman transitions on
171Yb+ with 355 nm pulses. Additionally, Fig. 4.10 shows the longitudinal spectrum
of a Doppler and EIT cooled multi-species chain where 171Yb+ is used to extract
information on the collective motion.
Using the tools demonstrated here, it should be possible to sympathetically
cool 171Yb+ atoms using 138Ba+ in between quantum logic gates for achieving higher
fidelity operations. Since the qubit used for communication purposes is re-initialized
before every entanglement trial, possible crosstalk between refrigerant and communi-
96
(∆’ − ∆) / 2π (MHz) T (ms)
Av
er
ag
e 
Ph
on
on
 Q
ua
nt
a 
(n
)-
A
bs
or
bt
io
n 
(a
rb
.)
Ca
rr
ie
r
BS
B
RS
B
(a) (b)
Figure 4.9: (a) The frequency of pi polarized beam (∆
′
) is scanned, while keeping ∆
fixed, to reveal Fano-like spectrum. A simple threshold method is used for defining
absorption units, where the vertical axis depicts the fraction of events in which
the detected number of 493 nm photons from 138Ba+ in a time window of 200 µs
is above the background level. When ∆ = ∆
′
, carrier and blue sideband (BSB)
transitions are suppressed and red sideband (RSB) transitions dissipate phonons
from the crystal. With the broad spectrum spanning a few MHz, multiple modes
can be cooled simultaneously. (b) The average phonon number in OP mode with
frequency ωOP/2pi ≈ 1.8 MHz is read out using 171Yb+ after an EIT cooling duration
of T on neighbor 138Ba+ . The final average phonon quanta reach n¯ ≈ 0.06 in about
300 µs.
97
⇑
P(
   
  )
⇑
P(
   
  )
(a)
(b)
Freq. (MHz)
Figure 4.10: Partial spectrum of 171Yb+ -138Ba+ chain and Rabi flops on 171Yb+
carrier transition with non-copropagating 355nm Raman beams on (a) Doppler and
(b) EIT cooled atoms. Duration of the Raman interaction is 30 µs for both scans.
Peaks around 1 MHz correspond to IP mode, ωIP, which is cooled to n¯ ≈ 0.1 with
the EIT technique. Weaker peaks around 0.8 MHz correspond to phonon exchange
frequencies between different modes, ωOP−ωIP. Since the phonon occupation num-
ber effects carrier Rabi frequency (see Eqn. 4.15), oscillation quickly vanishes in a
Doppler cooled atom due to timescale differences between oscillations corresponding
to different Fock states. In contrast, the oscillations persist for longer with the EIT
cooling employed.
98
cation 138Ba+ qubits would not pose the same problem encountered between memory
and communication qubits. Therefore, 138Ba+ atoms can be utilized for both tasks
in the same ion chain.
99
Chapter 5: Phonon Mediated Interactions between Ytterbium and
Barium Ions
The analysis in the previous chapter shows that the Coulomb force between
proximate ions results in quantized collective motion in a RF Paul trap. These
phonon modes can be utilized directly in the Cirac-Zoller (CZ) scheme as a bus to
transfer information between communication and memory qubits [28, 85] or spin-
dependent forces can be used in Mølmer-Sørensen (MS) interactions [78] for gener-
ating entanglement between 171Yb+ – 138Ba+ and several of these entangling gates
can be combined for information swap operations [67]. In this chapter, I analyze
these two different interactions and present our experimental results on a 171Yb+ –
138Ba+ chain.
5.1 Cirac-Zoller Interactions
In 1995, Cirac and Zoller proposed using common modes of motion as a bus
to introduce a conditional phase shift on qubit states for entanglement operations
between two ions on a string [85]. Stringent requirements of achieving ground state
occupation in the collective motional mode and individual addressing to qubits
makes this scheme less attractive compared to MS operations (to be discussed in
100
detail in next section) in quantum information processing. Nevertheless, the CZ
interaction picture offers a conceptually simple way of transferring information from
one qubit to the other and it is worth investigating.
5.1.1 Coherent Information Transfer Operation
Since the 355 nm and 532 nm Raman beams only drive transitions on their
respective 171Yb+ and 138Ba+ ions, the individual addressing requirement is easily
achieved without any particular effort on creating tightly focused beams. In addi-
tion, the EIT cooling described previously provides near ground state initialization
for better performance on CZ interactions.
Starting with 138Ba+ in a general superposition state that is to be transferred
to an initialized 171Yb+ qubit:
|ψ〉Ba+ = α|↓〉+ eiφ0β|↑〉 , |ψ〉Y b+ = |⇓〉 (5.1)
and the collective motion bus in the ground state, |n〉 = |0〉, gives the joint state of:
|Ψ1〉 = α|↓〉|⇓〉|0〉+ eiφ0β|↑〉|⇓〉|0〉. (5.2)
Application of a RSB pi rotation on the 138Ba+ qubit leads to:
|Ψ2〉 = α|↓〉|⇓〉|0〉 − ei(φ0−φ532)β|↓〉|⇓〉|1〉
= |↓〉|⇓〉(α|0〉 − βei(φ0−φ532)|1〉) (5.3)
where the 138Ba+ superposition is mapped to the collective motion with an addi-
tional optical phase difference between the two beams driving the transition (φ532 =
101
φRSB,1 − φRSB,2) imprinted on the state. Since the state |n〉 = |−1〉 does not ex-
ist, the α portion remains unchanged with this operation while β part undergoes
|↓〉|1〉 ↔ |↑〉|0〉 transition. Note that the realization of |n〉 = |0〉 is critical here for
perfect mapping of qubit state to motion. This state acquires a phase under time
evolution:
|Ψ2(τ)〉 = |↓〉|⇓〉(α|0〉 − βei(φ0−φ532+φτ )|1〉) (5.4)
where φτ = ωmτ with the motional mode frequency of ωm. The superposition is
mapped onto 171Yb+ by application of another RSB pi pulse, this time using 355 nm
Raman beams on the 171Yb+ :
|Ψ3〉 = |↓〉(α|⇓〉|0〉 − ei(φ0−φ532+φτ+φ355)β|⇑〉|0〉)
= |↓〉|0〉(α|⇓〉 − ei(φ0−φ532+φτ+φ355)β|⇑〉) (5.5)
where φ355 is the optical phase coming from the 355 nm laser fields. This step
finalizes the state transfer from 138Ba+ to 171Yb+ . In addition to amplitudes α and
β, the phase information needs to be transfered coherently for a full state mapping.
These phase issues will be investigated in the next section in conjunction with the
experimental setup.
5.1.2 Experimental Results
For the experimental demonstrations, we followed the procedure outlined in
Ref. [28] where the operation is performed between 9Be+ and 27Al+. In the first
experiment, the 138Ba+ qubit is prepared in a superposition state with the stimulated
102
R(T)
RSB
π
RSB
π
Readout
EIT
Ba+
Yb+
n
(a) (b)
⇑
P(
   
  )
Figure 5.1: (a) Experimental sequence. (b) The Rabi flop on 138Ba+ is readout on
171Yb+ after state mapping. Yellow squares and blue circles show measurements
from two different ion traps with average quanta of n¯ ≈ 0.1 and n¯ ≈ 0.5 after EIT
cooling, respectively. Poor cooling behavior in the trap with the higher phonon
occupation number is attributed to the observed excessive heating rate of ˙¯n ≈ 5
ms−1.
Raman transition using 532 nm laser beams with duration T:
|ψ〉Ba+ = R(T, φ0)|↓〉 = cos
(ΩT
2
)
|↓〉+ eiφ0sin
(ΩT
2
)
|↑〉 (5.6)
where Ω is the usual two photon Rabi frequency and φ0 is the optical phase. Hence,
this simply corresponds to the general picture with:
cos
(ΩT
2
)
→ α,
(ΩT
2
)
→ β . (5.7)
Afterwards, RSB pi pulses transfer the state from 138Ba+ to 171Yb+ where it gets
read out as shown in Fig. 5.1. Since the phase of the final state has no effect
on α and β coefficients, this measurement does not reveal coherence properties of
this transfer operation. The final temperature has a significant effect in qubit state
transfer fidelity, which is a big limitation of the CZ technique as demonstrated.
A separate experiment is conducted to verify that the phase written on the
original qubit is preserved after the transfer operation onto the destination qubit (see
103
R(π/2)
RSB
π
RSB
π
EIT
Ba+
Yb+
n
(a) (b)
⇑
P(
   
  )
R(π/2)
Delay
(T)
Figure 5.2: (a) Sequence of the Ramsey experiment. (b) A delay is imposed while
the information is held on the phonon bus. Therefore, Ramsey fringes oscillate
with frequency of the utilized mode, and measurement is in good agreement with
ωm/2pi ≈ 1.8 MHz of OP mode used here.
Fig. 5.2). Instead of continuously Rabi flopping the 138Ba+ qubit, phase evolution
ωmT is probed with fixed pi/2 pulses in a Ramsey experiment. Applying a pi/2
transition with phase φF to
171Yb+ in Eqn. 5.5 and replacing α = β = 1√
2
for the
initial rotation on 138Ba+ , the final state is:
|ΨF 〉 = 1
2
|↓〉|0〉(|⇓〉+ eiφF |⇑〉 − ei(φ0−φ532+φT+φ355)(|⇑〉 − e−iφF |⇓〉))
=
1
2
|↓〉|0〉[(1 + eiΦ)|⇓〉+ eiφF (1− eiΦ)|⇑〉] (5.8)
where Φ = φ0 − φ532 + φT + φ355 − φF . Since carrier pi/2 and sideband pi pulses
are driven by the same source for individual species, the differential phase between
them, φ0 − φ532 and φ355 − φF , is independent of optical phases. These differential
phases can be adjusted by changing the phase of the RF applied to the Raman
beam AOMs, however, they are kept constant for each experimental shot on this
case. Hence, the amplitude of the state |⇑〉 is expected to change with delay time
T due to the φT = ωmT dependence in
1
4
|(1− eiΦ)2|.
104
These experiments demonstrate that a superposition on the communication
qubit can be coherently transferred to the memory qubits with a CZ approach.
However, there are two major challenges in using this scheme on a multi-species
node of a network. First, the requirement for ground state cooled atoms would
require extra time be spent on sub-Doppler cooling procedures to handle heating
mechanisms, and to reduce the entanglement generation rate between modules as
a consequence. As an alternative, refrigerant atoms in the ion crystal can be uti-
lized for sympathetic cooling of communication qubits, although this might prevent
processing quantum information on memory qubits during the cooling cycle and
similarly introduce excessive down time on the network. Second, since the phase
introduced by the photonic entanglement (φ0) is a static phase [13], the transfer
operation described above might not preserve this phase information if the optical
phases are introduced by φ0−φ532. Stabilization of optical paths on the wavelength
scale or additional qubit rotations to cancel optical phases might be a solution to
this problem.
5.2 Mølmer-Sørensen Interactions
The MS interaction provides a robust alternative to CZ type state transfer
operations between communication and memory qubits. First, the atoms only need
to be in the Lamb-Dicke limit rather than the motional ground state [78]. This would
allow higher entanglement attempt rates since interruption for extra cooling steps
gets minimized. Second, phase information can be preserved with MS interactions
105
on the state transfer operation [67].
The MS interaction is gated by simultaneous excitation of red and blue side-
band transitions with a detuning of −δ and δ, respectively, from a selected motional
mode [37,78]. From Eqns. 3.17 and 3.16, this results in:
HMS =
h¯
2
∑
m=1,2
Ω
′
mσ
(m)
+ (ae
−i(δt−φm,r) + a†ei(δt+φm,b)) + h.c. (5.9)
where Ω
′
m = ηmΩmD′n,n′ is the sideband Rabi frequency and phases of the red (r) and
blue (b) transitions are φm,r(b) = ∆km,r(b)xm,0 −∆φr,(b) with equilibrium position of
the ions given by xm,0. The Lamb-Dicke parameter η is dependent on the wavevector
differences of two Raman beams and mode amplitude of the ion on the addressed
motion, which is plugged using Eqn. 3.13 modified for unequal ion masses:
ηm = ∆kmqm = ∆kmxm
√
h¯
2Mmω
(5.10)
where longitudinal motion is considered in this case and the mode frequency is ω.
The mode amplitudes xm are found analytically and Ω
′
m measurements verified these
findings in the previous chapter.
5.2.1 Spin-Dependent Forces
Before analyzing how two qubits can be entangled with this interaction, we
first look at the single ion case where Eqn. 5.9 simplifies to:
HMS =
h¯Ω
′
2
(σ+e
iφS − σ−e−iφS)(aeiφM e−iδt + a†e−iφM eiδt) (5.11)
where φS = (φr +φb)/2 is the spin phase and φM = (φr−φb)/2 is the motion phase.
106
The time evolution operator corresponding to this Hamiltonian can be found
by Magnus expansion:
U(t) = exp
(∑
i
Oˆi
)
(5.12)
and the expansion terms are:
Oˆ1(t) = − i
h¯
∫ t
0
H(t
′
)dt
′
(5.13)
Oˆ2(t) = − 1
2h¯2
∫ t
0
dt
′
∫ t′
0
[H(t
′
), H(t
′′
)]dt
′′
. (5.14)
Even though there are an infinite number of terms in this expansion, the series
terminates after the second term as will be shown here later. The first term of the
Magnus expansion with spin operator defined as Sˆ = (σ+e
iφS − σ−e−iφS):
Oˆ1(t) =
h¯Ω
′
2
Sˆ
∫ t
0
(aeiφM e−iδt + a†e−iφM eiδt) (5.15)
= Sˆ(α(t)a† − α∗(t)a) (5.16)
where
α(t) =
Ω
′
2δ
[1− eiδt]e−iφM . (5.17)
The second term is:
Oˆ2(t) = −1
2
(Ω′Sˆ
2
)2 ∫ t
0
dt
′
∫ t′
0
(e−iδ(t
′−t′′ ) − eiδ(t′−t′′ ))dt′′ (5.18)
= i
(Ω′Sˆ
2δ
)2
(δt− sin(δt)). (5.19)
The third and higher order Magnus expansion terms contain nested commutators
of the form [H(t), [H(t
′
), H(t
′′
)]]. Since harmonic oscillator raising and lowering op-
erators have a scalar commutator, the series terminates at second order. Therefore,
107
time evolution operator is:
UMS(t) = exp[Sˆ(α(t)a
† − α∗(t)a) + iSˆ2Φ(t)] (5.20)
Φ(t) = −
(Ω′
2δ
)2
(δt− sin(δt)) (5.21)
This interaction applies a spin-dependent force on the atomic ion which can
be seen easier on the eigenbasis of the spin operator. Considering the general case
of qubit spin states |0〉 and |1〉, the spin operator Sˆ has eigenstates:
|0φS〉 =
1√
2
(|0〉+ eiφs|1〉) (5.22)
|1φS〉 =
1√
2
(|0〉 − eiφs|1〉) (5.23)
with eigenvalues ±1. In this basis, the time evolution operator is:
UMS(t) = exp[(α(t)a
† − α∗(t)a)](|1φS〉〈1φS | − |0φS〉〈0φS |)
+ iΦ(t)(|1φS〉〈1φS |+ |0φS〉〈0φS |)]. (5.24)
The effect of this operator on atomic state |0, β〉 including both spin and motion is:
UMS(t)|0, β〉 = UMS(t)(|0φS〉+ |1φS〉)/
√
2
=
eiΦ(t)√
2
[Dˆ(α(t))|1φS , β〉 − Dˆ(−α(t))|0φS , β〉] (5.25)
with displacement operator defined as Dˆ(α(t)) = exp[α(t)a† − α∗(t)a]. The dis-
placement operator follows the Dˆ(α)Dˆ(β) = Dˆ(α + β)eiφD commutation rule with
φD = Im(αβ
∗) [73]. Therefore, the final state following the spin dependent force of
MS interaction is:
UMS(t)|0, β〉 = e
iΦ(t)
√
2
[eiφD |1φS , β + α(t)〉 − e−iφD |0φS , β − α(t)〉]. (5.26)
108
pxΩ′/√2δ
0φS
1φS
(a) (b)
⇑
P(
   
  )
Figure 5.3: (a) The motional state is displaced in the phase space by driving red
and blue sideband transitions with a detuning δ. The applied force is opposite for
different spin states, hence they get separated with this interaction. The trajectory,
shown in a frame rotating with the harmonic oscillator frequency, is a circle with
radius Ω
′
/
√
2δ that closes after T = 2npi/δ, where n is an integer indicating the
number of loops in the phase space. (b) Bi-chromatic fields are applied using the
355 nm field to realize MS interaction on a single 171Yb+ qubit initialized in |⇓〉.
The detuning, δ, from the red and blue sideband transition is scanned for a fixed
interaction duration of 50 µs. The spin and motion decouple when δ/2pi = n × 20
kHz and the 171Yb+ spin state goes back to |⇓〉.
The interaction can be visualized in the phase space (see Fig. 5.3(a)) where there
is no coupling between eigenstates of the spin operator, but motional states are
displaced and get entangled with the spin states. After an interaction time of T =
2npi/δ, with an integer n, the motional states overlap again at the starting point in
phase space (since α(T ) = 0 from Eqn. 5.17) and disentangle from the spin. From
Eqn, 5.21 the overall effect is an extra phase picked up by the spin states:
Φ(T ) = npi
( Ω′√
2δ
)2
(5.27)
which corresponds to a circular trajectory traced n times in the phase space.
This analysis is tested by measuring the 171Yb+ qubit state after applying a
spin-dependent force to the |⇓〉 state. This force is created by bi-chromatic 355 nm
109
fields and the expected spin rotation probability for a thermal distribution is given
by [37]:
P (|⇑〉)[t, δ] = 1
2
(1− e−(n¯+1/2)|2α(t,δ)|2) (5.28)
where n¯ is the average phonon occupation number. This theoretical curve is com-
pared to experimental measurements in Fig. 5.3(b). This experiment was conducted
on a 171Yb+ – 138Ba+ crystal, however the force is only applied to the 171Yb+ qubit
using the collective in-phase mode with ωIP ≈ 1 MHz. Hence, the 138Ba+ atom is
not involved in the dynamics and can be traced out from the joint state.
5.2.2 Multi-Species Entanglement
The same treatment can be extended for two atoms, specifically the 171Yb+
–138Ba+ crystal considered here. The Hamiltonian given in Eqn. 5.9 simplifies
to [73]:
H =
∑
m1=|⇓〉φS,1 ,|⇑〉φS,1
∑
m2=|↓〉φS,2 ,|↑〉φS,2
h¯
(Ω∗m1,m2
2
aeiδt +
Ωm1,m2
2
a†e−iδt
)
|m1m2〉〈m1m2|
(5.29)
where Ωm1,m2 = ±Ω′1e−iφM,1±Ω′2e−iφM,2 with positive (negative) signs corresponding
to m = |⇑〉φS,1 , |↑〉φS,2 ( m = |⇓〉φS1 , |↓〉φS2). Here, the spins are expressed in rotated
φS basis with eigenvectors of operator Sˆi = (σ+e
iφS,i − σ−e−iφS,i). The spin and
motional force phases are:
φS,i = (∆ki,rX0,i −∆φi,r + ∆ki,bX0,i −∆φi,b)/2 (5.30)
φM,i = (∆ki,rX0,i −∆φi,r −∆ki,bX0,i + ∆φi,b)/2 (5.31)
110
where X0,i is the equilibrium position.
The time evolution under this Hamiltonian is a displacement in phase space
similar to the single qubit case which is utilized for entangling two qubit internal
states via spin-dependent geometrical phase accumulations. After an interaction
time of T = 2pi/δ with Ω
′
1 = Ω
′
2 = Ω, the loops close in phase space and phases
acquired for the joint two qubit states are [67]:
Φe = Φ|⇓〉φS,1 |↓〉φS,2 ,|⇑〉φS,1 |↑〉φS,2 =
8piΩ2
δ2
sin2
(φM,1 − φM,2
2
)
(5.32)
Φo = Φ|⇓〉φS,1 |↑〉φS,2 ,|⇑〉φS,1 |↓〉φS,2 =
8piΩ2
δ2
cos2
(φM,1 − φM,2
2
)
(5.33)
where Φe and Φo represent the phase accumulated by even and odd spin parity
states, respectively. The entangling speed is maximum, for fixed sideband Rabi
frequencies, when the phase difference between geometric phases are maximized,
which happens at φM,1 − φM,2 = ppi with an integer p. This can be visualized
in the phase space by considering the displacement of individual motional packets
corresponding to internal spin states, which can add up or cancel depending on the
displacement direction (see Fig. 5.4). Therefore, application of the MS interaction
for T = 2pi/δ to qubits initialized to |⇓〉|↓〉 results in:
U(T )|⇓〉|↓〉 = U(T )
|⇓〉φS,1|↓〉φS,2 + |⇑〉φS,1|↑〉φS,2 − |⇓〉φS,1|↑〉φS,2 − |⇑〉φS,1|↓〉φS,2
2
= eiΦo
ei(Φe−Φo)(|⇓〉φS,1|↓〉φS,2 + |⇑〉φS,1|↑〉φS,2)− |⇓〉φS,1|↑〉φS,2 − |⇑〉φS,1|↓〉φS,2
2
.
(5.34)
The maximally entangled state is created when Φe − Φo = pi/2, which is realized
by changing the optical intensity of the Raman beams to adjust the sideband Rabi
111
px
(a)
x
⇓↓
⇑↑
⇑↓
⇓↑
⇑↓
⇓↑
⇑↑
⇓↓
p(b)
Figure 5.4: Addressing the out-of-phase mode of the 171Yb+ – 138Ba+ crystal where
the spin states are expressed in their corresponding φS basis, (a) [(b)] shows the
evolution of motional states under influence of a differential force. This is realized
by setting φM,1 = (2p + 1)pi + φM,2 [φM,1 = 2ppi + φM,2] where p is an integer. In
this case, the force is out-of-phase when the qubit internal states are in an even
[odd] parity state, therefore the mode is excited for only two out of four joint states.
Either setting can be used for generating a maximally entangled two qubit state
since a phase difference between pairs is evident in both cases.
frequencies:
U(T )|⇓〉|↓〉 = 1√
2
(|⇓〉|↓〉 − iei(φS,1+φS,2)|⇑〉|↑〉) (5.35)
where only the spin phase remains on the entangled state while the motion phase
mainly determines the required optical intensities.
5.2.2.1 Experimental Considerations and Results
There are several calibration steps for ideal experimental implementation of
the MS interaction where spin-dependent forces are applied using 355 and 532 nm
Raman beams from a pulsed laser. The out-of-phase longitudinal mode is used here
112
Yb Ba
ω
532 + ω
AOM3,r
ω
532 + ω
AOM3,b
ω 53
2 
+ ω
AO
M4
ω
355 + ω
AOM2
ω 35
5 
+ ω
AO
M1
,r
ω 35
5 
+ ω
AO
M1
,b
Figure 5.5: The red and blue sideband transitions are addressed by a bi-chromatic
setup where AOMs are used for frequency adjustments on the main Raman beams.
These beams have a ∆k component along the longitudinal axis to address the col-
lective motion in this direction. The RFs used on AOM’s 1 and 3 are produced with
the same arbitrary waveform generator which is essential in maintaining a well de-
fined phase relation between φM,1 and φM,2 as well as between single and two qubit
operations which involves φS,1 and φS,2. The effects of laser repetition rate fluctua-
tions on the beat-note frequency is canceled with feeding forward the corrections to
the AOM 2 [98] and a free running RF is applied to AOM 4.
with a frequency of ωOP ≈ 1.8 MHz to benefit from lower heating rates and a higher
mode frequency compared to in-phase mode. The fields are incident on the atoms
with a pi/4 angle to the motional axis as shown in Fig. 5.5, resulting in |∆k| ≈ √2k.
The off-resonant red and blue sideband transitions are driven by creating bi-
chromatic optical fields by modulating an AOM with two RF’s in conjunction with
113
another beam (see Fig. 5.5 and 5.6):
ωAOM1,r − ωAOM2 = ωHF − cωrep − ωOP − δ (5.36)
ωAOM1,b − ωAOM2 = ωHF − cωrep + ωOP + δ (5.37)
ωAOM3,r − ωAOM4 = ωZeeman − ωOP − δ (5.38)
ωAOM3,b − ωAOM4 = ωZeeman + ωOP + δ (5.39)
where ωrep/2pi is the repetition rate of the pulsed laser and c is an integer. Since
the Zeeman splitting of the 138Ba+ qubit is only a few MHz, the large bandwidth of
pulsed laser is not necessary to be utilized with 532 nm Raman beams.
For the multi-species entanglement generation, first the red and blue sideband
Rabi frequencies on each individual qubit need to be equalized. Due to diffraction
efficiency variance for different RF inputs and beam steering effects, the Rabi fre-
quencies might not be equal with the same RF power input. This calibration can be
done by measuring the sideband Rabi frequencies directly, and the single qubit MS
interaction can be used to verify and fine tune the optical intensities by observing
closure of the trajectories in phase space.
As analyzed in the previous section, the free parameters are δ, φS, φM and Ω for
each qubit. The duration of the entangling gate is constrained by chosen detuning
as T = 2npi/δ where n is an integer number indicating circle laps in the phase space.
The optical intensities need to be adjusted to realize Ω
′
1 = Ω
′
2 = Ω and
n× 8piΩ
2
δ2
=
pi
2
. (5.40)
Decoherence mechanisms, such as heating of the motional modes or magnetic field
noise, introduce errors during interaction, therefore it is beneficial to have faster
114
HP 8672A
@ 12.485 GHz
Fast
Photodiode
532 nm light
from pulsed laser
AWG
Microwave
Horn
MASTER OSCILLATOR
PLL
@ ~ 78.1 MHz
LO RF
IF
LO
RF
IF
AOM
1
AOM
2
1
2
ν  AWG,µ
ν  1,r
ν  2
AOM
3
ν  1,b
DDS @
100 MHz
AOM
4
ν  4
HPF
LPF
ν  3,r
ν  3,b
Figure 5.6: Schematic of RF circuitry used for driving coherent operations on 171Yb+
– 138Ba+ system. The output of an arbitrary waveform generator (AWG) can be
switched between microwave or stimulated Raman transitions. The waveform writ-
ten to the AWG for MS interactions includes all four RF signals (ν1,r, ν1,b, ν2,r, ν2,b
with ν = ω/2pi) and low/high pass filters (LPF, HPF) prevent output RFs from
going to the AOMs for other qubit. A phase-lock-loop (PLL) is used to lock the
beat-note frequency addressing 171Yb+ rotations [98].
115
entanglement operations. While a larger detuning can be used for this purpose, it
also introduces a requirement for larger Rabi frequencies as well from Eqn. 5.40. In
practice, the achievable entanglement speed is limited by available optical power on
the Raman beams.
After setting a detuning δ and fixing the interaction time correspondingly, and
equalizing the sideband Rabi frequencies between 171Yb+ and 138Ba+ , the motional
phase φM on one of the qubits is adjusted by changing the RF phase addressing
either the red or blue sideband transition. This way, maximum differential phase
accumulation between even and odd parity joint states can be realized (see Eqn.
5.33). One crucial aspect is maintaining the shot-to-shot phase relation between
these motional phases such that φM,1 − φM,2 is constant throughout experiments.
With the ∆ki,r −∆ki,b term in Eqn. 5.31, these phases are insensitive to changes in
optical path length fluctuations and therefore once set, they remain stable [33] and
usage of a single arbitrary waveform generator makes sure they have a well defined
phase relation between each other in every experiment.
In contrast, the spin phases φS are dependent on the optical path lengths and
they are written directly to the entangled state as in Eqn. 5.35. However, if the
spin rotations are driven with the same Raman beams, the phase coherence can be
maintained between single and two qubit operations. In a multi-species network,
photonic entanglement operations introduce extra complexity to the system since
the multi-species MS interaction is aimed to be used for transferring information
from communication to memory qubits, and an optical dependence might result
in decoherence between the photonic entanglement and information transfer oper-
116
ations. The method demonstrated in Ref. [33] to resolve this issue is not directly
compatible with a multi-species setup since canceling φS optical sensitivity transfers
this effect to φM which would result in variance of phase space trajectories and the
accumulated geometric phase. Instead, extra single qubit rotations can be included
to cancel any optical path dependence of spin phases [67, 73] but as will be shown
later, this condition can also be realized without resorting to them.
It should be noted that, since the 355 and 532 nm pulses do not follow the
same path, they do not apply the spin dependent force simultaneously to the qubits.
This might seem counterintuitive, but in the phase space picture it can be visualized
in terms of consecutive displacement on the motional state corresponding to spin
states which add or cancel depending on the motional phase of the Raman beams
and the parity of joint spin state.
Following these steps, the 171Yb+ and 138Ba+ qubits are entangled and the
results are shown in Fig. 5.7. The fidelity of this operation was ∼ F = 0.60 in
our system, and we attribute this low fidelity to an excessive heating rate ( ˙¯n ≈ 5
ms−1) in the trap where this operation was implemented [84]. Another multi-species
trap is assembled since these results were obtained, and a higher fidelity operation
is expected with a lower heating rate of ˙¯n ≈ 0.1 ms−1.
5.3 The Swap Gate
The entanglement generated via the MS interaction is not sufficient for trans-
ferring information from 138Ba+ to 171Yb+ by itself. However, a swap gate can be
117
⇑⇑ ⇑
⇑ ⇑
⇑ ⇑
⇑
Pr
ob
ab
ili
ty
(a) (b)
Figure 5.7: (a) Populations of 171Yb+ and 138Ba+ qubit states after MS interaction.
(b) Following the MS interaction, pi/2 pulses are applied to both qubits. The phase
of 138Ba+ pi/2 pulse is kept constant, while the 171Yb+ pi/2 rotation phase is scanned.
Maximum contrast points achieved in the resulting parity oscillation of the qubits
are shown here.
implemented by a sequence of two MS gates.
The full truth table of MS entangling gate is given by:
|⇓〉|↓〉 → 1√
2
(|⇓〉|↓〉 − iei(φS,1+φS,2)|⇑〉|↑〉)
|⇓〉|↑〉 → 1√
2
(|⇓〉|↑〉 − i|⇑〉|↓〉)
|⇑〉|↓〉 → 1√
2
(|⇑〉|↓〉 − i|⇓〉|↑〉)
|⇑〉|↑〉 → 1√
2
(|⇑〉|↑〉 − ie−i(φS,1+φS,2)|⇓〉|↓〉). (5.41)
In a multi-species apparatus, the aim is transferring a superposition state in 138Ba+
, α|↓〉+eiφ|↑〉, to the memory qubit. Applying two MS gates to the joint state, with
a pi phase shift on the total spin phase in the second operation with respect to the
first, results in transfer of 138Ba+ superposition to 171Yb+ :
UMS(φS,1 − pi, φS,2)UMS(φS,1, φS,2)[|⇓〉(α|↓〉+ βeiφ|↑〉)]→ |↓〉(α|⇓〉 − iβeiφ|⇑〉).
(5.42)
118
The required pi spin phase shift can be realized by advancing the red and blue
sideband RF phases by pi for the second MS operation, ∆φ1,r → ∆φ1,r + pi and
∆φ1,b → ∆φ1,b + pi, which results in:
φS,1 → φS,1 + pi (5.43)
φM,1 → φM,1 (5.44)
from Eqn. 5.30 and 5.31. This also preserves the motional phase, which results in
identical phase trajectories for both MS gates and produce the same truth table.
Importantly, the spin phases are not imprinted on the final state via this swap gate.
Hence, the phase information of 138Ba+ in addition to superposition amplitudes α
and β can be transferred to 171Yb+ without introducing decoherence.
To sum up, the MS approach provides a robust and coherent transfer method
compared to the CZ approach. However, with the low fidelity of the MS gate in our
system, transferring information with this method has not been attempted yet.
119
Chapter 6: Outlook
The basic ingredients of a multi-species node for realizing a network architec-
ture are demonstrated in this thesis. The first step toward validating this approach
of scaling would be combining these operations to show entanglement between a
138Ba+ photon and a 171Yb+ atomic qubit, and efforts are underway.
In this chapter, I describe how some of the additional key requirements can
be implemented in a network structure. First, the communication 138Ba+ qubits of
distant modules need to be entangled which can be done following the Bell state
measurement methods demonstrated in Refs. [13,15,25]. However, atomic structure
of a 138Ba+ atom is different from the atoms used in previous works, and requires
extra considerations on the experimental sequence and the setup.
Second, coherence across the network needs to maintained in this architec-
ture. These considerations are outlined in detail in Ref. [33] and I provide small
adjustments to this work to incorporate multi-species modules.
6.1 Entanglement of Remote Barium Qubits
The experimentally shown atom-photon entanglement in this thesis could be
utilized to connect separate modules. However, mixing between two orthogonal
120
polarization states at high collection solid angles [36] would result in considerable
error in this operation.
As an alternative, photons can be collected along the quantization axis, where
coupling of a pi polarized photon into a single mode fiber vanishes due to rota-
tional symmetry [71]. However, this approach requires adjustments of the photon
generation scheme and the optical beam setup.
The requirement for a circularly polarized beam for 138Ba+ qubit initialization
poses a challenge in this new setup. This beam needs to be along the quantization
axis for perfect polarization and therefore it propagates along the axis of collec-
tion and might damage the detectors that are used for the heralded entanglement
scheme. Even though the initialization beam is not incident when detection of the
single photons from 138Ba+ is attempted, intense optical beams on photon detectors
might introduce afterpulse that require additional delays in the sequence to mini-
mize overlap between this signal and the photons originating from the atom. Some
possible configurations to address this issue are shown in Fig. 6.1.
After initializing the 138Ba+ qubits to state |↓〉, a pi/2 rotation is applied to
create a superposition state and with a subsequent fast excitation pi pulse, the
atom–photon state is:
|↓〉 → |↓〉+ |↑〉 → (|↓〉|pi〉+ |↑〉∣∣σ−〉) + (|↑〉|pi〉+ |↓〉∣∣σ+〉) (6.1)
ignoring normalization. As mentioned before, the pi polarization does not couple
into the fiber along this direction, resulting in:
|Ψ〉 = |↑〉∣∣σ−〉+ |↓〉∣∣σ+〉 (6.2)
121
NA=0.6
Ions To 50/50 BS
and
Detectors
NA=0.36
NA=0.6 NA=0.6
(a)
(b) (c)
B
B B
95/5 Beam 
Sampler
95/5 Beam 
Sampler
Optical
Pumping
Optical
Pumping
Optical
Pumping
Figure 6.1: (a) The circular polarized optical pumping beam can be sent through the
opposite side of the single photon collection optics, but this light would couple into
the fiber and might have adverse effects on the single photon detectors. Additional
recovery time for detectors might result in a lower entanglement generation rate
with this configuration. (b) Instead, initialization light can be aligned from the
same side of collection optics by using a beam sampler such that most of the light
can be coupled into the fiber. Aberrations that might stem from the sampler can be
corrected with adaptive optics [59] or the sampler can be moved further downstream
as in part (c) if reflections that might go back to the detectors are not significant.
an atom–photon entangled state. A quarter wave plate rotates circular polarization
into linear, which is easier to distinguish in lab frame:
|Ψ〉 = |↑〉|H〉+ |↓〉|V 〉. (6.3)
The polarization state of the photons from two different ion traps are measured after
a 50/50 beam splitter where coincident detection of photons heralds the entangle-
ment of atomic qubits.
Another isotope of barium is under development for quantum information
purposes in the Campbell Group at UCLA. Ba-133 has nuclear spin 1/2 and con-
sequently 133Ba+ atomic level structure resembles 171Yb+ closely. However, natural
abundance of this isotope is too low for reliable trapping, hence enriched sources are
required. More importantly, radioactivity of this isotope requires additional safety
122
2S1/2
2P1/2
2S1/2
2P1/2
σ−
σ+π
π
(a) (b)
x
x
Figure 6.2: (a) After initialization in |↓〉, a pi/2 rotation prepares the qubits for
photon generation with fast excitation. (b) Spontaneous emission results in photons
entangled with their parent atom, and collection along the quantization axis prevents
pi polarized components from coupling to the fiber. However, there is a trade-off
between fidelity and entanglement generation rate with this approach since qubit
rotations need to be incorporated into each trial.
123
procedures. Its decay type is electron capture with an average emitted gamma
energy of 0.266 MeV and half life of 10.74 years.
Compared to 138Ba+ , faster entanglement generation, likely a factor of two,
can be achieved with 133Ba+ since qubit preparation step can be eliminated with
this isotope, similar to schemes demonstrated in Refs. [13,25]. Additionally, the fre-
quency of the photons rather than the polarization state can be utilized in heralded
entanglement schemes [15] to isolate environmental effects from flying photon qubits.
Other advantages would be deterministic single shot qubit state measurements and
longer coherence times, which are not strong requirements for communication qubits
in a multi-species module as mentioned before.
6.2 Maintaining Phase Stability in a Multi-Species Network
The phase stability in a single species network architecture with quantum
operations spread temporally and spatially is considered in great detail in Ref. [33].
All of the quantum gates are referenced to a common oscillator with sufficiently low
phase noise in this method (see Fig. 6.3).
This approach can be easily extended to a multi-species module. Since the
swap gate does not introduce additional phases and information from 138Ba+ to
171Yb+ can be transferred coherently, as long as operations involving the memory
qubits preserve phase information, the system can be scaled up with a multi-species
approach. There is also no limitation on which axis of motion needs to be used for
the 138Ba+ – 171Yb+ swap gate. Longitudinal modes can be used for this operation
124
HP 8672A
@ 12.566 GHz
Fast
Photodiode
532 nm light
from pulsed laser
AWG
Microwave
Horn
DDS @
160 MHz
Quantum
Gate AOMs
Copropagating
Carrier AOM
MASTER OSCILLATOR
PLL
@ ~ 83.5 MHz
LO RF
IF
LO
RF
IF
LO
RF
IF
AOM
1
AOM
2
1
2
3
1
2
1 2
2
ν  AWG,µ
ν  AWG,cp
ν    
  
AWG,r ν    
  
AWG,b
ν  1,r
ν  2
+
AOM
3
ν  1,b
DDS @
100 MHz
AOM
4
ν  4
HPF
LPF
ν  3,r
ν  3,b
Quantum Gate AOMs
Figure 6.3: Phase coherence between entangling operations on the memory qubits
and communication qubits can be maintained with some adjustments to Fig. 5.6.
AOM 1 can be swapped with a multi-channel device to address any pair of memory
qubits for realization of arbitrary quantum algorithms [14]
while the transverse modes can be used for the interactions between the memory
qubits. Following the techniques demonstrated in Ref. [14], AOM 1 in Fig. 6.3 can
be made multi-channel to do arbitrary logical gates between any pair of memory
qubits.
The signal from the master oscillator is distributed to other modules of the
network architecture for maintaining phase stability across the system. Additional
phase noise along the transmission lines might lead to decoherence, therefore pre-
ventive measures need to be taken for this purpose while distributing the clock
signal.
125
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