Quantum entanglement generation in trapped ions using

coherent and dissipative methods

by

Yiheng Lin

B.S., University of Science and Technology of China, 2009

M.S., University of Colorado Boulder, 2012

A thesis submitted to the

Faculty of the Graduate School of the

University of Colorado in partial fulfillment

of the requirements for the degree of

Doctor of Philosophy

Department of Physics

2015
This thesis entitled:

Quantum entanglement generation in trapped ions using coherent and dissipative methods

written by Yiheng Lin

has been approved for the Department of Physics

David J. Wineland

James Thompson

Date

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

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

mentioned discipline.
iii

Lin, Yiheng (Ph.D., Physics)

Quantum entanglement generation in trapped ions using coherent and dissipative methods

Thesis directed by Prof. Dr. David J. Wineland

Entangled states are a key resource in fundamental quantum physics, quantum cryptography,

and quantum computation. In this thesis, we focus on the demonstrations of two novel methods

to generate entanglement. First, we implement dissipative production of a maximally entangled

steady state on two trapped ions. Dissipative and coherent processes are combined and implemented in a continuous time-independent fashion, analogous to optical pumping of atomic states,

continuously driving the system towards the steady entangled state. With this method, we obtain

a Bell state fidelity up to 0.89(2). Second, we propose and demonstrate a novel coherent process

to confine quantum evolution in a subspace between an initial separable state and the target entangled state. We demonstrate this scheme on two and three ions obtaining a Bell state fidelity

up to 0.992(2). Both of these methods are robust against certain types of experimental noise and

decoherence. Additionally, we demonstrate sympathetic cooling of ion chains to near the ground

state of motion with an electromagnetically-induced-transparency (EIT) method. This results in

roughly an order of magnitude faster cooling time while using significantly lower laser power compared to the conventional resolved sideband cooling method. These techniques may be helpful for

scaled-up quantum computing.
Dedication

To my parents, grandparents, my wife, my teachers and mentors.
v

Acknowledgements

Many people deserves acknowledgement to this work and I apologize in advance in case I

unintentionally leave anyone out.

It has been a great honor to work in the ion storage group at National Institute of Standards

and Technology (NIST) at Boulder Colorado. When I joined the group at April 2010, I had

background on theoretical physics and nuclear magnetic resonance (NMR) quantum information

processing but I was a complete novice to trapped ion experiments, simply being interested in it. I

am very grateful to my advisor Dave Wineland who gave me an opportunity to work in the group.

There I met and learnt a lot from many talented and skilled people, and I also had a good time with

them. Dave Wineland is also a great advisor. We had a lot of freedom to work on the interesting

projects and can choose our own way to tackle the problems, yet he is always ready to help with his

massive knowledge and always able to directly or indirectly point out ways to solution. Together I

need to thank Dietrich Leibfried, another advisor in our projects who constantly gave us suggestions

and guidance, and he provides great help with his expertise in both experiments and theories.

I need to thank my colleagues who worked closely with me in the projects. Special thanks to

John Gaebler, who had been the only postdoc in our lab for very long time. He played a crucial role

in all the experiments I led or participated in, and he is always generously sharing his knowledge

and willing to help. During my first year in the lab, David Hanneke, Jonathan Home and John Jost

helped me learn how an ion trap works and operates, from the basics to details. Special thanks

to David who helped me become an experimentalist from mostly a theorist and get through this

difficult transition. I also need to thank Ryan Bowler, Ting Rei Tan and Yong Wan, with whom
vi

we helped each other on the experiments and beyond.

I also need to express my gratitude to our theoretical collaborators. Special thanks to Florentin Reiter and Anders S. Sørensen from University of Copenhagen, who collaborated on two

experiments out of the three I led. Special thanks to Florentin for his expertise and being considerate. Without John Gaebler seeing Florentin’s poster at the 2012 Gordon Research Conference and

we then met, you may see a very different and perhaps less interesting thesis. I also thank Adam

Keith, Emanual (Manny) Knill, Scott Glancy and Kevin Coakley for nice theoretical collaborations

at NIST. Special thanks to Manny for all the discussion with him and his insights.

I am also very fortunate meeting these people from whom I leant a lot of experimental

skills, and I appreciate their helps and the good time we together: Jason Amini, David Allcock, James C. Bergquist, Mike Biercuk, Brad Blakestad, Justin Bohnet, John Bollinger, Samuel

Brewer, Joe Britton, Kenton Brown, Shaun Burd, Peter Burns, Jwo-Sy Chen, James Chou, Kevin

Chou, Yves Colombe, Shon Cook, Robert Drullinger, Aaron Hankin, Dustin Hite, David Hume,

Wayne Itano, Robert J ̈ordens, Chris Langer, David Leibrandt, Yiheng Lin1, Katherine McCormick,

Kyle McKay, Christian Ospelkaus, Dave Pappas, Till Rosenband, Brian Sawyer, Daniel Slichter,

Raghavendra(Raghu) Srinivas, Carson Teale, Mike Thorpe, Susanna Todaro, Herman Uys, Aaron

VanDevender, Ulrich Warring, and Andrew Wilson.

I need to thank my wife Xinchen, my parents and my grandparents, for their care and

supports. Also I wanted to list all of my friends during my time in graduate school but here I

would specially thank a few of these people, without their support and encourage I could not make

it to the finish line: Dong Chen, Jiayan (Phoenix) Dai, Huiling Guo, Liqian Peng, Shuangwen

Sheng, Jianfeng Xie, Honghua Yang, Pen-li (Ben) Yu, Yanbao Zhang, and Xiaoge Zeng, etc.

Finally, I need to thank my undergraduate advisors Xinhua Peng and Jiangfeng Du, who

showed me the path to the interesting quantum phenomenon; and Luyuan Hao who showed me some

basics of optics experiments. I also need to thank my teachers who help building my foundations

of knowledge, specially: Guanghua Xu, Bingguang Wu, Liping Lin, Wenyan Sheng.

1 Ah you are reading it carefully
vii

Special thanks to Dave Wineland, Raghu Srinivas and Yong Wan who read this thesis in

detail and provided a lot of helpful comments.
viii

Contents

Chapter

1 Introduction 1

1.1 General description of quantum information and entanglement . . . . . . . . . . . . 1

1.2 Scalable quantum computing architecture and sympathetic cooling . . . . . . . . . . 4

1.3 General description of entanglement operation and novel methods . . . . . . . . . . . 6

1.4 Thesis organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2 Ion trapping 9

2.1 Idealised ion trapping theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.2 Trap configurations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.2.1 Trap wafers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.2.2 Trap RF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.2.3 The linear trap and the junction trap . . . . . . . . . . . . . . . . . . . . . . 14

2.3 Other components for ion trapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.3.1 Ion loading: atom source and photoionization . . . . . . . . . . . . . . . . . . 15

2.3.2 Vacuum system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.3.3 Imaging system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.3.4 Magnetic field coils . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.3.5 Experiment controls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
ix

3

9Be+ qubit 25

3.1 9Be+ energy levels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.2 Laser and microwave sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

3.3 State initialization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

3.4 Detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3.4.1 Photon collection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3.4.2 Maximal likelihood method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

3.4.3 Symmetric entangled state detections . . . . . . . . . . . . . . . . . . . . . . 42

4 Internal-motional state coupling 44

4.1 Quantization of motional modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

4.2 Resolved sideband coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

4.2.1 Electric dipole transition for a free ion . . . . . . . . . . . . . . . . . . . . . . 48

4.2.2 Raman transition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

4.2.3 Raman sideband coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

4.2.4 Raman sideband coupling for two ions: resonant case . . . . . . . . . . . . . . 52

4.2.5 Raman sideband coupling for two ions: detuned case . . . . . . . . . . . . . . 53

5 Sympathetic electromagnetically-induced-transparency(EIT) laser cooling 56

5.1 General description of sympathetic laser cooling . . . . . . . . . . . . . . . . . . . . . 57

5.2 Theoretical description of sympathetic EIT cooling . . . . . . . . . . . . . . . . . . . 58

5.2.1 EIT effect on a stationary ion . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

5.2.2 EIT effect with a trapped ion in motion . . . . . . . . . . . . . . . . . . . . . 59

5.3 Experiment sympathetic EIT laser cooling . . . . . . . . . . . . . . . . . . . . . . . . 62

5.3.1 EIT cooling on the 9Be+-

24Mg+ ion pair . . . . . . . . . . . . . . . . . . . . . 63

5.3.2 EIT cooling on the 9Be+-

24Mg+-24Mg+-9Be+ four-ion chain . . . . . . . . . 64

5.3.3 Technical notes for EIT laser cooling . . . . . . . . . . . . . . . . . . . . . . . 67

5.4 Conclusion and future applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
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5.5 Appendix: derivation for EIT cooling condition . . . . . . . . . . . . . . . . . . . . . 69

6 Steady state entanglement from dissipation 71

6.1 Theory and concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

6.2 Experimental setup and results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

6.2.1 Continuous application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

6.2.2 Stepwise Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

6.2.3 Numerical Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

7 Entanglement generation by confining dynamics in a subspace 89

7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

7.2 Two ion implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

7.3 Two ion implementation with a composite pulse sequence . . . . . . . . . . . . . . . 95

7.4 Three ion implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

8 Conclusion 99

Bibliography 101
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Tables

Table

3.1 Some field insensitive points with the relevant transitions and frequencies, and the

difference of second order sensitivities. Here we list only field-insensitive points for B

in the range of 11 mT to 180 mT. In this thesis we mainly work on fields for the first

two entries, with ∆mF = ±1. For the higher fields the insensitive points are located

at ∆F = 0 transitions, and the higher order field sensitivity is significantly lower than

the ∆F 6= 0 insensitive transitions. As this trend and Fig. 3.2 shows, for the high

fields, the Zeeman effect is dominant over the hyperfine coupling hAI·J, and the field

sensitivity between ∆F = 0 transitions becomes small. At higher fields of ∼819.439

mT, a 303 MHz field insensitive transition between |mI = −

3

2

, mJ =

1

2

i ↔ |−1

2

,

1

2

i

has been demonstrated in a Penning trap [1] for applications as a frequency standard. 29

3.2 List of 2s

2S1/2 ground states expanded in the |mI , mJ i basis, with coefficients

rounded to the third digit. Compared with the states at zero field, except for the

stretch states, the other states have modified coefficients. This table will be useful

when calculating the coupling coefficients for the electric dipole transitions, since the

selection rules are such that ∆mI = 0 and ∆mJ = 0, ±1. . . . . . . . . . . . . . . . 29

3.3 List of 2p

2P1/2

states expanded in the |mI , mJ i basis. Definitions are similar to

Table 3.2 for 2p

2P1/2

. These components will be useful when describing the laser

dipole coupling matrix element in this thesis. . . . . . . . . . . . . . . . . . . . . . . 30

3.4 typical π time of the Rabi flopping of the microwave transitions . . . . . . . . . . . . 36
xii

4.1 Normal motional modes for various ion chain configurations, with the same external

potential well for each case. The mode amplitudes are shown consistent with the ion

order. For the 9Be+ − 24Mg+ pair, due to the mass imbalance, the COM mode is

dominated by motion of the 24Mg+ ion, while the stretch mode is dominated by the

9Be+ ion. This reduces the sympathetic cooling efficiency of mix-species chains for

the modes which the coolant ion has small mode amplitudes. The mode information

shown in this table will be useful for the experiments described in the following

chapters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

6.1 list of parameters for repumping, cooling and heating processes . . . . . . . . . . . . 76

6.2 list of parameters for spontaneous photon scattering processes from Raman laser

beams; we use the same numbers for simulation in both the continuous and stepwise

cases. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

6.3 fit values from the Stark experiment decaying flopping . . . . . . . . . . . . . . . . . 88
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Figures

Figure

1.1 A scalable quantum information processing device with trapped ions [2]. The structure consists of linear trapping sections, and junctions. Ions are trapped in long

chains, separated, shuttled and recombined. Lasers beams are shown to entangle ion

pairs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.1 Schematic for trapped ion experiment, showing the major components and some

more details and requirements of technology therein. . . . . . . . . . . . . . . . . . . 10

2.2 An idealised ion trap with “four-rod” configuration. The gold rods are the RF

electrodes, and the segmented ones are DC electrodes. The two RF electrodes are

typically shorted together, and provide a quadruple confinement potential in the

radial plane. To form confinement of positively charged ions along the axial direction,

we apply positive potentials on the outer electrodes and lower potentials on the

center electrodes. The axial confinement is designed to be weaker than the radial

confinement, so that the ions are trapped along as a linear chain along ˆz. . . . . . . 11
xiv

2.3 View of the linear trap [3]. Only the top wafer is visible. The RF electrode and

the segmented DC electrodes are shown, and the DC electrodes are gold ribbon

bonded to the filter boards. The bottom figure represents a closed view of the

segmented electrodes. The widths of the electrodes from left (O1) to right (L2)

are {2,0.2,0.1,0.2,0.36,0.5,0.5,2}mm [3]. The small gaps between the electrodes are

0.02 mm. The electrodes O1-O2 form the experiment region; the electrodes L1-

L2 form the loading region. The gap between the DC and RF electrodes in the

experiment/loading region is 0.2/0.4 mm wide. . . . . . . . . . . . . . . . . . . . . . 16

2.4 Close view of the junction trap. Only the top wafer is visible. The RF electrode

and the segmented DC electrodes are shown, and the DC electrodes are gold-wirebonded to the filter boards. The bottom figure represents the scale of the segmented

electrodes. L, E, V, F, and C represents the loading zone, the experiment zone, the

vertical zone, the “far” zone, and the center of the junction, respectively. The width

of the electrode at E is 200 μm. The narrower electrode next to it is 100 μm wide,

specifically for ion separation. The gaps between the electrodes are 20 μm wide. Ions

chains can be separated and shuttled back and forth through two different routes,

either E → C → V → C → E or E → C → F → C → E, and thus enabling ion chain

reordering [4]. The inset shows a side view of the junction and connection of the RF

electrodes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.5 Schematic for the 9Be photoionization laser setup. The mirror mount in dash outlines

is a pick off mirror, which is not in the setup except for checking spectrum for 705

nm light. The mirror at the output of the tripler has finite transmission that leaks

a small portion of 235 nm light for wavelength monitoring. . . . . . . . . . . . . . . 19
xv

2.6 A mock up of the vacuum setup [3], showing relevant vacuum components. The

Quartz cell has six windows made from fused silica for UV light access, and four of

them are used for laser access and one is used for imaging. On the input end of the

resonator, there is a cylindrical component to connect the center conductor to the

ground. The output of the RF source after the amplifier is connected to a small wire

loop, mounted inside the can to couple RF power into the resonator. . . . . . . . . . 21

3.1 Schematic for the 9Be+ ion electronic transitions at 11.9446 mT magnetic field. Laser

beams are shown to address various levels, see text. The relevant energy splittings

are shown, and the transition |2, 0i ↔ |1, 1i with transition frequency 1207.495843

MHz is to first order insensitive to magnetic field fluctuations. A typical Raman

laser configuration driving the qubit transition is shown. At 11.9642 mT magnetic

field, there is another first order field insensitive transition |2, 1i ↔ |1, 0i of frequency

1207.352808 MHz. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.2 a) Energy levels in unit of (h GHz) by diagonalizing Hamiltonian in Eq. 3.1 over

the external magnetic field B in mT. The eigenstates are labeled in the format of

|F, mF i. For B 6= 0, F is no longer a good quantum number although mF is. Here

we just label the levels with the order of the energies, in conjunction with the low

field case. b) The energy sensitivity dE

dB in the unit of (h MHz/mT) over the external

magnetic field. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.3 Schematic for laser access to the trap, not to scale. The magnetic field coil is set so

that the magnetic field is 45◦ with respect to the trap axis. Be CO and 90 denote the

Raman laser beams to address the 9Be+ ion. PI denotes the photoionization beams.

Blue hollow squares denote the outline of the glass chamber and the windows. Figure

adapted from [5]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
xvi

3.4 Schematic for BD laser beams setup. The output of the doubler has a vertical polarization with respect to the laser table. Pos. D. represents a two dimensional position

detector (Newport Quadrant Cell); PZT mirror is a dual-piezo-driven mirror, capable of tipping both horizontal and vertical directions. Along the beam path, the

first(second) Pos. D. feeds back to the first(second) PZT mirror with proportionalintegrator loop. Thus the output beam pointing of the doubler is locked. The AOMs

are not shown with actual orientation angle. The dashed beam line forms the BD fardetuned Doppler laser, and the solid beam line after the first AOM forms the BDD

resonant/near detuned Doppler laser, with their frequency difference approximately

400 MHz. These two laser beams are combined at the BD det(detuned) SW(switch)

AOM, where the BD beam is diffracted on the first order, which is overlapped with

the undeflected BD beam. This configuration is due to the requirement of stronger

far-detuned beam power than the near detuned one. Here some of the aligning mirrors are not shown. The double pass BD Switch setup uses a 15 cm lens placed at

a focal length away from the center of the AOM. A corner cube retro-reflecting the

beam with a vertical displacement, placed at the location that forms a one to one

telescope with the lens. With this setup, the double pass alignment is insensitive

to the change of AOM input frequency. The output of this setup is combined with

the RD beams, then sent to the input of a UV fiber. The RF drive for BD switch

AOM is actively noise-eat when the pulse is on. For this purpose, a pick-off mirror

partially reflect the BD light to a detector, placed after the fiber, so that when the

BD light is on, its power is actively stabilized. . . . . . . . . . . . . . . . . . . . . . 33
xvii

3.5 Schematic for laser beam generation. “BBO doubling” denotes the doubling cavity

setup [6] with BBO as the nonlinear conversion medium. For the seed lasers, we use

NKT Koheras Adjustik lasers, and for the upper two amplifiers, we use NKT Koheras

Boostik HPA. For the lower two amplifiers, we use IPG Photonic Ytterbium/Erbium

Fiber Amplifiers. The respective frequencies of the Raman beams are controlled by

the AOM frequencies. The Doppler beams are recombined before sending through

a UV capable photonic crystal fiber [7]. To Doppler cool and image on both the

experimental zone and the load zone, we further split the Doppler beam on the

fiber output with a Glan-laser, with relative intensity controlled by a motorized λ/2

waveplate. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3.6 Schematic for RD laser beams setup. This setup is similar to the BD one, except

that the generated beam frequency difference is approximately 1110 MHz, matching

with the resonant frequency difference of the |F = 1, mF = 1i and |2, 1i states. The

λ

2 waveplates modifies the polarization of the laser beam, controlling the beam power

splitted at the Glan-laser. The Glan-laser is setup so that the lower output beam

is vertically polarized, and the upper output beam is horizontally polarized. The

waveplate after the Glan-laser rotates the polarization back to vertical. . . . . . . . 35

3.7 Schematic for Raman laser beams setup. This setup splits the Raman beam into the

“CO” beam and the “90” beam, as indicated in Fig. 3.3. We adjust the frequency

of the 90 AOM to drive transitions |F = 1, mF = 1i ↔ |2, 2i and |1, 1i ↔ |2, 0i. The

CO beam line has an optional 600 MHz AOM double pass setup, to generate a pair

of co-propagating CO beams to drive the carrier transition, which is not shown in the

figure. For the double pass setup for Raman beams, the corner cube is replaced with

three mirrors, to provide first order insensitive to AOM thermal refraction effect.

This method is proposed by T. Rosenband, as shown in Fig. 3.8. . . . . . . . . . . . 35
xviii

3.8 (a) Schematic for the “two mirror” double pass setup in a side-view. Here only

the diffracted beams are shown as the AOM outputs but the actual the diffraction

directions are perpendicular the page. The dashed orange line shows the deviation

angle θ due to the AOM thermal thermal refraction effect. This effect can result from

inhomogeneous distribution of RF power inside the AOM, or from inhomogeneous

heat dissipation along the vertical directions in the figure. In this case, after double

passing the AOM, the angular deviation is doubled. The lens is placed at a focal

length f away from the AOM, and the round trip path length between the lens and

mirrors is 2f. (b) Schematic for the “three mirror” double pass setup. For three

mirror setup, the deviated beam experiences a cancelling thermal refraction effect

and the double pass output is overlapped with the beam where the thermal refraction

effect is absent. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

3.9 Schematic for the dipolar antennas in the experiment, not to scale. The microwave

wavelength id denoted by λ. One arm of the antenna is soldered to the center

conductor of the SMA, and the other end is soldered to the outer conductor. . . . . 37

3.10 Typical histograms collected from a given detection process. (a) Histograms from

Ramsey experiments showing the count histograms for various states. From top to

bottom, histograms of approximately two, one, and zero ions in the bright states

are shown, corresponding to φ = {0, π/2, π} respectively. Gray vertical dashed lines

separate the histogram into sections of count ranges that can distinguish the number

of ions in the bright state. (b) Sections in (a) are summed up to form “rebinned”

histogram. The new bins are labeled as 1-3. (c) From the rebinned histograms

and the predicted populations from the Ramsey sequence, a population reference

histograms representing 0-2 ions in the bright state are obtained with the maximum

likelihood method. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
xix

4.1 (a) A simplified Λ configuration of Raman transition between internal states |ii and

|mi. (b) a Raman sideband transition of |m, ni → |i, n − 1i. . . . . . . . . . . . . . 50

4.2 State evolution subject to the sideband Hamiltonian of Eq. 4.30, starting from state

|↓↓, 0i. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

4.3 Eigenvalues of the detuned sideband. . . . . . . . . . . . . . . . . . . . . . . . . . . 55

5.1 24Mg+ energy levels, laser setup and a spectrum showing the EIT effect. (a) Relevant

energy levels for 24Mg+, in an applied magnetic field of 11.964 mT. The three levels

|g1i, |g2i and |ei serve as a Λ system for EIT cooling. Laser beams with σ

+ and π

polarizations couple the ground states to the excited state with Rabi rates Ω1 and Ω2

and common detuning ∆. Wavy lines show spontaneous emission from the excited

state to the ground states and the excited-level decay rate is denoted by γ ' 2π×41

MHz. The fourth level |e

0

i can perturb the EIT cooling when the π polarized laser

beam has frequency near the |g1i to |e

0

i resonance. (b) Beam configuration and a

depiction of the 9Be+−24Mg+ −24 Mg+−9Be+ ion, shown as balls from left to right.

This figure is viewed in an angle from top view.(c) Simulation of the absorption

spectrum of a stationary ion by a weak σ

+ probe beam for ∆ = 2π×96.7 MHz,

Ω1/2π = 30 MHz, and Ω2/2π = 12 MHz. For simplicity, the fourth level |e

0

i is

ignored. The probe detuning from the |g2i to |ei resonance is denoted by ∆P. This

Fano-like profile contains a narrow and broad feature corresponding to dressed states

|ψ+i and |ψ−i respectively [8], and the separation between the peak on the right side

and the null is labeled as δ. When ∆P = ∆, absorption vanishes due to coherent

population trapping. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
xx

5.2 Mean motional excitation number  ̄n for the I (red triangles, ω/2π = 2.1 MHz) and O

(blue squares, ω/2π = 4.5 MHz) axial modes of a 9Be+-

24Mg+ ion pair. (a)  ̄n after

800 μs of EIT cooling as a function of δ/2π. Optimal cooling for each mode occurs

when δ approximately equals the mode frequency. (b)  ̄n plotted as a function of

EIT cooling duration tc. From 0 to 74 μs, δ ' ωO, the out-of-phase mode frequency.

From 75 to 85 μs, δ ' ωI, the in-phase mode frequency. In both figures, error bars

represent statistical uncertainty of the sideband amplitude ratios. The solid lines are

simulations of the full dynamics including the |e

0

i level in 24Mg+, measured ambient

heating rates, detuning and beam intensities. In the simulations, we truncated the

motion to the first 6 Fock states for both modes for the steady-state simulation in

(a) and to the first 10 (6) Fock states for the I (O) mode for temporal simulation in

(b). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

5.3 Sympathetic EIT cooling on a 9Be+-

24Mg+-24Mg+-9Be+ ion chain. (a)Minimum  ̄n

values for each of the four axial modes of a 9Be+-

24Mg+-24Mg+-9Be+ ion chain as

a function of δ/2π after 800 μs of cooling to ensure steady state. Modes 1 to 4 are

shown in the legend, and labeled as red triangles, green squares, black circles and

blue diamonds, respectively. (b)  ̄n plotted as a function of EIT cooling duration tc.

From 0 to 40 μs, δ ' ω3, the mode 3 frequency. From 40 to 45 μs, δ ' ω1, the

mode 1 frequency. In both figures, error bars represent statistical uncertainty of the

sideband amplitude ratios. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
xxi

5.4 (a) Minimum values of  ̄n for the two-ion O mode vs. ∆/2π. The peak near 223 MHz

results from resonant scattering on the |g1i ↔ |e

0

i transition from the π polarized

light. Blue circles are the experimental data and red triangles are simulations based

on [8]. Green diamonds are simulations not including |e

0

i and the black solid line

shows  ̄n = (γ/4∆)2[9]. (b) Simulation of 1/e cooling time vs. the mass of ion X+

that is sympathetically cooled by 24Mg+ (axial modes), with fixed trap potential

such that ωI(O)/2π = 2.1 (4.5) MHz when X+ is 9Be+. Red triangles are for the I

mode; blue squares for the O mode. Optimum values of δ were chosen for each mode

and ion-mass combination, with Ω2/2π = 5.9 MHz and ∆/2π = 96.7 MHz. (Here

|e

0

i is neglected) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

6.1 Sideband interactions together with sympathetic cooling. The former is expressed

by blue double arrows and the latter green wavy arrows. Other than the |↑↑, n = 0i

state, all other spin states can be driven by the sideband interaction, which produces spin flips and motional excitation simultaneously. However, with sympathetic

cooling, the evolution would lead to pumping to the dark state of the sideband, the

|↑↑, n = 0i state. This state is dark since the sideband could not flip spin down and

subtract motional quanta at the same time. . . . . . . . . . . . . . . . . . . . . . . 74
xxii

6.2 Energy Levels and Entanglement Preparation Scheme a) The internal energy levels

(not to scale) of 9Be+ are shown as solid black lines for the ground motional state

and dashed lines for the first excited motional state. The couplings needed to produce steady-state entanglement are shown with blue double arrows for the strong

sideband coupling with Rabi rate Ωs and sympathetic cooling; the patterned and

dashed red arrows for the weak microwave coupling with Rabi rate Ωc, repumper,

and spontaneous emission from the |ei state. Wavy arrows depict the dissipative

processes. b) Four spin states that span the |↑i, |↓i qubit manifold of the two 9Be+

ions are shown as horizontal lines. Transfer processes that are accomplished by the

sideband drive and sympathetic cooling are shown as blue arrows, while processes

that occur by coupling the |↑i state to the auxiliary |ai state followed by excitation with the repumper and decay by spontaneous emission are shown as dashed

red arrows. Processes shown as thin lines are shifted out of resonance due to the

strong sideband coupling, leading to accumulation of population in the maximally

entangled state |Si in steady state. . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
xxiii

6.3 Steady State Entanglement. The measured populations of the singlet, triplet, |↑↑i

and | ↓↓i states are shown as squares, crosses, circles, and triangles, respectively,

as a function of the duration that all the elements of the dissipative entanglement

scheme are applied simultaneously. The system reaches a steady state with a 0.75(3)

population in the target singlet state after a few milliseconds. The solid lines are the

result of a simulation based on the experimental parameters. The slow decrease in

the singlet state fidelity at long times visible in the simulation is due to a leak of the

qubits to spin states outside the | ↑i, | ↓i, |ai manifold caused by spontaneous photon

scattering from the Raman laser beams that generate the sideband coupling. Strictly

speaking this depumping means that the state is only a quasi-steady state. For our

parameters there is, however, a clear separation of the preparation and depumping

time scales, justifying the description as steady state. Error bars represent standard

deviations of each point. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

6.4 Stepwise application of the dissipative entanglement scheme. a) We show the pulse

sequence for one step. In this case, we set the repump and cooling to be full power,

and we turn them on with enough duration so that they act as a complete pumping

step. These pulses remain short compared with the coherent pulse, thus not introducing too much gap time in between the coherent evolution. b) We assume initial

state to be the target |Si state, and we show a simulation with the sideband and

microwave drives are applied simultaneously. In this case, with Ωs/Ωc ≈ 7, similar

to the parameter in the stepwise experiment. We can see the microwave flops part

of the |Si state out to the auxiliary level but the population can be fully revived at

certain durations, as shown by an arrow in the figure. We set the duration of the

coherent pulse in a) to be at one of those durations, so we eliminate the leakage out

from the |Si state. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
xxiv

6.5 Entanglement With Stepwise Scheme. The measured populations of the singlet,

triplet, |↑↑i and | ↓↓i states are shown as squares, crosses, circles, and triangles,

respectively, as a function of the number of applied steps. Each step has a duration

of approximately 220 μs. The solid lines are the result of a model as explained in

the Methods section. Error bars represent standard deviations of each point. . . . . 84

7.1 Schematics for confined dynamics, for the cases of (a) two ions, and (b) three ions.

The thin black double arrows depict the relatively weak microwave coupling, and

the thick blue double arrow depicts the strong laser sideband coupling. With |↑ ... ↑i

states initially populated (red dots), the microwaves drive the state down the symmetric manifold, the states on the left, where |Wi =

|↑↓↓i+|↓↑↓i+|↓↓↑i

√

3

in (b). However,

the sideband excitations perturb the lower states, coupling them to other states, denoted by |Si and |Wci =

e

i2π/3

|↑↑↓i+|↑↓↑i+e−i2π/3|↓↑↑i

√

3

. Thus the microwave induced

dynamics are confined between the two highest energy states on the symmetric manifold. For (b) we only show the states that are relevant to the dynamics. . . . . . . 91

7.2 Two ion population evolution for confined dynamics with microwave and sideband

excitations applied simultaneously. Population mainly evolves between the |↑↑i and

the |Ti state, while other states have very small populations. The pink diamonds,

blue triangles, red squares and green circles represent the measured populations of

states with all spins down, one spin up, two spins up and the |Ti state, respectively.

The difference between P1 and P|Tiis due to the populations in the |Si state. The

solid lines shows the results of the numerical simulation taking into account known

experimental imperfections, with the same coloring convention as for the measured

populations. The black dashed line shows unit population. The error bars for P|Ti

state are indicated by the size of the symbols. All error bars are from data resampling

of a maximum likelihood analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
xxv

7.3 State evolution for confined dynamics of two trapped ions using a composite pulse

sequence. Similar to Fig. 7.2, populations are mainly confined to the |↑↑i and |Ti

states. The coloring and labeling conventions are the same as Fig. 7.2. The laser

phase and detuning are flipped 25.4 μs after the start of the experiment. Note that

the oscillations of |↓↓i are enhanced at t > 25.4 μs; however the maximal population

of |Ti state is increased compared to the single pulse used for the data in Fig. 7.2.

All error bars are from data resampling of a maximum likelihood analysis. . . . . . . 96
Chapter 1

Introduction

In this thesis we will describe the primary research accomplishments with trapped ions (1)

demonstrating sympathetic electromagnetic-induced-transparency (EIT) laser cooling on a mixed

ion chain, (2) generating steady state quantum entanglement with dissipation and (3) generating

quantum entanglement with novel engineered quantum dynamics. These works have different contexts and backgrounds, so it is hard to pick one of them to emphasize a motivation for the entire

thesis. Instead, we will briefly describe the overall background of quantum information processing

and its implementations on trapped ion experiments, and highlight the role of the three topics

above.

1.1 General description of quantum information and entanglement

Information technology is undoubtedly one of the most important breakthroughs in the 20th

century. Its development has been following the Moore’s law [10] for decades, where the number of

transistors in an integrated chip would double every eighteen months. This marked growth of the

level of integration that has led to an increase in the computational and storage capabilities. When

the size of the electronic devices on the chip approaches the size of the atoms, on the order of half

a nanometer, the quantum theory predicts a limit to the Moore’s law [11]. One possible solution

to this problem is to try to process information taking advantage of the quantum effects. It turns

out with those effects, computation can be substantially faster for solving particular mathematical

problems on a large scale, for example [11–13], known as quantum computing. Also for simulation
2

of quantum systems, use of quantum resources can significantly outperform those with classical

resources [14]. For these reasons, research on quantum computing and quantum simulation have

been hot topics for both theoretical [15] and experimental researchers [16–18].

The advantages of using quantum effects derive from superposition. Classically, high or low

electric potentials/currents of electric components are labeled as bit 0 and 1. However, the quantum

bits (qubits) are quantum states, labeled with |0i and |1i. In analog to the spin-1/2 system, we also

label {|0i, |1i} as {|↑i, |↓i} The qubits can be realized with various objects, for example electronic

states in neutral atoms, atoms in a cavity [19], Rydberg atoms [20], laser trapped atoms [21],

trapped ions [22], linear optics [23], photons coupled to ions [24] and the collective charge or flux

in a superconducting circuit [25]. The criteria for the choice of qubit is that they have well-defined

quantum mechanical states, which can be detected and distinguished, and with proper coherence

time be “alive” quantum mechanically, as described below. The use of quantum states to store

information enables quantum superposition, especially quantum entanglement, a unique property

of quantum mechanics giving the advantages. A quantum state |ψi, if we assume it is defined

within one-qubit states, can be generally be expressed by

|ψi = α|0i + β|1i, (1.1)

where α and β are complex numbers, representing the amplitude of |ψi being in each of the states;

|α|

2 and |β|2

represent the probabilities of being in each of the states, with their sum to be unity

by definition. We can remove the common phase between them since this phase is usually nondetectable. However, in general, |ψi represents a quantum state that is simultaneously in |0i and

|1i, and the complex phase difference between α and β is also well-defined, giving the so-called

quantum coherence, so we need 2 real numbers to fully characterize this state. It is convenient to

express the qubits in the vector form, so that |0i =





1

0





and |1i =





0

1





. We also denote the

complex conjugate of |Ψi as hΨ| = |Ψi

†

. Thus the one qubit rotation operation can be in general

expressed as R{θ, φ}, defined in matrix form as R{θ, φ} =





cos( θ

2

) −ie−iφ sin( θ

2

)

−ieiφ sin( θ

2

) cos( θ

2

)





. Inter-
3

estingly for N qubits, the superposition would involve in general 2N states, and one needs 2N+1 −2

real numbers to fully expand the state |ψN i in the |0i, |1i manifolds for each qubit,

|ψN i = α1|00...00i + α2|00...01i + α3|00...10i + ... + α2N |11...11i, (1.2)

where P2N

i=1 |αi

|

2 = 1. An example is that for 300 qubits, it would involve more than 1090 states.

This exponential scaling can in principle give a large win in computation. A simple comparison

[11, 26] is given by considering a classical machine that takes input x and gives output f(x) with

a certain function f. For N bits the input state can be any one of the classical inputs from

00...00, 00...01, 00...10, ..., 11...11, but this computation can only happen once at a time, suppose

we have only one machine. For the quantum case, we define the operation to be input state |xi|0i

and outputs state |xi|f(x)i. One can input an equal superposition of all the states

|ψini =

1

√

2N

(|00...00i + |00...01i + |00...10i + ... + |11...11i)|0i, (1.3)

which can be expressed as |ψini =

P

x

√

1

2N

|xi|0i, where x is the binary number range from 00...00

to 11...11. The output would be |ψouti =

P

x

√

1

2N

|xi|f(x)i, as if all the computations happen

simultaneously, giving the advantage known as so-called quantum parallelism1Interestingly, |ψouti

can not be expressed as a product state of different qubits in general. An example for two qubits

is what we call the triplet state in this thesis,

|Ti =

1

√

2

(|01i + |10i). (1.4)

The qubit states are highly correlated, and an attempt to express the states of each qubit separately,

i.e., writing |Ti = |ψ1i|ψ2i, is impossible mathematically. This interesting phenomenon is called

quantum entanglement, a unique feature of quantum mechanics compared with the classical counterparts. Entanglement is known to be the key property of quantum information processing [11],

such as quantum computing, quantum key distribution for secure communication, and quantum

simulation.

1 The information of f(x) can be further extracted by performing quantum Fourier transformation, to create

interference between states associated with |xi [27].
4

Entanglement also has many other applications, including but not limited to the following.

It can help to show the non-local fundamental feature of quantum mechanics, such as through

Bell’s inequality [28–31]. For spectroscopy, quantum entanglement could help suppressing the

uncertainty of measurement [32, 33], and improve the readouts of the atomic clocks [34–36]. Due

to its importance, generating quantum entanglement is one of the main topics in this thesis, and

we will describe both the traditional and novel methods developed during my thesis work. In the

following section we will describe scalable quantum computation and motivate the need of the

cooling technique we develop.

In practice, due to dissipative processes in the experiments, a quantum state needs to be

expressed as a density matrice ρ, in general represents a statistical mixture of different states |Ψii

with probability Pi, with ρ ≡ Pi|ΨiihΨi|. For a pure state |Ψi similar to the states above, the

corresponding density matrix is |ΨihΨ|. Thus given an experiment result ρ, the probability of

obtaining state |Ψi can be expressed as the fidelity, in this thesis we use the definition hΨ|ρ|Ψi.

1.2 Scalable quantum computing architecture and sympathetic cooling

There are many different ways to implement quantum computing. One way is to trap linear

chains of ions as elementary modular unit, and different modules are coupled by photons emitted

by the communication ion in the chains [37]. Another promising architecture is the scalable CCD

architecture for trapped ions [2, 22], as depicted by Fig. 1.1. In this scheme, ions are trapped in

different trap zones, and by controlling the electric potentials, ions can be separated, recombined

and shuttled between different zones. Ions can be individually addressed to perform single qubit

operations, for example the spin rotations and detections. And ions can also be entangled by pairs

in certain zones and then separated and reshuffled to entangle with other ions in the architecture.

In practice, the individual addressing can be realized by laser beams focused to single trap zone

or by microwave fields with ions moved to different locations in a field gradient [38]. To generate

entanglement, one way is to put pairs of ions together in the same trap zone and employ the

shared motional modes in the ion crystal [39]. An alternative way is to trap them in two close but
5

distinguishable zones [40, 41].

Figure 1.1: A scalable quantum information processing device with trapped ions [2]. The structure

consists of linear trapping sections, and junctions. Ions are trapped in long chains, separated,

shuttled and recombined. Lasers beams are shown to entangle ion pairs.

In the trap-array method, ions are frequently separated, shuttled and recombined. Traditionally, these operations impart significant motional excitation to the ions [42], which affects

subsequent entangling operations [43]. Direct cooling of the logic ions involves driving transitions

of the internal states accompanied by spontaneous emissions, which destroys the quantum states

that store information. For this reason, a technique called sympathetic cooling can be used [44],

where we co-trap a different species of “coolant” ion with the logical ion in the same trap, utilizing

the shared motional modes. We then apply cooling laser beams to the coolant ion to cool the

entire crystal to the motional ground state. Since the cooling laser beams are far off-resonant and

essentially decoupled from any transition of the logic ion, the stored information is preserved.

Traditionally, Doppler cooling and Raman sideband cooling are applied, but the typical

duration is on the order of hundreds of microseconds [42]. Even with the latest improvements

of diabatic and cold ion separation and shuttling [45], several rounds of Raman sideband cooling
6

are needed. One drawback is that to drive high fidelity Raman transitions, high Raman laser

power is needed for sufficient detuning [46]. To address these problems, we examine a technique

called sympathetic electromagnetically-induced-transparency (EIT) cooling. This technique uses

relatively low power and a near-detuned pair of laser beams to create dressed states that can

provide fast cooling and simultaneously cool a broad frequency range of motional modes. We apply

this technique to chains of 9Be+ −24 Mg+ and 9Be+ −24 Mg+ −24 Mg+ −9 Be+ ions, where 24Mg+

serves as coolant ions and we demonstrate near gound state cooling of all the axial modes with an

order of magnitude improvement on cooling speed while decreasing the applied laser power by two

orders of magnitude compared to experiments that use traditional sideband cooling.

1.3 General description of entanglement operation and novel methods

To generate entangled states, it is conceptually convenient to introduce the controlled-not(CNOT) operation. Here, if the state on the control qubit is at the |0i state, no operation would be

applied to the target qubit; if the control qubit is at the |1i state, then the target qubit would get

a spin flip, an operation known as a NOT logic gate. A special case is that if the control qubit is

initialized to an equal superposition of |0i and |1i, and the target qubit is initialized to spin down,

after this operation a triplet state is created as in Eq. 1.4.

Entangling operations require a mutual coupling between the two qubits. To entangle trapped

ions, a common way is to trap ions in the same electric potential, and due to the Coulomb interaction, ion motion consists of normal modes. For example, for the case of two equal mass ions,

the motional modes are the common mode, where the ions oscillate with the same phase; and the

stretch mode, where they oscillate out of phase. As detailed in Chapter 4, one can apply a motional

sideband transition, where a pair of non co-propagating laser is tuned to have frequency difference

equal to the qubit transition plus/minus a motional mode frequency. Starting from the ground

qubit state, this pair of laser beams would flip the qubit state while at the same time excite/deexcite one quanta of the relevant motional mode. One simple way to generate spin-spin coupling is

to apply the pair of laser beams focused on the two ions separately; in this case the spin-dependent
7

motional excitation on one ion can be used as a trigger for spin flip for the other one [15, 47]. Other

techniques have also been explored [48–53]. These also involve laser sideband couplings for qubit

spin flip of qubit phase accumulation.

In this thesis, we introduce two novel methods to generate entanglement. The first method

is to have the sideband transition as a perturbation and thus constrain the quantum dynamics,

known as the Quantum Zeno dynamics [54–56]. Here the qubit rotations can be implemented with

microwaves, which can have very high fidelities [57–59]. We can then drive a transition from an

unentangled initial state to an entangled state within the constrained manifold. Since the transition

is not directly driven by the laser, the requirement of precise control of the timing of the laser pulse,

laser intensity, laser detuning and motional mode frequency stability are not stringent, compared

to the quantum logic operations mentioned above. We demonstrate this scheme with two ions

and obtain a high fidelity triplet state creation. This scheme can be generalized to three ions in a

straightforward manner, and we demonstrate generation of the W-state, an equal superposition of

two spin up and one spin down states.

The second method is more complicated, where we generate a state that does not couple

to the sideband interaction. Furthermore, this method reaches a steady state using dissipative

techniques, where we induce cooling, optical pumping, microwave transitions, and laser sideband

interaction all together, and the system is pumped to an entangled state which does not couple to

any of the interactions above. Here the target state is a “dark” state, while the other states of the

system are pumped to and trapped in the dark state. In previous experiments, entanglement has

been generated by applying dissipation to atomic ensembles coupled by electromagnetic modes to

generate [60], and by applying quantum logic gates and dissipation processes to trapped ions [61].

Our experiment is the first demonstration of steady state production using dissipation for trapped

ions without using quantum logic gates, and can be further deployed as a stabilization technique,

where if the target state suffers an error, the continuous application of the interactions can actively

pump the state back to the target state [62].
8

1.4 Thesis organization

In Chapter 2 we describe the general features of our apparatus, giving some technical details,

including the structure of the ion trap, vacuum system, laser beam lines, microwave setup, imaging/detection setup and control systems. In Chapter 3 we describe the energy structure of the 9Be+

ion, the qubit ion in our experiments. We further describe the operations on the qubit, including

initializing and manipulating the qubit spin and motional states, with laser beams and microwave

fields, and detection of spin and motional state. In Chapter 4 we describe the spin-motional coupling induced by laser beams. In Chapter 5 we describe the EIT cooling method mentioned above.

We start with the general EIT theory, and we describe the experimental setup and the experiment

process. We further describe the technical details of our experiment and possibility of applying the

technique to other species of coolant ions. In Chapter 6, we describe the steady state entanglement

generation and stabilization using the dissipative method. In Chapter 7, we describe the theory

and experiment of the Zeno dynamics. We further present a composite pulse sequence technique to

our scheme for better performance, and also we present the application to three ions. For Chapters

6 and 7, we will describe our understanding of the error sources, the construction of numerical simulations, and we compare the experimental results with the simulations. Finally, we will conclude

this thesis with remarks on future applications of techniques in this thesis.

If you find this thesis interesting or have questions or remarks, please send me an email at

yiheng.lin@colorado.edu or yihenglin.ustc@gmail.com.
Chapter 2

Ion trapping

In this chapter we will describe various apparatus for the trapped ion experiments in this

thesis. The beauty/complexity of a trapped ion experiment is that it involves many different

technologies. In Fig. 2.1, a basic structure of the trapped ion setup is shown with some examples

of specific equipment. In this chapter we will briefly describe the various tasks and requirements in

Fig. 2.1 are satisfied. In the first and second sections we describe the general theory of ion trapping

and the trap configurations we are using, respectively. In the third section we describe a few other

components for the trapped ion experiment: the magnetic field coils, the vacuum system, imaging

system, and the experimental control system.

2.1 Idealised ion trapping theory

An idealised ion trap with a “four-rod” configuration is depicted in Fig. 2.2; to begin with

we describe the theory for trapping a single ion [22, 63]. An RF voltage with optional offset

V (t) = V0 cos(ΩTt) + Ur is applied to the RF electrodes. We ignore the phase of the RF because it

does not affect the trapping in this configuration. Near the center of the radial plane, the potential

expanded to second order has the form

Φ ≈

(V0 cos(ΩTt) + Ur)

2

(1 + x

2 − y2

R2

), (2.1)

where R is approximately the distance from the center to the nearest surface of the electrodes.

DC voltages U0 are applied to the high voltage DC electrodes and 0V applied to the low voltage
10

Figure 2.1: Schematic for trapped ion experiment, showing the major components and some more

details and requirements of technology therein.

electrodes. Note that the potential satisfies Laplace’s equation ∇2Φ = 0. Along the trap axis near

the center, to the second order a static potential is created of the form

Φs ≈ κU0(z

2 −

x

2 + y2

2

) = m

2q

ω

2

z

(z

2 −

x

2 + y2

2

), (2.2)
11

where m and q are the mass and charge of the ion, ωz =

p

2κqU0/m is the oscillation frequency

along the axial direction and κ is the geometrical factor determined by the actual trap geometry,

which can be numerically calculated with a finite element method. We call ωz the center-ofmass(COM) motional frequency. This frequency is independent of the number of a collection of

ions with identical mass and charge, as long as the expansion of Eq. 2.2 is valid.

trap axis z 

radial y 

RF electrode

DC electrode, 

high voltage

DC electrode, 

low voltage

DC electrode, 

high voltage

positive ions

radial x 

Figure 2.2: An idealised ion trap with “four-rod” configuration. The gold rods are the RF electrodes, and the segmented ones are DC electrodes. The two RF electrodes are typically shorted

together, and provide a quadruple confinement potential in the radial plane. To form confinement

of positively charged ions along the axial direction, we apply positive potentials on the outer electrodes and lower potentials on the center electrodes. The axial confinement is designed to be weaker

than the radial confinement, so that the ions are trapped along as a linear chain along ˆz.

We first describe the radial direction of motion. With Eq. 2.1, Eq. 2.2 and Newton’s second

law, equations of motion can be expressed in the form of the Mathieu equations

d

2x

d

2ξ2

+ (ax + 2qx cos(2ξ))x = 0, (2.3)

d

2y

d

2ξ2

+ (ay + 2qy cos(2ξ))y = 0, (2.4)

where ξ = ΩTt/2, ax = ( 4q

mΩ2

T

(

Ur

R2 − κU0)), ay = (−

4q

mΩ2

T

(

Ur

R2 + κU0)), and qx = −qy =

2qV0

mΩT

. Here

we ignore the radial potential from the DC electrodes. Note that Ur breaks the degeneracy of the

two equations. And the existence of the DC voltages can also break the degeneracy. To solve for

Eq. 2.3, the Floquet theorem can be applied due to the periodic nature of the equations. With

a proper choice of parameters the solution represents a non diverging trajectory, or the stable
12

trapping region [64, 65]. Typically with ai < q2

i  1, i ∈ {x, y}, a stable solution can be achieved

and can be described by an infinite series of harmonic oscillators with decreasing amplitudes for

higher orders. The two lowest orders are given by

ui(t) ≈ Ai(cos(ωit + φi)(1 + qi

2

cos(ΩTt) + q

2

i

32

cos(2ΩTt)) + βi

qi

2

sin(ωit + φi) sin(ΩTt)), (2.5)

where ui = {x, y}, Ai and φi depend on the initial conditions, and

ωi = βi

ΩT

2

, βi ≈ (ai + q

2

i /2)1/2

. (2.6)

The ωiis known as the secular motion frequency of the radial direction, and the motion for low

energy, corresponds to that of a harmonic oscillator. From Eq. 2.5 and ai < q2

i  1, i ∈ {x, y},

the motion is modulated by a fast oscillation with frequency ΩT, called the micromotion. However

with experimental imperfections such as stray charges, unbalanced DC electric voltages, the ion

can be displaced from the trap axis so the micromotion is not negligible. This can be compensated

by applying extra potentials on the DC electrodes, known as “shims”, as described in [5].

For linear traps, typically the radial confinement is made to be much stronger than the axial

confinement. If multiple ions are trapped in the same potential, the ions form a linear chain along

the axial direction. For two identical ions, the motion of the ions are coupled due to the strong

Coulomb repulsion. To solve for the equation of motion, typically one would start from solving

the equilibrium location of the ions, then perform an expansion to solve for the secular motions.

For identical ions, introducing the length scale s = q

2/3/(4π0mω2

z

)

1/3 would be convenient. For

9Be+ ion, with ωz = 2π × 3.58 MHz, s ≈ 3.12μm. For two ions, the separation of the two ions

is s2 = 21/3s; and for three ions the neighboring separation is s3 = (5/4)1/3s. For the motional

modes, the center of mass mode is not changed for small chain of ions, but more motional modes

would appear for multiple ions. More detailed analysis will be provide in Chapter 4.

2.2 Trap configurations

In recent years significant attention has been given to planar trap configurations [66], where

the four rods and other electrodes are flattened and can be fabricated onto a planar surface. This
13

enables the use of on-chip microwave lines for single- and multiple-ion logic gates, and in the future

can enable the use of on-chip detectors and laser waveguides [67, 68]. Here we focus on describing

the three dimensional traps used in the experiments in this thesis. We will first describe the basic

components including the trap wafer and trap RF, which form the trap. Then we will describe two

types of traps, the linear and the junction traps, which we use in the experiments.

2.2.1 Trap wafers

In the experiments described in this thesis, a three dimensional trap configuration is used,

similar to the four-rod configuration above. The rods have evolved to various geometries in the

experiments, such as four blades [69]; in our lab it was implemented as a wafer structure. The

wafer is constructed from laser machined polished alumina substrates. A through slot is laser

machined on each wafer to provide optical access with ions accessible through the middle of the

slots. The electrodes are formed by either patterned sputter-coated gold or electroplated gold

deposition [3, 70, 71] surrounding the slot. The separate wafers were stacked together by hand with

annealed stainless steel screws, with alumina spacers in between and gauge-pin holes for alignment.

In addition to the electrodes that form the “four rod” configuration, an extra wafer is needed to

provide a biased electric field, named as the “bias board”. The bias board is needed to provide DC

potentials along the direction of RF electrodes (radial y-direction in Fig. 2.2), since the two RF

electrodes are shorted together. All the electrodes on the wafer are extended to the edge of the

wafer, with larger widths near the edge of wafer for wire bonding or ribbon bonding to the back

plane.

The back plane is also called the filter board, providing low-pass filtering for the DC electrodes to block frequencies near the trap frequencies, on the order of a few MHz and higher. The

filter board has electrodes that at one end match with the size of the electrode on the wafer for

wire/ribbon bonding and the other side has larger size for soldering to leads that connect to voltage sources. For on board filtering, these electrodes also have gaps for on-chip resistors; one end

of the capacitors are soldered onto the electrode after the resistor and the other end connected
14

to the ground pad on the board. The electrodes are then further connected via feed throughs to

connections outside of the vacuum. The resistor is typically ∼ 250 Ω and capacitor ∼800 pF, to

have a corner frequency below 1 MHz.

To realise the basic components of scalable quantum information processing [2], multiple

segmented electrodes are fabricated on the trap, enabling multiple trap zones and control of ion

transport, including ion shuttling, combining ions from separated zones into the same zone and the

reversed processes(ion separating). Typically, a range of DC electrode voltages of -10 to 10 V is

enough to perform ion transport, as well as forming axial trapping potentials of ∼4 MHz for one

9Be+ ion, and performing micromotion nulling.

2.2.2 Trap RF

In our experiment, the RF source1ranges from -1 dBm to -4 dBm, which is further amplified2

to ∼2 Watts of power. As we describe above, we need a high voltage amplitude on the RF electrode

for stable trapping. Typically we capacitively couple the output of the amplifier to an RF λ/4

coaxial resonator [72]. One end of the center rod is connected to the grounded outer cylinder, and

the other end is connected to the trap RF electrode and is open circuited. The length of the outer

cylinder is longer than the inner rod, so it also functions as a shielding cover for the trap chip, with

holes to allow optical access. Also it functions as a support structure to hold the trap chip and

oven in place. With Q∼100 we can achieve a voltage on the electrode to be ∼200 V.

2.2.3 The linear trap and the junction trap

The first ion trap in my experiments is configured to trap ions in multiple multiple zones of a

linear trap, designed by Murray Barrett and built by John Jost [3]. The trap chip, filter board and

segmented electrode scales are depicted by Fig. 2.3. With this trap a number of experiments were

performed to demonstrate techniques for quantum information processing, including demonstration

of a magnetic field insensitive qubit for 9Be+ [73], entangled ion motion in separate trap zones [74],

1 Agilent Oscillator, Rohde & Schwarz signal generator SMT 02

2 Amplifier Research, 36 dB Gain
15

and a two qubit quantum processor with quantum logic gates and ion transport [42, 45, 75–77].

For this thesis, we demonstrate EIT cooling [78] and dissipative entanglement [79] with this trap.

Up to nine 9Be+ ions in a linear chain have been trapped and tested for separation waveforms [45].

In our experiments we trap with 9Be+ -

24Mg+ ion pair and 9Be+ -24Mg+ -24Mg+ -9Be+ ion

chains.

When we moved the apparatus to another building at NIST, we set up the experiment based

on a trap designed and built by R. Bradford Blakestad [4], shown in Fig. 2.4. The unique feature

of this trap is the “x” shape junction that provides capability of re-ordering the ions in a chain by

shuttling the ions through the junction. Such junctions are important building blocks of a scalable

architecture of trapped ion quantum computing [2], depicted by Fig. 1.1. In this architecture,

single qubit rotations, two/multiple qubit operations, and spin detections are dedicated to different

trap zones, and junctions are needed to extend this architecture to two dimensional. The junction

enables coupling between arbitrary pairs of ions, in contrast to linear trap which can only easily

couple neighboring pairs of ions. In the experiment described in this thesis, we trap ions only at

the load zone and the experiment zone.

2.3 Other components for ion trapping

Many other components are required for an ion trap experiment. Here we describe a few of

these. For loading ions in the trap, we describe the atom source and lasers used for photoionization.

We then briefly describe the vacuum system, imaging system, and the coils for the external magnetic

field. Finally we describe the laser and microwave sources for atomic state manipulation, and the

experiment controls.

2.3.1 Ion loading: atom source and photoionization

In our traps, a trapping zone is dedicated to loading ions. The loading process involves

emitting atoms form a thermal source (the oven), and atoms are subsequently ionized by electron

beam collision or by photoionization (PI). These processes can contaminate the nearby trap surface
16

Figure 2.3: View of the linear trap [3]. Only the top wafer is visible. The RF electrode and

the segmented DC electrodes are shown, and the DC electrodes are gold ribbon bonded to the

filter boards. The bottom figure represents a closed view of the segmented electrodes. The widths

of the electrodes from left (O1) to right (L2) are {2,0.2,0.1,0.2,0.36,0.5,0.5,2}mm [3]. The small

gaps between the electrodes are 0.02 mm. The electrodes O1-O2 form the experiment region; the

electrodes L1-L2 form the loading region. The gap between the DC and RF electrodes in the

experiment/loading region is 0.2/0.4 mm wide.

and increase the motional heating rate. Such a problem can be mitigated by dedicating a load zone
17 86

















	




















	



















	




















	












 

















 



	
 









Figure 4.5: a) Cross-sectional view of the two layers of electrodes in the X-junction array. b) Top

view of the electrode layout, with the rf electrodes indicated, and all other (control) electrodes held

at rf ground. A nearly identical set of electrodes sits below these electrodes, with rf and control

electrodes interchanged. Each control electrode is numbered, including the obscured electrodes on

the second layer. Forty-six control electrodes support 18 different trapping zones. The load zone

(L), the main experiment zone (E), the vertical zone (V), the horizontal zone (F) and the center

of the junction (C) are labeled. c) Schematic of the rf bridges from an oblique angle (not to scale).

Figure 2.4: Close view of the junction trap. Only the top wafer is visible. The RF electrode and

the segmented DC electrodes are shown, and the DC electrodes are gold-wire-bonded to the filter

boards. The bottom figure represents the scale of the segmented electrodes. L, E, V, F, and C

represents the loading zone, the experiment zone, the vertical zone, the “far” zone, and the center

of the junction, respectively. The width of the electrode at E is 200 μm. The narrower electrode

next to it is 100 μm wide, specifically for ion separation. The gaps between the electrodes are 20

μm wide. Ions chains can be separated and shuttled back and forth through two different routes,

either E → C → V → C → E or E → C → F → C → E, and thus enabling ion chain reordering [4].

The inset shows a side view of the junction and connection of the RF electrodes.
18

away from the experiment zone. A mask layer for shielding neutral atom flux to the experimental

zone also helped. After the ions are trapped and Doppler cooled in the load zone, they are shuttled

to the experiment zone by applying a voltage waveform on the DC electrodes.

We have ovens as sources of 9Be and 24Mg(25Mg) atoms. The 9Be oven is composed of a 9Be

wire twisted onto a tungsten wire, which is made into a spiral which is mounted to leads that are

connected to a current source which heats the oven. Resulting 9Be atoms that pass through the

load zone are ionized by either an electron gun, or using a 235 nm photoionization laser, detailed

below. The Mg ovens are composed of metal tubes that can be heated with current, with a slit

pointing towards the trap, filled with enriched/non enriched 24Mg atoms. The non enriched oven

contains 25Mg and 26Mg atoms, and the loading of isotope is determined by the exact wavelength

of the 285 nm photoionization laser. There is a oven shield to block unwanted atom emission to

the trap chip with a hole for the load zone.

We use a 235 nm pulsed laser source to ionize neutral 9Be atoms. The process is to drive

a transition from 1s

22s2

to 1s

22s2p, centered at 234.9329 nm [80], and then further excite to the

continuum for the 2p electron at a wavelength below resonance at 306.492 nm. We use one pulsed

laser at 235 nm to drive this two-step process. Additionally, the 313 nm Doppler cooling laser for

loading can also drive the second stage of transition. The setup, as depicted in Fig. 2.5 is mainly

composed of a 705 nm laser source and a frequency “tripler” producing 235 nm light. The 705 nm

source is composed of a 532 nm 5 W Verdi pumped Ti:Sapph mode locked laser from K-M labs3,

where we tune the center wavelength to 705 nm, the spectrum with a full width half maximum of

∼13 nm, repetition rate ∼100 MHz and pulse duration of sub-100 fs. This beam is then sent into

a frequency tripler from APE4, where the frequency is first doubled to 352.5 nm then sum with

705 nm to produce 235 nm light.

With normal operation of the Verdi V5 at 5.5 W, we typically obtain 600 mW average power

on the 705 nm laser, and 8 mW average power on the 235 nm output. We find that even with 4

3

the KM Labs Griffin Ti:Sapphire pulsed laser

4 APE GmbH Pr ̈ufprotokoll frequency trippler setup
19

mW average power before injecting to the chamber, we typically load a 9Be+ ion within tens of

seconds5.

For ionization of 24Mg and 25Mg atoms, we use a 285 nm CW source, frequency doubled from

a 570 diode laser source6. This source excites 24Mg atom from the 3s

2 1S0 state to the 3s3p1P1

and further to the continuum state [81, 82].

Verdi, 5W

Ti:Sapph

frequency 

tripler

532nm

705 nm

235 nm

periscope

Wave

meter

Wave

meter

loading zone

Figure 2.5: Schematic for the 9Be photoionization laser setup. The mirror mount in dash outlines

is a pick off mirror, which is not in the setup except for checking spectrum for 705 nm light. The

mirror at the output of the tripler has finite transmission that leaks a small portion of 235 nm light

for wavelength monitoring.

2.3.2 Vacuum system

The vacuum system is required to provide low background gas collision rates. The elastic

collisions can lead to ion motional heating, excitation to complicated motional trajectory (decrystalization). The worst effect of the collisions is that can lead to chemical effects and turn Be+ or

Mg+ ions into molecules. A typical example is for 9Be+ ion experiments, during detection, repumping or Doppler cooling, lasers are applied repeatedly to drive the 2S1/2 →2 P1/2,3/2

transitions, and

the excited P states of a 9Be+ ion are chemically reactive to collisions with an H2 molecule, and

5 Loading time is limited by the warm up time of the oven.

6 MPB Raman Amplifier system, seeded by Sacher diode laser ∼1140 nm and doubled by PPLN crystal. Output

power 1.6 W. Wavelength set to 570.5925 nm
20

can be converted to a BeH+ ion [22].7

A typical vacuum pressure for our system is 8×10−10 Torr ∼ 1×10−7 Pa. The vacuum

assembly consists of a quartz cell joined to stainless steel Conflat vacuum parts, depicted in Fig.

2.6. The windows for laser access are made from UV-grade fused silica. Other parts in the system

are: an ion pump8to maintain the vacuum environment; a titanium sublimation pump (TSP) to aid

in removing excess hydrogen molecules; a quartz cell to cover the trap and resonator; and a Conflat

interface for an ion gauge and electrical feed-throughs for the RF, DC and oven connections.

2.3.3 Imaging system

The imaging system is consist of magnifying imaging optics, an iris at the intermediate

focus for blocking stray scattered light, and a flipper mirror to direct fluorescence to either a UV

sensitive charge coupled device (CCD) camera9 or a photo multiplier tube (PMT)10 . The CCD

camera provides a relatively large field of view that is very helpful for beam alignment to the trap

features (corners, gaps and junctions), checking unwanted laser background scattering, and most

importantly for identification of individual ions loaded in the trap. The PMT functions as a single

pixel sensitive detection device that give counts for ion state dependent photon scattering, as the

main measurement device for ion spin states. Typically for 300 μs detection duration for a single

9Be+ ion, the imaging system gives ∼1 count if the ion is in the dark spin state, an ∼25 counts

in the bright state. The path of fluorescence to either the CCD camera or the PMT is switched

by the mechanical flipper mirror. A housing made from aluminum provide mounting for the CCD

camera, PMT and the flipper, also provides shielding from ambient room light.

For the linear trap setup, we use a two-step magnification for 9Be+ ion detection. The first

step is a 5x ∼f/1 lens, whose input surface is located ∼40 mm from the ion trap. A second stage

7 This turns out to be a significant problem for quantum simulation experiments in a Penning trap [83], where

hundreds of 9Be+ ions are trapped and the BeH+ ions are frequently observed. A technique is developed to dissociate

the molecule back to 9Be+ ion with a 157 nm UV laser, guided with nitrogen gas tube to prevent high laser power

loss in air [84].

8 Agilent Varian StarCell pump

9 Princeton Instrument PhotonMax 512

10 Hamamatsu R7600P
21

Port for electrical 

feedthroughs

Figure 2.6: A mock up of the vacuum setup [3], showing relevant vacuum components. The Quartz

cell has six windows made from fused silica for UV light access, and four of them are used for

laser access and one is used for imaging. On the input end of the resonator, there is a cylindrical

component to connect the center conductor to the ground. The output of the RF source after the

amplifier is connected to a small wire loop, mounted inside the can to couple RF power into the

resonator.

of ∼ 25x magnification is provided by a UV microscope objective. For the junction trap, since we

need to image both 9Be+ and 25Mg+ at the same time without moving the imaging optics, we use

a Schwardzchild imaging optics [85], design by P. Huang and D. Leibfried, and built by Sill optics.

The idea is to use curved reflective optics with proper coating for both the 313 nm and 280 nm

laser throughout. For this setup, the Schwarzschild imaging optics provides an amplification of 10x,
22

and the output is further amplified by 5x through a second stage of a curved-flat mirror set, built

by T. R. Tan. To compensate for the differential refraction of 313 nm and 280 nm light through

the fused-silica vacuum window, a combination of fused-silica and CaF2 optical components are

included between the vacuum window and the first reflective curved mirror.

2.3.4 Magnetic field coils

We apply an external magnetic field to lift the degeneracy of the hyperfine states of the

ion.11 The field coil is in a Helmholtz configuration, to provide a nearly homogeneous field near

the trap center. With this external field, the lasers have a well-defined quantization axis, where σ

±

polarized light propagates along the field direction, and π polarized light propagates perpendicular

to the field direction with the electric field pointing along the magnetic field direction. Details are

described in the Sec. 3.2. Laser beam access to the trap determines the design of the magnetic

field coil so that the field is oriented 45◦ with respect to the trap axis, and the gap of the two

Helmholtz coils provides laser beam access to the π light. The shape of the coil is designed to be

approximately square (about the coil axis) for ease of installation.

For 9Be+ ions, we apply an 11.9446 mT field, to minimize the magnetic field sensitivity on

two of the ground hyperfine states, which we select as qubits. Details are described in Sec. 3.1.

To generate this relatively high field, we need to apply 62.38 A static current12 to the coil (4.8

V) made with 9 turns of copper tubing through which water is circulated [3]. In addition to the

main coil, we also have other smaller coils whose fields are mutually orthogonal. The ones along

the direction of the main quantization field provides fine tuning of the field strength to match the

qubit insensitive point; other coils that are orthogonal to the main quantization field, to fine tune

the magnetic field direction to be aligned with the axis of the σ

± laser beams, and have the option

to compensate for unwanted field gradients across the ion chain. More importantly, these coils can

11 The external magnetic field could be generated by a permanent magnet, but the thermal stability of the field

strength from these magnets was a concern. Despite this drawback, by bringing the magnet close enough to the ion

location, a significant magnetic field gradient can be generated, which can be used to generate spin-motion coupling

[86]. For experiments where decoherence due to the magnetic field stability is not a major issue, a simple coil with

moderate current to generate a few Gauss is enough to resolve the hyperfine states.

12 Agilent 5682A
23

provide compensation of the slow magnetic field drift, by recalibrating the currents after every 5

min13.

2.3.5 Experiment controls

For the experiments, we need to (1) dynamically control the duration, phases, frequency and

amplitudes of laser and microwave pulses, (2) collect photons and (3) we need to dynamically control

the voltages on the electrodes for ion shuttling, separation and recombination. This control system

is based on various field-programable-gate-arrays (FPGAs) and a graphical-user-interface (GUI)

[5, 87]. An FPGA consists of reconfigurable integrated circuits that can form user-specified digital

logic operations, between the inputs and outputs. The first requirement is achieved by control

of the input radiofrequency, to the AOMs for the laser and to the dipolar antenna, described in

previous sections. The second requirement is achieved by taking the output of the PMT as a

transistortransistor logic (TTL) signal which is sent to the master FPGA. The last requirement

is satisfied with a waveform generator with fast digital-analog-convertors (DACs) and a onboard

FPGA described in detail at [5].

The RF is generated by Direct-Digital-Synthesizers (DDS)14 that outputs frequencies up

to approximately 480 MHz, referenced to a 1 GHz clock. This clock is derived by the 10 MHz

Hydrogen maser source at NIST. We amplify the DDS RF signal up to ∼2 W to drive the AOMs.

To generate microwave signals, we take the DDS outpupt signal centered at 250 to 300 MHz and

double twice to reach the ion transition frequencies at 1 to 1.2 GHz for 9Be+.

15 To generate a

pulse, the RF signal is connected to a TTL controlled RF switch. We use 16 independent TTL

channels and 16 independent DDS channels, controlled by a FPGA16 with a commercial evaluation

board17 clocked at 62.5 MHz. The FPGA receives the commands sent by the user through the

13 These smaller coils are either driven by Agilent U8001A with typical current of 0.6 A or computer controlled with

Agilent 6614C with General Purpose Interface Bus (GPIB) to Universal Serial Bus (USB) interface. Experimentally

we observe the field on the ion has a noticeable dependence with the location of the aluminum lens mounts near the

trap.

14 The DDS we use are AD9858. A newer versions AD9914 is deployed for the ion clock experiments.

15 Similar procedure with an extra doubling step is done for 25Mg+ ion for other experiments.

16 Xilinx Virtex-IV

17 Xilinx Xtreme DSP kit
24

GUI, then sends out signals that changes the status of the TTL and DDS modules.

The laser amplitude control is done by using a Newport LB1005 servo box, where shortly after

the laser pulse is turned on, the servo box is triggered to start servoing the RF amplitude to stabilize

the laser intensity. We partially reflect the UV laser beam out of the UV fiber to photodetectors.

Thus the laser beam pointing fluctuations before the fibers are converted to intensity fluctuations

at the output. So this system takes care of beam pointing fluctuations as well as laser power

fluctuations. The lock point of the intensity on the servo lock is either set by hand, or can be set by

a GPIB controlled voltage source. Thus we can scan the laser intensity if needed. When the pulse

is turned off, the servo box is set to the holding stage, keeping the output voltage to be constant

without active servoing. 18

18 This process can be simplified by using a digital servo system, designed by David Leibrandt at NIST. This system

uses a local FPGA, with Xilinx Spartan Spartan 6 LX150 with Opal Kelly XEM6010-LX150 evaluation board. The

photodetector signal is digitized by a analog-digital-convertor (ADC), processed by the FPGA, and the servo signal

is generated by DAC. This system is much more versatile than the analog counterpart, and custom servo functions

and filters can be programmed.
Chapter 3

9Be+ qubit

A qubit for a trapped ion can be implemented with two internal hyperfine states. For the

case of 9Be+ ion, the magnetic dipole radiative decay lifetime between the eight hyperfine ground

states is very long, on the order of 1015 s [22]. The coherence time between the two hyperfine states

can be affected by fluctuations in the ambient magnetic field; however frequency fluctuations can be

made equal between specific pair of hyperfine states by properly setting the magnetic field. Using

such states as qubits, the coherence time can be made very long, up to tens of seconds with proper

choice of magnetic field [73]. To perform quantum information processing, before the experiment,

we need to initialize one qubit state with high probability. Also after the experiment, we need to

detect the populations of the qubit states. In this chapter, we will describe these processes.

3.1 9Be+ energy levels

Beryllium, and in general the alkaline earth atoms, have two valence electrons. For 9Be+

ion, the valence electronic structure is similar to alkali atoms which is commonly used in trapped

neutral atom experiments. The 9Be+ ion ground electronic configuration is the 1s

22s2S1/2

states,

and for excited states we normally only take into account the 1s

22p2P1/2 and 1s22p2P3/2

states.

As depicted in Fig. 3.1, the electric dipole transition wavelength between the S to P states is

approximately 313 nm, and the frequency difference between the two P states is approximately

197 GHz [88]. The hyperfine splitting between the 2s

2S1/2

states is 1.2 GHz, due to the coupling

between the 9Be+ nuclear spin I =

3

2

and the spin of the valence electron. The hyperfine levels are
26

labeled as F=1 and F=2, the former with higher energy.

F = 2

F = 1

mF = –2

mF = –1

102.8 86.7 76.4 69.1

86.5 76.2

1207.495,843

F = 2

F = 1

mF = –2

mF = –1

39.9 28.5 23.2 20.1

28.2 23.1

261.8

2

P3/2

197 GHz

2

P1/2

313.20 nm

2

S1/2

mJ= 3/2

mJ= 1/2

mJ= –1/2

mJ= –3/2

mF = –3

mF = +3

223.4

repumper Blue (RD Blue)

repumper Red (RD Red)

Doppler/detection (BD) Doppler detuned (BDD)

Be CO

Be 90

Figure 3.1: Schematic for the 9Be+ ion electronic transitions at 11.9446 mT magnetic field. Laser

beams are shown to address various levels, see text. The relevant energy splittings are shown, and

the transition |2, 0i ↔ |1, 1i with transition frequency 1207.495843 MHz is to first order insensitive

to magnetic field fluctuations. A typical Raman laser configuration driving the qubit transition

is shown. At 11.9642 mT magnetic field, there is another first order field insensitive transition

|2, 1i ↔ |1, 0i of frequency 1207.352808 MHz.
27

To further split and resolve the hyperfine manifolds, we apply an external magnetic field to

induce the Zeeman effect. A typical field strength in our experiment is 11.9446 mT to obtain first

order insensitivity to field amplitude fluctuations for certain transitions. For the 2s

2S1/2

states,

the Hamiltonian considering the electron spin, nuclear spin under the influence of their interactions

and the magnetic field B is

HZeeman = hAI · J − μ · B (3.1)

= hAI · J + μB(gJJ + gI I) · B, (3.2)

where h is the Planck constant, J = {σx, σy, σz}/2 is the angular momentum operator for the

2s

2S1/2

states, σx,y,x are the Pauli operators, μB = h13.996 MHz/mT is the Bohr magneton,

A = −625.008837048(10) MHz is the hyperfine constant [89] and gJ/I is the corresponding Land ́e gfactor. The measured values are gJ=2.00226206(42) and gI

gJ

=2.134779853(2)×10−4[89]. In general,

gJ ≈ 1 + J(J+1)+S(S+1)−L(L+1)

2J(J+1) . I = {Ix, Iy, Iz} is the nuclear spin operator, where

Ix =

1

2





0

√

3 0 0

√

3 0 2 0

0 2 0 √3

0 0 √3 0





, Iy =

1

2





0 −

√

3i 0 0

√

3i 0 −2i 0

0 2i 0 −

√

3i

0 0 √3i 0





, (3.3)

and Iz =

1

2





3 0 0 0

0 1 0 0

0 0 −1 0

0 0 0 −3





. (3.4)

Here we choose the z-axis along the magnetic field with B = Bz = |B|, and the z projections

of I and J are labeled as mI and mJ , respectively. At B = 0, the total angular momentum

F = I + J is conserved, and thus |F, mF = mI + mJ i represent the eigenstates. The eigenenergies

and eigenstates can be found by using the Breit-Rabi formula [90]. At low field, the energy shifts for

the |F, mF i states are well approximated by (hgF μBBmF ), where gF =

F(F +1)+J(J+1)−I(I+1)

2F(F +1) = ±

1

4
28

for F = 2 and F = 1, respectively. These energy shifts can be obtained by treating the second

term of Eq. 3.1 as a perturbation. However, for higher magnetic fields, the perturbation theory is

no longer valid, and we need to diagonalize Eq. 3.1 to obtain the correct energy levels and eigenenergy E, as depicted in Fig. 3.2, where we also plot the energy sensitivity dE

dB . At certain B field

amplitudes, the sensitivities of the levels are equal, so that any fluctuations of the B field amplitude

will be first order homogeneous to these levels. Thus, transition frequency between these levels are

to the first order insensitive to B field fluctuations. In Table. 3.1 we summarize the field insensitive

points, the relevant transitions between the |ii and |ji states, and the difference of second order

sensitivities ∆d

2E

dB2 = |

d

2Ei

dB2 −

d

2Ej

dB2 |.

20 40 60 80 100

B/mT

- 1.5

- 1.0

- 0.5

0.5

1.0

1.5

2.0

Energy/(h GHz)

|2,2⟩

|2,1⟩

|2,0⟩

|2,-1⟩

|2,-2⟩

|1,1⟩

|1,0⟩

|1,-1⟩

a)

5 10 15 20 25 30

- 10

- 5

5

10

b) E. Sens. /(h MHz/mT)

B/mT

|2,2⟩

|2,1⟩

|2,0⟩

|2,-1⟩

|2,-2⟩

|1,1⟩

|1,0⟩

|1,-1⟩

Figure 3.2: a) Energy levels in unit of (h GHz) by diagonalizing Hamiltonian in Eq. 3.1 over the

external magnetic field B in mT. The eigenstates are labeled in the format of |F, mF i. For B 6= 0,

F is no longer a good quantum number although mF is. Here we just label the levels with the

order of the energies, in conjunction with the low field case. b) The energy sensitivity dE

dB in the

unit of (h MHz/mT) over the external magnetic field.

Normally we operate at a magnetic field B ≈11.944618 mT. At this field, the field sensitivity

dE

dB of the |F = 2, mF = 2i ↔ |1, 1i transition is 17.644091 (h MHz)/mT. This is important for the

|1, 1i ↔ |2, 2i transition for state initialization and detection, as we describe in Sec. 3.3. At this

field, the eigenvalues for the hyperfine states are listed in Table 3.2, compared with the case of zero
29

transition frequency (GHz) B (mT) ∆d

2E

dB2 (h MHz/mT2

)

|F = 2, mF = 0i ↔ |1, 1i 1.207496 11.944618 0.609734

|2, 1i ↔ |1, 0i 1.207353 11.964241 0.609765

|2, 1i ↔ |1, 1i 1.082547 22.307269 0.725158

|1, 0i ↔ |1, 1i 0.322552 160.181925 0.001059

|2, 0i ↔ |2, 1i 0.324548 174.719221 0.000625

Table 3.1: Some field insensitive points with the relevant transitions and frequencies, and the

difference of second order sensitivities. Here we list only field-insensitive points for B in the range

of 11 mT to 180 mT. In this thesis we mainly work on fields for the first two entries, with ∆mF = ±1.

For the higher fields the insensitive points are located at ∆F = 0 transitions, and the higher order

field sensitivity is significantly lower than the ∆F 6= 0 insensitive transitions. As this trend and Fig.

3.2 shows, for the high fields, the Zeeman effect is dominant over the hyperfine coupling hAI·J, and

the field sensitivity between ∆F = 0 transitions becomes small. At higher fields of ∼819.439 mT, a

303 MHz field insensitive transition between |mI = −

3

2

, mJ =

1

2

i ↔ |−1

2

,

1

2

i has been demonstrated

in a Penning trap [1] for applications as a frequency standard.

field.

states expanded at B = 11.944618 mT expanded at B = 0

|F = 2, mF = 2i |mI = 3/2, mJ = 1/2i |mI = 3/2, mJ = 1/2i

|2, 1i 0.609|3/2, −1/2i + 0.793|1/2, 1/2i

1

2

|3/2, −1/2i +

√

3

2

|1/2, 1/2i

|2, 0i 0.793|1/2, −1/2i + 0.609|−1/2, 1/2i √

1

2

|1/2, −1/2i + √

1

2

|−1/2, 1/2i

|2, −1i 0.912|−1/2, −1/2i + 0.410|−3/2, 1/2i

√

3

2

|−1/2, −1/2i +

1

2

|−3/2, 1/2i

|2, −2i |−3/2, −1/2i |−3/2, −1/2i

|1, 1i 0.793|3/2, −1/2i − 0.609|1/2, 1/2i

√

3

2

|3/2, −1/2i − 1

2

|1/2, 1/2i

|1, 0i 0.609|1/2, −1/2i − 0.793|−1/2, 1/2i √

1

2

|1/2, −1/2i − √

1

2

|−1/2, 1/2i

|1, −1i 0.410|−1/2, −1/2i − 0.912|−3/2, 1/2i

1

2

|−1/2, −1/2i −

√

3

2

|−3/2, 1/2i

Table 3.2: List of 2s

2S1/2 ground states expanded in the |mI , mJ i basis, with coefficients rounded

to the third digit. Compared with the states at zero field, except for the stretch states, the other

states have modified coefficients. This table will be useful when calculating the coupling coefficients

for the electric dipole transitions, since the selection rules are such that ∆mI = 0 and ∆mJ = 0, ±1.

For the 2p

2P1/2 and 2p2P3/2

states, gJ ≈ { 2

3

,

4

3

}, respectively. The zero-field hyperfine

splittings in the 2p

2P1/2

states are 237 MHz [88]; the 2p

2P3/2 hyperfine splitting is < 1 MHz [91].

At the field of 11.9446 mT, the Zeeman splitting of 2p

2P3/2 can be approximated by (hgJμBBmJ ) ≈

hmJ × 223 MHz with eigenstates approximated by |mI , mJ i, since the Zeeman effect is dominant

over the hyperfine couplings. For 2p

2P1/2

states, a calculation similar to the 2s

2S1/2

is performed.
30

This is done by using Eq. 3.1 A1

2

≈ −118.6(3.6) MHz, and gI = 0.784955 μB

μN

[88], where μN is the

nuclear magneton. More energy splittings are shown in Fig. 3.1.

states expanded at B = 11.944618 mT

|F = 2, mF = 2i |mI = 3/2, mJ = 1/2i

|2, 1i 0.69458|3/2, −1/2i + 0.719415|1/2, 1/2i

|2, 0i 0.844109|1/2, −1/2i + 0.536171|−1/2, 1/2i

|2, −1i 0.934293|−1/2, −1/2i + 0.356505|−3/2, 1/2i

|2, −2i |−3/2, −1/2i

|1, 1i 0.719415|3/2, −1/2i − 0.69458|1/2, 1/2i

|1, 0i 0.536171|1/2, −1/2i − 0.844109|−1/2, 1/2i

|1, −1i 0.356505|−1/2, −1/2i − 0.934293|−3/2, 1/2i

Table 3.3: List of 2p

2P1/2

states expanded in the |mI , mJ i basis. Definitions are similar to Table

3.2 for 2p

2P1/2

. These components will be useful when describing the laser dipole coupling matrix

element in this thesis.

3.2 Laser and microwave sources

From previous subsection, we have seen that the 2S1/2 and 2P1/2 hyperfine states can be

expanded in the |mI , mJ i basis. Transitions can be driven with laser fields or microwaves. For

9Be+ and 24Mg+, we do not consider metastable states. For laser dipole transitions, the selection

rule is that the laser can drive transitions that satisfy ∆L = 1, ∆mI = 0 and ∆mJ = {0, ±1}. Here

L is the orbital angular momentum quantum number, the second condition is due to the fact that

laser field interacts very weakly, and the latter depends on the laser polarization. From Table 3.2,

when expanding the eigenstates in |mI , mJ i basis, at the field for our case the Zeeman effect only

shifts the coefficients of the expansion compared with B = 0, without involving extra |mI , mJ i

components. Thus we can safely view laser dipole transitions between ∆mF = {0, ±1} for the

selection rule, consistent with ∆mF = ∆mI + ∆mJ = {0, ±1}. However it is important to note

that the eigenstates are not precisely |F, mF i, since F is not a good quantum number, and should

be addressed by treatments leading to Table 3.2 and 3.3.

We use laser beams for cooling, detection, and spin and motional state manipulation. To

simplify the laser setup, for transitions that are close to each other, we split a laser beam and mod-
31

ify their frequencies with Acousto-Optic Modulators (AOM) [92]. For a 9Be+ ions, the transitions

are depicted in Fig. 3.1. The 2p

2P state linewidth is Γ=2π×19.4 MHz. The “BD” (blue-Doppler)

near detuned/detection laser, with wavelength approximately 313.13 nm, drives the cycling transition between the 2s

2S1/2

|F = 2, mF = 2i state and the 2p

2P3/2

|F = 3, mF = 3i excited state,

to provide far and near detuned Doppler cooling, and to drive a cycling transition [93] to scatter

photons for detection. The 400 MHz red detuned BD laser addressing 2p

2P3/2

|3, 3i state is named

with “BDD” (blue Doppler detuned), and is typically much stronger than other Doppler lasers, to

power broaden the transition. It provides Doppler cooling to ions in case they are in a high-energy

motional state due to background collisions or during the loading process. Since it’s far red detuned, this laser also provides optical pumping to clear the populations from other than the ground

|2, 2i state. More detail is described in Sec. 3.3

The repumper laser beams are named with “RD” (red Doppler). They repump the ground

|2, 1i and |1, 1i states, are further named with RD blue and RD red respectively, with wavelength

approximately 313.2 nm.1 These RD lasers provide fine pumping to the |2, 2i state with high

probability, typically ∼0.999 in a few μs.

To drive the transition between the ground states, we apply two laser beams setup in a “Λ”

configuration, as shown in Fig. 3.1, to implement two-photon stimulated Raman transitions. For

the two beams with different incident angles, these beams can also serve to drive spin-motional

excitation. These two beams are named as “Be CO” and “Be 90”, with frequency difference ∼1.2

GHz2. The Raman beam wavelengths are approximately either 313.29 nm or 313.36 nm for different

experiments in this thesis. Details will be described in Chapter 4.

The laser beam access is depicted in Fig. 3.3. All the Doppler laser beams are ideally purely

σ

+ polarized, to minimize depumpings from the |2, 2i state. The Raman laser beams are set so

that the Be 90 beam contains both σ

± polarizations and the Be CO beam has π polarization3

. To

1

It is a historical reason for this naming although the RD lasers are not mainly setup to provide Doppler cooling.

2 The Raman beams can be set to have frequency difference ∼1.0 GHz to drive motional sidebands on the

|2, 2i ↔ |1, 1i transition. The “CO” beam is also used as one of the co-propagating beams to drive Raman transitions

in other experiments [92]. The “90” beam is 90 degree from the “CO” beams.

3

In the case of driving co-carrier transitions, the CO beam can contain simultaneously π polization and σ

±
32

Imaging System

trap 

axis 

RF resonator

Field Coils

ion

trap

Be CO

Be 90 Mg PI

Be PI

Be Doppler resonant/detection

Be Doppler Detuned

Be Repumper Blue

Be Repumper Red

Mg BD

Mg Detuned

B

Figure 3.3: Schematic for laser access to the trap, not to scale. The magnetic field coil is set so

that the magnetic field is 45◦ with respect to the trap axis. Be CO and 90 denote the Raman laser

beams to address the 9Be+ ion. PI denotes the photoionization beams. Blue hollow squares denote

the outline of the glass chamber and the windows. Figure adapted from [5].

generate the laser beams above, we first generate 626 nm laser beams by frequency summing 1550

nm and 1050 nm fiber laser beams [94], then frequency doubling to 313 nm with a bowtie cavity

with a Barium borate (BBO) crystal, as depicted in Fig. 3.5. To reference the frequency of the

Doppler beams, we lock the 626 nm Doppler lasers to Doppler-free absorption lines from I2 vapor

cells, feeding back to piezos in the Doppler IR fiber seed lasers. The AOM setups to generate the

BD and RD laser beams are shown in Fig. 3.4 and Fig. 3.6. After these beams are generated, they

are combined with a 50/50 beam splitter and coupled through the UV fiber and sent to the trap.

The Raman laser setup is shown in Fig. 3.7.

We also use microwave fields to drive transitions within the ground state manifold shown

polarizations by setting the beam to be linearly polarized with an angle between 0◦and 90◦from the magnetic field.
33

+200 MHz

BD split 

AOM

+200 MHz

BD Switch 

AOM 

(2pass)

+200 MHz

BD det SW 

AOM

1

st Ord.

0

th Ord.

1

st Ord.

1

st Ord.

to combine 

with RD

pol. Doubler

output 

313nm

Pos. D.

Pos. D.

PZT 

mirr.

PZT 

mirr.

Figure 3.4: Schematic for BD laser beams setup. The output of the doubler has a vertical polarization with respect to the laser table. Pos. D. represents a two dimensional position detector

(Newport Quadrant Cell); PZT mirror is a dual-piezo-driven mirror, capable of tipping both horizontal and vertical directions. Along the beam path, the first(second) Pos. D. feeds back to the

first(second) PZT mirror with proportional-integrator loop. Thus the output beam pointing of

the doubler is locked. The AOMs are not shown with actual orientation angle. The dashed beam

line forms the BD far-detuned Doppler laser, and the solid beam line after the first AOM forms

the BDD resonant/near detuned Doppler laser, with their frequency difference approximately 400

MHz. These two laser beams are combined at the BD det(detuned) SW(switch) AOM, where the

BD beam is diffracted on the first order, which is overlapped with the undeflected BD beam. This

configuration is due to the requirement of stronger far-detuned beam power than the near detuned

one. Here some of the aligning mirrors are not shown. The double pass BD Switch setup uses a 15

cm lens placed at a focal length away from the center of the AOM. A corner cube retro-reflecting

the beam with a vertical displacement, placed at the location that forms a one to one telescope

with the lens. With this setup, the double pass alignment is insensitive to the change of AOM

input frequency. The output of this setup is combined with the RD beams, then sent to the input

of a UV fiber. The RF drive for BD switch AOM is actively noise-eat when the pulse is on. For

this purpose, a pick-off mirror partially reflect the BD light to a detector, placed after the fiber, so

that when the BD light is on, its power is actively stabilized.

in Fig. 3.1. The microwave transitions used in this thesis are summarized in Table 3.4. These

transitions are used for various purposes, such as spin flopping and transferring from the |2, 2i

state to the spin state. For detection, we transfer one of the “bright” spin states to the |2, 2i state

for fluorescence detection.

Microwave transitions are induced in a brute-force manner with a dipole antenna, depicted in

Fig. 3.9, that is external to vacuum system. This antenna is constructed by soldering two sections

of copper wires to a SMA female connector. The length of these wires are set to 1/4 of the desired

microwave wavelength. For 9Be+ and 25Mg+, two antennas are built for center frequencies of 1.2
34

1050.876 nm seed + 

amplifier

1549.879 nm seed + 

amplifier

1051.275 nm seed + 

amplifier

1551.800 nm seed + 

amplifier

Summing

crystal

Summing

crystal

Summing

crystal

BBO

doubling

BBO

doubling

BBO

doubling

RD

BD

Raman

AOM

setup

AOM

setup

AOM

setup

RD red

RD blue

BD 

detuned

BD near 

detuned/

detection

Be CO

Be 90

Figure 3.5: Schematic for laser beam generation. “BBO doubling” denotes the doubling cavity

setup [6] with BBO as the nonlinear conversion medium. For the seed lasers, we use NKT Koheras

Adjustik lasers, and for the upper two amplifiers, we use NKT Koheras Boostik HPA. For the

lower two amplifiers, we use IPG Photonic Ytterbium/Erbium Fiber Amplifiers. The respective

frequencies of the Raman beams are controlled by the AOM frequencies. The Doppler beams are

recombined before sending through a UV capable photonic crystal fiber [7]. To Doppler cool and

image on both the experimental zone and the load zone, we further split the Doppler beam on the

fiber output with a Glan-laser, with relative intensity controlled by a motorized λ/2 waveplate.

GHz and 1.8 GHz respectively. The antennas are placed vertically above the trap, on top of the

quartz cell. We further impedance match this antenna to the microwave source with a stub-tuner.

The microwave frequency is generated by doubling the signal from the direct-digital-synthesizer

(DDS) twice or three times, filtered, and then amplified with a 30 W amplifier4. With this

configuration, the Rabi flopping π time for 9Be+ with various transitions is summarized in Table

3.4.

4 Mini Circuit ZHL-30W-252+
35

+400 MHz

RD Blue 

AOM 

(2pass)

-155 MHz

RD Red 

AOM 

50/50 (2pass) λ

2

to combine 

with BD

Glan-laser

λ

2

pol. Doubler

output 

313nm

Pos. D.

Pos. D.

PZT 

mirr.

PZT 

mirr.

Figure 3.6: Schematic for RD laser beams setup. This setup is similar to the BD one, except

that the generated beam frequency difference is approximately 1110 MHz, matching with the resonant frequency difference of the |F = 1, mF = 1i and |2, 1i states. The λ

2 waveplates modifies the

polarization of the laser beam, controlling the beam power splitted at the Glan-laser. The Glanlaser is setup so that the lower output beam is vertically polarized, and the upper output beam is

horizontally polarized. The waveplate after the Glan-laser rotates the polarization back to vertical.

-220 MHz

CO AOM 

(2pass)

300 MHz

90 AOM 

(2pass)

λ/2

-200 MHz

CO AOM 

(1pass)

UV fiber

UV fiber

λ/2

ions

λ/2

Glan-laser

λ/2

Doubler

output 

313nm

Pos. D.

Pos. D.

PZT 

mirr.

PZT 

mirr.

Figure 3.7: Schematic for Raman laser beams setup. This setup splits the Raman beam into the

“CO” beam and the “90” beam, as indicated in Fig. 3.3. We adjust the frequency of the 90 AOM

to drive transitions |F = 1, mF = 1i ↔ |2, 2i and |1, 1i ↔ |2, 0i. The CO beam line has an optional

600 MHz AOM double pass setup, to generate a pair of co-propagating CO beams to drive the

carrier transition, which is not shown in the figure. For the double pass setup for Raman beams,

the corner cube is replaced with three mirrors, to provide first order insensitive to AOM thermal

refraction effect. This method is proposed by T. Rosenband, as shown in Fig. 3.8.
36

θ

−θ

−2θ

AOM

double

pass

f vertical

(a)

AOM

double

pass

θ

θ

angle cancelled f

(b)

Figure 3.8: (a) Schematic for the “two mirror” double pass setup in a side-view. Here only the

diffracted beams are shown as the AOM outputs but the actual the diffraction directions are

perpendicular the page. The dashed orange line shows the deviation angle θ due to the AOM

thermal thermal refraction effect. This effect can result from inhomogeneous distribution of RF

power inside the AOM, or from inhomogeneous heat dissipation along the vertical directions in the

figure. In this case, after double passing the AOM, the angular deviation is doubled. The lens is

placed at a focal length f away from the AOM, and the round trip path length between the lens

and mirrors is 2f. (b) Schematic for the “three mirror” double pass setup. For three mirror setup,

the deviated beam experiences a cancelling thermal refraction effect and the double pass output is

overlapped with the beam where the thermal refraction effect is absent.

transition frequency(MHz) π time(μs)

|F = 2, mF = 2i ↔ |1, 1i 1018.007400 7.4

|1, 1i ↔ |2, 0i 1207.495843 7.2

|2, 0i ↔ |1, 0i 1294.042400 11.7

|2, 0i ↔ |1, −1i 1370.290005 88.8

|2, 1i ↔ |1, 1i 1120.806368 4.4

|2, 1i ↔ |1, 0i 1207.352925 10.6

Table 3.4: typical π time of the Rabi flopping of the microwave transitions

3.3 State initialization

As we have seen in previous section, the 9Be+ ion has multiple ground states. To have a

repeatable starting point for the experiment, we need to transfer, or reset, the ion population to a
37

Figure 3.9: Schematic for the dipolar antennas in the experiment, not to scale. The microwave

wavelength id denoted by λ. One arm of the antenna is soldered to the center conductor of the

SMA, and the other end is soldered to the outer conductor.

fixed state. This operation is referred to as state initialization or state preparation. We apply optical

pumping, a dissipative process that takes advantage of the selection rules of the transition between

the ground and excited states. Consider transitions between 2S1/2 and 2P3/2states. For a σ

+

polarized5 BD detuned laser beam due to power broadening, it can off-resonantly drive transitions

from several of the 2S1/2states to the excited 2P3/2states, with selection rule ∆mJ = +1. The

excited state decays back to the ground manifold with selection rule of ∆mJ = 0, ±1. Thus for

one round of excitation-decay process, the net transition rule at the ground manifold is ∆mJ =

0, +1, +2. Thus with one BD detuned beam, repeated pumping within the ground manifold occurs

so that the mJ quantum number is being increased, until reaching the maximum number mJ = 2,

corresponding to the 2S1/2|F = 2, mF = 2i state. This state is sometimes called the “stretched”

state.

Due to the imperfections of the polarization of the BDD detuned beam, there is a finite probability that the wrong polarization can drive transitions with an overall change ∆mF = −2, −1, 0, a

5

controlled by a combination of Glan-polarizer, λ/4 − λ/2 waveplate, mounted on a tip-tilt stage with waveplate

rotations
38

depumping process. The equilibrium of the pumping and small depumping process results in a majority of the population in the stretched state, but there can still be a finite population6

remaining

in the 2S1/2|F = 2, mF = 1i and |1, 1i states.

To increase the pumping success probability, we apply the σ

+ polarized “repumper” beams,

the RD Red and RD Blue beams, to resonantly drive transitions of

2S1/2

|F = 2, mF = 1i →2 P1/2|F = 2, mF = 2i, and 2S1/2|F = 1, mF = 1i →2 P1/2|F = 2, mF = 2i

respectively, and pump to the desired 2S1/2|F = 2, mF = 2i. These beams will not drive transitions from the 2S1/2

|F = 2, mF = 2i states, since mF = 3 states do not exist in the 2P1/2 manifold.

For imperfections of the polarizations of the RD Blue repumper, the nearest depumping transitions 2S1/2

|F = 2, mF = 2i →2 P1/2|F = 2, mF = {2, 1}i are ∼100 and ∼ 60 MHz detuned from

resonance. Thus the depumping process is suppressed.

With the application of the BD detuned beam, then simultaneously the RD blue and red

beams, we typically achieve a population ∼0.999 in the 2S1/2|F = 2, mF = 2i state.

For experiments where we want to initialize the ion to other states, we drive from the

|F = 2, mF = 2i to the desired state with either Raman transitions or microwave transitions. A

typical desired initial state is the |F = 1, mF = 1i state. Due to the magnetic field sensitivity of

this transition, which leads to transition frequency fluctuations, we can apply the composite pulse

technique [95]. One sequence we used for the experiment is R{

π

2

, 0} − R{

3π

2

,

π

2

} − R{

π

2

, 0}, where

R denotes the rotation operation, θ and φ denotes the rotation angle and the azimuthal angle of a

vector in the spin x-y frame about which the spin is rotated.

3.4 Detection

3.4.1 Photon collection

To detect the presence of the 2S1/2|F = 2, mF = 2i state, we use the BD resonant beam,

which drives the “cycling” transition between this state to the excited 2P3/2|mI = 3/2, mJ = 3/2i

state. The name cycling comes form the fact that the excited state can only decay back to the

6

typically < 1% of population
39

2S1/2

|F = 2, mF = 2i state. With the BD detection beam tuned close to resonance with the transition, when the ion is in the 2S1/2

|F = 2, mF = 2i state, photons are repeatedly scattered, and thus

the ion appears to be “bright”. When the ion is in the other states, the number of the scattered

photons will be significantly less, and the ion appears to be “dark”. However, the BD detection

beam can off-resonantly repump other 2S1/2states to the |F = 2, mF = 2i state, and the ion subsequently becomes bright. This effect is significant for the F = 2 states, and less significant for

the F = 1 states, due to the ∼1 GHz splitting in energy. The other effect is the lower the value

of mF , the more repump steps it takes to reach the bright mF = 2 state. Thus for detecting of

the spin manifold |F = 2, mF = 0i and |F = 1, mF = 1i states, we first use a microwave π pulse to

drive |F = 1, mF = 1i → |F = 2, mF = 2i, so the bright state represents the |F = 1, mF = 1i ≡ |↑i.

When a photon scattering event occurs, any superposition of the spin states will be “collapsed”

into either the bright or dark state, and ideally remain therein through out the detection. We also

apply a “shelving” technique [63, 96], to essentially make the “dark” |F = 2, mF = 0i ≡ |↓i state

even darker by transferring it to the |1, −1i state. However, experimentally we find this transition

is much weaker than other transitions as shown in Table 3.4, and the transfer fidelity is not high due

to magnetic field sensitivity. Thus we apply another microwave pulse before detection to transfer

the residual populations in the |2, 0i state to |1, 0i state.

3.4.2 Maximal likelihood method

The photons collected can be shown as a histogram constructed over many repeated identical

experiments; typical ones for two ions are shown in Fig. 3.10 (a). The histogram approximately

follows Poisson distributions. The Poisson distribution describes the collection of photons, for an

ion with a constant photon scattering rate, for a fixed duration of detection. With i ions in the

bright state, the mean photon count is approximately proportional to i, and mean number of bright

counts is typically ∼40 counts for a detection duration of 330 μs, as shown in Fig. 3.10 (a). Another

example is the photon scattering of the 1

2

(|↑↑i + |↑↓i + |↓↑i + |↓↓i) state. A simple procedure is to

fit the histograms with with Poisson distributions to extract the probability of different ions in the
40

bright state [87]. However, due to experimental imperfections, e.g. optical pumping, the photon

scattering rate may not be constant since the dark state has a finite probability to be repumped

into the bright state and start scattering photons. Similarly the bright state has a finite probability

to be depumped and scatter less photons over time. Thus the histogram does not precisely consist

of Poissonian components, and to fit it with high precision we need to model the repumping and

depumping processes [87, 97, 98].

Recently, an alternative detection program has been developed to provide analysis independent of the model of the photon count distribution. This program7

is coded in Python and Matlab

developed by A. Keith, S. Glancy, K. Coakley and E. Knill at NIST, using maximum-likelihood

method [99]. We first obtain a reference histogram with a microwave “Ramsey” pulse sequence of

R{

π

2

, 0} − R{π, 0} − R{

π

2

, φ} flopping between the |F = 2, mF = 2i and |F = 1, mF = 1i states8,

where φ is a scanned variable. Thus we obtain histograms shown in Fig. 3.10 (a) To mitigate the

effects from finite number of experiments and speed up the analysis process, we apply a “rebinning”

method, to sum up events in certain count ranges. As an example, the histograms in Fig. 3.10 (a)

are divided into three sections, shown with vertical dashed lines. Each of the sections are summed

up to form the histograms in Fig. 3.10 (b), respectively. We can see this rebinning process reflects

the populations of Pi for each of the histogram, where Pi denotes probability of i ions in the bright

state. For each of the reference histograms, we predict the coefficients for linear combinations of Pi

from the Ramsey flopping, which we assume with negligible errors9. Then we numerically find the

optimal histograms for Pi = 1, i = 0 − 2 to match with the coefficients and the rebinned reference

histograms, as shown in Fig. 3.10 (c). This numerical process follows the maximum-likelihood

method. This optimal histogram is used as a new reference to extract spin population from data

of other experiments. For the entangled state experiments, to obtain the population of the target

state |ψi, we also use maximum-likelihood method to find the best fit of density matrix ρ to both

7 We compared the analysis given by this program with the model with the repumping and depumping processes

and they give similar result.

8 we use this sequence to echo out spin processions between the pulses due to slow magnetic field drifts

9 To obtain the histogram for all ion bright, a more precise way is to directly measure the population after

repumping to the |2, 2i state; and for all ion dark, a possible improvement is to apply a composite π pulse followed

by shelving and ion fluorescence.
41

0 20 40 60 80 100 120 0

200

400

bin 1 bin 2 bin 3

P2≈1

counts

events

0 20 40 60 80 100 120 0

200

400

P2≈1/4,P1≈1/2,P0≈1/4

counts

events

0 20 40 60 80 100 120 0

1000

2000

P0≈1

counts

events

(a)

1 2 3 0

2000

4000

6000

P2≈1

bin number

events

1 2 3 0

1000

2000

3000

P2≈1/4,P1≈1/2,P0≈1/4

bin number

events

1 2 3 0

2000

4000

6000

P0≈1

bin number

events

(b)

1 2 3

0

0.5

1

P0 = 1

bin number

P

1 2 3

0

0.5

1

P1 = 1

bin number

P

1 2 3

0

0.5

1

P2 = 1

bin number

P

(c)

Figure 3.10: Typical histograms collected from a given detection process. (a) Histograms from

Ramsey experiments showing the count histograms for various states. From top to bottom, histograms of approximately two, one, and zero ions in the bright states are shown, corresponding

to φ = {0, π/2, π} respectively. Gray vertical dashed lines separate the histogram into sections

of count ranges that can distinguish the number of ions in the bright state. (b) Sections in (a)

are summed up to form “rebinned” histogram. The new bins are labeled as 1-3. (c) From the

rebinned histograms and the predicted populations from the Ramsey sequence, a population reference histograms representing 0-2 ions in the bright state are obtained with the maximum likelihood

method.
42

the histograms with and without the “analysis pulse”, detailed in the following section, according

to the reference histogram. Then we obtain the population Pψ = hψ|ρ|ψi.

3.4.3 Symmetric entangled state detections

In this subsection, we will describe a general method of detection state |ψi, particularly for

the symmetric entangled state for two and three ions. This is useful since for some experiments we

have not had the capability to individually address the ions for detection. Here we set the dark

and bright states to correspond to the states in vector form of |↑i =





1

0





and |↓i =





0

1





respectively. For an experimental outcome density matrix ρ of two ions, after we apply a unitary

rotation U known as the “analysis pulse”, the detection of population Pi for i ions in the bright

state can be expressed as

Pi,U = Tr(UρU

†Ai) = Tr(ρU†AiU), (3.5)

where A0 = |↑↑ih↑↑|, A1 = |↑↓ih↑↓|+|↓↑ih↓↑|, A2 = |↓↓ih↓↓| are the measurement operators to give

the probability of i ions bright and T r denotes the trace. Here we assume the unitary rotation and

detection are global, and the populations stay within the symmetric states. Thus Pi,U contains

information of the experiment outcome. On the other hand, a theoretical measurement of the

overlap of ρ and |ψi is given by

P|ψi = hψ|ρ|ψi = Tr(ρ|ψihψ|). (3.6)

We can see |ψihψ| is an observation of the state |ψi. Thus with the linear property of the matrix

trace, we can find linear combinations of U

†AiU to form the observation |ψihψ|. Here we apply U as

single qubit rotation R{θ, φ} =





cos( θ

2

) −ie−iφ sin( θ

2

)

−ieiφ sin( θ

2

) cos( θ

2

)





. With varied θ and φ. Changing

notation, we write operator R†{θ, φ}AiR{θ, φ} = Ri{θ, φ}. Here we give some examples of the

pulse sequence and observations for different entangled states.
43

|ψφi = √

1

2

(|↑↑i + e

iφ|↓↓i):

|ψφihψφ| =

1

2

(A0 + A2

+ (R2{

π

2

,

φ

2

+

π

2

} + R0{

π

2

,

φ

2

+

π

2

} − R1{

π

2

,

φ

2

+

π

2

})

− (R2{

π

2

,

φ

2

} + R0{

π

2

,

φ

2

} − R1{

π

2

,

φ

2

}))

|Ti = √

1

2

(|↑↓i + |↓↑i): |TihT| =

1

2A1 +

1

2

1 − hR1{

π

2

, φ}iφ

|Si = √

1

2

(|↑↓i − |↓↑i): |SihS| =

1

2A1 −

1

2

1 + hR1{

π

2

, φ}iφ

Here 1 denotes the identity matrix, h·iφ denotes an average of measurements over φ. For three

ions, we have

|Wi = √

1

3

(|↑↑↓i + |↑↓↑i + |↓↑↑i): |WihW| =

2

3A3 + A2 +

1

2A1 +

1

3A0 −

3

2

hR2{cos−1(

1

3

), φ}iφ.

For |Ti, we can give an intuitive explanation of the measurement operators. Since |Ti is in

the 1-ion bright manifold, we only need to distinguish it from |Si, the other state in this manifold,

in which case a measurement for A1 can provide the population of |Ti (as well as |Si). In general,

the key process of symmetric measurement is to distinguish the symmetric state (here |Ti) with

the asymmetric states (here |Si). We insert a microwave π/2 “analysis” pulse with variable phase

before the collection of histograms. This pulse rotates the |Ti state into superpositions of |↑↑i and

|↓↓i states, while the |Si state is invariant. Measurement of populations in |↑↑i and |↓↓i states

allow us to extract the |Ti population. To make the measurement simple, we only need to subtract

the residual population in the 1-ion manifold. Thus to get the |Ti population subtract R1{

π

2

, φ}

from the combination of A0 and A2, and to get the |Si population, the R1{

π

2

, φ} is added to A0

and A2.
Chapter 4

Internal-motional state coupling

The ions are typically coupled through their shared motional modes. Thus through this

motional bus, the electronic states (spins) can be effectively coupled through an internal-motional

state coupling. One method to realize this coupling is to use the position dependence of the laser

or microwave fields, so that the motional information is imprinted to the spin transitions. In this

chapter, we will start with describing the quantization of motional states. Then we will describe the

laser induced spin-motion interaction, with various aspects that are important to the experiments

in this thesis.

4.1 Quantization of motional modes

We continue the description in Sec. 2.1, and provide a quantum mechanical treatment for

the axial motional modes. Here we ignore the micromotion. For one ion, from Eq. 2.2, the axial

component of the motional Hamiltonian is

Hm =

p

2

2m

+ qΦs (4.1)

=

p

2

2m

+

1

2

mω2

z

z

2

(4.2)

= ~ωz(a

†

a +

1

2

), (4.3)

where m is the ion mass, and the last step results from a transformation of z =

q ~

2mωz

(a + a

†

),

p = i

q

mωz~

2

(a

† − a), and the commutation relation [a, a†

] = 1. The eigenstates of Hm are labeled
45

as “Fock” states |ni [100], with n > 0 and a

†a|ni = n|ni, and properties a|ni =

√

n|n − 1i and

a

†

|ni =

√

n + 1|n + 1i.

With more than one ions in the same well, the above process is modified to include the

Coulomb repulsion between the ions. For two ions of charge q and masses m1 and m2, the potential

energy is

V =

1

2

m1ω

2

z,1

z

2

1 +

1

2

m2ω

2

z,2

z

2

2 +

q

2

4π0|z1 − z2|

, (4.4)

where 0 is the permittivity, zi and ωz,i are the position and single-ion motional frequency for the i

th

ion. Since the trapping potential is independent of the ion mass, quantity V0 = miω

2

z,i is a constant.

Assuming z1 > z2, and solving ∂V

∂z1,2

= 0, we can obtain the equilibrium position of the ions {z

0

1

, z0

2

}.

For two ions, these equations can be solved analytically, z

0

1 = −z

0

2 =



q

2

16π0V0

 1

3

. For a typical

ωz = 2π × 3.58 MHz, for two 9Be+ ions, we find the equilibrium positions z

0

1 = −z

0

2 ≈ 1.97μm.

With zi = z

0

i + z

0

i

, i = 1, 2, the potential energy can be expanded to the second order in z

0

i

as

V ≈ V0((z

0

1

)

2 + (z0

2

)

2 − z0

1

z

0

2

), (4.5)

where we ignore the constant term. Combined with the kinetic energy, the Hamiltonian is thus

H ≈

P

2

1

2m1

+

P

2

2

2m2

+ V0((z

0

1

)

2 + (z0

2

)

2 − z0

1

z

0

2

). (4.6)

Similar to the one ion case, we need to diagonalize the Hamiltonian. Here we apply a transformation

z

0

1 = cos θ

r

~

2m1ω1

(a1 + a

†

1

) − sin θ

r

~

2m1ω2

(a2 + a

†

2

) (4.7)

z

0

2 = cos θ

r

~

2m2ω2

(a2 + a

†

2

) + sin θ

r

~

2m2ω1

(a1 + a

†

1

) (4.8)

P1 = cos θ

r

~m1ω1

2

(ia†

1 − ia1) − sin θ

r

~m1ω2

2

(ia†

2 − ia2) (4.9)

P2 = cos θ

r

~m2ω2

2

(ia†

2 − ia2) + sin θ

r

~m2ω1

2

(ia†

1 − ia1), (4.10)

where aiis the annihilation operators for normal motional mode i = 1, 2, θ satisfies cos 2θ =
46

√ m1−m2

m2

1+m22−m1m2

, and

ω

2

1 =

3V0

m1 + m2 +

p

m2

1 + m22 − m1m2

(4.11)

ω

2

2 =

3V0

m1 + m2 −

p

m2

1 + m22 − m1m2

. (4.12)

Thus the Hamiltonian is diagonalized such that

H = ~ω1(a

†

1

a1 +

1

2

) + ~ω2(a

†

2

a2 +

1

2

). (4.13)

The eigenmodes expressed interms of the two ions motion is thus

a1 + a

†

1 =

r

2ω1

~

(cos θ

√

m1z

0

1 + sin θ

√

m2z

0

2

), (4.14)

a2 + a

†

2 =

r

2ω2

~

(cos θ

√

m2z

0

2 − sin θ

√

m1z

0

1

), (4.15)

we can see the eigenmodes contains motion of both ions, and the component of the two ion’s motion

in mode 1 and 2 is represented by unit vectors {cos θ,sin θ} and {− sin θ, cos θ} respectively, the

mode amplitudes. They combined with the q2ωj

~

√mi factor fully express the j

th eigenmode in

terms of the motion of the i

th ion. For the two ions with same sign of mode amplitudes, they are

oscillating with the same phase; this mode is designated the common (COM) mode. The other

mode with opposite signs of amplitudes is called the stretch mode. The motional mode analysis

above can be extended to more ions numerically. In Table 4.1, we show the mode amplitudes and

mode frequencies for the case of one, two and three 9Be+ ions, a 9Be+ − 24Mg+ ion pair and a

9Be+ − 24Mg+ − 24Mg+ − 9Be+ ion chain, in the same potential well.

4.2 Resolved sideband coupling

In chapter 3 we describe laser beams driving transitions of ions’ electronic states. Here we will

start with describing laser driven electric dipole transitions for a stationary ion. We then include

the ion motion. Intuitively, in the ion’s frame, the laser beam frequencies are modulated by the ions’

motion through the Doppler effect, thus creating laser “sidebands” in the frequency domain. These

sidebands are spaced by the motional frequency, and the amplitudes are proportionally to the Bessel
47

config ion chain mode frequency

(2π×MHz)

mode amplitudes

case 1 9Be+ COM 3.58 1

9Be+ − 9Be+ COM 3.58 { √

1

2

, √

1

2

}

9Be+ − 9Be+ stretch 6.20 { √

1

2

, − √

1

2

}

9Be+ − 9Be+ − 9Be+ COM 3.58 { √

1

3

, √

1

3

, √

1

3

}

9Be+ − 9Be+ − 9Be+ stretch 6.20 { √

1

2

, 0, − √

1

2

}

9Be+ − 9Be+ − 9Be+ “Egyptian” 8.62 { √

1

6

, − √

2

6

, √

1

6

}

case 2 9Be+ COM 2.97 1

9Be+ − 24Mg+ COM 2.10 {0.378,0.926}

9Be+ − 24Mg+ stretch 4.46 {0.926,-0.378}

case 3 9Be+ COM 2.71 1

9Be+ − 24Mg+ − 24Mg+ − 9Be+ COM 1.96 {0.322,0.629,0.629,0.322}

9Be+ − 24Mg+ − 24Mg+ − 9Be+ stretch 4.12 {0.466,0.532,-0.532,-0.466}

9Be+ − 24Mg+ − 24Mg+ − 9Be+ “Egyptian” 5.53 {0.629,-0.322,-0.322,0.629}

9Be+ − 24Mg+ − 24Mg+ − 9Be+ “FUNK” 5.78 {0.532,-0.466,0.466,-0.532}

Table 4.1: Normal motional modes for various ion chain configurations, with the same external

potential well for each case. The mode amplitudes are shown consistent with the ion order. For the

9Be+−24Mg+ pair, due to the mass imbalance, the COM mode is dominated by motion of the 24Mg+

ion, while the stretch mode is dominated by the 9Be+ ion. This reduces the sympathetic cooling

efficiency of mix-species chains for the modes which the coolant ion has small mode amplitudes. The

mode information shown in this table will be useful for the experiments described in the following

chapters.

function Jn(η) classically, where n denotes the n

th order sideband, and η is the modulation index

equals to a product of the ion motion amplitude and the laser k-vector along the motion direction,

which is also known as Lamb-Dicke parameter detailed below. Thus with proper configurations as

detailed below, a laser detuned by the motional frequency can drive an electronic transition at the

same time excite/de-excite the ion’s motion, known as a sideband transition. This transition will

be used to induce spin-motion coupling1.

1 There are other ways to implement internal-motional state coupling, such as applying a strong static field

gradient [86], or applying a microwave field gradient. [101]
48

4.2.1 Electric dipole transition for a free ion

Here we treat the laser beam as a classical oscillating electric field and ignore the magnetic

components, since the frequencies of relevant magnetic dipole transitions are far off-resonance

from typical laser frequencies. In a one dimensional model, the laser field can be modeled as

E(z, t) = E0 cos(kz − ωLt + φ), where E0 = ˆE0 contains the field polarization ˆ and amplitude E0,

k =

2π

λ

is the wavenumber of the laser, λ is the wavelength of the laser, z is the ion position relative

to equilibrium, ωL is the laser frequency and φ is the optical phase. For 9Be+ in a magnetic field

as in chapter 3, to drive a transition from the 2s

2S1

2

to 2p

2P1

2

state, the laser frequency needs to

be near resonant with the transition frequency ω0. The Hamiltonian is thus

H = H0 + V (4.16)

V = −μd · E(z, t), (4.17)

where H0 represents the electronic energy levels, μd = −|e|r is the electric dipole operator for the

transition, e is the electron charge, and r is the operator representing position from the nuclei to

the electron. To express H as a matrix in the hyperfine manifold, we typically only need to obtain

the matrix elements between the hyperfine levels |ii and |ei that are relevant to the transition.

Thus

H0 = ~ωi|iihi| + ~ωe|eihe| (4.18)

V = 2~(Ωie|iihe| + Ωei|eihi|) cos(kz − ωLt + φ), (4.19)

where ~ωi,e is the energy of the state |ii and |ei, ωL is the laser frequency, and Ωie = Ω∗

ei =

−

1

~

|e|E0hi|r · ˆ|ei/2 can be calculated by integrating with the state |ii and |ei expressed with

wavefunctions in radial and angular coordinates. The angular coordinates for 9Be+ states can be

obtained from description in Table 3.2 and Table 3.3.

To obtain the dynamics, it is useful to transform Eq. 4.19 to an interaction picture for the internal states. For the Hamiltonian H = H0+V , in the interaction picture, Hint = e

iH0t/~V e−iH0t/~

,

corresponding to viewing the wavefunction |Ψi in a frame with transformation |Ψ0i = e

iH0t/~

|Ψi.
49

Here we have

Hint = e

iH0t/~V e−iH0t/~

(4.20)

= 2(Ωie|iihe|e

i(ωi−ωe)t + Ωei|eihi|ei(ωe−ωi)t

) cos(kz − ωLt + φ). (4.21)

By rewriting cos(kz −ωLt+φ) = 1

2

(e

i(kz−ωLt+φ) +e−i(kz−ωLt+φ)

), the time dependent part contains

two frequencies, |ωi − ωe ± ωL|. If we assume |ii and |ei are the ground/excited states and we

set ∆ ≡ ωL − (ωe − ωi), with ∆  (ωe − ωi), thus one of the frequency difference is ∆ but

the other frequency is very high and is approximately twice the frequency difference between the

ground/excited states. In this case we can make the rotating wave approximation (RWA) to neglect

the high frequency component. Thus the Hamiltonian in the interaction picture is

Hint = ~Ωie|iihe|e

i(∆t−kz−φ) + H.C., (4.22)

where H.C. denotes Hermitian conjugate.

4.2.2 Raman transition

To drive transitions optically between two hyperfine states, we need a Λ configuration, depicted in Fig. 3.1 and Fig. 4.1. This Λ involves two dipole transitions with the close detuning from

the same excited state. In this case, the Hamiltonian in the interaction picture is

Hint,Λ = ~Ωie|iihe|e

i(∆t−kz−φ) + ~Ωme|mihe|ei(∆0t−k

0z−φ0

) + H.C., (4.23)

where the ground hyperfine states are denoted |ii and |mi, and ∆ and ∆0 denote the detunings of

the lasers driving the |ii → |ei and |mi → |ei transitions. With ∆ much larger than Ωie and Ωme,

a second order perturbation theory is performed [102] and we obtain

Heff = ~

|Ωie|

2

∆

|iihi| + ~

|Ωme|

2

∆0

|mihm| + ~(

ΩieΩ

∗

me

∆

|iihm|e

i(νt+(k

0−k)z+φ0−φ) + H.C.), (4.24)

assuming the frequency difference of the laser beams ν = ∆ − ∆0is such that |ν|  |∆|. The

first two terms of this effective Hamiltonian represent the Stark shifts of the ground states by

the two laser beams, and can be nulled by setting the difference frequency of the laser beams to
50

be ν

0 = ν − (

|Ωme|

2

∆0 −

|Ωme|

2

∆

). The third terms represent an effective coupling between the ground

states, with Rabi rate Ω0 = |

ΩieΩ∗me

∆

|. With the Stark shift nulled, this drives a resonant two-photon

stimulated-Raman transition between ground states |ii and |mi.

|Ωme |Ω |

ie|

|m⟩

|i⟩

|e⟩

Δ Δ

′

|Ωme |Ω |

ie|

|m⟩

|i⟩

|e⟩

...

...

|0⟩

|1⟩

|n − 1⟩

|n⟩

|0⟩

|1⟩

|n − 1⟩

|n⟩

(a) (b)

ω

Figure 4.1: (a) A simplified Λ configuration of Raman transition between internal states |ii and

|mi. (b) a Raman sideband transition of |m, ni → |i, n − 1i.

The major draw back of using a Raman transition is the off-resonant couplings from the

ground states to the excited states which are subject to decay; this can destroy coherence and may

also cause incoherent hyperfine state changes through spontaneous emission [46]. With the excited

state decay rate Γ, the spontaneous emission rate γSE ∝ Γ

|Ωie,je|

2

∆2 . Since the Raman transition

Rabi rate scales as 1

∆

, we can decrease γSE relative to the Rabi rate by increasing ∆, at the cost

of proportionally increasing the laser intensities to maintain the Raman transition Rabi rate.
51

4.2.3 Raman sideband coupling

With ∆k = k

0 −k 6= 0, the two laser beams create a phase gradient in space, and thus the ion

motion needs to be taken into account. With one ion, the motional Hamiltonian is Hs = ~ωa†a,

neglecting the constant term. From Sec. 4.1, we have z = z0(a+a

†

)+z

0

, where z

0

is the equilibrium

position of the ion, z0 is the spread of the wavepacket, and a is the annihilation operator. We define

the quantity ∆kz0 ≡ η, the Lamb-Dicke parameter. For η  1, we can expand Heff to the orders

of η, and in the interaction picture of Hs, Heff becomes

Heff = ~Ω0|iihm|e

i(ν

0

t+φ

0−φ+∆kz0

)

(1 + iη(e

−iωta + eiωta†

) + ...) + H.C.. (4.25)

The interesting case is for ν

0 − ω ≡ δ  1, where the laser beam frequency difference is offset

from the hyperfine state energy difference by a motional frequency, called the first sideband of the

hyperfine transition2. In this case the Raman transition is de-exciting the motion while driving

the transition from |mi to |ii

3

. From Eq. 4.25, we need to include all terms in the expansion of

e

iη(a+a

0

)

that have time dependence equal to e

−iωt [22, 103, 104]. We have

Heff = ~

X

n

Ωn−1,n|i, n − 1ihm, n|e

i(φ

0−φ+∆kz0

) + H.C., (4.26)

where |m, ni includes the motional Fock state |ni,

Ωn−1,n ≡ Ω0|hn − 1|e

iη(a+a

†

)

|ni| = Ωexp(−η

2/2)ηL1

n−1

(η

2

)/

√

n, and

L

α

n

(X) = Pn

m=0(−1)m





n + α

n − m





Xm

m!

is the generalized Laguerre polynomial. To the first order

of η, we can still approximate the Hamiltonian as the Jaynes-Cummings type

Heff = ~Ω|iihm|a ei(δt+φ

0−φ+∆kz0

) + H.C., (4.27)

where Ω = Ω0ηe−η

2/2

. This Hamiltonian serves as a starting point to understand the couplings

between the spin and motion in this thesis. With this coupling, transition |m, ni ↔ |i, n − 1i is

driven. This coupling can be used to provide cooling with hyperfine state repumping of |ii → |mi,

2 The analysis for ν0 + ω ≡ δ  1 is similar

3 motional excitation would happen for the opposite case driving from |ii to |mi
52

to generate a repeated cooling cycle |m, ni ↔ |i, n − 1i → |m, n − 1i → ...|m, 0i. This is the case of

resolved Raman sideband cooling, where dissipation of the internal states is transferred to motion

with the internal-motional state coupling. The cooling can be applied in a continuous manner,

where the sideband coupling and repumping are applied simultaneously. With repumping rate

γ  Ω, the effective cooling rate is proportional to Ω2/γ. It can also be applied in a stepwise way,

where sideband and repumping pulses are alternatively applied. The process starts from repumping

to the |mi state, then ideally, every step contains a sideband π transition from |m, ni → |i, n − 1i

followed by a repump pulse. Assuming the ion is at Fock state |Ni initially, the i

th step sideband

pulse duration corresponds to the Rabi rate for Fock state |N − i + 1i. Typically, N is chosen to

be large to cover enough Fock states after Doppler cooling [105].

4.2.4 Raman sideband coupling for two ions: resonant case

From the previous section, the sideband Hamiltonian for two ions can be written

HSB = ~Ω(|ii

1

hm|e

i∆kz0

1 + |ii

2

hm|e

i∆kz0

2 )a ei(δt+φ

0−φ) + H.C., (4.28)

where we assume equal Rabi rates on the two ions, labeled as ion i = 1, 2, and z

0

i

represents the

equilibrium position of ion i. From now on we label |ii ≡ |↓i and |mi ≡ |↑i. By choosing the

location of ion 1 as the origin, and defining ∆z = z

0

2 − z

0

1

, we have

HSB = ~Ω(σ

−

1 + σ

−

2

e

i(∆k∆z)

)a ei(δt+φ

0−φ) + H.C., (4.29)

where σ

− = |↓ih↑| is the Pauli lowering operator. By controlling the separation ∆z of the two

ions, the relative phase between the σ

− operators can be controlled. This corresponds to a relative

phase added to the states and is important when creating different entangled states, such as |Ti =

√

1

2

(|↑↓i + |↓↑i) versus |Si = √

1

2

(|↑↓i − |↓↑i). To analyze the eigensystem of HSB, we first choosing

∆k∆z = 2Mπ, where M is an integer number. And we set δ = 0 and φ

0 − φ = 0. Thus the

Hamiltonian is simplified as

HSB = ~Ω(σ

−

1 + σ

−

2

)a + H.C., (4.30)
53

eigenenergy/~ eigenstate

0 |↑↑, 0i

0 |S, ni

0 |Ψ0, ni = √

1

2n+3 (

√

n + 2|↓↓, ni − √n + 1|↑↑, n + 2i)

±

√

2Ω √

1

2

(|T, 0i ± |↑↑, 1i)

±

√

4n + 2Ω |Ψ±, ni = √

1

4n+2 (

√

n + 1|↑↑, n + 1i ± √2n + 1|T, ni +

√

n|↓↓, n − 1i).

The eigenenergy and eigenstates are the following:

Here we point out some properties of this Hamiltonian.

1. The eigenenergies of states |↑↑, 0i, |S, ni and |Ψ0, ni are 0 since they do not couple to HSB, and

are thus “dark” states.

2. The state |↓↓, 0i can be expressed in a superposition of eigenstates as |↓↓, 0i = √

1

6

(|Ψ+, 1i +

|Ψ−, 1i+ 2|Ψ0, 0i. Thus the evolution starting from this state is thus |↓↓, 0i → √

1

6

(|Ψ+, 1ie

−i

√

6Ωt +

|Ψ−, 1ie

i

√

6Ωt + 2|Ψ0, ni). The half period of the evolution is thus tπ = π/(

√

6Ω). The population

of |Ti is maximized to 1

3

at t =

2M+1

2

tπ, where M is an integer, and the population of |↑↑i is

maximized to 8

9

at t = (2M + 1)tπ, as shown in Fig. 4.2. Thus a resonant sideband alone is not

sufficient to produce |Ti. This is due to the Fock state dependence of the coupling strengths.

3. The above analysis is relevant to the dissipative entanglement experiment in this thesis. For

∆k∆z = (2M + 1)π, the results are similar except for an exchange between |Ti and |Si. This

sideband interaction is also important for the experiment of confined dynamics in this thesis.

4.2.5 Raman sideband coupling for two ions: detuned case

As described in previous section, for the resonant case, the |↓↓, 0i state contains the |Ψ0, 0i

dark state. For the confined dynamics described in this thesis in Chapter 7, the desired interaction

would only have dark states of |↑↑, 0i and |T, 0i. It turns out that the sieband interaction with

δ 6= 0 can realize this case. As we will see below, in a suitable frame the non zero detuning is
54

0 0.5 1 1.5 2

0

0.2

0.4

0.6

0.8

1

duration t / t

π

populations

P|T,1〉

P|↑↑,2〉

P|↓↓,0〉

Figure 4.2: State evolution subject to the sideband Hamiltonian of Eq. 4.30, starting from state

|↓↓, 0i.

equivalent to introducing extra shifts on different Fock state, and thus |Ψ0, 0i is dressed and energy

shifted since it contains more than one Fock states. Here we provide detailed analysis for this case

and find the optimal δ that gives the maximal energy shift of the |↓↓, 0i state from its unperturbed

value.

To simplify the discussion we choose φ

0 − φ = 0 and ∆k∆z = (2M + 1)π. Thus the sideband

Hamiltonian becomes

HSB = ~Ω(σ

−

1 − σ

−

2

)a eiδt + H.C.. (4.31)

To make the Hamiltonian time independent, we move to an interaction frame for the motion defined

by U = e

iδa†at, equivalent to a transformation of wavefunction |Ψ0

i = U|Ψi. The Hamiltonian in

this frame is

H0

SB = −iU ∂t(U

†

) + UHSBU

†

(4.32)

= −~δa†a + (~Ω(σ

−

1 − σ

−

2

)a + H.C.). (4.33)
55

To analyse the eigenenergies and eigenstates of this Hamiltonian, we express HSB in terms of the

subspace {|↓↓, 0i, |S, 1i, |↑↑, 2i}. We have

H0

SB =





0

√

2Ω 0

√

2Ω −δ −2Ω

0 −2Ω −2δ





. (4.34)

The eigenvalues versus the detuning are shown in Fig. 4.3. All three eigenstates have components

of the |↓↓, 0i state.

-3 -2 -1 1 2 3

δ/Ω

-6

-4

-2

2

4

6

ħΩ

Figure 4.3: Eigenvalues of the detuned sideband.

As described in Chapter 7, we will want to find δ values, where all eigenvalues are shifted

maximally from zero. From Fig. 4.3, this condition is fulfilled when two of the eigenenergies are

equal but opposite sign with δ 6= 0. Imposing this condition, we find δ = ±

q

7

3Ω, and for the

positive case the eigenenergies and eigenstates are

eigenenergy/~ eigenstate

λ1 =

q

4

3Ω, |Ψ1i = −

q

3

118 (19 + √

7)|↓↓, 0i − q

1

59 (19 + √

7)|S, 1i +

q 3

23+5√7

|↑↑, 2i

λ2 = −

q

4

3Ω, |Ψ2i =

q

3

118 (19 −

√

7)|↓↓, 0i − q

1

59 (19 −

√

7)|S, 1i +

q

1

118 (23 + 5√

7)|↑↑, 2i

λ3 = −

√

21Ω, |Ψ3i = −

q

2

59 |↓↓, 0i +

q

21

59 |S, 1i + √

6

59 |↑↑, 2i
Chapter 5

Sympathetic electromagnetically-induced-transparency(EIT) laser cooling

To achieve high fidelity multi-qubit operations with trapped ions, the qubit ions should be

laser cooled as close as possible to the ground state of motion. At the same time, the laser beams

that provide the cooling should not perturb information stored in the qubits. To achieve this, one

needs to trap ions specifically for cooling, the “refrigerants”, in the same trap as the qubit ions.

The regrigerant ions can be same species as the qubit ions and be cooled with laser beams focused

strongly enough to not perturb the qubit ions, but in practice this isolation is difficult to achieve.

An alternative procedure is to make the refrigerant ions to be a different species that has resonant

wavelengths substantially different from the resonant wavelengths of qubit ions. This process,

known as sympathetic cooling [44], can be implemented with Raman sideband cooling [106]. An

alternative method is to use electromagnetically-induced-transparency (EIT) laser cooling. We

will see that this method significantly reduces the cooling duration and the required laser power

compared with sideband cooling, by applying near-detuned lasers to induce the EIT effect.

In this chapter, we describe experiments and simulations applying EIT laser cooling on the

motional modes of a linear ion chain [78]. We first describe the general aspects of sympathetic

laser cooling of trapped ions and the conventional method of Raman sideband cooling. Second, we

discuss principles of EIT laser cooling, and experimental implementations on sympathetic cooling

9Be+ ions by 24Mg+ ions in 9Be+ -24Mg+ ion pairs and 9Be+ -24Mg+ -24Mg+ -9Be+ ion

chains. We then compare cooling speeds and final temperatures of Raman sideband cooling and

EIT laser cooling in our experiments. We achieve approximately an order-of-magnitude reduction
57

in the duration required to cool the motional modes to near the ground state; we also achieve

approximately two orders-of-magnitude reduction in the required laser beam intensity, compared

to previous implementations of conventional Raman sideband cooling. Finally, we discuss further

aspects of cooling 24Mg+ ions to sympathetically cool qubit ions of different mass, and possible

improvements on future EIT-cooling experiments that use 24Mg+ ions.

5.1 General description of sympathetic laser cooling

One proposal for building a quantum information processor is to use trapped, laser-cooled

ions [17, 47, 107]. The internal states of the ions serve as individual qubits that can be manipulated

by laser beams or microwave radiation. The Coulomb coupling between ions establishes normal

modes of motion; sideband transitions involving both the qubit states and motional modes enable

entangling-gate operations between multiple qubits. For high-fidelity deterministic entangling gates,

the extent of the motion of the relevant modes should be in the Lamb-Dicke regime [107] where

the amplitude of the ions’ motion from equilibrium is much less than the effective wavelength of

the coupling radiation1. Thus for most experiments, the motion needs be cooled to near the

quantum-mechanical ground state, which has typically been achieved with sideband laser cooling

[39, 107, 108].

Ion motion can be excited by ambient noisy electric fields [22]. Therefore, a method for

recooling the ions is needed for implementation of lengthy quantum algorithms. One recooling

strategy is to trap the qubit ions together with “refrigerant” ions. With Coulomb coupling between

these ions, the qubits in the cooling process can be “sympathetically” cooled without disturbing the

internal states [2, 3, 22, 34, 74–76, 109]. Demonstrations of this technique in information processing

have so far used stimulated sideband cooling [3, 34, 39, 74–76]. While effective, stimulated Raman

sideband cooling can typically cool only one mode at a time because of the differences in mode

frequencies and narrowness of the sideband transitions. Furthermore, the laser-beam intensities and

detuning must be sufficiently large to avoid heating from spontaneous emission. In experiments

1 Probabilistic schemes for entanglement do not require such strong confinement[37]
58

performed in a scalable configuration, the duration required for recooling has been an important

limiting factor [42, 45, 75]. The larger recooling duration for multiple recooling steps can enhance

errors due to qubit dephasing [73].

A technique that can mitigate these problems is electromagnetically induced transparency

(EIT) laser cooling, described theoretically in [8, 9] and demonstrated on a single ion in [110–112].

For EIT cooling, the required laser beam intensities are relatively small, and the cooling bandwidth

is large enough that multiple modes can be cooled simultaneously, which speeds up the overall

cooling process for multiple motional modes. To demonstrate these features, we investigate EIT

cooling of multiple modes of linear ion chains containing 9Be+ and 24Mg+ ions. EIT cooling is

applied to the 24Mg+ ions and cools all modes along the axis of the chain to near the ground state.

This process thereby sympathetically cooling the 9Be+ ions.

5.2 Theoretical description of sympathetic EIT cooling

5.2.1 EIT effect on a stationary ion

The complete theoretical description of EIT laser cooling is given in [8]. Consider the threelevel Λ system comprised of the bare states |g1i, |g2i and |ei shown in Fig. 5.1(a), where the excited

state |ei has a rate γ to decay back to the ground state manifold {|g1i, |g2i}. For an ion at rest,

laser beams with resonant Rabi rates Ω1 and Ω2 and equal detunings ∆1 = ∆2 ≡ ∆ > 0 dress the

bare states such that the system relaxes to the “dark” steady state |ψDi = (Ω2|g1i−Ω1|g2i)/Ω with

Ω ≡

p

Ω2

1 + Ω22

. This results from the Λ system Hamiltonian in the rotating frame [8] Hstatic =

H0 + V0, where H0 = ~∆(|g1ihg1| + |g2ihg2|), and V0 =

~

2

(Ω1|eihg1| + Ω2|eihg2|) + H.c. represents

the laser induced interactions. With a basis transformation from {|g1i, |g2i} to {|ψDi, |ψCi =

(Ω1|g1i + Ω2|g2i)/Ω}, we have Hstatic = −~∆(|ψDihψD| + |ψCihψC|) + ~

2Ω(|eihψC| + |ψCihe|). We

can see |ψCi is coupled to the excited |ei state, while |ψDi is not coupled and remains “dark”.

Thus applying the laser beams in the Λ configuration pumps the system into the “dark” state.
59

Additionally, Hstatic can be further diagonalized as

Hstatic =

~

2

(∆ + p

∆2 + Ω2)|ψ+ihψ+| (5.1)

+

~

2

(∆ −

p

∆2 + Ω2)|ψ−ihψ−|

+ ~∆|ψDihψD|,

where |ψ+i = cos θ|ei + sin θ|ψCi, |ψ−i = sin θ|ei − cos θ|ψCi, and tan θ =

√

∆2+Ω2+∆

Ω

. This basis

{|ψDi, |ψ±i} is called “EIT basis”. Assuming ∆  Ω, sin θ ≈ 1 −

Ω2

8∆2 and cos θ ≈

Ω

2∆ . Thus |ψ−i

state is approximately given by the |ei state, while |ψ+i has a component of |ei with amplitude

on the order of Ω

2∆ in population. Both |ψ±i states contain an excited state amplitude and are

subject to decay to the ground state. Thus if we added a weak laser beam that probes the system,

it would create an absorption spectrum as depicted in Fig. 5.1(c), where the narrow/broad peak

correspond to the transition driving |ψDi → |ψ±i, respectively. The frequency shift between the

absorption null and the relatively narrow absorption peak on the right side of the spectrum is

δ = (p∆2 + Ω2 − ∆)/2, (5.2)

corresponding to the energy difference of |ψDi and |ψ+i in Eq. 5.1.

5.2.2 EIT effect with a trapped ion in motion

If the difference in k-vectors for the two dressing beams has a component along the direction

of a motional mode, excitations in ion’s motion will prevent it from being in the dark state. As we

will see below, the transition of |ψD, ni → |ψ+, n−1i is driven, where |ni denotes the motional Fock

state. |ψ+, n − 1i subsequently decays back to the ground state manifold and is quickly pumped

to |ψD, n − 1i, neglecting photon recoil heating. Thus ideally an effective close cooling cycle of

|ψD, ni → |ψD, n − 1i is formed until the ground state is achieved. The final phonon occupation

number  ̄n is limited by the off-resonant motional heating transition of |ψD, ni → |ψ−, n + 1i. Here

we provide more detailed analysis of the above process.

The Hamiltonian is H = H0+Hmec+V , where Hmec = ~ω



a

†a +

1

2



, a =

P

n

√

n|n − 1ihn| is

the annihilation operator or, ω is the motional frequency, and V is now modified from the stationary
60

δ

Figure 5.1: 24Mg+ energy levels, laser setup and a spectrum showing the EIT effect. (a) Relevant

energy levels for 24Mg+, in an applied magnetic field of 11.964 mT. The three levels |g1i, |g2i and

|ei serve as a Λ system for EIT cooling. Laser beams with σ

+ and π polarizations couple the

ground states to the excited state with Rabi rates Ω1 and Ω2 and common detuning ∆. Wavy

lines show spontaneous emission from the excited state to the ground states and the excited-level

decay rate is denoted by γ ' 2π×41 MHz. The fourth level |e

0

i can perturb the EIT cooling when

the π polarized laser beam has frequency near the |g1i to |e

0

i resonance. (b) Beam configuration

and a depiction of the 9Be+−24Mg+ − 24 Mg+−9Be+ ion, shown as balls from left to right. This

figure is viewed in an angle from top view.(c) Simulation of the absorption spectrum of a stationary

ion by a weak σ

+ probe beam for ∆ = 2π×96.7 MHz, Ω1/2π = 30 MHz, and Ω2/2π = 12 MHz.

For simplicity, the fourth level |e

0

i is ignored. The probe detuning from the |g2i to |ei resonance

is denoted by ∆P. This Fano-like profile contains a narrow and broad feature corresponding to

dressed states |ψ+i and |ψ−i respectively [8], and the separation between the peak on the right

side and the null is labeled as δ. When ∆P = ∆, absorption vanishes due to coherent population

trapping.

V0 to take account the motion

V =

~

2



Ω1e

ik1z cos φ1

|eihg1| + Ω2e

ik2z cos φ2

|eihg2| + H.c., (5.3)

where z is the position operator of the ion along the axial direction, ki cos φiis the projection of

the laser beam ki vector along the trap axis. We will assume the ion is in the Lamb-Dicke regime,

so we expand the term eikiz cos φi up to the first order to show the physics of the cooling process.
61

Higher order expansions are considered in the full simulations. Thus we have V ≈ V0 + zV1. The

latter can be expressed in the basis of {|ψDi, |ψCi},

V1 =

i~

2

X

j=1,2

ki cos φjΩj (|eihgj | − |gj ihe|) (5.4)

=

i~

2

Ω1Ω2

Ω

∆k(|eihψD| − |ψDihe|)

+

Ω

2

1

k1 cos φ1 + Ω2

2

k2 cos φ2

Ω

(|eihψC| − |ψCihe|)



, (5.5)

where the difference k vector ∆k = (k1 cos φ1 − k2 cos φ2), with z = z0(a + a

†

). When the EIT

condition is fulfilled with ∆1 = ∆2, population in |ψCi would be pumped out quickly and relax to

the dark state |ψDi, thus we will ignore the second term of V1. Turning to EIT eigenbasis for the

first term, we have

V1 =

i~

2

Ω1Ω2

Ω

(∆k)



cos θ|ψ+ihψD| + sin θ|ψ−ihψD| − H.c.. (5.6)

The total Hamiltonian can now be written as H = H0 +V0 +Hmec +zV1, and as shown above

in (5.1), H0 + V0 is diagonal in the EIT eigenbasis. One can now transform zV1 to the interaction

picture zV1,int = e

i(H0+V0+Hmec)t/~

zV1e

−i(H0+V0+Hmec)t/~

, where2

zV1,int =

i~

2

Ω1Ω2

Ω

(z0∆k)

X

n

(

√

ne−iωt|n − 1ihn| +

√

neiωt|nihn − 1|) ×

(e

i

−∆+√∆2+Ω2

2

t

cos θ|ψ+ihψD| + e

i

−∆−

√

∆2+Ω2

2

t

sin θ|ψ−ihψD| − H.c.). (5.7)

Here ∆ is defined to be positive for blue detuned laser in contrast with [8]. As defined above, the

Stark shift δ =

−∆+√

∆2+Ω2

2

. By imposing the condition δ = ω and applying the rotating wave

approximation(RWA), V1 will drive the transition from |ψDi|ni to |ψ+i|n − 1i.

3 Combining this

coherent transition with the EIT dark state re-establishing process, the cooling cycle is complete.

With −∆+√

∆2+Ω2

2 = ω, we have Ω2 = 4ω(ω + ∆), which can be called the EIT cooling condition4

,

consistent with [8]. Intuitively, this condition corresponds to match the energy shift δ in Fig. 5.1(c)

2 Detailed derivation is in the appendix.

3 For a red detuned ∆ < 0, from Eq. 5.7 the transition |ψDi|ni to |ψ−i|n+ 1i is driven with motion adding. Thus

cooling can only occur for blue detuned laser beam configurations. This is different from the requirements of Doppler

cooling.

4 The sign of ∆ is changed between this thesis and [8]
62

with the motional frequency ω. We note that the width of the right-hand peak can be made broad

enough5that the condition for cooling is met for multiple modes for the same value of δ. This may

facilitate experiments involving many ions such as simulations where the mode frequencies have a

relatively narrow distribution [113–116].

5.3 Experiment sympathetic EIT laser cooling

We trap 9Be+ and 24Mg+ ions in a linear radio-frequency Paul trap described in [3, 74] and

depicted schematically in Fig. 5.1(b). The ions form a linear chain along the axis of the trap, the

axis of weakest confinement. We perform experiments on either a single 9Be+ -

24Mg+ pair or a fourion chain with the ions in the order of 9Be+-

24Mg+-24Mg+-9Be+ [3, 74]. A single trapped 9Be+ ion

has a motional frequency ωz = 2π × 2.97 MHz along the trap axis and {ωx, ωy} = 2π × {12.4, 11.7}

MHz, along the transverse directions. An internal-state quantization magnetic field B is applied

along a direction 45◦to the trap axis in Fig. 5.1(b). This breaks the degeneracy of the magnetic

sublevels of 9Be+ and 24Mg+, denoted by quantum number mJ . As in Fig. 5.1(a), mJ indicates

the projection of the 24Mg+ ion’s angular momentum along the direction of B. For B = 11.964 mT,

the energy splitting of the qubit states 2s

2S1/2

|F = 2, mF = 1i and |F = 1, mF = 0i of 9Be+ is

first-order insensitive to changes in B, leading to long coherence times of superposition states [73].

We apply two laser beams near the 3s

2S1/2

to 3p

2P1/2

transition in 24Mg+ at approximately

280.353 nm [Fig. 5.1(a)]. These two beams are derived from the same laser and frequency shifted by

acousto-optic modulators [106]. As indicated in Fig. 5.1(b), one of the beams propagates along the

direction of B with σ

+ polarization to couple |g1i to |ei with resonant Rabi rate Ω1 and detuning

∆1 from the excited state. The other beam has π polarization and couples |g2i to |ei with resonant

Rabi rate Ω2 and detuning ∆2. To meet the EIT prerequisite, we set ∆1 = ∆2 = ∆, where ∆/2π

can be set to a precision of approximately 1.5 MHz. To maximize the cooling, we set the difference

wave-vector of the two beams parallel to the trap axis, so that the net momentum from the lasers

5 From the Fano profile, the width is estimated to be γω/∆, assuming the Rabi rates meet the EIT condition. Thus

lowering ∆ helps broadening the cooling bandwidth, however at the cost of cooling rates proportional to 4ω(ω+∆)

γ

and the final phonon occupation number  ̄n ≈ (

γ

4∆ )

2

[9].
63

have maximum projection on trap axis. To meet the EIT cooling condition Eq. 5.2, we control the

values of Ω1 and Ω2 by adjusting laser beam powers with acousto-optic modulators, and the values

of Ω1 and Ω2 are determined from measurements of the Rabi rate for Raman carrier transitions

and the AC Stark shift from the σ

+ polarized beam when the laser is detuned from resonance.

The EIT laser cooling experiment includes the following processes. We first apply Doppler

cooling to 9Be+, which initializes the temperatures of the axial modes of motion to near the Doppler

limit given approximately by ~γBe/(2kB), where γBe is the 9Be+ excited-state decay rate, and kB

is Boltzmann’s constant). We then apply the EIT cooling beams to 24Mg+ for a duration tc. To

determine the final mean motional-state quantum number  ̄n of the normal modes, we use a pair of

313.220 nm laser beams [39, 106, 117] to compare the strength of the red and blue Raman sideband

transitions in the 9Be+ ions on the |2, 1i → |1, 0i transition, as a temperature measurement for

the sympathetically-cooled qubit ion [106]. We measure temperature on 9Be+ is that the 24Mg+

S1/2 → P1/2transition is not a cycling transition and therefore not useful for state discrimination.

5.3.1 EIT cooling on the 9Be+-

24Mg+ ion pair

The ion pair has two axial motional modes: a mode where the two ions oscillate in-phase

(I) with frequency ωI/2π = 2.1 MHz and an out-of-phase mode (O) with frequency ωO/2π = 4.5

MHz. The relevant ground state mode amplitudes for the 24Mg+ ion are denoted by z

I

0

and z

O

0

respectively. The contribution of 24Mg+ vibration to the mode is given by the unitless Lamb-Dicke

parameters, defined as ηI,O = ∆kzz

I,O

0

. This parameter reflects the effectiveness of 24Mg+ laser with

the mode; here, ηI = 0.294 and ηO = 0.083. The EIT cooling condition δ ' ω cannot be satisfied

exactly for both modes simultaneously, since the mode frequencies are substantially different.

To investigate steady state performance of the EIT cooling, we EIT cool 24Mg+ in the pair

for 800 μs, long enough for the system to reach equilibrium. We set ∆/2π = 96.7 MHz and

Ω2/2π = 12.5 MHz and we scan the value of Ω1 to vary δ, as shown in Eq. (5.2). The minimum

values of  ̄nI = 0.08(1) and  ̄nO = 0.04(1) are obtained when δ closely matches the respective mode

frequency, as shown in Fig. 5.2(a). We observe a deviation ' 10% of the value of δ needed for
64

optimum cooling compared to the mode frequency. The deviation can be explained by additional

AC Stark shifts and photon scattering from the π-polarized beam that couples |g1i to |e

0

i ≡

2P1/2

|mJ = −1/2i in 24Mg+, see Fig. 5.1(a). We perform a numerical simulation of the full

dynamics, including state |e

0

i . We include the effects of heating rates of both modes, n ̄ ̇

I = 0.38

quanta/ms and n ̄ ̇ O = 0.06 quanta/ms. The simulation shows the average occupation numbers

as solid lines in Fig. 5.2(a) and are in good agreement with our experimental results. For these

simulations, we use the treatment of [8], which is valid in the Lamb-Dicke regime, adjusted for the

relevant modes and mode amplitudes of the 24Mg+ ions.

To investigate the temporal dynamics of the cooling, we set δ to be near a mode frequency

and measure  ̄n vs cooling duration tc. We first Doppler-cool both modes with 9Be+ reaching  ̄nI ∼

5 and  ̄nO ∼ 2. We find that at the experimentally determined optimum values of δ/2π of 2.55(5)

MHz and 4.87(5) MHz, the 1/e cooling time for the I mode is 4(1) μs and for the O mode is 15(1)

μs. The faster cooling rate for the in-phase mode is expected because of its larger 24Mg+ LambDicke parameter. This difference in equilibration times can be use to efficiently cool both modes,

as shown in Fig. 5.2(b). We first set δ ' ωO and apply cooling for 75 μs, yielding  ̄nO = 0.04(1).

During this stage, the I mode is cooled to  ̄nI = 0.36(3). We then set δ ' ωI and apply the cooling

beams for an additional 10 μs, reaching  ̄nI = 0.08(1). In this second cooling stage, the O mode

begins to heat to its equilibrium value of  ̄n = 0.19(3), shown in Fig. 5.2(a) for this value of δ.

However in 10 μs, this heating is small, leading to a final value of  ̄nO = 0.07(2). Therefore, this

two-stage cooling enables cooling of both modes to near their minimum  ̄n values in 85 μs

6

.

5.3.2 EIT cooling on the 9Be+-

24Mg+-24Mg+-9Be+ four-ion chain

We also investigate sympathetic EIT cooling for the four-ion chain. We label the four-ion

axial modes {1, 2, 3, 4}. They have mode frequencies ' {2.0, 4.1, 5.5, 5.8} MHz and corresponding

24Mg+ Lamb-Dicke parameters {0.21, 0.12, 0.063, 0.089}. Fig. 5.3 shows the final  ̄n of each mode vs.

6 To achieve this result we roughly tune the laser power to be approximately 50% larger than expected, then

stabilize the laser power and use the set points as a fine control of the laser power, to satisfy the EIT condition. In

our previous experiments on 9Be+-

24Mg+ ion pair, the duration of the conventional Raman sideband cooling to cool

both axial modes are ∼300 μs.
65

Figure 5.2: Mean motional excitation number  ̄n for the I (red triangles, ω/2π = 2.1 MHz) and

O (blue squares, ω/2π = 4.5 MHz) axial modes of a 9Be+-

24Mg+ ion pair. (a)  ̄n after 800 μs of

EIT cooling as a function of δ/2π. Optimal cooling for each mode occurs when δ approximately

equals the mode frequency. (b)  ̄n plotted as a function of EIT cooling duration tc. From 0 to 74 μs,

δ ' ωO, the out-of-phase mode frequency. From 75 to 85 μs, δ ' ωI, the in-phase mode frequency.

In both figures, error bars represent statistical uncertainty of the sideband amplitude ratios. The

solid lines are simulations of the full dynamics including the |e

0

i level in 24Mg+, measured ambient

heating rates, detuning and beam intensities. In the simulations, we truncated the motion to the

first 6 Fock states for both modes for the steady-state simulation in (a) and to the first 10 (6) Fock

states for the I (O) mode for temporal simulation in (b).

δ/2π after 800 μs of cooling to ensure steady state. We set ∆/2π = 96.7 MHz, Ω2/2π = 9.6 MHz,

and scan Ω1/2π from 17 to 73 MHz. The EIT cooling bandwidth, estimated by δγ/∆ ≈ 2π×2.5

MHz , is sufficient that modes 2, 3, and 4 can be simultaneously cooled to near their minimum

n < ̄ 0.15 by setting δ/2π = 6.1 MHz; however, at this value, mode 1 is cooled only to  ̄n = 0.7(1).
66

Figure 5.3: Sympathetic EIT cooling on a 9Be+-

24Mg+-24Mg+-9Be+ ion chain. (a)Minimum  ̄n

values for each of the four axial modes of a 9Be+-

24Mg+-24Mg+-9Be+ ion chain as a function of

δ/2π after 800 μs of cooling to ensure steady state. Modes 1 to 4 are shown in the legend, and

labeled as red triangles, green squares, black circles and blue diamonds, respectively. (b)  ̄n plotted

as a function of EIT cooling duration tc. From 0 to 40 μs, δ ' ω3, the mode 3 frequency. From 40

to 45 μs, δ ' ω1, the mode 1 frequency. In both figures, error bars represent statistical uncertainty

of the sideband amplitude ratios.

We therefore cool in two stages, similar to the 9Be+-

24Mg+ case: (1) we first apply 40 μs of cooling

with δ/2π = 6.1 MHz to cool modes 2, 3, and 4, (2) then we cool 5 μs with δ/2π = 2.4 MHz tuned

to match mode 1. After these two steps, we achieve  ̄n = {0.11(2), 0.20(5), 0.14(5), 0.18(5)}. In our

experiments, laser beam power of the pi(sigma)-polarized beam ranges between 3 and 10 μW (3
67

Figure 5.4: (a) Minimum values of  ̄n for the two-ion O mode vs. ∆/2π. The peak near 223 MHz

results from resonant scattering on the |g1i ↔ |e

0

i transition from the π polarized light. Blue

circles are the experimental data and red triangles are simulations based on [8]. Green diamonds

are simulations not including |e

0

i and the black solid line shows  ̄n = (γ/4∆)2[9]. (b) Simulation

of 1/e cooling time vs. the mass of ion X+ that is sympathetically cooled by 24Mg+ (axial modes),

with fixed trap potential such that ωI(O)/2π = 2.1 (4.5) MHz when X+ is 9Be+. Red triangles are

for the I mode; blue squares for the O mode. Optimum values of δ were chosen for each mode and

ion-mass combination, with Ω2/2π = 5.9 MHz and ∆/2π = 96.7 MHz. (Here |e

0

i is neglected)

and 17 μW), respectively7. In previous implementations of sequential-Raman-sideband cooling

[42, 74–76], cooling of these modes from Doppler temperatures to  ̄n ∼ 0.1 for each mode required

∼ 600 μs with approximately an order-of-magnitude higher laser intensities8.

5.3.3 Technical notes for EIT laser cooling

5.3.3.1 Off-resonant photon scattering from extra excited states

The |g1i to |e

0

i transition frequency is 223.3 MHz higher than that of the |g2i to |ei transition.

Thus, EIT cooling will be strongly affected for ∆ near 223.3 MHz because of recoil from scattering

on the |g1i to |e

0

i transition. To illustrate this effect, we measure the minimum value of  ̄n for

cooling the O mode of the 9Be+−24Mg+ ion pair as a function of the detuning ∆ depicted in Fig.

5.4(a). For each value of ∆, we optimize the EIT cooling by varying δ. The height of this recoil

peak depends on the Rabi rate ratio Ω2/Ω1, with higher ratios leading to higher values of  ̄n. For

7 We did not measure the beam waists to high precision but they can be estimated to be approximately 100×50 μm

in 1/e2intensity.

8 with similar beam waists
68

data shown in Fig. 5.4(a), the ratio was held at 0.24. We note that for large detuning ∆, higher

laser intensity is needed to maintain values of δ near the mode frequencies.

5.3.3.2 Induced transverse direction heating

When ∆kz is aligned along the trap axis, the motional modes along the transverse axes

are heated by photon recoil. To study this effect we first cool one of the transverse modes of a

9Be+-24Mg+ pair (frequency ' 4.6 MHz) to near its ground state with Raman sideband cooling

on 9Be+. We then apply an EIT cooling pulse on the O mode, with similar laser beam conditions

as above. After 60 μs the O mode is cooled from the Doppler temperature  ̄n ' 2 to  ̄n = 0.04(1),

while the transverse mode is heated from  ̄n = 0.20(6) to 0.9(2). Once the O mode is cooled to

near its minimum value, the heating rate of the transverse mode decreases because the ion becomes

approximately trapped in the dark state for spin and the ground state of axial motion. This

relatively low transverse excitation should cause a negligible error on a two-qubit gate which is

affected by the transverse modes only through second-order coupling to the axial mode frequencies

due to Coulomb repulsions between the ions9[118, 119]. Furthermore, Doppler cooling of all modes

before EIT cooling would prohibit any cumulative effect of the heating in experiments requiring

many rounds of sympathetic cooling.

5.3.3.3 Sympathetic cooling efficiency on other ion pairs

To study the efficiency of EIT sympathetic cooling on other commonly used qubit ion species,

such as 27Al+,

43Ca+ and 171Yb+, etc., we simulate cooling of an ion pair 24Mg+ - X+, where X+

is the sympathetically cooled ion of different mass, as shown in Fig. 5.4(b). Smaller differences in

ion mass lead to more balanced mode amplitudes and a reduction in the difference of cooling rates

for individual modes. Large mass imbalances lead to at least one motional mode having a small

24Mg+ amplitude and thus a long cooling time [109].

9

In [118, 119], an effect is described where a frequency shift of the axial mode would be induced
69

5.4 Conclusion and future applications

In summary, we have described sympathetic cooling of 9Be+ ions by EIT cooling of 24Mg+ ions

held in the same trap. We investigated the cooling for both an ion pair and a four-ion chain crystal

that can be used as a configuration for performing entangling gates between pairs of 9Be+ ions

in a scalable architecture [74–76]. By taking advantage of the different cooling rates for different

modes of motion, we demonstrate a two-stage EIT-cooling scheme that can bring all modes to

near their ground states. Compared to previous implementations of conventional Raman sideband

cooling, sympathetic EIT cooling provides a broad cooling bandwidth, requires less laser power,

and is technically easier to implement. This method may also be useful for sympathetic cooling

of molecular ions, and for use in quantum logic spectroscopy [34], trapped-ion quantum simulation

[113–116], strongly-confined neutral atoms [120], and nano-mechanical resonators [121].

5.5 Appendix: derivation for EIT cooling condition

Here, we present more detail on the derivation of Eq. 5.7.

zV1,int = e

iHmect/~

ze−iHmect/~e

i(H0+V0)t/~V1e−i(H0+V0)t/~

. (5.8)

From the definition of Hmec in the main text, we obtain for the motional part of zV1,int,

e

iHmect/~

ze−iHmect/~(5.9)

=z0e

iHmect/~

(

X

n

(

√

n|n − 1ihn| +

√

n|nihn − 1|))e

−iHmect/~

=z0

X

n

(

√

nei(n−1)ωt|n − 1ihn|e

−inωt +

√

neinωt|nihn − 1|e

−i(n−1)ωt)

=z0

X

n

(

√

ne−iωt|n − 1ihn| +

√

neiωt|nihn − 1|), (5.10)
70

and for the internal part, with the help from (5.1),

e

i(H0+V0)t/~V1e−i(H0+V0)t/~

=

i~

2

Ω1Ω2

Ω

(∆k)e

i(H0+V0)t/~



cos θ|ψ+ihψD| + sin θ|ψ−ihψD| − H.c.e

−i(H0+V0)t/~

=

i~

2

Ω1Ω2

Ω

(∆k)



cos θei

∆+√∆2+Ω2

2

t

|ψ+ihψD|e

−i∆t + sin θei

∆−

√

∆2+Ω2

2

t

|ψ−ihψD|e

−i∆t − H.c.

=

i~

2

Ω1Ω2

Ω

(∆k)(e

i

−∆+√∆2+Ω2

2

t

cos θ|ψ+ihψD| + e

i

−∆−

√

∆2+Ω2

2

t

sin θ|ψ−ihψD| − H.c.) (5.11)

With (5.9) and (5.11),

zV1,int =

i~

2

Ω1Ω2

Ω

(z0∆k)

X

n

(

√

ne−iωt|n − 1ihn| +

√

neiωt|nihn − 1|)×

(e

i

−∆+√

∆2+Ω2

2

t

cos θ|ψ+ihψD| + e

i

−∆−

√

∆2+Ω2

2

t

sin θ|ψ−ihψD| − H.c.). (5.12)

Thus with −∆+√

∆2+Ω2

2 ≈ ω, the |ψD, ni → |ψ+, n−1i transition is driven. For ∆  Ω, cos θ ≈

Ω

2∆ ,

and the transition Rabi rate is approximately Ω1Ω2z0∆k

4∆

√

n.
Chapter 6

Steady state entanglement from dissipation

Quantum entanglement is a key phenomenon of quantum mechanics and plays an important

role in quantum information processing, quantum simulation and quantum computation. Traditionally in trapped ion systems, entanglement can be generated by deterministic interactions such

as coherent quantum operations [53, 122]. These operations can function as quantum logic gates,

the work horse for quantum computation. Similar operations can also produce effective interactions

that simulates interesting quantum systems such as Ising models [83, 123, 124]. High operation

fidelity has been continually pursued, and although it’s a technically difficult task, impressive

progress has been made [43, 125]. In recent years, researchers have also studied different ways of

generating quantum entanglement. One way is to use photons to link ions in separate traps [126],

which is useful for distributed quantum computation, one way quantum computation and quantum

networks. By this method, the entangled pair generation rate has been increased up to several

Hertz [127]. On the other hand, entanglements can also be generated with measurement based

methods, for example by photons interference [128, 129], by performing quantum-non-demolition

measurements [33, 130] and by applying heralded methods [131] to atoms in a cavity. Another way

is to introduce dissipation, where ideally all other states are pumped to the desired entangled state.

In this case, previous experimental works include using quantum logic gates and optical repumping

on auxiliary qubits, to pump to a Bell state [61], and using photons leaking between cavities to

couple two atom ensembles together [60].

Here we study novel methods to generate entangled states without using entangling quantum
72

logic operations. In this chapter, we introduce a dissipative steady state preparation method,

where we do not use auxiliary qubits, but continuously apply spin-motion coupling, microwave

fields, optical pumping and sympathetic cooling, which leads to a steady state highly overlapped

with the desired entangled state.

6.1 Theory and concepts

Generating entanglement with dissipation has recently become a hot topic. Traditionally, a

successful quantum information system implied extreme isolation from the environment because

couplings with the environment destroys the fragile quantum information. For example, consider

atoms trapped by optical lattices in a cavity, where atoms are coupled with the cavity photon mode.

Because the cavity mirrors have finite reflectivity, photons have a non negligible possibility to be

lost by escaping the cavity or being absorbed by the mirror. In this case, the environment is the

space or materials out of the cavity, and the coupling to the environment is the loss of the cavity

mirrors. However, a recent proposal [132, 133] suggests that the loss of photons can actually lead to

effects analogous to optical pumping of the system, and a dark entangled state exists where all other

states will eventually be pumped to that state, and entanglement is established and maintained

as steady state. This proposal can be translated into the trapped ion context because of their

similarity in couplings, and we implemented it here, with theoretical collaboration with F. Reiter

and A. S. Sørensen from University of Copenhagen.

The experiment uses four trapped ions, in a 9Be+ −24 Mg+ −24 Mg+ −9 Be+ ion chain. The

9Be ions the quantum bits, and the goal is to form the entangled singlet state |Si ≡ √

1

2

(|↑↓i −

|↓↑i), where the spins are defined in the hyperfine S1/2 ground states of the 9Be ions, as |↑i ≡

|F = 1, mF = 1i and |↓i ≡ |F = 2, mF = 2i. We use a pair of Raman sideband laser beams to

couple the ions via the motion, or phonons, in analogy to how the atoms in a cavity are coupled

via photons. We also induce a repumping laser and microwave drive to create selective spin state

pumping, as detailed below. The 24Mg+ ions serves as coolants. We can apply 280 nm laser beams

to drive Raman sideband transitions and provide repumping to ground state cool the 24Mg+ ions.
73

In this case, we cool the entire ion chain without scattering photons on the qubit 9Be+ ions. This

technique is called sympathetic cooling, and such a loss of phonons is analogous to the loss of

photons from a cavity, however in the ion experiment, we can turn the cooling on and off and tune

the strength conveniently while one cannot control the cavity mirror loss.

The two 9Be+ ions spins can be expressed in a basis spanned by the symmetric |↑↑i, |Ti, |↓↓i

and asymmetric |Si states, where |Ti ≡ √

1

2

(|↑↓i + |↓↑i) is the triplet state. The core of our scheme

is trying to induce dissipation to all the symmetric states, and pump the ions to the |Si state, while

trying to have minimize leakage from |Si state, as illustrated by Fig. 6.2 a). In other words we try

to minimize dissipation on the |Si state. It is a common misunderstanding that the scheme creates

entanglement directly from the dissipation components. In our case, the dissipative components

do not couple the ion spins. In fact, the entanglement is indirectly induced by the sideband laser;

however this interaction is reshaped by a combination of dissipation and other coherent processes to

create the targeted state. None of the ingredients of the scheme other than the sideband interaction

are able to entangle the two qubit ions.

The dissipation is comprised of two distinct parts. The first part is a combination of sympathetic cooling and Raman sideband transitions, to pump within the symmetric subspace from

|↓↓i → |Ti → |↑↑i, as depicted with Fig. 6.1. We apply the “blue” sideband, the motion adding

sideband from the |↓i state point, denoted by Hs = Ωs(σ

+

1 + σ

+

2

)b

† + H.C., where Ωs is the Rabi

rate, σ

+ = |↑ih↓|, subscript i denotes the ith ion and b is annihilation operator. The plus sign

between the two σ

+ operators in Hs is crucial to our experiment to define the target singlet dark

state, and thus need to be handled carefully, as detailed below. If spin is in the |↓↓i and |Ti states,

the spin down component will get a spin flip up and gain one quantum of motion, which will be

removed by the cooling process. However, with ground state cooling, if the spin state is |↑↑i, the

sideband cannot flip spin down and subtract a quanta from motion, so the |↑↑i state is a “dark”

state for both the cooling and sideband coupling.

The second part of dissipation is a combination of microwave and optical pumping, to pump

from the remaining symmetric state |↑↑i to the final targeted state |Si. This process is tricky
74

| ↑↑ 〉|2⟩

| ↓↓ 〉|0⟩

|T〉|1⟩

|T〉|0⟩

| ↑↑ 〉|1⟩

| ↑↑ 〉|0⟩

Figure 6.1: Sideband interactions together with sympathetic cooling. The former is expressed by

blue double arrows and the latter green wavy arrows. Other than the |↑↑, n = 0i state, all other spin

states can be driven by the sideband interaction, which produces spin flips and motional excitation

simultaneously. However, with sympathetic cooling, the evolution would lead to pumping to the

dark state of the sideband, the |↑↑, n = 0i state. This state is dark since the sideband could not

flip spin down and subtract motional quanta at the same time.
75

because any direct optical pumping on the spin level will result in destruction of the target state.

To prevent that, we use an auxiliary state |ai, which the optical pumping acts on with rate γ, and a

microwave carrier drive Hc = Ωc(|↑i1ha|+|↑i2ha|)+H.C., from spin up to the auxiliary for both ion

1 and 2. We would like to have state-selected drive only on the |↑↑i state while keeping the |Si state

untouched. This is accomplished by a side effect of the continuous Raman sideband interaction

mentioned above. The microwave drives transitions |↑↑i ↔ √

1

2

(|↑ ai + |a ↑i) ↔ |aai. These spin

states at the ground state of motion are dark to the sideband excitation; therefore the microwave

transitions occurs without being perturbed. In this case, repumping the |ai state returns the ions

back to the qubit manifold, with a probability of creating the |↑↓i or |↓↑i states, which contain the

targeted |Si state and the |Ti state. Note that the |Ti state is dissipatively pumped to the |↑↑i

state. Considering the microwave transition |Si ↔ √

1

2

(|↓ ai−|a ↓i), the |Si state is unperturbed by

the sideband excitation while the √

1

2

(|↓ ai−|a ↓i) state is a superposition of two dressed eigenstates

resulting from the sideband interaction, with eigenenergy of ±Ωs in the interaction frame of the

strong sideband. The eigenstates are 1

2

((|↓ ai − |a ↓i)|n = 0i ± (|↑ ai − |a ↑i)|n = 1i), respectively.

Thus if the microwave carrier Rabi rate Ωc is weak enough, i.e. Ωc  Ωs, the transition from |Si

is out of resonance, while the |↑↑i state is driven to auxiliary state resonantly and then pumped to

the |Si state. Thus we have all the ingredients depicted in Fig. 6.2 b).

6.2 Experimental setup and results

We trapped a 9Be+ −24 Mg+ −24 Mg+ −9 Be+ ion chain in the linear trap described above.

For the 9Be+ ion, we label the 2s

2S1/2

internal states |F = 1, mF = 1i ≡ |↑i, |2, 2i ≡ |↓i and

|2, 1i ≡ |ai. With such a choice of spin definitions, we can conveniently apply the red-Doppler

“blue” laser beam as in Fig. 3.4 mentioned above to repump the |ai state back to the qubit manifold,

with laser setup depicted in Fig. 6.2, and we set the strength of depletion for the |ai state with 1/e

repump time to be 88 μs. The drawback of this selection is the unwanted depumping on the |2, 2i

states if the polarization of the repump beam is not purely σ

+, and the unwanted transition is ∼140

MHz red detuned from the laser frequency [87]. One way to get around this issue is interchanging
76

the |↑i and |ai, and use the red-Doppler “red” beam Sec. 3.2 as repumper instead, however in our

case we did not have Raman laser frequency and microwave coil setup to drive transitions between

the |2, 2i and |2, 1i states, at the time of our experiment. For other choices of three ground state

levels as the spin and auxiliary levels, however, one can use the field insensitive transition between

|2, 1i and |1, 0i or |2, 0i and |1, 1i by selecting relevant magnetic field and have longer coherence

time. However that induces difficulty on the repumping. For example define the spins as |2, 1i and

|1, 0i states and auxiliary state to be the |2, 0i state. The repumping process from the |2, 0i state

would also have a probability to populate the unwanted |2, 2i and |1, 1i states, which would require

extra repumping laser beams to pump these states back to the manifold.

We measure the axial motional mode frequency to be 2π×{2.0, 4.1, 5.5, 5.8}MHz, labeled

as mode ν{1−4}respectively. The related mode amplitudes, Lamb-Dicke parameters [22] are in the

Table 6.1. We apply our scheme to the stretch mode ν3 to couple the 9Be ions. This mode is

chosen because it has the highest Lamb-Dicke parameters for the 9Be+ ions for better efficiency

of Raman laser coupling, while having the lowest Lamb-Dicke parameters for the 9Be+ ions for

low recoil during cooling, and it has relatively low ambient heating rate of the axial modes, ∼0.06

quantum/ms.

mode frequency

/(2πMHz)

amplitudes λ(

9Be+) λ(24Mg+)

1 2.0 {0.322,0.629,0.629.0322} 0.155 0.207

2 4.1 {-0.466,-0.532,0.532,0.466} 0.155 0.121

3 5.5 {0.629,-0.322,-0.322,0.629} 0.180 0.063

4 5.8 {0.532,-0.466,0.466,-0.532} 0.149 0.089

Table 6.1: list of parameters for repumping, cooling and heating processes

6.2.1 Continuous application

For the Raman sideband, we use a pair of 313 nm laser beams, set up as Sec. 3.2. We tune the

sideband laser frequency difference to be f0+ν3, where f0 is the qubit resonant frequency 2π×1.018

GHz. These two lasers are set to be ∼270 GHz red detuned from the 2s

2S1/2

to 2p

2P1/2

transition.
77

Figure 6.2: Energy Levels and Entanglement Preparation Scheme a) The internal energy levels (not

to scale) of 9Be+ are shown as solid black lines for the ground motional state and dashed lines for the

first excited motional state. The couplings needed to produce steady-state entanglement are shown

with blue double arrows for the strong sideband coupling with Rabi rate Ωs and sympathetic

cooling; the patterned and dashed red arrows for the weak microwave coupling with Rabi rate

Ωc, repumper, and spontaneous emission from the |ei state. Wavy arrows depict the dissipative

processes. b) Four spin states that span the |↑i, |↓i qubit manifold of the two 9Be+ ions are shown

as horizontal lines. Transfer processes that are accomplished by the sideband drive and sympathetic

cooling are shown as blue arrows, while processes that occur by coupling the |↑i state to the auxiliary

|ai state followed by excitation with the repumper and decay by spontaneous emission are shown

as dashed red arrows. Processes shown as thin lines are shifted out of resonance due to the strong

sideband coupling, leading to accumulation of population in the maximally entangled state |Si in

steady state.
78

More details of Raman sideband is described in Chapter 4. We measure the sideband Rabi rate to

be Ωs = 2π ×7.8(1)kHz, by scanning the laser pulse duration of the sideband transition, and fit the

flopping curve to a cosine function. To control the sign between the two ions in Hs, since mode ν3

has same sign of the mode amplitudes between the two 9Be+ ions, we need to set the laser phases

on the ions to be also the same. For this purpose this we set the ion spacing so that the laser

phase difference is integer m of 2π. The laser phase difference can be calculated with ∆kz, where

∆k ≈

2π

313nm

√

2 as the difference k vector along the trap axis, with laser beams incident 45◦ along

the trap axis, but 90◦ between each other. For our confinement and for ν3, z ≈ 11μm, and m is

near 50. We optimized the ion spacing by maximizing the creation of the target state. Better ways

of doing it would be to apply the Zeno dynamics to compare the laser phase with the microwave

uniform phase across the ions as described in Chapter 7, and if possible the most sensitive way is

to apply the σz two ion phase gate [52].

We implement the microwave transition by using a dipolar antenna described in Sec. 3.2, and

we set the power of the microwave input such that the carrier Rabi rate is Ωc = 2π × 0.543(6) kHz,

which is determined by scanning the microwave pulse duration and fitting the curve to sinusoidal

function. This gives Ωs/Ωc ∼14.

We continuously laser-cool the 24Mg+ ions to sympathetically cool the ion chain. We apply

a pair of red sideband Raman lasers beams, and a repump beam simultaneously, setup as in Fig.

3.3. These lasers, with wavelength approximately at 280 nm and total laser power less than a few

mW, interact negligibly with the spins of the 9Be+ ions. With the 24Mg sideband Rabi rate Ωs,Mg

and repump rate γMg, controlled by laser power. We apply the sideband and repump laser beams

simultaneously for continuous cooling. We set sideband laser beams resonant with the red sideband

of mode 3, with Ωs,Mg ≈ 2π × 11.9kHz and γMg ≈ 2π × 625 kHz, measured by observing the 1/e

repumping time of 1.6μs, giving a 1/e cooling time to be 203 μs, and cool to  ̄n = 0.11(1). We also

observe off resonant cooling on modes 1, 2 and 4, with 1/e cooling times to be 1300 μs, 294 μs and

181 μs, with steady state  ̄n to be approximately 2.9, 0.95 and 0.12, respetively. We also tried using

EIT cooling [78] for continuous sympathetic cooling, however we didn’t achieve as good fidelity
79

for the |Si state. One speculation is that due to the strong sideband couplings on both 9Be+ and

24Mg+ ions during the EIT cooling, the internal spins between 9Be+ and 24Mg+ ions is coupled

via exchange of motional quanta, thus adds extra depumping channel for the targeted state. Also

the EIT cooling is less controllable in terms of making the sideband coupling small but repumping

to be strong, compared with the Raman sideband cooling case, since the same beam is providing

both repumping and sideband coupling.

We apply all the interactions simultaneously and after about 6 ms, a steady state is obtained

and with 0.75(3) fidelity obtaining the targeted |Si state, as shown in Fig. 6.3. Definition of fidelity

is described in Sec. 1.1. This fidelity can be explained by numerical simulation, where we modeled

the system with experimental parameters and imperfections, as explained in Sec. 6.2.3. The fidelity

is obtained by averaging measurements between 6 and 12 ms. To measure spin population, we drive

the cycling transition |↓i ↔ 2p

2P3/2

|3, 3i for both 9Be+ ions for 250 μs, and by fitting the detected

photon count histogram to Poisson distributions, we can distinguish the number of ions populated

on the |↑i state, i.e., |↑↑i, |↓↓i and the combination of |Si and |Ti states, which contains one spin

up and one spin down. To further distinguish the latter two states, we need to insert microwave

π/2 pulses on the spin transition, with varied phases. This detection method is explained in Sec.

3.4.

6.2.2 Stepwise Application

For the continuous application, we apply strong Raman sidebands, weak microwave drive,

repumping laser beams, and sympathetic cooling simultaneously. We expect to achieve a steady

entangled state with error on the order of Ω2

c/Ω

2

s

, due to the finite ratio between Ωs and Ωc, κ

and γ, assuming the latter rates are on the same order of magnitude. Experimentally, this case

also suffers from the spontaneous emission from the Raman sideband lasers as a significant source

of error and the error rate will increase proportional to the duration of the laser application, as

detailed below in Sec. 6.2.3.

One way to mitigate these issues is to apply the interactions in a stepwise manner, as shown
80

Figure 6.3: Steady State Entanglement. The measured populations of the singlet, triplet, |↑↑i and

| ↓↓i states are shown as squares, crosses, circles, and triangles, respectively, as a function of the

duration that all the elements of the dissipative entanglement scheme are applied simultaneously.

The system reaches a steady state with a 0.75(3) population in the target singlet state after a few

milliseconds. The solid lines are the result of a simulation based on the experimental parameters.

The slow decrease in the singlet state fidelity at long times visible in the simulation is due to a

leak of the qubits to spin states outside the | ↑i, | ↓i, |ai manifold caused by spontaneous photon

scattering from the Raman laser beams that generate the sideband coupling. Strictly speaking this

depumping means that the state is only a quasi-steady state. For our parameters there is, however,

a clear separation of the preparation and depumping time scales, justifying the description as steady

state. Error bars represent standard deviations of each point.
81

in Fig. 6.4 a). In this case, we use a pulse sequence to create the |Si state, where each round of

pulses contains one coherent segment and one dissipative segment. The coherent segment turns on

the sideband and microwave, and the dissipative segment turns on the repumping and sympathetic

cooling. The following gives a qualitative explanation of how this works by a comparison with the

continuous application. In the continuous scheme, due to the stochastic property of the dissipative

elements, the coherent evolution due to the combination of sideband and microwave is interrupted

by repumping and cooling events. With finite Ωc/Ωs ratio, the |Si state is off-resonantly coupled

to √

1

2

(|↓ ai+|a ↓i) state, and the evolution, from the |Si state to superposition of the |Si state and

the √

1

2

(|↓ ai + |a ↓i) state, is interrupted by the dissipative repump event at random time stamp,

causing a finite leak rate from the |Si state. However, in the stepwise application, we time the

coherent segment to be √

2π

Ω2

s+Ω2

c

, so that we stop at the time where the evolution causes a return to

the |Si state, as shown in Fig. 6.4 b), and at the same time we try to optimize the transition from

the |↑↑i state by adjusting Ωc. The combination of these two processes minimizes the leak rate from

the |Si state and at the same time optimizes the creation rate of the |Si state. Therefore we can in

theory eliminate the requirement of Ωc/Ωs  1, increase Ωs and make the entire application much

shorter and less susceptible to spontaneous emisson error. In this case we lose some advantage of

the continuous pumping where we can obtain a true steady state, but we can control Ωc/Ωs to be

still not so large, and during the coherent evolution the qubits are not too far driven from the |Si

state, thus on average it is an approximate steady state. Also for |n 6= 0i, the oscillation period is

dependent on n, and for non perfect ground state cooling this will give raise to extra errors. The

true steady state would still be the most insensitive to laser power fluctuation and pulse duration

imprecisions.

In the stepwise application, we set Ωs = 2π × 8.4(1) kHz and Ωc = 2π × 1.24(6) kHz, and we

turn on the coherent subpulse with both sideband and microwave carrier for 106.5 μs, matching

approximately with 2π/p

Ω2

s + Ω2c

. For the dissipative subpulse, we set the repumping beam 1/e

time to be approximately 3μs and turn on for 6 μs; and then we turn on sympathetic cooling for

the axial modes for approximately 100 μs. In each cooling cycle, we cool each mode of motion by
82

microwave

repump cooling

Coherent 

duration

sideband

× 

a)

— singlet

Ω

Ω

~7

Coherent duration 

(arbitrary unit)

b)

Figure 6.4: Stepwise application of the dissipative entanglement scheme. a) We show the pulse

sequence for one step. In this case, we set the repump and cooling to be full power, and we turn

them on with enough duration so that they act as a complete pumping step. These pulses remain

short compared with the coherent pulse, thus not introducing too much gap time in between the

coherent evolution. b) We assume initial state to be the target |Si state, and we show a simulation

with the sideband and microwave drives are applied simultaneously. In this case, with Ωs/Ωc ≈ 7,

similar to the parameter in the stepwise experiment. We can see the microwave flops part of the

|Si state out to the auxiliary level but the population can be fully revived at certain durations, as

shown by an arrow in the figure. We set the duration of the coherent pulse in a) to be at one of

those durations, so we eliminate the leakage out from the |Si state.
83

applying a red sideband π pulse followed by a repump beam to remove one motional quanta. The

cooling cycles cools mode 1 first, which has the highest heating rate on the order of one quanta per

ms; then we cool mode 2 and 4, and we cool mode 3 last to minimize heating on mode 3 between

the cooling and the subsequent application of the coherent subpulses. We repeat the above cycle

and we observe creation of steady state at about 30 steps. Note that in this case, the duration

for the sideband laser at 30 step is approximately 3 ms, which is half of the continuous case. We

measure the population of the |Si state, and we measure |Si state fidelity 0.89(2).

6.2.3 Numerical Simulation

With a numerical model, we simulation the scheme, including various experimental imperfections, such as (1) beam pointing fluctuations, which induces Rabi rate imbalance between the

two ions (2) spontaneous photon scattering from the Raman laser beams which causes couplings

of sideband laser to upper levels, which induces random spin flip and pumping to states out of the

manifold of {|↑i, |↓i and |ai} (3) imperfect cooling, and ambient heating process, (4) off resonant

coupling to other axial motional modes.

We assume a Markovian process of the system-environment coupling, and we use the Lindblad

master equation for modeling [134, 135]

ρ ̇ = −i[Hcoh, ρ] +X

j

(LjρL†

j −

1

2

(L

†

jLjρ + ρL†

jLj )), (6.1)

where Hcoh is the Hamiltonian for the coherent interactions, as detailed below; Lj are different

dissipative components for the process; and ρ is the system density matrix. Here we model ρ

including the |↑i, |↓i, and |ai for each 9Be+ ion. Additionally, two more 9Be+ levels |oi ≡ |F =

1, mF = 0i and |ti ≡ |F = 2, mF = 0i are included, to capture the effect of spontaneous emission

that pumps the system out of the desired manifold. For the motional levels, we include the first

7 motional levels for mode 3, the motional mode for Raman sidebands; and we include the first 3

motional levels for mode 4, where the Raman sideband has a small off resonance coupling.
84

Figure 6.5: Entanglement With Stepwise Scheme. The measured populations of the singlet, triplet,

|↑↑i and | ↓↓i states are shown as squares, crosses, circles, and triangles, respectively, as a function

of the number of applied steps. Each step has a duration of approximately 220 μs. The solid

lines are the result of a model as explained in the Methods section. Error bars represent standard

deviations of each point.
85

6.2.3.1 Dissipative Components

We assume the dissipative components are the same on both ions. We model the repumping

in the desired manifold by Lγj,a =

√γj,a|aih↑ |, where j denotes the |↑i, |↓i and |aistates. We

model the continuous sympathetic cooling on 24Mg+ ions to be Lκ =

√

κb, measured by observing

the 1/e cooling time for mode 3, which was 203 μs, determined by exponential fit of the  ̄n from

Doppler-cooled value of  ̄n ≈ 2.5 to the sideband coupling steady state  ̄n = 0.11(1). And we account

for a heating process that limits the cooling represented by Lκh =

√

κhb

†

, where κh = κ

n ̄

1+ ̄n

, with

experimentally measured  ̄n from sideband cooling. The parameter κ and κh include the ambient

motional heating rate of 0.06 quantum/ms. We similarly define κ4 and κ4,h for mode 4, with

annihilation operator c. The parameter κ4 and κ4,h include the ambient motional heating rate

of mode 4 of 0.0085 quantum/ms. For the stepwise case, we model the sympathetic cooling by

simulating the sideband π pulse with Hamiltonian HMg = Ωs,Mg(σ

+

1,Mg +σ

+

2,Mg)b+H.C., taking into

account the imperfection of driving a π sideband pulse of two 24Mg+ ions for motional population

|n ≥ 2i. This is followed by a repump pulse with Lindblad operator Lrep,Mg =

p

γMg|↑Mgih↓Mg|.

In this case, the κ and κh mostly reflected the ambient heating. The above parameters for the

continuous and stepwise application are in Table 6.1 and Table 6.2. For the stepwise case, we turn

up the laser powers for the repumping and cooling so that we apply the repumping and cooling in

much shorter pulses than the continuous case. The imperfection of motional mode cooling affects

the creation rate for the |Si state, reducing its steady state population, or giving an error by 0.02 for

the continuous application. The |Si state creation was also affected by beam pointing fluctuation,

detailed below, giving an error of 0.02 for the continuous case. For the step wise application, if we

eliminates the heating processes the error in the numerical simulation decreases by 0.04. This error

is a combination of the above effects and the motional Fock state dependence of the oscillation

from |Si to the √

1

2

(|↓ ai − |a ↓i) state.

We model the spontaneous photon scattering processes from Raman laser beams as random

spin flips from the state i to the state j with Lindblad operator Lj,i =

p

Γj,i|jihi|, where Γj,i
86

can be calculated using the Kramers-Heisenberg formula [46]. We experimentally measure Γj,i

by preparing at state i and observing the population build up at state j over time. Since the

spontaneous emission rate is on the order of 10−4Ωs, we can to a good approximation ignore multi

scattering events during the measurements, and use a linear fit to extract the spontaneous emission

rates for short durations. The parameters we used in the simulation are shown in Table 6.2. The

similarity of continuous and stepwise cases are due to the similar Raman sideband Rabi rates. For

the simulation, we found the error due the the spontaneous emission to be approximately 0.07 for

the continuous and 0.04 for the stepwise applications. To make the error smaller, one possibility

is to create the sideband coupling with near-field microwave [136] or static field gradients [86], but

the require field strength may be difficult to achieve for this scheme.

property values property values

Γ↓,↑/(2π), Γ↑,↓/(2π) 1.32 Hz Γa,↑/(2π) 0.39 Hz

Γt,↑/(2π) 0.49 Hz Γo,↑/(2π) 0.28 Hz

Γa,↓/(2π) 0.78 Hz Γ↓,a/(2π) 0.39 Hz

Γt,a/(2π) 0.83 Hz Γo,a/(2π), Γa,o/(2π) 0.49 Hz

Γ↑,t/(2π) 0.29 Hz Γa,t/(2π) 0.83 Hz

Γ↑,o/(2π) 0.49 Hz

Table 6.2: list of parameters for spontaneous photon scattering processes from Raman laser beams;

we use the same numbers for simulation in both the continuous and stepwise cases.

6.2.3.2 Coherent Components

We model the coherent Hamiltonian as

Hcoh = H0

s + Hc + H4, (6.2)

Hs = Ωs((1 −

r

2

)σ

+

1 + (1 + r

2

)σ

+

2

)b

† + H.C., (6.3)

Hc = Ωc(|↑i1ha| + |↑i2ha|) + H.C. (6.4)

H4 = Ωs

η4

η3

ζ4

ζ3

(|↑i1ha| − |↑i2ha|)c

†

e

−iδt + H.C. (6.5)
87

where H4 denotes the off resonant coupling to mode 4. In the following we will describe and define

components of the above equations.

In Eq. 6.3, r is a parameter to represent Rabi rate imbalance between ions, due to the finite

Gaussian beam waist and beam pointing fluctuations. To determine the r value induced by each

of the Raman sideband beams, we perform a Stark shift experiment on each of the beams. We

apply two π/2 microwave pulses sandwiched with one of the off resonant Raman laser beam, and

observe the decay of sinusoidal flopping of spin populations due to a combination of laser intensity

fluctuation on the ion and spontaneous emission. Laser beam intensity fluctuation can be caused

by the overall laser power fluctuations and beam pointing fluctuations. To extract each component,

we perform the same measurements on a single 9Be+ ions and a 9Be+ −24 Mg+ −24 Mg+ −9 Be+

chain and compare the difference, since the single ion experiment will be more sensitive to power

fluctuations and the ion chain experiment will be more sensitive to beam position fluctuations. We

model the decay spin flopping for one and two 9Be+ ion cases as

P1 =

1

2

+

1

2

e

−ηtRe[heiΩ0t− 1

4

δ

2

1

t

2

e

−2ξ

2/w2

0 i

ξ∼e

−ξ2/δ2

2

], (6.6)

P2 =

1

2

+

1

2

e

−ηtRe[heiΩ0t− 1

4

δ

2

1

t

2

e

−2(ξ−x0)

2/w2

0 i

ξ∼e

−ξ2/δ2

2

] (6.7)

respectively, where P1 is the probability of one ion at spin up after the sequence of the Ramsey experiment, and P2 is the average value of P1 on each 9Be+ ion in the 9Be+ −24 Mg+ −24 Mg+ −9 Be+

chain; η represents the decay due to spontaneous emission, Ω0 is the free flopping rate for the

Stark experiment, δ1 represents the overall laser intensity fluctuation assuming a Gaussian noise

(Ω ∼ e

−(Ω−Ω0)2/δ2

), w0 is the beam waist (intensity profile ∼ e

−2x

2/w2

0 ), x0 is the position of ion

away from center, δ2 represents the beam pointing position fluctuation assuming Gaussian noise,

and the notation hf(x)ix∼g(x) means the average of function f(x) over a distribution of parameter

x over function g(x). The expression r = 2√

2x0δ2 relates position fluctuation to r. We fit the data

of the CO and 90 Raman beams, and we extract a list of relevant values, as shown in Table. 6.3.

We determine the final root-mean-square of r values to put in Eq. 6.3 as r90√

+rCO

2

= 0.014. In the

full simulation, we compare the steady state fidelity with and without the effect from r6= 0, and
88

we observe an error change by 0.02 for the continuous and 0.01 for the stepwise application of the

scheme. In addition to our thoughts in the supplementary information in [79], we found a good way

to reduce the beam pointing fluctuation by applying the photonic crystal structure UV wavelength

fibers [7], which converts beam pointing noise into amplitude noise, and combines this with laser

power stabilization using a detector after the fiber.

property CO beam 90 beam

x0/ω 0.11 0.07

δ1/Ω0 0.02 0.01

δ2/ω 0.04 0.04

r 0.012 0.008

Table 6.3: fit values from the Stark experiment decaying flopping

In Eq. 6.5, δ ≈ 2π × 250kHz is the mode frequency difference between modes 3 and 4, and

{η3 = 0.180, ζ3 = 0.629} and {η4 = 0.155, ζ4 = 0.532} are the Lamb-Dicke parameters and mode

amplitude for 9Be+ ion of modes 3 and 4, respectively. Note that since the mode amplitudes for

modes 3 and 4 are opposite on the 9Be+ ions, the signs between the ion operators are opposite

between Hs and H4. Thus for H4, the |Si state is no longer a dark state, and is coupled to

|↑↑i|n4 = 1i, with motional excitation in mode 4, which is then dissipated by the sympathetic

cooling for this mode during the scheme. We find this effect creates a drop of fidelity in the

simulations to be 0.006 and 0.016 for the continuous and stepwise applications, respectively. The

reason why this effect is more important for the stepwise case is not completely understood. One

possible explanation is that in the stepwise case, the duration for the coherent pulse makes the

transition from the |Si state more favorable than the continuous case.
Chapter 7

Entanglement generation by confining dynamics in a subspace

7.1 Introduction

Entangled states are a crucial resource in quantum-based technologies such as computers and

communication systems [27]. With current errors in entanglement generation, it is important to

explore new methods. Here we restrict the number of energy states available to global rotations

with applied laser fields, in an approach that has become known as quantum Zeno dynamics [54–

56]. Under suitable microwave driving fields, a separable state evolves into an entangled state.

With two trapped ions, we obtain Bell state fidelities up to 0.992(2); with three ions, a W-state

[137] fidelity of 0.905(5). The procedure is relatively insensitive to certain imperfections such as

fluctuations in laser intensity and frequency, and motional mode frequency. Since the function of

the laser beams here is to provide strong restriction of the microwave driven dynamics, the exact

laser parameters are not critical. This technique can be generalized to other systems.

The quantum Zeno effect has usually been referred to as the inhibition of quantum dynamics

due to frequent measurements [55, 138]. This basic idea can be generalized to restrict the dynamics

to within an unmeasured subspace of the overall system. Recent proposals [139–147] show that

the subspace isolation can also be created by coupling the remainder of the system to auxiliary

levels. Dynamics within the isolated subspace1can be implemented even though the system may

undergo unitary evolution, without measurement. For sufficiently large perturbations, the system

dynamics become confined to the unperturbed subspace. For measurement-based Zeno dynamics,

1

also known as quantum Zeno dynamics
90

there will be a finite probability of escaping from the desired subspace disspatively. However, if

the restriction is brought about by coherent interactions, there will be no dissipation, and thus the

state amplitudes that leak from the restricted subspace remain coherent and can in principle be

recovered with unitary operations.

Dynamics in the restricted subspace have recently been demonstrated with atoms in BoseEinstein condensates [148], atoms in a cavity [149], atoms with Rydberg-Rydberg interactions [150]

and photons in superconducting cavities [151]. In this work, we apply coherent laser fields to confine

the quantum evolution of effective spin- 1

2

particles realized with trapped ions between an initial

product state and an entangled state of the internal states of the ions. This represents a novel

technique to create entanglement, which can have important applications to quantum computing

and quantum communication systems [27, 152]. We also demonstrate an advantage of the coherent

subspace creation where, with application of composite pulse sequence, amplitudes escaped from

the desired subspace can be brought back, thus improving the production of entangled state.

In our implementation, the dynamics within the confined subspace is driven by single-qubit

unitary rotations operations, which can achieve high fidelity as demonstrated experimentally in

[57–59, 153, 154]. However, these operations alone cannot directly generate entanglement between

multiple qubits. For example, when applying a global rotation to an initial state with N spin- 1

2

systems in the spin up state |↑i, each spin rotates independently and remains separable. In the Dicke

states basis [155], the spins evolve in the symmetric angular momentum manifold |J = N/2, mJ i

[156], where J is the total angular momentum quantum number and mJ is the projection of angular

momentum along a quantization axis. All individual |J, mJ i states are entangled states except the

maximal spin states, |↑↑ ... ↑i = |J, Ji and |↓↓ ... ↓i = |J, −Ji. To generate entanglement, one

can perturb specific |J, mJ i states in the manifold to alter the dynamics. A simple case is to

apply a perturbation to shift the |J, J − 2i state out of resonance, as depicted in Fig. 7.1. In

this case, the dynamics are restricted within the |J, Ji and |J, J − 1i states. Thus starting from

|J, Ji, with one effective π-pulse, the entangled |J, J − 1i state [137, 157], is prepared. For two

and three spins, these states are the triplet Bell state |Ti = √

1

2

(|↑↓i + |↓↑i); and W-state [137]
91

|Wi = √

1

3

(|↑↑↓i + |↑↓↑i + |↓↑↑i), respectively.

| ↑↑〉|n = 0〉

|T〉|n = 0〉

| ↓↓〉 |n = 0〉

|S〉|n = 1〉

| ↑↑〉|n = 2〉

Ωd

ΩS

| ↑↑↑〉|n = 0〉

|W〉|n = 0〉

|W〉 |n = 0〉

Ωd

ΩS

|Wc〉|n = 1〉

(a) (b)

Figure 7.1: Schematics for confined dynamics, for the cases of (a) two ions, and (b) three ions.

The thin black double arrows depict the relatively weak microwave coupling, and the thick blue

double arrow depicts the strong laser sideband coupling. With |↑ ... ↑i states initially populated

(red dots), the microwaves drive the state down the symmetric manifold, the states on the left,

where |Wi =

|↑↓↓i+|↓↑↓i+|↓↓↑i

√

3

in (b). However, the sideband excitations perturb the lower states,

coupling them to other states, denoted by |Si and |Wci =

e

i2π/3

|↑↑↓i+|↑↓↑i+e−i2π/3|↓↑↑i

√

3

. Thus the

microwave induced dynamics are confined between the two highest energy states on the symmetric

manifold. For (b) we only show the states that are relevant to the dynamics.

7.2 Two ion implementation

We experimentally demonstrate this scheme with trapped 9Be+ ions aligned along the axis

of a linear Paul trap. At an external magnetic-field of 11.946 mT, applied approximately 45◦to the

axis, as shown in Fig. 3.3, the energy splitting ω0 ≈ 2π ×1.2075 GHz of the 2S1/2 hyperfine ground

states |F = 2, mF = 0i ≡ |↓i and |F = 1, mF = 1i ≡ |↑i, is first-order insensitive to magnetic field
92

fluctuations [73]. The effective rotation in the restricted subspace is produced by a uniform resonant

microwave field, while the perturbations are provided by a laser-induced coupling between ions via

a shared motional mode.

With two ions in the trap, the center-of-mass “COM” and the “stretch” axial normal modes of

motions has frequencies {ωc, ωs} ≈ 2π × {3.58, 6.20} MHz, as described in Sec. 4.1. The microwave

field couples the unperturbed Dicke states with the Hamiltonian

Hd = ~Ωd

X

i=1,2

σ

x

i =

√

2~Ωd(|↓↓ihT| + |Tih↑↑|) + H.C., (7.1)

where ~ is the reduced Planck constant, Ωd is the single-ion Rabi rate, and σ

x

i

is the Pauli operator

on the i

th ion. If the spins are initially in a product state, evolution under this Hamiltonian will

not generate entanglement. The perturbation from laser couplings effectively remove the terms

|↓↓ihT| + H.C. from Hd, thus an effective dynamics is implemented between |↑↑i and |Ti states

with Rabi rate √

2Ωd. The effective π time for the |↑↑i ↔ |Ti transition is thus tπ = π/(2√2Ωd).

To generate the desired dynamics, we induce the sideband interaction by applying a pair of

laser beams (Be CO and BE 90) such that the difference of their momentum vectors ∆k aligned

along the trap axis and their frequency difference set to ω0 + ωs + δ, as shown in Fig. 3.3. Setting

the equilibrium position of ion 1 to be the origin of the axis, we denote the equilibrium position

of ion 2 to be z. The lasers induce a near-resonant “blue sideband” coupling described in the

interaction frame by

Hs = ~Ωs(σ

−

1 − e

i|∆k|zσ

−

2

)ae−iδt+iφ + H.C., (7.2)

where e

i|∆k|x

is the differential optical laser phase. The minus sign between the σ operators results

from the opposite motional mode amplitudes of the ions in the stretch mode. The common phase

of the laser beams at the origin for t = 0 is denoted as φ, and will become important for the

scheme based on composite pulse sequence. To obtain the coupling of Eq. 7.3, we adjust the axial

confinement such that the ion-spacing x is near M 2π

|∆k|

, where M is an integer number here with

M = 18. Since φ is not important in the single pulse application, here we set it to zero. With this
93

setting, we have

Hs = ~Ωs(σ

−

1 − σ

−

2

)ae−iδt + H.C., (7.3)

where Ωs is the Rabi rate, a is the annihilation operator of the stretch mode, and σ

− = |↓ih↑|.

This interaction does not couple to the |T, ni state. However, it couples the states |↓↓i|ni ↔

|Si|n + 1i ↔ |↑↑i|n + 2i, where |ni denotes a stretch mode Fock state, and |Si = √

1

2

(|↑↓i − |↓↑i).

The energies of the coupled states are shifted by approximately ±~Ωs and 4~Ωs, if the δ is set to

q

7

3Ωs, as shown in Sec. 4.2.5. In addition, Hs couples |↑↑, ni to |S, n − 1i for n > 0, therefore

we want to initialize the stretch mode in the ground state. Thus for n=0, if the shifts are much

larger than ~Ωd, the system evolves as an effective two-level system between |↑↑i|0i and |Ti|0i

under the combined influence of Hs and Hd. However, |↑↑, n > 0i states will also couple to the

laser interaction and the desired subspace will not be isolated, therefore, high fidelity ground state

motional preparation is crucial.

To initialize the spin and motional states, we first cooled both axial modes of the ions to near

the ground state, achieving average motional occupation of  ̄n < {0.01, 0.006} for the COM and

stretch mode after 30 repeatitions of Raman sideband cooling [39]. Optical pumping prepares both

ions in the |F = 2, mF = 2i atomic state. We then apply a global composite microwave π-pulse to

initialize to the |↑↑i state [95]. We set the laser and microwave intensities such that Ωs = 2π ×17.7

kHz and Ωd = 2π × 1.52 kHz; and set δ = 2π × 27.1 kHz. We simultaneously apply microwaves

and laser beams for a variable duration t, followed by detection pulses. We observe coherent Rabi

flopping between the |↑↑i and |Ti states as shown in Fig. 7.2, where the populations in the |↑↑i,

|Ti, and |↓↓i are determined as described in Sec. 3.4.

We observe a maximal fidelity of the |Ti state of 0.980(2) at a duration of tπ =116 μs,2

matching to the theoretical prediction of tπ = π/(2√

2Ωd). Several effects lead to imperfect

preparation of the |Ti state in our experiment. (1) Spontaneous emission from laser-excited electronic states, which projects the spin state to product states. Through simulation we estimate the

2

In the experiment, we turn on/off the laser beams approximately 0.4 μs before/after the microwave field. Here

we only refer to durations when the laser beams and microwave field are applied simultaneously.
94

0 50 100 150

0

0.2

0.4

0.6

0.8

1

duration t / μs

populations

P0

P1

P2

P|T〉

Figure 7.2: Two ion population evolution for confined dynamics with microwave and sideband

excitations applied simultaneously. Population mainly evolves between the |↑↑i and the |Ti state,

while other states have very small populations. The pink diamonds, blue triangles, red squares

and green circles represent the measured populations of states with all spins down, one spin up,

two spins up and the |Ti state, respectively. The difference between P1 and P|Tiis due to the

populations in the |Si state. The solid lines shows the results of the numerical simulation taking into

account known experimental imperfections, with the same coloring convention as for the measured

populations. The black dashed line shows unit population. The error bars for P|Tistate are

indicated by the size of the symbols. All error bars are from data resampling of a maximum

likelihood analysis.

reduction of the maximal |Ti state caused by this effect is approximately 0.008. This infidelity can

be reduced by tuning the laser frequencies further from the excited states at the cost of reduced

coupling strength, which may be compensated using increased laser power [46]. Another potential

possibility would be to use a magnetic-field gradient to directly couple the spins to the motion

instead of using lasers [86, 136]. (2) Imperfect ground state cooling. The |↑↑, n 6= 0i state is not a

dark state of the laser interaction and will be coupled to other states. We estimate this infidelity to

be less than 0.006. In simulations of thermal states with mean motional occupation  ̄n ∈ {0, 0.05},

the infidelity of the scheme is approximately equal to  ̄n. This infidelity could be reduced if the lasers

were set to a frequency difference near ω0 + 2ωs ± δ

0

such that they only couple states separated by

two motional quanta. In this case the scheme will work as long as the motional mode is prepared in
95

either the |n = 0i or |n = 1i state, however, the corresponding Rabi rate will be smaller by a factor

of the Lamb-Dicke parameter. Ambient heating of the motional mode can cause extra infidelities;

however, we apply the sideband near the resonance of the stretch mode, which is insensitive to

uniform electric field noise and therefore has a low heating rate, leading to a negligible infidelity.

(3) Imperfect initialization of the |↑↑i spin state. This can be due to imperfect optical pumping

to the initial |2, 2i states or state-preparation π-pulses. We estimate this infidelity to be less than

0.002. (4) Not satisfying the condition Ωs  Ωd. In this case, off-resonant excitations from the

|T, 0i state to the {|↓↓, n = 0i, |S, n = 1i and |↑↑, n = 2i} states will play a role. In simulations with

Ωs/Ωd ≈ 12, we observe a reduction in the maximal value of the |Ti state population of 0.0096 due

to this effect. This infidelity can be reduced by making the ratio larger, which requires reducing the

Rabi rate of the microwaves or increasing the laser power. In both cases this will lead to increased

infidelity due to spontaneous emission. Thus, for a given spontaneous emission rate, there is an

optimal Ωs/Ωd, which is around 12 for our conditions. In Fig. 7.2, we plot the numerical simulation

as a solid line in Fig. 7.2 and find good agreement with the data. Detailed theoretical modeling

and analysis of the infidelity sources have been done by F. Reiter and A. S. Sørensen [158].

7.3 Two ion implementation with a composite pulse sequence

The evolution is ideally unitary and thus the state amplitudes outside the intended subspace

can in principle be recovered. To demonstrate this, we apply a suitable composite pulse pair which

allows us to return the population of the undesired states |↓↓, n = 0i, |S, n = 1i and |↑↑, n = 2i

into the isolated subspace and thereby increase the population of |Ti. The idea of the composite

pulse sequence is originated from an observation that the optimal confinement of Hilbert space

can be achieved for both δ = ±

q

7

3Ωs and any value of φ. Thus switching the sign of δ and/or φ

during the evolution would not heavily affect the dynamics within the confined subspace, however

the dynamics involving the undesired states are significantly affected. To construct the composite

pulse, for simplicity we only consider a two-segment sequence, where the signs of δ and/or φ are

flipped between the two segments. Numerically we find that it works best to switch the sign of
96

both δ and φ at 1

3

of the π time for the |↑↑i → |Ti transition [158]. Thus we split the laser pulse

into two segments of duration t1 and t2, flipping the sign of δ and changing the laser phase by π at

t1. We set Ωs = 2π × 17.4 kHz, Ωd = 2π × 2.55 kHz, δ1 = −δ2 = 2π × 26.8 kHz, t1 = 25.4 μs, and

t2 = 47.3 μs to obtain a |Ti state population of 0.992(2). Fig. 7.3 shows the population evolution

during the composite pulse sequence, in agreement with numerical simulations (solid lines). Higher

fidelity is achieved despite weaker isolation for Ωs/Ωd ≈ 7. Reducing Ωs/Ωd has the beneficial effect

of reducing spontaneous emission infidelity to 0.0048, with negligible infidelity from the finite ratio

of Ωs/Ωd. Other infidelities are less than 0.002 from state preparation and detection, and less than

0.005 from initial cooling.

0 20 40 60 80 100

0

0.2

0.4

0.6

0.8

1

duration t / μs

populations

P0

P1

P2

P|T〉

Figure 7.3: State evolution for confined dynamics of two trapped ions using a composite pulse

sequence. Similar to Fig. 7.2, populations are mainly confined to the |↑↑i and |Ti states. The

coloring and labeling conventions are the same as Fig. 7.2. The laser phase and detuning are

flipped 25.4 μs after the start of the experiment. Note that the oscillations of |↓↓i are enhanced at

t > 25.4 μs; however the maximal population of |Ti state is increased compared to the single pulse

used for the data in Fig. 7.2. All error bars are from data resampling of a maximum likelihood

analysis.
97

7.4 Three ion implementation

We also demonstrate the confined dynamics on three 9Be+ ions. We tune the laser frequency

close to the center-of-mass (COM) mode sideband, which has equal mode amplitudes on each ion.

Similar to the case of two ions, we set the center ion equilibrium position to zero, and the outer

ions’ positions are ±z

0

. Thus the sideband interaction can be expressed as

H0

s = ~Ω

0

s

(e

i|∆k|z

0

σ

−

1 + σ

−

2 + e

−i|∆k|x

0

σ

−

3

)ae−iδ0

t + H.C., (7.4)

where δ

0

is the detuning from the sideband resonance. We adjust the inter-ion spacing x

0 near

M0 2π

|∆k|

, here with M0 = 15 + 1

3

. With this we obtain a sideband interaction

H0

s = ~Ω

0

s

(e

i2π/3σ

−

1 + σ

−

2 + e

−i2π/3σ

−

3

)ae−iδ0

t + H.C.. (7.5)

The microwave coupling can be expressed with the Hamiltonian

H0

d = ~Ωd

X

i=1,2,3

σ

x

i

=

√

3Ωd(|↑↑↑ihW| +

√

3|WihW| + |Wih↓↓↓|

− |WacihWac| − |WcihWc|) + H.C.,

(7.6)

where Ωd is the single-ion Rabi rate, and |Wi =

|↑↓↓i+|↓↑↓i+|↓↓↑i

√

3

; and the asymmetric “cyclic”

states |Wci =

e

i2π/3

|↑↑↓i+|↑↓↑i+e−i2π/3|↓↑↑i

√

3

, the “anti-cyclic” |Waci =

e−i2π/3|↑↑↓i+|↑↓↑i+e

i2π/3

|↓↑↑i

√

3

. The

definition of |Wci and |Waci are similar except that the individual spin directions are reversed.

Thus the sideband interaction shifts the energy of the |Wi state out of resonance, but does

not couple to the |Wi and |↑↑↑i|n = 0i states. Compared to Hs in the two ion case, the three

ion case is special because with detuning δ

0 = 0, H0

s

is sufficient to shift |W , n = 0i state out of

resonance, by coupling it to only the cyclic asymmetric state |Wci|1i. Thus with Ω0

s  Ωd, and

cooling of the axial modes to near the ground state, the weak microwave excitation only couples the

initial state to the |Wi state, while further coupling to |Wi state is off resonance. Similar to the

two-ion case, the maximum population of the |Wi state is obtained at the duration of π/(2√

3Ωd).
98

After ground state cooling and preparing the ions in the |↑↑↑i state we apply the sideband

interaction and resonant microwave radiation to couple the Dicke states. We set the sideband and

microwave Rabi rate Ω0

s = 2π×17.1 kHz and Ωd = 2π×1.22 kHz. We denote population for i spins

in the |↑i state as Pi, i= 0-3. We observe flopping between the |↑↑↑i and |Wi states, in agreement

with numerical simulations. We obtain the |Wi state fidelity to be 0.905(5) after a duration of

114.1 μs.

We determine that the infidelity of |Wi state has contributions of 0.016 from imperfect

isolation, 0.010 from spontaneous emission and less than 0.005 from state preparation. Since we

need to apply the scheme on the common mode for the three ions, this mode experiences significant

ambient heating rate, which also results in poorer ground state cooling. We estimate the heating

rate to be approximately 136 quanta/s, giving an infidelity of 0.011 and with initial  ̄n ≈ 0.02, giving

an infidelity of 0.018.

Also, due to the finite laser beam waist across the ions, the outer ions will experience different

Stark shifts from the center ion, in turn leading to slightly different microwave resonant frequencies.

We estimate the infidelity from this effect to be 0.023. This effect can be mitigated by use of laser

beams with larger beam waists, and/or reduced spacing between the ions. Combining all known

effects, simulation predicts a maximum population of |Wi state population of 0.915, approximately

agreed with the experimental result. Since the infidelity for three ions is dominated by other sources

than considered in Sec. 4, we expect the gain from using a composite pulse to be limited and we

do not investigate it here.

For more than three ions in a chain, simulations indicate that unwanted dark states appear

so that straightforward application of the sideband interaction does not yield an effective two-level

system between the first two Dicke states. However, by using a combination of multiple sideband

laser interactions and engineering the phases of the sideband couplings on each ion, the scheme may

be scaled up to isolate an effective an effective two-level system. Our scheme can be generalized to

other experimental platforms, for example superconducting qubits and atoms in a cavity, since the

interaction Hamiltonians are in the same form.
Chapter 8

Conclusion

In this thesis, we have mainly described two novel methods of generating entanglement, with

subspace engineering and including dissipation to the system. Both of these methods use laser

sideband interactions to induce strong energy shifts to reshape the system evolution determined by

the symmetry of the coupling across the ions, which provides certain robustness to fluctuations of

laser intensity, laser frequency and trapping frequency. Compared to the conventional entangling

quantum logic gates, this robustness may help to improve fidelities in the future. Since the worldrecord entangling gate fidelity with trapped ions using Raman lasers is currently 0.999(1) [125],

we can see that our implementations still have lower fidelities. The primary reason is that in our

schemes we need to have strong laser sidebands compared with the evolutions which introduces

more spontaneous emission infidelities. In other systems this problem can be suppressed. For

example consider the scheme implemented with two 171Yb+ ions, where very low spontaneous

emission can be achieved [159, 160]. Given the same trapping potential as in our experiment, the

spontaneous emission infidelity may be suppressed by roughly an order of magnitude using a high

intensity mode-locked pulsed laser operating in the weak pulse regime [159, 160]. In this case,

for the subspace engineering scheme, simulation predicts an infidelity of the |Ti state preparation

below 0.001 with weaker microwave drive. In this case the simple scheme without composite pulses

would be sufficient to achieve high fidelity, with robustness to fluctuations.

The subspace engineering described here relies on high-fidelity ground state cooling, which

is not an issue for the state-of-the-art Mølmer-Sørensen gate as long as the Lamb-Dicke limit is
100

satisfied [50]. Despite these drawbacks of the scheme, we still see that for the first demonstration

of the technique with subspace engineering we have already achieved fidelity up to 0.99. Future

schemes combining these techniques to more tasks of quantum information processing could be

very interesting, not limited to using Raman laser sidebands to induce entanglement or even not

limited to trapped ion systems. The idea of our scheme is more naturally extendable to systems

with strong couplings, which may be noisy but do not induce projective spin errors.

For the dissipative entanglement, we have introduced dissipation as part of the processes

of generating entanglement. Robustness is inherent for the dissipations we introduce purposely;

however there was still uncontrolled dissipation such as spontaneous emission, and at the time of

the experiment, the beam pointing fluctuations. The optimal case for generation of entangled state

by dissipation is that major dissipation sources are all accounted as part of the scheme. Indeed,

our scheme was originally proposed for atoms in optical cavity [132], where the photon loss from

cavity is used as part of the dissipative process as a fundamental loss mechanism for that system.

In our case, this photon loss process is replaced by sympathetic cooling - a controlled phonon loss

process.

Thanks for reading my thesis! If you find this thesis interesting or have questions or remarks,

please send me an email at yiheng.lin@colorado.edu or yihenglin.ustc@gmail.com.
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