Quantum Engineering of a Diamond Spin Qubit with Nanoelectromechanical Systems

Quantum Engineering of a Diamond Spin Qubit With Nanoelectromechanical Systems

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Citation Sohn, Young-Ik. 2018. Quantum Engineering of a Diamond Spin Qubit With Nanoelectromechanical Systems. Doctoral dissertation, Harvard University, Graduate School of Arts & Sciences.

Citable link http://nrs.harvard.edu/urn-3:HUL.InstRepos:42015089

Terms of Use This article was downloaded from Harvard University’s DASH repository, and is made available under the terms and conditions applicable to Other Posted Material, as set forth at http:// nrs.harvard.edu/urn-3:HUL.InstRepos:dash.current.terms-of- use#LAA Quantum Engineering of a Diamond Spin Qubit with Nanoelectromechanical Systems

a dissertation presented by Young-Ik Sohn to The Department of John A. Paulson School of Engineering and Applied Sciences

in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the subject of Applied Physics

Quantum Engineering of a Diamond Spin Qubit with Nanoelectromechanical Systems

Abstract

Quantum emitters are indispensable building blocks for quantum computers and networks.

By entangling multiple individual quantum systems, it is possible to make overall system exponentially more powerful. Quantum emitters play a key role in this regard, since they offer an optical interface between a flying qubit (photon) and a stationary qubit (spin) for a long distance. Among those, solid-state emitters are an appealing candidate for its scalability. Among many kinds, we study color centers in diamond: nitrogen-vacancy (NV) and silicon-vacancy

(SiV) centers. Being trapped atom in a solid, a color center provides both unique opportunities and challenges. Using dynamic interaction between phonons and spin qubits, it is possible to build an on-chip universal quantum bus. On the other hand, the host material causes inhomogeneous distribution of emitters by its material strain and exposes color centers to thermal lattice vibrations. In this work, we use nanoelectromechanical systems (NEMS) to address both issues. First, we make nanocantilevers with embedded NV centers and use its flexural motion for parametric coupling. Both electron spin resonance and spin-echo measurements are performed. As a result, we deduce the single-phonon coupling rate of approximately 1.8 Hz,

iii Thesis advisor: Professor Marko Lončar Young-Ik Sohn which is still many orders of magnitude smaller than the minimum requirement for a quantum node. Therefore, it is necessary to further scale down the device without deteriorating other parameters. In this context, we fabricate on-chip dynamic actuator that is compatible with cantilevers of small mode volume and high quality factor. We measure resonant frequencies of fundamental flexural modes on the order of tens of MHz, with mechanical quality factors on the order of thousands. Finally, we present electrostatically actuated diamond cantilever with implanted SiV centers. By deflecting beams, we control the electronic structure of SiV centers, which is revealed by taking optical spectra at different strain conditions. Furthermore, we probe the dynamics of the spin qubit while controlling strain. By applying strain on the order of 10−4 to SiV centers, we improve the spin coherence time by sixfold at 4K, until it is limited by a next dominant dephasing mechanism. We conclude with an outlook of phononic quantum nodes with SiV center.

iv Contents

0 Introduction 1

1 Fundamentals of the crystal strain and its effect on quantum emitters 10 1.1 Crystal strain in diamond ...... 11 1.2 Strain Hamiltonian of quantum emitters ...... 22 1.3 Application of the strain Hamiltonian ...... 27 1.4 Physics of the quantum nodes ...... 31

2 Phononic quantum node with NV center in cantilever 37 2.1 Motivation ...... 37 2.2 Requirements for strong spin-phonon coupling ...... 39 2.3 Device fabrication and experimental setup ...... 41 2.4 Strain Hamiltonian of the ground state triplet of NV center ...... 42 2.5 AC strain induced ESR broadening ...... 49 2.6 Temporal dynamics of the mechanically driven spin ...... 53 2.7 Conclusion and outlook for strong coupling ...... 57

3 Dynamic actuation of single-crystal diamond nanobeams 61 3.1 Background and motivation ...... 61 3.2 Device description ...... 62 3.3 Device characterization ...... 66 3.4 Discussion of results ...... 69 3.5 Additional capabilities: resonance tuning and parametric actuation ...... 70 3.6 Conclusion and outlook ...... 72

4 Controlling the coherence of a diamond spin qubit through strain engineering 74 4.1 Background and motivation ...... 74 4.2 Electronic structure of SiV ...... 81 4.3 Device description ...... 83 4.4 Effect of strain on electronic structure ...... 91 4.5 Controlling electron-phonon processes ...... 106 4.6 Spin dynamics with controlled strain environment ...... 114 4.7 Investigation of double-dip CPT signal ...... 120 4.8 Conclusion ...... 125

v Appendix A Technical informations 127 A.1 Chapter 2 ...... 127 A.2 Chapter 4 ...... 129

Appendix B SiV Hamiltonian 130 B.1 Matlab code for the calculation of SiV’s electronic structure ...... 130

References 165

vi Listing of figures

1.1 Schematic illustration of the stress tensor. Stress tensor in general needs nine components to fully describe the applied stress to infinitesimally small hypothetical box inside the material...... 15 1.2 (a) Object without strain. (b) Object with ε11. (c) Obejct with ε12...... 18 1.3 (a) NV center. Blue sphere is a nitrogen atom and black ones are carbons. (b) SiV center. Red sphere is a silicon atom and black ones are carbons...... 22 1.4 Illustration of tuning two emitters together spectrally with the strain gradient. Emis- sion spectra of the two emitters are initially different in a, and are tuned together by an appropriate strain gradient between the two emitters in b...... 30

2.1 (a) Representative scanning electron microscope (SEM) image of the angle-etched diamond cantilevers used. (b) Representative confocal microscope scan of a section of the cantilever showing fluoresecence from NV centers. (c) Driven response of the fundamental out-of-plane flexural mode (right inset) of the triangular cross- section (left inset) cantilevers studied in this work. For this particular device, we have w=580 nm, t=170 nm, and l=19 μm. The mode frequency is 937.2 kHz, and it has a Q-factor of 10,000. Measurements were taken in high vacuum (1e-5 torr) at room temperature. (d) Hyperfine structure of the ms = 0 to ms = +1 electron spin transition in the NV ground state indicating the three allowed microwave transitions. (e) AC strain induced broadening of the ms = 0 to ms = +1 hyperfine transitions near the clamp of the cantilever with gradually increasing mechanical amplitude. The mechanical mode is inertially driven at its resonance frequency with a piezo stack in all measurements. Open circles indicate measured data, and smoothed solid lines serve as a guide to the eye. Legend shows values of piezo drive power for each measurement. 0 dBm of drive power corresponds to an amplitude of 559 ± 2 nm at the tip of the cantilever...... 43 2.2 (a) Driven response of the cantilever at a piezo drive power of -12 dBm. The drive frequencies used for frequency dependent broadening measurements are indicated with numbers 1-5. (b) ESR spectra at the same location in the cantilever at mechanical drive frequencies 1-5 indicated in (a). (c) ESR spectra at the tip and clamp of the cantilever for -6 dBm drive power. Strain profile of the mechanical mode from an FEM simulation for the corresponding displacement amplitude is also shown. Open circles in each ESR spectrum are measured data, and smoothed lines serve as a guide to the eye ...... 52

vii 2.3 (a) Experimental pulse sequence for spin echo measurement of dispersive spin-cantilever interaction due to axial strain (b) Spin echo signal from NVs in the cantilever at a piezo drive power of 0 dBm (tip amplitude of 559±2 nm) for the mode at ωm = 923.4 kHz, showing two periods of the modulation due to axial strain coupling. The solid line is a fit to Eq. 2.17. Vertical error bars correspond to photon shot noise in the measurement. Inset shows schematic of dispersive interaction between the qubit and mechanical mode due to axial strain...... 55 2.4 Spin echo at the same location in the cantilever for varying piezo drive powers, offset along y-axis. One period of the signal is plotted in each case. Solid lines are fits to the zero order Bessel function form indicated in the text, and vertical error bars correspond to photon shot noise. The scale bar on the side is a guide for the y-axis indicating maximum (1.0) and minimum (0.3) possible population in ms = 0 as dictated by Eq. 2.17. The legend on right side indicates the piezo drive powers used for each measurement...... 56 2.5 Variation of driven spin-phonon coupling rate due to axial strain (G) with the calibrated displacement amplitude of the mechanical mode. The five data points correspond to the piezo drive powers used in plot (a) in increasing order. Vertical error bars correspond to the error in each G estimate from fitting to the spin echo function given by Eq. 2.17. Solid line is a linear fit, which yields dG/dx = 4.02 ± 0.40 kHz/nm...... 58

3.1 SEM images of (a) 4 μm long cantilever and (b) 7 μm doubly clamped beam. (c) Finite element method (FEM) simulations are used to calculate the force applied to suspended nanobeams with a given geometry and electrostatic environment. The color map indicates potential with respect to the right-hand Au electrode and the streamlines show the corresponding electric field. (d) Vertical force per unit length applied to such beams in the case of 20V of DC voltage is plotted as a function of beam width and distance above the electrode. Separation between electrodes is the sum of beam width and 50 nm margin on either side. Beam height is the distance between top surface of the beam and the electrode center in vertical axis. . . . . 63 3.2 (a) Schematic illustration of angled-etching nanofabrication approach used in this work:(i) Electron beam lithography mask is deposited, (ii) top-down reactive ion etching of diamond is performed, followed by the (iii) angled-etching step and (iv) mask removal. (v) New electron beam resist is spin coated, and (vi) electron beam lithography followed by (vii) metal evaporation and (viii) lift-off are used to define electrodes. (b) High magnification SEM image of 4 μm cantilever shows that good alignment can be achieved. (c) SEM image of device array sharing electrodes. . 64 3.3 Optical characterization setup...... 67

viii 3.4 Fundamental out-of-plane resonant response of devices shown in Fig. 3.1(a) and (b) are given in (a) and (b), respectively. Lorentzian frequency responses are shown at low driving power, and both beams start to enter nonlinear regime at higher driving power...... 68 3.5 (a) Tuning of mechanical resonance of doubly clamped beam using DC bias. With applying ±9V, frequency tuning range that can be achieved is approximately 260 linewidths. (b) Typical tongue shape of parametric instability was observed. . . 70

4.1 (a) Electronic level structure of the SiV showing the mean zero phonon line (ZPL) wavelength, frequency splittings between orbital branches in the ground state (GS) and excited state (ES) (Δgs and Δes respectively) at zero strain, and the four optical transitions A, B, C, and D. Also shown are single-phonon transitions in the GS and ES manifolds. (b) PLE spectrum of a single SiV center. Each peaks in the spectrum corresponds to optical transitions marked in (a)...... 82 4.2 (a) Scanning electron microscope (SEM) image of a representative diamond NEMS cantilever. Dark regions correspond to diamond, and light regions correspond to metal electrodes. (b) Confocal photoluminescence image of three adjacent cantilevers. The array of bright spots in each cantilever is fluorescence from SiV centers. . . 83 4.3 Simulation of the displacement of the cantilever due to the application of a DC voltage of 200 V between the top and bottom electrodes. The component of the strain tensor along the long axis of the cantilever is displayed using the colour scale. Crys- tal axes of diamond are indicated in relation to the geometry of the cantilever. Ar- rows on top of the cantilever indicate the highest symmetry axes of four possible SiV orientations, and their colour indicates separation into two distinct classes upon application of strain. SiVs shown by blue arrows are oriented along [111¯ ], [111¯ ] directions, are orthogonal to the cantilever long-axis, and experience strain predominantly in the plane normal to their highest symmetry axis. SiVs shown by red arrows are oriented along [111], [1¯11¯ ] directions, and experience appreciable strain along their highest symmetry axis. Inset shows the molecular structure of a blue- labelled SiV along with its internal axes, when viewed in the plane normal to the [110] axis...... 85 4.4 (a) Schematic of oxygen-plasma assisted ion-milling process for angled-etching of diamond cantilevers. The ion beam is directed at the diamond sample, with a vertically- etched device pattern. The tilted stage is continuously rotated during the etching process. After the cantilevers are freely standing, the etch-mask is stripped. (b) Fab- rication process for the placement of electrodes. First, the coarsely aligned bonding pad is defined with a bi-layer PMMA process followed by gold evaporation. Then the same process is repeated to define tantalum electrodes near cantilevers, but with better alignment precision. Conductive layer (ESPACER 300Z) on top of the cantilever is helpful for precise alignment. (c) SEM image of the complete chip showing connection between the bonding pad and electrodes on top of the cantilevers. 86

ix 4.5 (a) Illustration of a parallel plate capacitor with one freely movable plate (top), and one fixed plate (bottom). The top electrode can be actuated by applying a voltage. (b) Potential energy of the system in (a) with the different voltages. The stable minimum in the potential disappears, when the system reaches the condition of pull- in instability at a voltage of 4Vo. (c) FEM simulation of the strain-component along the long-axis of the cantilever (most dominant strain-tensor component) near the clamp of the cantilever (inset). Turnaround points in the graph represent pull-in instabilities...... 90 4.6 Tuning of optical transitions of ransverse-orientation SiV (red in Fig. 4.3). Volt- age applied to the device is indicated next to each spectrum...... 97 4.7 Tuning of optical transitions of ransverse-orientation SiV (blue in Fig. 4.3). Volt- age applied to the device is indicated next to each spectrum...... 98 4.8 Normalized strain-tensor components experienced by (a) transverse-orientation SiV (red in Fig. 4.3), and (b) axial orientation SiV (blue in Fig. 4.3) in the SiV co-ordinate frame upon deflection of the cantilever. (c) Variation in orbital splittings within GS (solid green squares) and ES (open blue circles) upon application of Eg-strain. Data points are extracted from the optical spectra in Fig. 2(a). Solid curves are fits to theory in text. (d) Tuning of mean optical wavelength with A1g strain. Data points are extracted from the optical spectra in Fig. 2(b). Solid line is a linear fit as predicted by theory in text. (e) Dominant effect of Eg-strain on the electronic levels of the SiV. (f) Dominant effect of A1g-strain on the electronic levels of the SiV. . 101 4.9 Change in orbital transition rates γup and γdown as a function of the ground state splitting Δgs...... 110 4.10 Time-resolved fluorescence signal in pump-probe measurement for a delay τ = 50 ns between the two laser pulses. The laser is resonant with the D transition, and optically pumps the GS population into the lower orbital branch |1⟩ over a timescale of few ns. After time τ, the fluorescence signal from the probe pulse has a leading edge determined by the population in the upper orbital branch |2⟩. The decay rates | ⟩ | ⟩ between levels 1 and 2 - γup due to phonon-absorption, and γdown due to phonon- emission - are also shown...... 111 4.11 Fluorescence time-traces for various pump-probe delays between τ=5 ns to 70 ns taken at GS-splitting Δgs=46 GHz. x-axis is time in ns, and y-axis is photon counts integrated over multiple iterations of the pulse sequence...... 112 4.12 (a) Fluorescence time-trace for τ=50 ns from Fig. 4.11 showing relevant quantities related to the population in level |2⟩. (b) Thermalization curve constructed by extracting the normalized change in photon-counts for various pump-probe delays τ. Solid line is an exponential fit...... 113

x 4.13 Transitions relevant to the time scale of spin dynamics. Blue arrows represent transition processes relevant to spin decoherence. Red arrows are transitions relevant to spin relaxation. Note that spin relaxation involve spin-flipping transitions which are much slower than spin-conserving counterparts. The labels |1 ↓⟩, |1 ↑⟩, |2 ↓⟩, |2 ↑⟩ refer to the eigenstates in the high-strain regime. Reduction of both decoherence and relaxation rate comes from the suppression of resonant absorption of a single phonon...... 115 4.14 (a) SiV level structure in the presence of strain and external magnetic field. A spin qubit is defined with levels |1 ↓⟩ and |1 ↑⟩ on the lower orbital branch of the GS. This qubit can be polarized, and prepared optically using the Λ-scheme provided by transitions C1 and C2. Phonon transitions within ground- and excited-state manifolds are also indicated. The upward phonon transition (phonon absorption process) can be suppressed at high strain, thereby mitigating the effect of phonons on the coherence of the spin qubit. (b) Coherent population trapping (CPT) spectra probing the spin transition at increasing values of the GS orbital splitting Δgs from top to bottom. Bold solid curves are Lorentzian fits. Optical power is adjusted in each measurement to minimize power-broadening. (c) Linewidth of CPT dips (estimated from Lorentzian fits) as a function of GS orbital splitting Δgs indicating improvement in spin coherence with increasing strain. (d) Power dependence of CPT-linewidth at the highest strain condition (Δgs=467 GHz). Data points are estimated linewidths from CPT measurements, and the solid curve is a linear fit, which reveals linewidth ± ∗ ± of 0.64 0.06 MHz corresponding to T2 = 0.25 0.02μs...... 117 4.15 Spin relaxation rate change as a function of the ground state splitting Δgs. Relax- ation is progressively suppressed as a function of increasing ground state splitting due to reduced single phonon absorption...... 119 4.16 Second-order correlation measurement of the SiV− centre investigated. The measured data points are plotted as grey dots. A fit based on a three-level model and accounting for timing jitter is plotted as a blue curve...... 120 4.17 Dependence of CPT dip separation on magnetic field orientation. The angle plotted on the x-axis is measured with respect to the vertical direction on the sample. 0◦ corresponds to the [001] axis of diamond, while 90◦ corresponds to the [110] axis of diamond, along which the cantilever long-axis is aligned. The SiV investigated is a transverse SiV, so its internal Z-axis is either [111¯ ] or [111¯ ]. Error bars correspond to the standard deviation on the CPT dip frequencies estimated from Lorentzian fits...... 122

xi Science is incredible.

xii Acknowledgments

Roots of giant trees can be as big as its stems, branches, and leaves altogether, the only parts we see above the ground. Similarly, beautiful things in the world, everything that stands out, and anything in the limelight has its roots we do not see often. The root of my career is what I would like to write about.

Obtaining a doctor’s degree is one of the most significant achievements in my life. It is not the best thing in the world nor the free pass to a happy life, however, I know it means something. Although I have made serious efforts to arrive here, it would not have been possible without the love and support from people surrounding me. I would like to express my gratitude to a few people among them during this journey.

First, I would like to thank my dear advisor Prof. Marko Lončar. In life, we make choices at every moment. Results of those decisions are sometimes small but other times tremendous.

When it comes to making a big choice, such as choosing a research group for Ph.D., its consequence can be so monumental that it is not easy to make one. In 2011, I was making such a choice and finally decided to join Marko’s group. In retrospect, I believe this was the best choice I have ever made in my adulthood of all aspects of my life. By the time I contacted

Marko and we decided to work together, I did not expect I would have such a great time over the next five and half years.

xiii Years of Ph.D. study in Marko’s lab was the prime time of my life. I could focus on amaz- ing projects without worrying about anything unnecessary. Marko let me choose the subject

I loved, supported consistently and believed in me even when the progress was slow. Every morning when I wake up, I feel super delighted due to the excitement of the science. Marko is the single most important person who made me feel this way, and I have been continuously feeling lucky to have my longest but the most significant academic career in his lab.

I also would like to show my gratitude to Prof. Donhee Ham, who is a professor I came to know at the beginning of my Ph.D. As Koreans, we had a lot to share about our home country, and we have been resonating so well in many aspects. Donhee was my invaluable mentor both in the scientific career and personal life.

In 2016, I had stayed at the University of Cambridge for two months to take a measurement.

It was one of the best experiences I have had during Ph.D., and I would like to thank Prof.

Mete Atatüre for giving me such an opportunity. He used to say that a critical measurement takes place right before the deadline and I first regarded it as a joke. However, it turned out to be the case for my experiment. I could get the data two days before when I was supposed to finish and wrap up. After all, I learned that it was not a mere coincidence. Perseverance was the key to my small success, and I will never forget this lesson throughout my life.

I cannot thank enough my lab partner Srujan Meesala for introducing me to the great project.

This thesis would never have been possible without his contribution. I love the topic we studied together where device engineering and experimental physics are married uniquely. Srujan is the one who has brought this idea to the lab and got me interested. I will remember that he

xiv is the one who made my Ph.D. study so enjoyable.

I would like to say a few things about a transcendent man, my dear friend Yinan Zhang. We overlapped only a year at the lab, however, that one year was enough for building a foundation of our invaluable friendship. Even though our time zones differ by twelve hours and we do not communicate on a daily basis, our friendship makes me feel great about life. After Yinan left, we have had at least one trip together every year, and they are one of the happiest moments during my Ph.D. I sincerely consider him as my real brother, my real family.

My great friend Nan Niu is the one who has always been there for me throughout my Ph.D.

Over many years, I have been going through personal difficulty, and it made me feel quite dark and depressed sometimes. Now I have overcome all of those problems, and in retrospect,

I realize that it is entirely thanks to the friendship with him. Nan’s energy and enthusiasm towards life are beyond description. Over the years, his positive spirit flowed over and influenced me profoundly, making me a happier, stronger and better person. We both see each other as real brothers, and I sincerely hope our friendship continues for the lifetime.

Finally, I would like to write about my parents. They are the biggest reason why I am standing here. Their continuous love and support led me this far. I used to find it quite intriguing that how parents love their children so much without any strings attached. After becoming a parent myself, I realize why they love me so much, but more importantly, in fact, it is not that easy to become a good parent like them. After this epiphany, I came to understand that nothing in my life would have been possible without my parents’ patience, perseverance, gen- erosity, love and so on.

xv Especially, I would like to mention a few things about my mother. As a thirty-something- year-old Korean woman in the 1980s, it was a natural choice for her to quit a teaching career and devote the life to raising two children. It seems to be her choice, but in fact, it was not, given the social environment she was living in. As a consequence, over many decades, she had stayed home and taken such good care of me. In contrast, I was born in the 1980s in a well-established family and was lucky enough to have enormous support and opportunities from the family as well as the society. Sometimes I think about what if my mother was given this much resource and opportunities? With her intelligence and perseverance, I believe her career could have been no less great than my own, and I feel complicated about how the odds in life form our destinies. Given the amount of time my mother spent with me, there is no doubt that she is the person who influenced me the most and who also contributed the most to my doctor’s degree. With all my heart, I believe at least the half my degree should belong to my great mother, if not everything.

xvi 0 Introduction

Quantum mechanics is considered one of the greatest achievements of the physics in 20th century, and by far the most accurate theory giving us precision of the physical constant better than one part in a trillion.84 Numerous Nobel awards were given to great scientists working in this field, and their efforts laid the foundations of the modern society. Most notably, the inven- tion of transistor and laser would not have been possible without the knowledge of quantum

1 mechanics. These two great inventions of the mid 20th century have led to the information revolution in the later part of the century, and have enabled the information age that we currently live in.

Although quantum mechanics gave us fundamental understanding of the world we live in, it is notoriously counterintuitive. It is so strange that great Niels Bohr once said: “Anyone who is not shocked by quantum theory has not understood it,” during his conversation with Werner

Heisenberg. Two most bizarre features of quantum mechanics are ‘superposition’ and ‘entanglement.’ Feynman famously quoted “nobody really understands it,” however, physicists have established good models of superposition and entanglement so that we are able to use those to understand many in a phenomenological level. Models of quantum mechanics have led to rev- olutions in science and technology, however, the direct application of these two features have been limited until now.

In particular, superposition is the most well known strange feature of the quantum mechanics even known to the public by the famous thought experiment - “Schrödinger’s cat.” The direct manipulation of superposition has been well developed and led to the nuclear magnetic resonance (NMR) and magnetic resonance imaging (MRI). However, other than this one exception, direct application of superposition is not readily found outside academic research.

Although less well known outside the physics community, entanglement is considered by far the most shocking feature of the quantum mechanics by many physicists. The great Ein- stein even called it “Spooky Action at a Distance” in his famous paper on EPR paradox.31

Strangeness of entanglement has caused heated debates among physicists as well as incessant

2 probing of locality and reality of our universe. After many decades from the EPR paradox, the model of entanglement has survived numerous attacks and still stands solid. Despite the fact that entanglement being at the foundation of our universe, the fragility of quantum system has stopped entanglement from being utilized for useful applications for the society.

Around the beginning of the 21st century, there has been serious progress in building quantum systems. On the theory side, there have been a few breakthroughs that proved the power and feasibility of quantum computation. Most notably, Shor’s algorithm has proven how the factorization of a large prime number brings NP (nondeterministic polynomial time) problem of the classical into the domain of P (polynomial time) problem by using the power of quantum parallelism, which directly uses both superposition and entanglement.99 Despite the fact that the power of quantum parallelism was theoretically proven, it was not until the proposal of the quantum error correction98 that the science community started believing that building a quantum computer is a real possibility.

On the experiment side, there has been serious breakthroughs which happended concur- rently across many research fields. The superconducting qubit community started demonstrat- ing simple quantum logic operations using Josephson junctions.124 Similar level of progress has been made with trapped ion systems as well.79 Around the same time, quantum computing with linear optics has also become a competing platform with the theoretical breakthrough.63

To bring these proof-of-concept lab demonstrations into practical realizations, it is essential to scale up the system until the quantum system exceeds the performance of the state-of-the- art classical system. For the scaling up of quantum systems, it is crucial to link them in a quan-

3 tum mechanical way. This ‘quantum connectivity’ is essentially the entanglement of separate quantum system.61

The most popular scheme to scaling up quantum systems is to use photons as ‘flying qubits.’

Photons at telecom and visible wavelengths are the only robust flying qubit that can travel a far distance without losing its quantum information. To build a quantum interface, it is necessary to entangle a photon and the node. The natural physical system of a choice is an atom because when an electron makes a quantum jump between separate orbitals, it shoots out a photon that is entangled with orbitals.20 However, building systems with floating atoms in a scalable manner is challenging because of the challenges in trapping them. An attractive alternative is to use ion traps or to use the solid-state quantum emitter, which is the main subject of this study.

Solid state quantum emitters used in our work, diamond color centers, are an emerging quantum system with a few key benefits that distinguish them from others. A color center has an optical quantum interface just like an atom so it meets the basic requirement for the node of quantum networks. However, it is technically much easier to build a scalable system with them because they are trapped inside the solid crystals. Unlike neutral atoms floating in a vacuum chamber, a confocal microscope is all that is needed to interface color centers with photons.

Years of effort in the field led to progress along many fronts. The most widely studied one, the nitrogen vacancy color center, can now be easily made at the single emitter level with a high yield and the spatial precision on the order of tens of nanometers.4 Although trapping

4 and isolation of single emitter promises the scalable platform, the very nature of the trapping caused challenges to the optical interface. First of all, extraction of light from the inner side of the solid to air is challenging due to the Snell’s law. For example, a significant amount of emission from the emitter is lost to total internal reflection because most of the materials have the refractive index larger than that of air. Fabrication of optical micro-/nanostructures have partially overcame these issues by extracting light much more efficiently compared to unpro- cessed bulk material with a flat face.5 Even with a much better extraction efficiency, still there is an outstanding issue in the optical interface. Because of an interaction with bulk phonon modes, optical transitions incur the so-called phonon sideband at the different wavelength from its zero-phonon line (ZPL) and lose its coherent nature. Researchers have been trying to overcome this problem by taking advantage of Purcell enhancement.48 By placing optical cavity at the ZPL frequency, it is possible to selectively enhance the ZPL compared to phonon sideband because the emitter sees an enhanced photon density of states (DOS) (∝ Q) and the √ larger quantum fluctuations (∝ 1/ V). Several types of optical cavities have been made for this purpose in the early stage,47,48 and most recently there was the work that combined the efficient photon extraction and the optical cavity.13,100

To be qualified for quantum networks, it is not sufficient to just have a good optical interface. To establish an entanglement between neighboring quantum nodes, it is essential to keep the coherence alive while the flying qubit, a photon, reaches a neighboring node. Such long coherence time is not possible with a qubit from ground and excited state from the emitter because of the short life time of the excited state, which typically amounts to a few nanoseconds.

5 Therefore, the most useful scheme is to use the spin degree of freedom of the quantum emitter.

In a simplistic picture, an electron spin can be seen as a small magnet, and the superposition of up and down spin can be used as a single qubit. Hence, it is clear that we need a very low level of magnetic field noise to have a long coherence time of the spin qubit, and this is why the diamond has been emerging as a host material for the color centers.

Pure single-crystalline diamond is mostly composed of 12C atoms forming the diamond lattice. Since 12C nuclei has zero spin, they do not contribute to the magnetic field bath. Fur- thermore, isotopically purified diamond can reduce the contents of 13C, which has the nuclear spin S = 1/2, giving a big improvement in suppressing magnetic field noise.6 More importantly, advancements in engineering of the growth of diamond by chemical vapor deposition

(CVD) have led to ultra pure diamond with less than 5 ppb nitrogen concentration. Diamond is also a great host for color centers because its wide bandgap (∼5.5 eV) allows a great variety of color centers.2 Here, we study two of the most popular color centers that have been studied for the last decade, nitrogen vacancy (NV) and silicon vacancy (SiV) color centers. However, we would like to emphasize that the potential of diamond color centers are not just limited to these two.

Despite the progress made in the past, there are remaining challenges to build a practical quantum network. First, how do we make an interface of the diamond spin qubit and other systems? Quantum emitters are the only possible platform now for implementing long-distance quantum network, however, its development for the quantum computer has been lagging be- hind others such as superconducting qubits and trapped ions. Consequently, in the current

6 stage it makes sense to put an effort in implementing the quantum interface between electron spin of the quantum emitter and other quantum systems. It is natural to use magnetic field of the microwave to make a quantum interface but the magnetic moment of the electron imposes a fundamental challenge on building such a system.94 Viable alternative is a spin with spin-orbit coupling where either electric field or acoustic wave can produce effective magnetic fields that can couple to the spin. In this work, we use acoustic wave to make a coherent interface between phonon modes and electron spin quibt of the quantum emitter. Furthermore, quantum interface using acoustic wave is strongly desirable because of its universality. Over the years, it has been demonstrated that acoustic waves can couple to various quantum systems.66 Yet the spin qubit of the solid-state emitter has not been a part of that list. Hence making a quantum interface between electron spin and phonons will pave the way to realize the coherent interface from the variety of quantum computers to spin and traveling photons. Here,

NV center in diamond is used to pursue the coherent coupling to phonons, and the characteristic of the system is discussed.

There is another important problem that exists for all solid-state emitters, which is caused by the solid state lattice. Unlike atoms, solid state quantum emitters are not identical because they experience slightly different environment from one another. External perturbations that can affect electronic structure of the quantum emitters are electric, magnetic and strain fields.

Electric and magnetic field fluctuations are typically dynamic and mitigating these noises involves material level engineering or active feedback.1 In the case of the strain, it imposes both static deformation of the quantum emitter and dynamic fluctuations. For example, though

7 diamond growth has seen impressive progress, still the material stress across the chip is not uniform.39 As a result, emitters at different locations are distinguishable due to crystal de- formations nearby, caused by static stress. Therefore, oftentimes emitters give photons with different wavelengths and polarizations. Not all but many quantum systems require indistinguishable photons, most notably linear optical quantum computing (LOQC), and hence the inhomogeneous stress of the material needs to be solved. In addition to the static environment, there is a problem caused by dynamic fluctuations as well. All bulk materials support phonon modes and they can be thermally occupied. When there are thermal phonons that couple to the quantum emitter, it is constantly exposed to random fluctuations that causes decoherence.

Here we study the way to resolve both problems by using SiV color center in diamond combined with nanoelectromechanical systems (NEMS).

In Chapter 1, basics of the crystal strain and the response of the color centers is presented using simple mathematics and a group theory. We discuss what implications there are in the effect of the strain to quantum emitters. Foundations of the strain Hamiltonian for both NV and SiV centers are laid out. After that, we discuss the desirable features of quantum emitters by incurring the physics of the strong coupling and Purcell enhancement.

In Chapter 2, theory and experimental results of the coupling between NV center and acoustic wave using dynamic actuation of cantilever are presented. In particular, we use the parametric coupling and quantify the strength of it.

In Chapter 3, we present the fabrication and characterization results of the dynamic actuation of cantilever made in single-crystal diamond. Pros and cons of the fabricated platform are

8 discussed in detail, with an emphasis on the application for the quantum interface.

In Chapter 4, SiV color center in the single-crystal diamond cantilever is presented. Here we induce the static strain locally to SiV center and measure the change in electronic structure as well as the life and coherence time of the spin. We conclude the chapter with the discussion of the outlook of coupling SiV spin to phonon modes.

9 1 Fundamentals of the crystal strain and its

effect on quantum emitters

In this Chapter, simple mathematical description of the basics of strain tensor and its effects on the electronic structure of both NV and SiV center are discussed. After going through the physics of strain response of color centers, we discuss the implications of it on both challenges

10 and opportunities.

1.1 Crystal strain in diamond

Most studies of diamond color centers are performed with single-crystal diamond. Therefore, it is important to have a mathematical description of the crystal strain and the elastic response of the diamond.

1.1.1 Tensors

What are the physical properties of the material? Simply speaking, it is the relationship between two measured quantities.82 Let’s take an example of Hooke’s law.

F = kx (1.1)

Here, F (force) and x (displacement) are the measured quantities and k (spring constant) is the property of the material. This equation is a simple one dimensional example, however, the full description of the three dimensional material needs to take the anisotropy into account. To mathematically describe the physical properties with anisotropy, tensor is used.

The easiest way to define the tensor is to use the transformation between two orthogonal axes such as Cartesian coordinate systems. Specifically, assume that we have the coordinate

′ ′ ′ system of (Z1, Z2, Z3) and it is converted to another coordinate system (Z1, Z2, Z3). Transfor-

11 mation is defined as below.

       Z′   a a a   Z   1   11 12 13   1               ′  =     (1.2)  Z2   a21 a22 a23   Z2        ′ Z3 a31 a32 a33 Z3

Here we restrict the transformation to keep the length of the vector the same and therefore,

′ aij is the cosine of the angle between Zi and Zj. aij are called the direction cosines. With the concept of the transformation, tensors can be classified depending on how they transform from one coordinate to another. To help understanding, let’s take familiar examples. A scalar is simply a number such as a temperature and is a tensor of rank 0. In the case of a scalar, transformation does not change the representation of the quantity. Let’s take another example. A

T vector is simply an arrow in three dimensional space, such as a velocity: v = (v1, v2, v3) . Un- der transformation of the coordinate system, the new representation of the velocity follows the

Eq. (1.2) as below.

∑ ′ vi = aijvj (1.3) j

Note that the vector was specified by 31 real numbers and and each component after transformation is the weighted sum of 31 components from the original frame. A vector is a tensor of rank 1. In general, a tensor of rank N is described by 3N components and the transformed each component is the weighted sum of 3N components from the original frame. In general,

12 following equation holds for the tensor component of rank N.

′ | | Tijk... = a ailajmakn...Tlmn... (1.4)

′ where we have used Einstein notation, and Tlmn... and Tijk... are tensor components of the old and new system, respectively. |a| is the determinant of the direction cosine matrix in Eq. (1.2) and its value is ±1 depending on the whether the handedness of reference frame changes or not. When a = −1, it is called the axial tensor and the examples include Hall effect and py- romagnetism. Qualitatively, it applies to the physical property involves the handedness. The other case where a = 1 is called the polar tensor and all tensor quantities being studied in the rest of the thesis belong to this class.

Specifically, here we deal with only three kinds of tensors: stress, strain and elasticity tensors. As mentioned earlier, tensor is useful to study ‘physical property,’ especially when there is anisotropy. It is important to note that not all three tensors represent physical properties. Re- visiting Hook’s law in the Eq. (1.1), it is clear that F is an ‘input’ and x is an ‘output’ whereas k is a ‘response’ of the system. Among these three only k is the physical property and the other two are measured values. Coming back to tensor quantities, we realize three tensor quantities, stress, strain and elasticity, correspond to force, displacement and spring constant in one dimensional case. In general, Hooke’s law in solid is expressed as below.

(σ) = (c)(ε) (1.5)

13 where (σ) is the stress tensor expressed in 9 × 1 matrix, (c) is the elasticity tensor in 9 × 9 matrix and (ε) is the strain matrix in 9 × 1. It is clear that elasticity tensor is the physical quantity of the material and stress and strain tensors are measured quantities each corresponding to input and output. Tensors like the elasticity tensor are called ‘property tensors.’ These tensors represent physical properties of the system and therefore they are closely tied to the symmetries of the system itself. Tensors like strain or stress tensor are called ‘field tensors.’ These quantities are controlled by the experiment so they are at experimenters liberty.

In the case of generalized Hooke’s law in the Eq. (1.4), stress and strain tensors are second rank tensors, which are to be studied in more detail in the next sections. To describe the relationship between two second rank tensors, the elasticity tensor needs to be a fourth rank tensor. In general, the tensor rank N has the meaning of the number of directions involved in measuring the physical property. In the case of elasticity in particular, there are two directions involved in applying stress and two more directions need to be added to take the strain measurement into account. Adding all directions necessary in the measurement, we get a total of four which is the same with the rank of an elasticity tensor.

1.1.2 Stress tensor

Stress tensor is a field tensor of second rank. Its physical interpretation can be understood easiest by referring to the Fig. 1.1. A hypothetical infinitesimal cubic is subject to force on its six faces. Since we do not consider translational motion or acceleration of the box, only three out of six faces are enough for the study as seen in the Fig. 1.1. Each force component is can-

14 Z3

σ33

σ13 σ23 σ31

σ32

σ 11 σ12

σ21

σ22

Figure 1.1: Schematic illustration of the stress tensor. Stress tensor in general needs nine components to fully describe the applied stress to inﬁnitesimally small hypothetical box inside the material. celed by the force with equal magnitude and opposite direction on the parallel face. Therefore, tensor can be expressed in the matrix form as below.

   σ σ σ   11 12 13    [ ]   2 (σ) =   N/m (1.6)  σ21 σ22 σ23    σ31 σ32 σ33

15 where indices 1,2,3 correspond to orthogonal directions of axes in the reference frame. Note that unit signifies that stress should be interpreted as the local pressure. Referring to the Fig.

1.1, it is straightforward to interpret each component. For example, σ22 is normal to the face and it is called the ‘tensile’ stress when its sign is positive. It is the force that stretches the material. When the sign is negative, it is the force that pushes in the material and called ‘compressive’ stress. σ21 is the shear stress and it is the force applied in parallel to the face on which it is applied to. A simple example of the shear force is the scissors where shear forces are used to cut papers. In the theory of continuum mechanics, this infinitesimal cubic box in Fig. 1.1 does not rotate such that the total torque applied to it should be zero. This condition is met when σij = σji is true. Therefore, nine components in the Eq. (1.6) are reduced to six. Conven- tionally, stress tensor is rewritten as a 6 × 1 matrix as below.

     σ1 = σ11           σ2 = σ22     σ11 σ12 σ13           σ3 = σ33    ≡   (1.7)  σ21 σ22 σ23           σ4 = σ23    σ31 σ32 σ33      σ = σ   5 13    σ6 = σ12

Both notations are used in the literature. It is instructive to see the mathematical form of the commonly used stress in experiment. A uniaxial stress is where the normal force is applied along only one axis, for instance axis Z1, where all components of the stress tensor are zero

16 except for σ11. Likewise, when only σ11 and σ22 are not zero, then it is called biaxial stress.

Note that when σ11 = σ22 holds, it is the physical compression or stretching in Z1 − Z2 plane.

Furthermore, if σ11 = σ22 = σ33 < 0 holds true and all others are zero, it is called the hydrostatic pressure which happens when the material is placed inside the pressurized chamber or in the liquid. It is worth noting that hydrostatic pressure, whether it be in the two or three dimensions, represents the force applied to reduce or expand the overall volume or area.

Also, let’s briefly look into symmetry of stress tensor. Limiting applied pressure only in

Z1 − Z2 plane, following decomposition is possible for arbitrary biaxial stress.

       σ 0 0   (σ + σ ) 0 0   −(σ − σ ) 0 0   11   11 22   22 11          1   1     =  ( + )  +  ( − )  (1.8)  0 σ22 0  2  0 σ11 σ22 0  2  0 σ22 σ11 0        0 0 0 0 0 0 0 0 0

One can note that it is similar to decomposing arbitrary one dimensional well-define function into the sum of even and odd parts. Interpretation of the equation (refeq:Biaxial stress decomposition) is the following: we can decompose any biaxial stress into hydrostatic pure normal (hydrostatic pressure) and pure shear components. As seen in the later part of the paper, stress components with different natures of symmetry cause corresponding effects on electronic structures of diamond color centers. Proper decomposition needs to be done with the consideration of the group theory, however, this simplified example shows the core idea of the full treatment in the later part of the thesis.

17 1.1.3 Strain tensor

Strain simply means the fractional change in its shape. In the simplest case of one dimensional rod of length L, its strain is defined as

ΔL ε = (1.9) L when its length is changed by ΔL. In the case of three dimensional elastic material, strain is defined as the second rank tensor shown below.

a b c Z 2 Z2 Z2

ε21 = ε12 ε11 ε12 Z 1 Z1 Z1

Figure 1.2: (a) Object without strain. (b) Object with ε11. (c) Obejct with ε12.

   ε ε ε   11 21 31  ( )   ∂ui   εij = =   (1.10) ∂Z  ε21 ε22 ε23  j   ε31 ε32 ε33

18 where ui is the displacement along the axis Zi. Similarly to stress tensor, εii represents tensile strain as shown in the Fig. 1.2b. Likewise, excluding the rotation of infinitesimal cubic box, we get ε12 = ε21 where it represents the shear strain similarly to that of stress (Fig. 1.2c).

Similarly to the Eq. (1.7), strain tensor can also be written in 6 × 1 matrix form.

     ε1 = ε11           ε2 = ε22     ε11 ε21 ε31           ε3 = ε33    ≡   (1.11)  ε21 ε22 ε23           ε4 = 2ε23    ε31 ε32 ε33      ε = 2ε   5 13    ε6 = 2ε12

Note the factors of two in front of the original shear components.

Let’s take a look at the example that is useful to remember for the later part of the thesis.

Consider the following example of the strain tensor, where its 3 × 3 matrix from is diagonal.

   ε 0 0   11        (1.12)  0 ε22 0    0 0 ε33

Keeping on the first order when the strain is a small number, fractional change in volume is

19 given by

ΔV = ε + ε + ε (1.13) V 11 22 33

Similar equation holds true for the fractional change in area when only ε11 and ε22 are considered. This quantity does not vary under unitary rotation and that is consistent with the physical interpretation of fractional volume change (the fractional volume change does not depend on the coordinate system used to represent the tensor).

1.1.4 Elasticity tensor

Given the formal definitions of stress and strain tensors, we can define elasticity tensor prop- erly. As noted previously, elasticity tensor is qualitatively the same with the spring constant in one dimensional Hooke’s law, but with distinguished by its anisotropy. Mathematically, it is defined as below.

σij = cijklεkl (1.14) where Einstein notation is used. Since there are four indices, there can be 34 = 81 tensor components in principle. However, recalling that both stress and strain tensor components can be reduced to six components, 62 = 36 components are enough to fully characterize the elasticity of any solid system. In general, Hooke’s law in three dimension can be written as the

20 following.

             σ1   c11 c12 c13 c14 c15 c16   ε1                     σ2   c21 c22 c23 c24 c25 c26   ε2                     σ3   c31 c32 c33 c34 c35 c36   ε3          =     (1.15)        σ4   c41 c42 c43 c44 c45 c46   ε4                     σ   c c c c c c   ε   5   51 52 53 54 55 56   5        σ6 c61 c62 c63 c64 c65 c66 ε6

82 By using energy argument, it can be shown that cij = cji, further lowering the number of independent components from 36 to 21. Since elasticity tensor is the property tensor, it is possible to even reduce more the number of independent components by using Nemann’s Prin- ciple.82 Taking into account the symmetry of the diamond lattice, elasticity matrix is given as below.

             σ1   c11 c12 c12 0 0 0   ε1                     σ2   c12 c11 c12 0 0 0   ε2                     σ3   c12 c12 c11 0 0 0   ε3          =     (1.16)        σ4   0 0 0 c44 0 0   ε4                     σ   0 0 0 0 c 0   ε   5   44   5        σ6 0 0 0 0 0 c44 ε6

21 1.2 Strain Hamiltonian of quantum emitters

Central theme of this thesis is to study how color centers respond when arbitrary strain is im- posed and the application of this principle to quantum information technology. The best tool to understand how electronic structures of color centers behave with an external strain is the group theory.

a b

Z Z

Y X Y X

Figure 1.3: (a) NV center. Blue sphere is a nitrogen atom and black ones are carbons. (b) SiV center. Red sphere is a silicon atom and black ones are carbons.

Color centers in this study are NV and SiV centers and their schematics are shown in the

Fig. 1.3. As shown, these color centers are made by replacing a few carbons with other atoms or leaving the site vacant. Typically dangling bonds from the nearest carbons, sp3-orbitals, participate in forming the electronic structures of the color centers and therefore, carbon atoms nearby need to be included when developing the group theory model. For this reason both

22 color centers are the subgroups of the diamond’s point group. We start the discussion from the crystal symmetry of the diamond, the host material for both color centers.

Group theory is essentially about the symmetry of the crystal structure and its implication on physical properties. In the case of three dimensional materials, crystal structures can be studied either by space group or the point group. Space group is for studying spatial configuration throughout the infinite space. In contrast, point group is used to study the symmetry of the local structure around a particular origin. Although point group itself assumes the local structure, it is useful to study the property of the single-crystal material because the repeat of the unit cell, a local feature, can span out the entire space from its definition. There are 32 classes of point groups and single-crystal diamond is known to have full octahedral symme-

82 try, which is called Oh point symmetry group using Schoenflies symbols. Oh point symmetry is equivalent to m3m group in Herman-Mauguin notation. In the rest of the thesis, we use

Schoenflies symbol.

A Particular point group refers to a set of elements where elements correspond to a specific symmetry operation, such as a rotation, reflection or inversion. Oh point group has 48 symmetry elements in the group. All the point groups are required to fulfill certain mathematical requirements.49

When the point defect forms inside the diamond crystal, such as NV or SiV center, in many cases it has the symmetry axis while reducing the point group symmetry to the subgroup of the original. For both NV and SiV centers, symmetry axis points along ⟨111⟩ crystal axis and these are classified as ‘trigonal centers.’ Effect of the external strain on these defects are well

23 studied in the literature52 which we briefly overview in the following.

The goal of the rest of this section is to show how we decompose arbitrary stress or strain into the sum of components in a symmetry adapted form. In doing so, we can get physical intuition of how strain acts on electronic structure of color enters. The stress-response of the electronic levels of trigonal point-defects in cubic crystals was treated theoretically.52

When applied stress is small, the perturbation Hamiltonian Hstrain can written in terms of stress-tensor components.52

∑ strain H = Aijσij (1.17) i,j

Here Aij are operators for each stress-tensor component, and act on the color centers. i, j are indices for the axes of the reference frame. For the rest of the thesis, we use (x, y, z) as our reference frame for diamond crystal axes, ⟨100⟩. Stress and strain are linearly related via the elasticity tensor of the diamond cijkl as shown in Eq. (1.14).

Thus the Hamiltonian Eq. (1.17) can also be written in terms of strain-tensor components as

∑ strain H = Bklεkl (1.18) kl

where the operators Bkl are related to the operators Aij according to

∑ Bkl = cijklAij (1.19) ij

24 Either stress or strain tensor could be used because the elasticity tensor of the diamond cijkl is well known.

Group theory can be used to rewrite the strain Hamiltonian Hstrain in terms of linear combinations of stress-tensor (strain-tensor) components adapted to the symmetries of the color center of interest. Each of these combinations can be viewed as a particular ‘mode’ of stress

(strain). For instance, hydrostatic pressure part of the stress can be written as (σxx + σyy + σzz), and it can be seen as a projection from an arbitrary stress tensor onto a particular irreducible representation in the point group of interest. Knowing the type of irreducible representation of hydrostatic pressure in particular point symmetry, its effect on the electronic levels of the defect can be quite readily deduced using group theory. Mathematically, we rewrite strain

Hamiltonian as the following.

∑ Hstrain = AΓr σΓr (1.20) Γ,r

{ } where σΓr is the linear combination of σij and transforms according to the rth basis vector of

the irreducible representation Γ. AΓr should transform in the same way as σΓr .

For NV and SiV centers, introduction of the defect reduces the cubic symmetry Oh to C3v and D3d point groups, respectively. Both point groups belong to Tetragonal centers in the material of Oh crystal symmetry and the symmetry adapted Hamiltonian of the stress (or strain) tensor can be found in the literature.52 The final result is repeated in the following.

25 Hstrain A A′ = 1(σxx + σyy + σzz) + 1(σyz + σzx + σxy) √ + EX (2σzz − σxx − σyy) + 3EY (σxx + σyy) (1.21) √ ′ ′ + EX (2σxy − σyz − σzx) + 3EY (σyz − σzx)

where the Hamiltonian is decomposed into six terms. Matrix elements of operators for each

term can be calculated using Wigner-Eckhart theorem. For an operator AΓr that transforming as the rth basis vector of the irreducible representation Γ, the matrix element for the basis pair

|κp⟩ , |λq⟩ are given in Eq. 1.22, where each transform as pth and qth row of irreducible representation Κ and Λ, repsectively.

⟨ | | ⟩ ∥ ∥ ⟨ | ⟩ κp AΓr λq = [κ AΓr λ] Γrλq κp (1.22)

Here, ⟨Γrλq|κp⟩ is the Clebsch-Gordan coefficient that projects the basis vector |λq⟩ out of the product basis set |Γr⟩ |κp⟩. Basis set will differ depending on the point symmetry group of the color center. Therefore, the details of stress (or strain) Hamiltonian will be presented separately, where Chapter 2 is dedicated to NV center while Chapter 4 covers SiV center.

26 1.3 Application of the strain Hamiltonian

In previous sections, we have shown that how strain at the site of the solid-state quantum emitter fundamentally affects its electronic structure. Here, we show how we can use this feature to engineer the quantum emitter to overcome existing challenges or make devices with novel functionalities.

To engineer the strain environment of the quantum emitter, we need to implement a reliable control knob. This is where nanoelectromechanical systems (NEMS) play an important role.

A predecessor of NEMS is microelectromechanical systems (MEMS), and the difference of those two only come from the minimal device dimensions. Both work under the same princi- ples and their minor details are not covered here. In the rest of the thesis, NEMS is chosen as the representative terminology.

NEMS devices have been developed in the last few decades for numerous applications, which can be classified into two big categories according to their functionalities: actuation and sensing. As the name suggests, NEMS is all about the interface between electricity and mechanical motion. Actuators induce motions by using electricity as a control. One of the most successful applications is the digital mirror device (DMD), where the movable mirrors work as optical switches, and the technology is widely used in displays. Sensors measure physical quantities that are coupled to the motion of the proof mass and use electricity to read out the motion. NEMS sensors are easily found in many modern electronics and the list of those include accelerometer, gyroscope, pressure sensor and microphone.

27 In this work, we focus on its use as an actuator. Traditionally, the purpose of actuation is the movement of the target object in a controllable fashion. However, here we pay attention to a new aspect of it, the stress and strain of the deformed material. Motion of a clamped object inevitably causes deformation of the material and strain inside as a result of the elasticity.

Combining this idea with a physics of the strain control of electronic structure is the foundation of this study.

Technically, actuation itself can be divided into two categories depending on its behavior in time domain. Static actuation is for stationary displacement of the material by applying

DC voltage. Dynamic actuation is usually the generation of an oscillatory motion by applying sinusoidal voltage. In this work, we explore both of these applications to color centers and the motivations of these are given in the following.

1.3.1 Generation of spectrally indistinguishable photons from different emitters

Importance of quantum emitters as a building block of quantum computer or network comes from the fact that they are single photon sources. For many quantum technology architectures, single photons play foundational role such as a flying qubit or entangled photon pairs. Flying qubits generate quantum entanglement between distant nodes in the network, by working as carriers of the quantum informations from stationary qubits in the nodes. On the other hand, single photon sources also play a crucial role in linear optical quantum computing (LOQC) where the principle of computation originates from entanglement between two photons. In both examples, one of the key requirements for the single photon source is to produce indis-

28 tinguishable photons. With two indistinguishable photons, it is possible to produce entangled photon pair using Hong-Ou-Mandel effect.50

Conditions for indistinguishable photons include almost perfect match of space, time, energy and polarization of the photon. Among these, energy of a single photon, ℏω is extremely difficult challenge unlike others because it requires a nonlinear system with a high efficiency.

Therefore, it is important to produce single photons at exact same frequencies from quantum emitters in the very beginning.

Unlike neutral atoms or ions, solid-state quantum emitters inevitably are distinguishable because of the difference in their environments. As for diamond color centers, even with the recent advancements in the growth techniques of synthetic diamond, it is difficult to produce it without any internal stress.39 Inhomogeneous stress across the chips results different strain seen by emitters at different site and therefore they emit photons at slightly different frequencies. To solve this problem, it is easiest to counter the shift of frequency caused by the exter- nally applied strain. Electric fields are useful for systems with non-zero Stark effect, such as

NV, however it is not universally applicable to all quantum emitters. For example, SiV center is known to have no first order Stark effect.101

We propose a solution to this problem using NEMS devices made on single-crystal diamond. By using static actuators made locally, it is possible to control the local strain of the material by applying voltages. In doing so, it is possible to bring the colors of two separate emitters onto the same frequency.74 This concept is schematically drawn in Fig. 1.4. It is important to notice that NEMS is scalable technology so that it does not undermine the scalabil-

29 a b

Figure 1.4: Illustration of tuning two emitters together spectrally with the strain gradient. Emission spectra of the two emitters are initially different in a, and are tuned together by an appropriate strain gradient between the two emitters in b. ity of solid-state quantum emitters.

1.3.2 Acoustic phonon as a universal quantum bus

NEMS devices in its use for harmonic motion has a unique application for building an on- chip quantum network. The concept of the strong coupling at a single node in the context of quantum network proposed61 is the necessary requirement for all quantum systems. Imple- mentation of these quantum nodes have been mostly done with quantum emitters and optical cavities in the past. However, recent proposals9,89 aroused considerable interest of realizing such a system based on phonons rather than photons. Therefore, a quantum emitter with two- level system placed inside the mechanical resonator serve as such a node which can couple to traveling phonons. In the context of solid state emitters, mechanical hybrid systems were first

30 proposed123, and subsequently demonstrated78,125 with the electronic states of quantum dots.

Benefit of having such system is that phonons couple ‘universally’ to most of the quantum systems. It has been shown that phonons can coherently interact with most of existing quantum systems.66 Using the fact that phonons couple to the diversity of quantum systems, one can imagine building ‘universal quantum bus’ with traveling acoustic phonons for the interface between systems with completely different physical implementations.

1.4 Physics of the quantum nodes

To have a coherent quantum interface between flying and stationary qubits, it is important to meet a requirement of having the cooperativity larger than one.77 In this Section, we explain the definition of the cooperativity and the implications of such requirement when building quantum nodes.

1.4.1 Fermi’s golden rule

Fermi’s golden rule is a good starting point to develop the discussion. Its simplest from is given in Eq. (1.23), which predicts the transition rate between two energy eigenstates.

2π ⟨ ⟩ 2 Γi→f = ℏ f Hp i ρ(ω) (1.23)

where |i⟩, |f⟩ are initial and final eigenstates, Hp represents the perturbation operator, ρ(ω) is the density of the final state at radial transition frequency ω. Conventional application of the

31 Fermi’s golden rule is the calculation of the absorption or emission of light in various conditions.

1.4.2 Purcell effect

Based on Fermi’s golden rule, Edward Purcell explained enhanced spontaneous emission rate when the emitter is placed inside the cavity whose resonance frequency matches the transition frequency. The amount of enhancement is given as below.

( ) 3 Q F = (λ /n)3 (1.24) P 4π 2 c V

where λc is the wavelength of emitted photon in vacuum and n is the refractive index of the homogeneous medium. Q is the quality factor of the cavity and V is the mode volume.

From the viewpoint of device engineering, the key result of Purcell effect is that the enhancement is proportional to Q/V. Purcell effect can be physically interpreted as the following. First of all, Q is responsible for increased density of states because we have added a single mode with the spectral width of only 1/Δν. Therefore, high quality factor of the cavity contributes to enhanced density of states in Eq. (1.23). Although V can be seen as the part of the density of states (recalling that the unit of photon density of states is typically ( ) 1/ Hz · m3 ), we get further insight by incorporating its effect into the transition matrix element in Fermi’s golden rule.

32 1.4.3 Physics of cavity quantum electrodynamics (cQED)

Let’s assume we have an electromagnetic cavity with mode volume V and the radial frequency

ω = 2πν. We assume the electromagnetic cavity and mainly use electric field from below, however, the following argument can be applied to any physical energy domain. The root- mean-square electric field caused by vacuum fluctuation at the anti-node of the cavity can be estimated as

∫ 1 ε 2 ε 2 ℏω = ⃗E(⃗r) dV = E⃗ V (1.25) 4 2 2 0

where ⃗E(⃗r) is the electric field at the position⃗r and E0 is the amplitude of the electric field at the position of the emitter. ε is the permittivity of the cavity material. The Eq. (1.25) is from the interpretation that a single mode cavity can be seen as a harmonic oscillator. Since all harmonic oscillator has non-zero ground state energy, which induces the vacuum fluctuation, its energy should show up as both electric and magnetic fields. Since both fields equally divide the energy in the cavity, half of the ground state energy should be in the form of electric fields. √ From the result of the equation, it can be noted that E0 ∝ 1/V. Therefore, placing a cavity with a small mode volume around the emitter induces an enhanced vacuum fluctuations at the location of the emitter.

This leads to a further insight of the emitter itself. Referring back to Fermi’s golden rule in

Eq. (1.23), one can notice that the quantity ⟨f Hp i⟩ is an abstract term yet. In the case of the

33 spontaneous emission, the transition can be interpreted as the stimulated emission by vacuum fluctuation.43 In such a case,

⟨f Hp i⟩ = ⃗μ · E⃗0 (1.26) where ⃗μ = ⟨f |⃗r| i⟩ is the transition dipole moment. It is important to note that the transition dipole moment is entirely given by the electronic orbital of an emitter such that it is essentially a given property of it. From these quantities, the vacuum Rabi frequency is introduced as below.

⃗μ · E⃗0 g = ℏ (1.27)

Vacuum Rabi frequency g represents the rate of coherent interaction between the emitter and the cavity mode when they are on resonance. g has an important role in realizing a quantum node because quantum state transfer protocol22 only works when it meets the following condition.

2g2 C = > 1 (1.28) κγ where γ is typically a decoherence rate of the emitter, κ ≡ Δν, and the quantity C is called the single-photon cooperativity. Qualitatively, C means the ratio of the rate between useful (coherent) interaction to useless (decoherent) interaction. Cooperativity is the most important figure

34 of merit for all systems that work under the same principle as this particular example. Evi- dently, it is important to have large cooperativity for the strong coherent quantum interaction and that leads to the following goals in designing quantum systems of this type.

1.4.4 Goal of quantum engineering

Here we discuss desirable features of parameters to get as large cooperativity as possible.

There are four relevant parameters: Q, ⃗μ, V and γ.

First, let’s discuss Q and V together which comprise the Purcell effect FP ∝ Q/V. Purcell effect is an enhanced spontaneous emission rate from its value when the emitter is in the homogeneous medium. When Purcell effect is strong enough such that vacuum Rabi frequency, whose squared value is proportional to spontaneous emission rate, is as strong as the overall decoherence of the system, we reach the regime g ≫ κ, γ. Under this condition, photon is coherently exchanged between the cavity and the emitter before it gets lost to the reservoir through irreversible process. Large quality factor Q plays the role to keep photons inside the cavity-atom system as long as possible. As discussed, small mode volume V is important to have large g, which is caused by large field fluctuations. Chapter 3 shows a progress made towards high Q/V cavity.

Recalling that g = ⃗μ · E⃗0, one can realize that |⃗μ| should be a large number for a better cooperativity. As noted, the transition dipole moment is a given feature of an emitter, so it is important to choose such one from the first place. For example, one of the reasons why superconducting qubit has been popular is that its transition dipole moment is humongous due

35 to its large physical size. Likewise, for the application of cavity QED physics for phonons, it is important to identity the quantity that corresponds to the transition dipole moment in the cavity QED. For this reason, we pick SiV center in Chapter 4 because the transition dipole moment of its spin qubit is one of the largess among all known spin systems.

Lastly, the decoherence rate γ of a quantum emitter needs to be small to show coherent interaction. Typically, the decoherence rate of quantum emitters working in optical transition frequency is limited by its spontaneous emission life time in the excited state. Yet other quantum systems such as spin qubit or superconducting qubit are not limited by the spontaneous emission because their transition frequency is much smaller (on the order of a few GHz) compared to that of emitters at optical frequency (on the order of hundreds of THz). Therefore, coherence time for spin qubits vary a lot depending on the limiting factor for decoherence process. In this regard, in Chapter 2, we make an attempt to build a phononic quantum node with

NV center, where we take advantage of long spin coherence time due to low magnetic noise environment in diamonds. On the other hand, in Chapter 4, we investigate the physical origin of the decoherence of SiV’s spin qubit and suppress it using NEMS device.

36 2 Phononic quantum node with NV center

in cantilever

2.1 Motivation

Quantum two level systems (qubits) strongly coupled to mechanical resonators can function as hybrid quantum systems with several potential applications in quantum information sci-

37 ence.104,105,121 The physics of these systems can be well described with the tools of cQED, in which an atom is strongly coupled to photons in an electromagnetic cavity. As pointed out in

Section 1.3.2, phonons or mechanical vibrations couple to a wide variety of well-studied quantum systems, and therefore, are considered a promising means to coherently interface qubits across disparate energy scales.104,105,121 Such a mechanical hybrid quantum system with negatively charged nitrogen vacancy (NV(-), hereafter referred to as NV) centers in diamond, in particular, would benefit from their long spin coherence times.7 It has been proposed that in the strong spin-phonon coupling regime, phonons can be used to mediate quantum state transfer, and generate effective interactions between NV spins.89 Strong coupling of an NV spin ensemble to a mechanical resonator can also be used to generate squeezed spin states,9 which can enable high sensitivity magnetometry.97

Seminal experiments on coupling NV spins to mechanical oscillators relied on magnetic field gradients.3,51,64 More recently, owing to the development of single crystal diamond nanofabrication techniques,12,46,48,57,58,87,113 the effect of lattice strain on the NV ground state spin sublevels has been exploited to couple NVs to mechanical modes of diamond cantilevers,8,70,86,114 and bulk acoustic wave resonators fabricated on diamond.68,69,70 Strain- mediated coupling is experimentally elegant since its origin is intrinsic to a monolithic device, and it does not involve functionalization of mechanical resonators, or precise and stable posi- tioning of magnetic tips very close to a diamond chip. However, current demonstrations are far from the strong coupling regime due to the small spin-phonon coupling strength provided by strain from relatively large mechanical resonators. In this work, we present an important

38 step towards strong coupling by incorporating photostable NVs in a diamond cantilever with nanoscale transverse dimensions, and demonstrate a single phonon coupling rate of ∼ 2 Hz from dispersive interaction of NV spins with the resonator. This is a ∼ 10 − 100× improvement over existing NV-strain coupling demonstrations. In our experiments, we first detect the effect of driven cantilever motion on NVs as a broadening of their electron spin resonance

(ESR) signal, and through follow-up measurements, establish this to be strain-mediated coupling to the mechanical mode of interest. Subsequently, we use spin echo to probe the temporal dynamics of NVs in the cantilever, and precisely measure the spin-phonon coupling rate.

In the conclusion, we discuss subsequent device engineering options to further improve this coupling strength by ∼ 100×, and reach the strong coupling regime.

2.2 Requirements for strong spin-phonon coupling

As explained in Section 1.4.3, the key requirement for applications that rely on strong qubit- phonon coupling is that the co-operativity of the interaction exceed unity,64 but with a minor correction.

2g2 C = > 1 (2.1) (1 + nth)κγ

Here, g is the single phonon coupling rate, nth is the thermal phonon occupation of the mechanical mode of interest, κ is the intrinsic mechanical damping rate, and γ is the qubit dephasing rate. The only difference from the Eq. 1.28 is that the (1 + nth) factor in denominator. This

39 is difference comes from that the energy of a resonant phonon in the cantilever is far smaller than kBT and hence the resonator mode is filled with nth number of thermal phonons.

For strain-mediated linear coupling, the single phonon coupling rate is given by g = dεZPM, where εZPM is the strain due to zero point motion, and d is the strain susceptibility, an intrinsic property of the qubit. The spin triplet ground state of the NV has a relatively small d ≈ 10−20

GHz/strain,86,114 since the three spin sublevels share the same orbital wavefunction. The effect of strain on these levels is proposed to be a filtered down effect from stronger perturbations to the orbitals themselves, particularly from spin-orbit coupling to the excited state,27 and a change in spin-spin interaction energy in the deformed ground state orbital.29 Thus, engineering the mechanical mode to provide large εZPM is essential to achieve large g. For instance, for the fundamental out-of-plane flexural mode of a cantilever of width w, thickness t, and length l, we can use Euler-Bernoulli beam theory56 to show that

1 εZPM ∝ √ (2.2) l3w

This sharp inverse scaling of εZPM with cantilever dimensions highlights the importance √ of working with small resonators. It is analogous to the 1/ Veff scaling of the single photon Rabi frequency in cQED, where Veff is the electromagnetic mode volume. To achieve the strong coupling condition in Eq. (1.28), assuming an NV spin coherence time T2 = 100 ms, a mechanical quality factor Q = ω/κ = 106, and cryogenic operation temperatures (4K or lower), cantilevers of width w ∼ 50 − 100 nm and length l ∼ 1 μm (corresponding mechanical

40 frequency, ωm ≈ few hundred MHz) that provide g ≈ few hundred Hz are required. Towards this end, our devices in this work have w of the order of a few hundred nm. This is typically the length scale at which proximity to surfaces begins to deteriorate the photostability of the

NV charge state.

2.3 Device fabrication and experimental setup

Incorporating photostable NV centers close to surfaces,23,40 particularly in nanostructures with small transverse dimensions such as nanophotonic cavities,34,48 has been found to be a considerably challenging task in recent years. To prevent charge state blinking and photo- ionization of NVs under optical excitation,102 high quality surfaces with low defect density, and appropriate surface termination are necessary. Recent advances in annealing and surface passivation procedures18 have significantly improved the ability to retain photostable NV centers generated by ion implantation even after fabrication of nanostructures around them.25

Using these techniques in combination with our angled reactive ion etching (RIE) fabrication scheme,12 we were able to generate photostable NVs in diamond nano-cantilevers with a triangular cross-section (Fig. 2.1a,b). Previously, we have demonstrated high Q factor mechanical modes (Q approaching 100,000) with frequencies ranging from <1 MHz to tens of

MHz in cantilevers, and doubly clamped nanobeams fabricated using the same angled etching scheme.14

Our measurements are carried out at high vacuum (10−5 torr), and room temperature in a vacuum chamber with a view port underneath a homebuilt scanning confocal microscope for

41 addressing NV centers. Microwaves for ESR measurements are delivered with a wire bond positioned close to the devices of interest. The diamond chip is mounted on a piezo actuator for resonant actuation of cantilevers. Mechanical mode spectroscopy performed via optical interferometry65 is used to characterize the modes of the cantilevers. For the experiments described in this paper, we used a triangular cross-section cantilever with w=580 nm, t=170 nm, and l =19 μm. The mechanical mode of interest (Fig. 2.1c) is the out-of-plane flexural mode, which was found to have a frequency, ωm = 2π × 937.2 kHz, and a quality factor,

Q ∼ 10, 000.

2.4 Strain Hamiltonian of the ground state triplet of NV center

We continue the derivation of the strain Hamiltonian from the Section 1.2. Rewriting Eq.

(1.21) in their symmetry adapted form (C3v point group for NV center) as below is useful to get physical intuition.

Hstrain = Hstrain + Hstrain + Hstrain A1 Ex Ey (2.3) where Hstrain transforms as the irreducible representation Γ . ΓIR IR

Matrix element of each operator in Eq. (2.3) can be calculated by applying Wigner-Eckart theorem (Eq. (1.22)) to Eq. (1.21). In doing so, we can write the Hamiltonian in the matrix form as below.

42 a c ) w 100 t 50 l

amplitude (nm) 0 936 936.5 937 937.5 938 938.5 4 μm Cantilever displacement Driving frequency (kHz) e 0 dBm -6 dBm b -12 dBm -21 dBm

4 μm Contrast (%)

d ms = +1 mI = 0 mI = +1(-1)

mI = -1(+1) D0+γBz

mI = 0 2.875 2.88 2.885 2.89 m = 0 m = ±1 s I Microwave frequency (GHz)

Figure 2.1: (a) Representative scanning electron microscope (SEM) image of the angle-etched diamond cantilevers used. (b) Representative confocal microscope scan of a section of the cantilever showing fluoresecence from NV centers. (c) Driven response of the fundamental out-of-plane flexural mode (right inset) of the triangular cross-section (left inset) cantilevers studied in this work. For this particular device, we have w=580 nm, t=170 nm, and l=19 μm. The mode frequency is 937.2 kHz, and it has a Q-factor of 10,000. Measurements were taken in high vacuum (1e-5 torr) at room temperature. (d) Hyperfine structure of the ms = 0 to ms = +1 electron spin transition in the NV ground state indicating the three allowed microwave transitions. (e) AC strain induced broadening of the ms = 0 to ms = +1 hyperfine transitions near the clamp of the cantilever with gradually increasing mechanical amplitude. The mechanical mode is inertially driven at its resonance frequency with a piezo stack in all measurements. Open circles indicate measured data, and smoothed solid lines serve as a guide to the eye. Legend shows values of piezo drive power for each measurement. 0 dBm of drive power corresponds to an amplitude of 559 ± 2 nm at the tip of the cantilever.

43 ( ) 2 Hstrain = M S2 − (2.4) A1 A1 z 3 ( ) Hstrain 2 − 2 Ex = MEx Sy Sx (2.5) ( ) Hstrain Ey = MEy SxSy + SySx (2.6)

where components MA1 , MEx , MEy are given by the linear combinations

MA1 = A1(σxx + σyy + σzz) + 2A2(σyz + σzx + σxy)

− − − − MEx = B(2σzz σxx σyy) + C(2σxy σyz σzx) (2.7) √ √ − − MEy = 3B(σxx σyy) + 3C(σyz σzx)

Wigner-Eckhart theorem (Eq. (1.22) was used to find operators in {|A1⟩ , |Ex⟩ , |Ey⟩} basis.

28 Clebsch-Gordon coefficients for C3v point symmetry group can be found in the literature.

Also, the four coefficients A1, A2, B, C are defined as below.

44 A1 = [κ∥A1∥λ] [ ] ∥A′ ∥ A2 = κ 1 λ 1 B = √ [κ∥E∥λ] (2.8) 2 1 [ ] C = √ κ∥E′∥λ 2 where |κ⟩ and |λ⟩ are one of three ground state triplets.

The two terms multiplied by A1 and A2 that make up MA1 are stress-modes that transform as

A1, and lead to a split in the energy between ms = 0 and ms = ±1 spin sublevels. On the other

− hand, the terms in MEx and MEy mix ms = +1 and ms = 1 spin sublevels.

To gain more physical intuition for these stress-modes, we can convert MA1 , MEx , MEy in crystal coordinate system (x, y, z) to the SiV basis coordinate system (X, Y, Z). (σIJ) is stress ( ) tensor expressed in SiV coordinate system and σij is the one in crystal coordinate system.

◦ ◦ They are related by the unitary coordinate transformation matrix Q = Qy(54.7 )Qz(45 ), where Qz(θ), and Qx(φ) correspond to two dimensional coordinate transformation rotations by

θ and φ about the z- and x-axes respectively.

( ) T (σIJ) = Q σij Q (2.9)

Substituting expressions for individual components of σ from the above formula in Eq. (2.7),

45 Stress term Effect Stress susceptibility σXX + σYY Splitting ms = 0 and ms = ±1 (A1 − A2) σZZ Splitting ms = 0 and ms = ±1 (A1 + 2A2) − − σXX σYY Mixing ms = +1 and ms = 1 √(B + C) σZX Mixing ms = +1 and ms = −1 2(C − 2B) − − σXY Mixing ms = +1 and ms = 1 √ 2(B + C) σYZ Mixing ms = +1 and ms = −1 2(C − 2B)

Table 2.1: Various stress-modes, and their effects on the ground state spin sublevels. we get

− MA1 = (A1 A2)(σXX + σYY) + (A1 + 2A2)σZZ √ − − MEx = (B + C)(σXX σYY) + 2(C 2B)σZX (2.10) √ − − MEy = 2(B + C)σXY + 2(C 2B)σYZ

Table (2.1) describes the effects of various modes of stress on the ground state spin sublevels, and the stress-response coefficients (susceptibilities) of these modes.

We now use Eq. (1.16) to write the relations in Eq. (2.7) in terms of strain-tensor components. Note that Eq. (1.16) is in the basis of crystal coordinate system. Therefore, we tem- porarily switch back to (x, y, z) coordinate system to use Hooke’s law as it is expressed in Eq.

75 (1.16). For diamond, we have c11 = 1075 GPa, c12 = 139 GPa, c44 = 567 GPa. After conversion, we obtain the following strain-relations similar in form to the stress-relations, but with different coefficients.

46 A A MA1 = 1(εxx + εyy + εzz) + 2 2(εyz + εzx + εxy)

B − − C − − MEx = (2εzz εxx εyy) + (2εxy εyz εzx) (2.11) √ √ B − C − MEy = 3 (εxx εyy) + 3 (εyz εzx)

The coefficients A1, A2, B, C in terms of the coefficients A1, A2, B, C in Eq. (2.8) are given by

A1 = (c11 + 2c12)A1

A2 = c44A2

B = (c11 − c12)B (2.12)

C = c44C

( ) Just as we did for the stress tensor, we can rotate the strain tensor εij into the SiV basis to get relations similar to those in Eq. (2.10). We write these below.

47 Strain term Effect Susceptibility Relation to stress susceptibilities εXX + εYY Splitting ms = 0 and ms = ±1 t⊥ A1 − A2 = (c11 + 2c12)A1 − c44A2 εZZ Splitting ms = 0 and ms = ±1 t∥ A1 + 2A2 = (c11 + 2c12)A1 + 2c44A2 − − B C − εXX εYY Mixing ms = +1 and ms = 1 d √ + = (√c11 c12)B + c44C εZX Mixing ms = +1 and ms = −1 f 2(C − 2B) = 2(c44C − 2(c11 − c12)B) εXY Mixing ms = +1 and ms = −1 −2d εYZ Mixing ms = +1 and ms = −1 f

Table 2.2: Decomposition of strain susceptibilities. Relation to stress susceptibilities is given.

MA1 = t⊥(εXX + εYY) + t∥εZZ

− MEx = d(εXX εYY) + fεZX (2.13)

− MEy = 2dεXY + fεYZ

where t⊥, t∥, d, f are the four strain-susceptibility parameters that completely describe the strain-response of the ground state of the NV center. Note that the transformation from stress to strain Hamiltonian could have been done as well by directly using Hooke’s law but with the elasticity tensor in NV basis coordinate system. Table 2.2 shows the strain-susceptibilities in terms of coefficients used earlier in this Section, and defines the effect of corresponding strain-modes on the ground state triplet of the NV center.

48 2.5 AC strain induced ESR broadening

Including Eq. (2.3) in the Hamiltonian of the ground state triple of NV center, we have the following expression.9,27,86,114

( ) ( ( 2 2 1 2 2 H = (D + M ) S − + γS · B − S M + iM ) + S−(M − iM )) (2.14) 0 A1 z 3 2 + Ex Ey Ex Ey

Here Si are the S = 1 Pauli spin operators. D0 = 2.87 GHz is the zero-field splitting between ms = 0 and ms = ±1 levels due to spin-spin interaction, γ = 2.8 MHz/G is the gyromagnetic ratio for the NV ground state. At small B-fields (≪ D0/γ), the NV-axis is the spin quantization axis (z−axis in the above Hamiltonian). The perturbative strain terms lead to frequency shifts in the ms = ±1 levels respectively given by

√ 2 Δω± = M ± (γB ) + M2 + M2 (2.15) A1 z Ex Ey

Physically, Eq. (2.15) reveals that A1 strain leads to a linear modification of the zero-field splitting, while E strain mixes the ms = ±1 states, thereby causing a quadratic splitting between them. In a mechanical resonator driven at the frequency ωm, the local strain components of the tensor (εIJ) oscillate at the frequency ωm. From the strain susceptibilities measured,86,114 and finite element calculations on our structures, we anticipate a frequency modulation by A1g strain comparable to the ESR linewidth, when the mechanical mode is driven to

49 an amplitude of ≈500 nm.

At the chosen nitrogen ion implantation density, we expect ∼ 10 NV centers within our confocal laser spot. ESR measurements are performed on such an NV ensemble at a fixed position in the cantilever, and simultaneously, the flexural mode shown in Fig. 1c is driven by supplying an RF voltage to the piezo actuator at the resonance frequency ωm. A small static magnetic field Bz= 4 G is applied with a bar magnet placed outside the cryostat, and only the ms = 0 to ms = +1 transition is probed. The external magnetic field is aligned exactly vertically to ensure that all four NV classes experience the same projection Bz along their re- spective axes. The cantilever itself is fabricated such that its long axis is aligned to the ⟨100⟩ crystal axis to within a few degrees as determined by electron back scatter diffraction (EBSD).

As a result, all four NV classes are symmetrically aligned with respect to the dominant strain component of the flexural mode, which occurs along the cantilever long axis. Thus, at a given location in the cantilever, all four NV classes experience the same axial and transverse strain amplitudes, and hence experience identical transition frequency modulation. Effects of inhomogeneous coupling strength due to implantation straggle, and varying lateral position within the confocal laser spot are addressed in Appendix A. Low microwave power was used to prevent power broadening, and retain near native linewidths in the ESR.

Fig. 2.1e shows ESR spectra at the same location in the cantilever for progressively increasing mechanical amplitude. At the lowest piezo drive power of -21 dBm, we observe three dips spaced equally by 2.2 MHz corresponding to the hyperfine structure arising from interaction between the NV electron spin and the 14N nuclear spin (Fig. 2.1d). This was found to

50 be identical to the ESR spectrum with no piezo drive (not shown). For each of the hyperfine transitions, we measure a linewidth of ≈2 MHz. As the piezo drive power is increased to -12 dBm, we observe a broadening of the hyperfine features to the point where the hyperfine structure is barely resolvable. At -6 dBm, the hyperfine structure is washed out, and at 0 dBm, the overall ESR dip is even broader. Such broadening of the ESR signal with progressively larger mechanical amplitude is expected, since the measurement sequence involves dwelling at each microwave frequency sample for many (> 106) cycles of the mechanical oscillation period.

As a result, we would expect to average over the AC modulation of the microwave transition, and detect an overall broadening determined by the modulation amplitude.

To verify that the ESR broadening arises from the mechanical mode, we perform the same measurement at a fixed drive power of -12 dBm, and multiple drive frequencies around resonance. The slight asymmetry in the driven mechanical response (Fig. 2.2a) can be attributed to the onset of a Duffing-type nonlinearity at this drive power. When driven far off the mechanical resonance as in points 1 and 5 in Fig. 2.2b, the ESR spectrum retains three clear hyperfine dips with linewidths close to the native linewidth. At smaller detunings as in points 2 and 4 in 2.2b, we observe a broadening of the individual hyperfine features. When driven exactly on resonance as in point 3 in Fig. 2.2d, the hyperfine features are on the verge of being washed out. Thus, the ESR broadening effect follows the frequency response of the mechanical mode. To further confirm that this is a strain-induced effect, we repeat the measurement at multiple points along the length of the cantilever for a fixed piezo drive power of -6 dBm.

From the strain profile of the flexural mode (Fig. 2.2c), we expect a roughly linear variation

51 a b 250 0 3 1 200 2 150 3 1 2 4 Contrast (%) 4 100 2.88 2.885 2.89 1 5 50 0 amplitude (nm) 0 1 936.9 937 937.1 937.2 937.3 937.4 937.5 2 Cantilever displacement Driving frequency (kHz) 3 2 Contrast (%) 4 c 2.88 2.885 2.89 0 0 1 2 1 Tip 2 3 3 3 Contrast (%) 4 Contrast (%) 2.88 2.885 2.89 2.88 2.885 2.89 MW frequency (GHz) 0 1 2 3 4 Contrast (%) 4 0 2.88 2.885 2.89 1 0 Clamp 2 1 3 2 Contrast (%) 2.88 2.885 2.89 3 5

MW frequency (GHz) Contrast (%) 4 2.88 2.885 2.89 MW frequency (GHz)

Figure 2.2: (a) Driven response of the cantilever at a piezo drive power of -12 dBm. The drive frequencies used for frequency dependent broadening measurements are indicated with numbers 1-5. (b) ESR spectra at the same location in the cantilever at mechanical drive frequencies 1-5 indicated in (a). (c) ESR spectra at the tip and clamp of the cantilever for -6 dBm drive power. Strain proﬁle of the mechanical mode from an FEM simulation for the corresponding displacement amplitude is also shown. Open circles in each ESR spectrum are measured data, and smoothed lines serve as a guide to the eye

52 in AC strain amplitude from its maximum value near the clamp of the cantilever to zero at the tip of the cantilever. This effect is observed in the form of ESR broadening for NVs near the clamp, and retention of native linewidths for NVs at the tip (Fig. 2.2c).

2.6 Temporal dynamics of the mechanically driven spin

The ESR broadening measurements in Figs. 2.1e and 2.2 provide strong evidence of strain from the driven mechanical mode coupling to the NV spin. From the washing out of hyperfine structure in the measurements, we can deduce driven coupling rates of the order of the hyperfine splitting (2.2 MHz). In order to probe the temporal dynamics of the NV spin due to mechanical motion, and precisely measure the coupling strength, we employ spin echo measurements. It has been shown in previous demonstrations that the two distinct modes of level shifts generated by axial and transverse strain can be used to achieve dispersive86 and resonant interactions8,68,69,70 of the spin with mechanical motion, respectively. In our work, the frequency of our mechanical mode (≈1 MHz) is smaller than the ESR linewidth, and we will focus on the dispersive regime provided by axial strain. We apply a moderate static magnetic field, and suppress the effect of transverse strain to first order as evinced by Eq. (2.15). In this regime, if we work with the effective qubit defined by the ms = 0 and ms = +1 levels, driven motion of the mechanical resonator can modulate the phase of our effective qubit at the frequency ωm, analogous to an AC magnetic field. This is described by the time dependent

53 Hamiltonian

Hint(t) = 2πGcos (ωmt + φ) Sz (2.16)

Here G = MA1 is the AC strain coupling rate from the driven motion, φ is an arbitrary phase offset, and Sz is the corresponding S = 1/2 Pauli spin operator.

In our spin-echo measurements, we apply an external static magnetic field Bz= 27 G. As in the case of ESR measurements, the magnetic field is aligned to ensure equal Zeeman splittings for all four NV orientations. Our experimental sequence is shown in Fig. 2.3a, wherein the piezo drive signal, and hence the strain field, has an arbitrary phase φ with respect to the microwave pulses that varies over multiple iterations of the sequence. The spin echo signal obtained from this measurement (Fig. 2.3b) at a piezo drive power of 0 dBm shows a periodicity corresponding to twice the time period of the mechanical mode. The theoretically expected spin echo signal in this measurement has the form of a zero order Bessel function with a periodic argument.64,86

[ ( ( ))] 1 3 8πG ω τ −(2τ/T2) 2 m p(2τ) = 1 + e J0 sin (2.17) 2 ωm 4

The exponential damping term multiplying the periodic function corresponds to dephasing of the NV electron spin due to interactions with the surrounding 13C nuclear spin bath in dia-

16 mond, and T2 is the dephasing time. In our experiments, T2 ≫ the mechanical oscillation period (2π/ωm), and the effect of spin decoherence is relatively small. A fit to Eq. (2.17) yields

54 a 532 nm Microwave τ τ

ms=0 to +1 π/2 π π/2 Mechanical drive Photon counting Readout Reference b ms = +1 G||=d||εzz 1

=0 D +γB

s 0 z m

0.8 ms = 0

0.6

Population in 0.4

0.2 0 0.5 1 1.5 2 2.5 3 3.5 4 Total time, T = 2τ (μs)

Figure 2.3: (a) Experimental pulse sequence for spin echo measurement of dispersive spin-cantilever interaction due to axial strain (b) Spin echo signal from NVs in the cantilever at a piezo drive power of 0 dBm (tip amplitude of 559±2 nm) for the mode at ωm = 923.4 kHz, showing two periods of the modulation due to axial strain coupling. The solid line is a ﬁt to Eq. 2.17. Vertical error bars correspond to photon shot noise in the measurement. Inset shows schematic of dispersive interaction between the qubit and mechanical mode due to axial strain.

55 0 dBm -3 dBm -6 dBm -9 dBm -12 dBm

1.0

0.3 =0 s

Population in m

0 0.5 1 1.5 2 Total time, T = 2τ (μs)

Figure 2.4: Spin echo at the same location in the cantilever for varying piezo drive powers, offset along y-axis. One period of the signal is plotted in each case. Solid lines are ﬁts to the zero order Bessel function form indicated in the text, and vertical error bars correspond to photon shot noise. The scale bar on the side is a guide for the y-axis indicating maximum (1.0) and minimum (0.3) possible population in ms = 0 as dictated by Eq. 2.17. The legend on right side indicates the piezo drive powers used for each measurement.

ωm = 2π × 918.7 ± 5.6 kHz, which is in reasonable agreement with the driving frequency of

923.4 kHz used in the experiment. The extracted driven coupling rate G = 2.10 ± 0.07 MHz is of the order of the hyperfine splitting as observed in our ESR broadening measurements.

Finally, we performed spin echo measurements at the same location on the cantilever for

56 varying piezo drive powers (Fig. 2.4). The no-drive spin echo signal (not shown) is flat, indicating that this measurement is not sensitive to the thermal motion of the cantilever mode

(estimated to have an amplitude of 0.53 nm). As the cantilever is driven, the spin echo signal begins to show a dip when the evolution time 2τ equals the mechanical oscillation period

(2π/ωm). At larger amplitudes, the spin precesses by more than one full rotation on the equa- tor of the Bloch sphere, and we observe higher order fringes within one period of the signal.

These drive-power dependent measurements further allow us to verify that the axial strain coupling is linear in the displacement amplitude (Fig. 2.5). From the linear fit, we infer a displacement sensitivity dG/dx = 4.02 ± 0.40 kHz/nm. By estimating the zero point motion of the mode from its effective mass, this displacement sensitivity yields a single phonon coupling strength, g = 1.84 ± 0.18 Hz for an NV at the clamp of this cantilever. Compared with recent demonstrations of NV-strain coupling, this is about two orders of magnitude larger than the value first measured,86 and an order of magnitude larger than another result in the literature.8,114

2.7 Conclusion and outlook for strong coupling

In conclusion, we demonstrate nanoscale diamond cantilevers for strain-mediated coupling of NV spins to mechanical resonators. The relatively small dimensions of our devices offer a significant improvement in the single phonon coupling strength compared to previous work with mechanical modes at similar frequencies.

With further developments in the future, magnetometry applications can benefit from an

57 2.5

1.5

1 G from spin echo (MHz) 0.5

0 0 100 200 300 400 500 600 Cantilever displacement amplitude (nm)

Figure 2.5: Variation of driven spin-phonon coupling rate due to axial strain (G) with the calibrated displacement amplitude of the mechanical mode. The five data points correspond to the piezo drive powers used in plot (a) in increasing order. Vertical error bars correspond to the error in each G estimate from fitting to the spin echo function given by Eq. 2.17. Solid line is a linear fit, which yields dG/dx = 4.02 ± 0.40 kHz/nm.

NV ensemble coupled to a mechanical resonator.9 In particular, collective enhancement from a dense spin ensemble can boost the co-operativity by a factor of the number of spins N,53 allowing one to work with less demanding device dimensions favorable for NV photostability and high mechanical Q factors.

To realize phononic quantum node, it is essential to meet the basic requirement in Eq. (2.1).

Shorter cantilevers will boost g even further according to the scaling behavior shown in Eq.

58 (2.2). This will also increase the mechanical frequencies, and allow operation in the sideband resolved regime with access to the resonant spin-phonon interaction provided by transverse strain coupling.8,68,69,70 As suggested by our estimates in the introduction, nanostructures that allow strong coupling have mechanical frequencies in the few hundreds of MHz range.

However, further scaling of NEMS accompanies significant engineering challenges on many fronts. First of all, we need to maintain photostable NVs in nanostructures with widths on the order of hundreds of nanometers or smaller. It is well known that etched surface is in general produce ‘noisy environment’ for NV centers.25 With smaller dimenions, NV centers inevitably are positioned near etched surfaces and its photostability may be questionable. For the same reason, spin coherence time is known to deteriorate near the surface so that it can be a potential problem.59,60

Smaller mechanical structures cause challenges in actuation capability as well. As for actuation, it is difficult to use an stand-alone piezoelectric stack attached to the chip for high frequency application. Furthermore, this configuration is not a scalable solution for the future phononic quantum network. At the same time, scaling mechanical resonators typically compromises its quality factor and hence the cooperativity as well.54 In Chapter 3, we demonstrate the diamond NEMS devices with dynamic actuation while trying to keep small mode volume and high quality factor.

Alternatively, since the framework of strain coupling outlined above is fairly general, the same devices may be used, but with a different qubit, whose energy levels have a larger strain response. Use of NV center’s excited electronic state has been used to demonstrate large strain

59 susceptibility.67 Although this system has large strain susceptibility, it compromises coherence time of the qubit due to the large energy gap. In this contexts, we study another system,

SiV center, in Chapter 4 whose spin qubit has about 5 orders of magnitude larger strain susceptibility compared to that of NV center.

60 3 Dynamic actuation of single-crystal

diamond nanobeams

3.1 Background and motivation

Owing to its large Young’s modulus, excellent thermal properties, and low thermoelastic dissipation, single-crystal diamond (SCD) is a promising candidate for realization of high fre-

61 quency (f ) and high quality factor (Q) mechanical resonators. SCD is also a promising platform for applications in quantum information science and technology due to the color centers which can be embedded inside.89 In particular, the negatively charged nitrogen vacancy (NV) color centers can be used as qubits with optical readout due to their long coherence times (milliseconds) even at room temperature.6 For example, coupling between an NV center and a mechanical resonator may enable high fidelity control of NV center spin state via rapid adiabatic passage,69,70 and potentially the remote coupling of distant NV centers via mechanics.89 As discussed in Section 1.3.2, mechanical resonators may enable coherent coupling between systems with degrees of freedom possessing dramatically different properties and energy scales.

3.2 Device description

3.2.1 Operating principle

Recalling the discussion in Sections 1.4.3 and 2.7, one of the key challenges to realize phononic quantum node is to scale down NEMS devices while keeping high quality factor. Here, we demonstrate nanoscale resonators with high f Q product in SCD. To drive the resonators we use dielectrophoretic actuation,117 which allows us to realize NEMS at a frequency range of

1-50 MHz with flexural mechanical modes. Dielectrophoresis has been used in the past to achieve mechanical resonance tuning,91 coherent control of classical mechanical resonators,36 cavity electromechanics,35 and nonlinear mechanics.116 In our approach, on-chip metal electrodes are fabricated on either side of SCD nanobeam cantilevers (Fig. 3.1a) and doubly

62 a b

1 µm 1 µm

c [V] d nm]

y [ Au Au

x [nm]

Figure 3.1: SEM images of (a) 4 μm long cantilever and (b) 7 μm doubly clamped beam. (c) Finite element method (FEM) simulations are used to calculate the force applied to suspended nanobeams with a given geometry and electrostatic environment. The color map indicates potential with respect to the right-hand Au electrode and the streamlines show the corresponding electric ﬁeld. (d) Vertical force per unit length applied to such beams in the case of 20V of DC voltage is plotted as a function of beam width and distance above the electrode. Separation between electrodes is the sum of beam width and 50 nm margin on either side. Beam height is the distance between top surface of the beam and the electrode center in vertical axis. clamped nanobeams (Fig. 3.1b). Fringing electromagnetic fields of an RF drive the diamond devices (Fig. 3.1c), with optimal actuation occurring when the RF frequency is resonant with the mechanical mode. Our numerical modeling indicates that it is crucial that the vertical distance between the metal electrodes and diamond nanobeam is small in order to achieve efficient actuation (Fig. 3.1d).

Other actuation schemes for nanomechanical resonators have been demonstrated previously, including electrostatic and piezo-electric actuation approaches.32 These, however, require deposition of a conductive thin film or electronic doping on the moving part of nanomechanical structure, because undoped diamond is neither conductive nor piezoelectric. These

63 can reduce mechanical Q-factors54 and negatively impact spin and optical degrees of freedom of color centers embedded inside diamond. The latter are known to be sensitive to the fabrication imperfections and surface terminations.18 Forces resulting from gradient electromagnetic fields, on the other hand, do not require any modifications to the diamond mechanical resonator. Therefore, the dielectrophoresis scheme does not add additional mechanical loss chan- nels. One caveat is that careful design of device geometry is required for the dielectrophoretic actuation, because its force at a given voltage is much weaker than other methods.

a Diamond HSQ (i) (ii) (iii) (iv) PMMA MMA (v) (vi) (vii) (viii) Gold

b c

1 µm 5 µm

Figure 3.2: (a) Schematic illustration of angled-etching nanofabrication approach used in this work:(i) Elec- tron beam lithography mask is deposited, (ii) top-down reactive ion etching of diamond is performed, followed by the (iii) angled-etching step and (iv) mask removal. (v) New electron beam resist is spin coated, and (vi) electron beam lithography followed by (vii) metal evaporation and (viii) lift-off are used to deﬁne electrodes. (b) High magniﬁcation SEM image of 4 μm cantilever shows that good alignment can be achieved. (c) SEM image of device array sharing electrodes.

64 3.2.2 Device fabrication

The fabrication scheme for realizing diamond NEMS is shown in Fig. 3.2a. Diamond nanocantilevers and doubly clamped nanobeams are first fabricated using our recently developed angled-etching technique, described in detail elsewhere.12 Briefly, angled-etching employs anisotropic oxygen plasma etching at an oblique angle to the substrate surface, yielding suspended triangular cross-section nanobeams directly from single-crystal bulk diamond sub- strates. To ensure efficient actuation by dielectrophoresis, the diamond nanobeam width and distance between the substrate and the bottom apex of the triangular nanobeam cross-section must be carefully chosen (Fig. 3.1c and d). Once free-standing diamond nanobeams are fabricated, metal electrodes are patterned on the diamond substrate via lift-off process. First, the diamond substrate is spin coated with a polymethylmethacrylate-copolymer (MMA/PMMA) bi- layer resist, where the MMA copolymer thickness is chosen to be slightly thicker than the distance between the nanobeam top surface and the substrate. After resist coating, exposure and alignment are done with electron beam lithography. After developing the resist, an adhesion layer of 50 nm titanium and a 200 nm thick gold layer are evaporated on the surface by electron beam evaporation. Lift-off in Remover PG completes electrode patterning. Fig. 3.2b is a top-down SEM image of a diamond nanobeam cantilever with gold electrodes fabricated on either side. We observe very good alignment of the electrodes to the diamond nanobeam, with alignment errors on the order of tens of nanometers. In fact, the slight misalignment enables the actuation of diamond nanobeam in-plane motion.91 Fig. 3.2c shows an array of fabricated

65 diamond doubly clamped nanobeam mechanical resonators that share driving electrodes. This configuration allows us to characterize in parallel a large number of resonators having slightly different geometry and hence different mechanical resonance frequencies.* Our diamond nanomechanical resonators had a width range between 200 nm and 300 nm and lengths between 1 μm and 20 μm, corresponding to fundamental flexural resonance frequencies ranging from a few MHz to hundreds of MHz. We note that due to the nature of our angled-etching fabrication technique, the width and thickness of the nanobeam triangular cross-section are correlated.14

3.3 Device characterization

All experiments were performed at room temperature, with our wire bonded diamond substrate in a vacuum chamber, held at a pressure below 10−4 Torr. Fig. 3.3 shows a schematic of the optical interferometry characterization setup32 used to read out the nanomechanical motion.† Sending RF signals for actuation and read-out at corresponding frequencies was done in a transmission measurement with a Vector Network Analyzer (VNA). The VNA was replaced with a real-time spectrum analyzer for those measurements (e.g. measuring thermal fluctuation of the nanobeam) which did not involve actuation, and the parametric actuation measurement that we discuss later. A bias-tee was also included to combine a DC bias with the RF *This array of devices made on a separate chip with different electrode configurations from other devices presented in this work. †One particular experiment (Fig. 3.4) was performed with the separate setup, which is a path- stabilized Michelson interferometer. The reason is to have the same sensitivity with less incident laser power, which makes the measurement more stable. In this setup, He–Ne laser with wavelength (λ ≈ 632 nm) was used.

66 Telecom Laser Flip White Light mirror

BS Vacuum Objective chamber Bias Tee PD CCD

Motorized VNA stages

Figure 3.3: Optical characterization setup. drive signal to ensure proper actuation, as the dielectrophoresis actuation force is proportional

2 117 to the square of applied voltage, F ∝ (VDC + VRF cos ωt) .

For the most of fabricated diamond nanobeams, both the fundamental out-of-plane mechanical flexural modes were characterized. Resonant responses of the fundamental out-of- plane motion of devices shown in Fig. 3.1a and b are plotted in Fig. 3.4 a and b, respectively.

Curves are the raw data, with both figures showing the expected resonant responses at low driving power as well as nonlinear response at higher driving power. 10V of DC voltage was

67 applied for the both measurements. The resonance frequency of out-of-plane mode that we could measure on 4 μm long cantilever was ∼18.3 MHz with the mechanical quality factor of

4.4 × 104. In the case of 7 μm long doubly clamped nanobeam, measured resonant frequency was ∼22.7 MHz with the mechanical quality factor of 2.0 × 103. For both devices, root mean square (RMS) amplitude of motion was thermomechanically calibrated by measuring thermal fluctuations.45 To do so, the value of effective mass is estimated from nanobeam’s geometry and analytic theory presented elsewhere.45 The width and length of nanobeams are measured via SEM, and the thickness is estimated from the ratio of out-of-plane and in-plane resonance frequencies of the cantilever.14

a b

Figure 3.4: Fundamental out-of-plane resonant response of devices shown in Fig. 3.1(a) and (b) are given in (a) and (b), respectively. Lorentzian frequency responses are shown at low driving power, and both beams start to enter nonlinear regime at higher driving power.

Highest resonant frequency measured with driven motions is as large as ∼50 MHz, although its thermal fluctuation was not detectable. Unfortunately, in our current experiments, we were not able to measure devices with resonances >50 MHz, due to the limited sensitivity

68 √ of our measurements (∼ 0.5 pm/ Hz). In our current characterization setup, the noise floor of our detection was affected by three different instruments: shot noise from laser source, dark current and thermal noise from the photodetector and thermal noise from the receiver. De- pending on the settings of instruments, any of these three could be the limiting factor for the detection noise floor. ‡

3.4 Discussion of results

In many MEMS / NEMS applications, a high f Q product is the key figure of merit. For example, in the case of mass sensing based on mechanical resonator, sensitivity scales with the square of its frequency and the quality factor determines the minimum detectable frequency shift. State-of-the-art flexural NEMS device can reach f Q product of 6.8 × 1012 Hz.118 In our devices, the maximum f Q product that we measured was 8.1 × 1011 Hz in the case of a 260 nm wide and 4 um long diamond nanobeam cantilever (Fig. 3.4).

For applications in quantum information science, coupling of NV center with mechanical resonator has been studied recently with various platforms.69,70,86,114 Assuming our device in

Fig 3.4 is used for such experiment, we estimate the relevant physical quantities below. When

RF power of −10 dBm is applied on its resonance, assuming that NV is implanted near the clamp at 10 nm depth, applied strain at the site of it is estimated to be 7.4 × 10−5 from FEM

‡When maximum telecom laser power of 20 mW was used,√ amount of power reaches the sample is < 5mW. In such a case, the setup had a sensitivity of ∼ 0.5 pm/ Hz. For the path-stabilized Michel-√ son interferometer, when < 400 μW reached the sample, it gave a similar sensitivity of ∼ 0.5 pm/ Hz. Signal-to-noise ratio can be always improved with the higher laser power, however, it accompanies the heating of the device and unstable measurement.

69 modeling. Estimated strain is large enough to induce significant coupling of NV ground-state spin with mechanical vibrations. For example, if z-axis of NV is perpendicular to the length direction of the cantilever, estimated strain corresponds to the coupling of 1.6 MHz.86

a b

Figure 3.5: (a) Tuning of mechanical resonance of doubly clamped beam using DC bias. With applying ±9V, frequency tuning range that can be achieved is approximately 260 linewidths. (b) Typical tongue shape of parametric instability was observed.

3.5 Additional capabilities: resonance tuning and parametric actuation

In addition to basic actuation capability, the dielectrophoretic actuation scheme can be used to tune the mechanical resonance frequency.117 This is because the actuation force has dependence on the displacement of the diamond nanobeam. Since the force has quadratic dependence on applied voltage, the amount of shift in the resonance frequency has quadratic dependence as well. Fig. 3.5a shows power spectral density (PSD) of the thermal fluctuations of a doubly clamped nanobeam (250 nm wide, 100 nm thick, 19 μm long), as the applied DC bias was changed from −9 V to +9 V. Bright spots observed in each data column correspond

70 to the resonance frequencies. The solid black line is a quadratic fit for applied DC bias and shows an excellent match with the data. In the given range of applied DC bias, the mechanical resonance could be tuned over roughly 260 full widths at half maxima of the resonance peak. We observed a blue shift of the diamond nanobeam resonance when DC voltage is applied which differs from observed red shift in the similar work.117 Upon further inspection, it was observed that our nanobeams are buckled down due to considerable amount of residual compressive stress (due to the diamond growth process).39 Therefore, the central part of nanobeams are positioned quite close to the top surface of the electrodes than design value, in which case the blue shift in resonance is expected.91

Since the resonance frequency is easily parametrically tuned much more than a linewidth, parametric excitation is also expected. When the spring constant of nanobeam is a function of the displacement, its motion can be modeled by Mathieu’s equation as shown below:

[ ] d2 Ω d + 0 + Ω2 (1 + α − 2Γ sin 2Ω t) x(t) = 0 (3.1) dt2 Q dt 0 0

where x(t),Ω0, Q, F(t) and m are the beam displacement, the mechanical resonance frequency, mechanical Q-factor, external driving force and effective mass of the resonator, respectively.

α is the detuning from parametric excitation and Γ is proportional to the parametric excitation

71 amplitude. The criteria for the onset of parametric instability is Ω0/Q = Γ. Mathieu’s equation can be analytically solved and the solution predicts its stability on a phase plane, axes of which are detuning and driving amplitude. Here, we show an “instability tongue”81 when a

71 doubly clamped diamond nanobeam is parametrically excited. In Fig. 3.5b the measured instability tongue is shown when the nanobeam (250 nm wide, 100 nm thick, 16 μm long) was excited around twice its natural frequency of ∼8.36 MHz, with 10 V of DC voltage applied together. In this experiment, excitation was applied by an RF signal generator and the response was measured with spectrum analyzer, with the amplitude of the driven motion thermomechanically calibrated.45 Parametric excitation is particularly interesting for NEMS devices since it can circumvent electric cross talk, which can be detrimental for nanoscale systems,37 and can be used to realize a NEMS oscillator119 and mechanical memory element.71

3.6 Conclusion and outlook

In summary, we have realized a resonant actuator based on dielectrophoresis for SCD nanomechanical resonators. Actuation of both the cantilever and doubly clamped diamond nanobeams was achieved for the fundamental out-of-plane vibrations. Our driving frequency range spanned from a few MHz to nearly 50 MHz, though higher frequency actuation is expected to be measured by a displacement read-out scheme with better sensitivity. Additional functionalities of the system are frequency tuning with DC bias and parametric excitation. The SCD actuation scheme we developed here is expected to be an excellent platform for coupling NV energy levels to mechanical degree of freedom. Additionally, control over diamond nanobeam mechanical motion by dielectrophoresis forces may be applied in the resonance tuning and modulation of recently demonstrated diamond optical cavities, in a manner similar to what has previously been demonstrated with silicon nanophotonic devices.10,26,38

72 Although the system presented in this chapter has shown some promising route towards high frequency actuation, small acoustic mode volume and reasonable quality factor, there are a couple of limitations. Since it is using flexural mode, reading out its motion when it scales down gets more challenging. Integration with piezoelectric films allows the actuation and readout even for hundreds of MHz flexural modes,71,72 however, it typically involves cryogenic temperature and complex electronics besides the overload of fabrication. Furthermore, when it comes to the integration with color centers, sensitivity to electric field needs to be carefully considered. Specifically, when fringing electric field is used to induce cantilever motions, electronic structure of the color center in the system should not be affected by it to avoid complications. This requirement is relieved when color centers with lower sensitivity to electric field. As will be seen, SiV center in Chapter 4 has zero first-order Strak effect so that it is a good candidate to be incorporated with our system presented here.

73 4 Controlling the coherence of a diamond

spin qubit through strain engineering

4.1 Background and motivation

The realization of a distributed quantum network requires long-lived memory qubits such as electron spins efficiently interfaced with indistinguishable optical photons.88 Perfect quantum

74 system that meets both requirements are atoms and ions, which are building blocks given by the nature. However, to implement a practical quantum system, it is necessary to control individual emitters in a scalable manner. Although incredible progresses have been made in the field of cold atoms and ion traps, scalable platform is a daunting challenge for these systems.

Viable alternative is solid-state quantum emitters with spin degree of freedom such as NV or

SiV centers.4 As an artificial atom trapped in solid, the system naturally offers extraordinary convenience for controlling the system in a scalable manner. As a price for scalability, solid- state quantum emitters have its own challenges because of the same reason why it is useful - trapping inside the solid-state environment.

In this Chapter, we are showing how two outstanding problems of solid-state emitters can be resolved. First one is the inhomogeneous distribution, where emitters emit photons at different wavelengths. Second issue is the decoherence of the spin qubit of SiV center, where the dominant mechanism is the thermal vibration of the host diamond lattice.55 In this work, we use a SiV center in diamond nano-electro-mechanical systems (NEMS) to solve both problems.

4.1.1 Inhomogeneous distribution

Identical particles in quantum mechanics result fundamentally important consequences such as exchange interaction43 or entanglement.50 In applications of quantum emitters, oftentimes it is crucial to produce identical photons from multiple emitters, or more adequately, ‘indistinguishable photons.’ Indistinguishability is absolutely necessary in linear optical quantum

75 computing (LOQC) as well as in building distributed quantum networks.

Indistinguishable photons must have the exact same features in many aspects, such as the complete matching of wavelength, space and polarization. Although polarization is easily controlled by letting photons pass through linear optical components, changing the wavelength of a single photon in a highly efficient manner is extremely challenging. Therefore, it is highly desirable to make the quantum emitters emit at precisely the same wavelength.

Atoms are natural building blocks of the universe and they are completely indistinguishable.

Therefore, photons from atoms have precisely the same frequency too. In contrast, solid-state quantum emitters usually emit photons at similar wavelengths but with a distribution over a small range. This is the so-called ‘inhomogeneous distribution,’ and it is a fundamental challenge that every solid-state quantum emitter has.

In the case of SiV center, it is an excellent system compared to others when it comes to inhomogeneous broadening. Statistics of the SiV center in high quality single-crystal diamond showed that inhomogeneous distribution of its optical transition is on the order of 10 GHz,33 which is the best of all known solid-state quantum emitters. Still, non-zero distribution of transition frequencies need to be overcome to generate indistinguishable photons in a scalable fashion.

Origin of the inhomogeneity is the different environment that each SiV sees, where the root of it is the spatial atomic arrangement near the emitter. This can be thought as the local strain of SiV and they are caused by internal stress of the material. In practice, it is very challenging to grow the material with zero built-in stress and diamond is not an exception. Stress of

76 the diamond sample by birefringence imaging shows stress distribution of the diamond sample.39 Non-zero stress in diamond sample can cause inhomogeneous distribution for quantum emitters as well as buckling of doubly-clamped diamond nanobeams.14

4.1.2 Fluctuating environment

To have a long coherence time, quantum emitters need to be well isolated form the environment. In the environment of solid-state qubits, there are multiple noise sources that can cause decoherence of the quantum system: electric, magnetic field fluctuations and lattice vibrations.44 Due to these dynamic fluctuations in the solid-state environment, it remains a challenge to identify a solid-state emitter that simultaneously offers excellent optical properties and long-lived qubit coherence.

There exist well-developed techniques to mitigate the impact of fluctuating electric18,23,59,60 and magnetic fields24,30,106,111 from the environment. However, one may ask more fundamental question: rather than countering or suppressing these fluctuations, can we just get rid of the source of the noise? Or can we use the system that is immune to noise? Here we use SiV in diamond that has beneficial features in both regards.

Let’s discuss magnetic field noise first. Typically, solid-state quantum emitters can be used for two quantum systems that is coherently combined. Orbital degrees of freedom with optical transition frequency offers interface for building long-distance quantum network by emitting or absorbing photons, which can travel through the optical fiber or the free space. In contrast, spin degree of freedom in the ground orbital can serve as a long-lived quantum memory. Typi-

77 cally, a single spin qubit has the transition frequency on the order of GHz compared to optical transition frequency on the order of hundreds of THz. Since spontaneous decay rate of the emitter is proportional to the cubic of the transition frequency in homogenous material, lifetime of excited state orbital is typically limited to be on the order of a few of nanoseconds.

However, spontaneous decay rate of spin qubit is extremely slow such that its lifetime is typically limited by other noise sources.

Since a spin can be seen as a small magnet, oftentimes fluctuating magnetic fields are the dominant cause for the relaxation or decoherence. In general, inside the solid there are numerous number of atoms that may have spin degree of freedom that can randomly fluctuate. This is why diamond can be an excellent host material for quantum emitters. Pure single-crystal diamond is made of mostly 12C atoms. Since 12C has zero spin, it does not generate magnetic fields, and therefore we expect zero magnetic field in pure single-crystal diamond with only

12C atoms. In practice, however, there are many defects in the material especially the nitrogen atoms that occur naturally as well as 13C isotopes that carry nuclear spin. Therefore, realizing a good magnetic field environment for quantum emitters boils down to making pure single- crystal diamond with as little defects and isotopes as possible.

There has been a great progress in chemical vapor deposition (CVD) of the single-crystal diamond and currently diamonds with the nitrogen concentration below 5 ppb are commercially available. It can be further engineered to reduced the amount of 13C as well. Natural abundance of 13C limits the 12C content to be 98.9%, and nuclear spins from those work as a decoherence source for the ground level spin of the NV centers. By using isotopic enrichment

78 techniques it is possible to synthesize the single-crystal diamond with the 99.7 % composi- tion of 12C, where the spin coherence time of NV center is extended compared to the diamond sample with natural abundance of the 13C isotopes.6 Among all materials, single-crystal diamond is one of the purest host in terms of magnetic field environment and such feature led to the fast growth in the research of NV center. Nitrogen vacancy centers give the excellent spin coherence time reaching a second even at room temperature, when the dynamic decoupling is applied.7

Although NV center offers the best electron spin qubit trapped in solid at room temperature, its orbital degree of freedom has been posing a challenge that is extremely difficult to overcome. Due to a small fraction of its zero-phonon line even at a cryogenic temperature,17 it is important to place the NV center in the optical cavity whose resonance frequency matches the optical transition to use Purcell enhancement. Otherwise, the emission of photons with coherent quantum information is too slow to use for a practical quantum network. There has been attempts to build the quantum node with NV center in the diamond optical cavities, however, etched surface of diamond has been posing a challenge to realize a stable NV emission.18

The root of such challenge is twofold. First, there is electric field noise source that is extremely difficult to control. There has been efforts to remove the noise source,59,60 however, it only works up to a ceratin degree. Second root of the problem is that NV center itself. NV center has non-zero first order Stark effect for its dipole transition, fluctuating electric fields led to random changes in orbital transition frequency.18

In contrast, SiV center is known to have zero first order Stark effect for all of its orbital

79 levels.49 This feature of SiV center makes it an appealing platform for the optical interface in quantum networks. Even without getting rid of electric field fluctuations, the quantum system itself is insensitive to such environmental changes. The key difference between NV and

SiV centers come from their geometric configurations seen in the Fig. 1.3. In the case of NV center, nitrogen atom sits at one of the vacancy site where a carbon atom used to be located.

In contrast, a Si atom of SiV center is located in the middle of two vacancies due to the large

49 atomic size compared to that of the carbon. As a result, the NV center belongs to the C3v point symmetry group while SiV center belongs to the D3d point symmetry group. This difference leads to non-zero and zero permanent dipole moments for NV and SiV centers, respectively. Therefore, SiVs center are much more robust to electric field fluctuations compared to

NV centers.

Although both magnetic and electric field fluctuations are significantly mitigated in the case of SiV center in diamond, there is a remaining noise source that is unique to the solid- state emitters: thermal vibrations of the lattice. There are acoustic modes supported in solids from zero to optical frequency range in all bulk materials, and the concept of the particle in a quantized Hamiltonian is called phonon.62 Phonons are conceptually very similar to photons which are quantized particles of electromagnetic waves. In microscopic scale, phonon modes are simply collective motions of atoms. Furthermore, from the local defect’s point of view, phonon modes can be interpreted as temporal change of strain tensor. As pointed out in

Chapter 1.2, effect of strain on color centers induce changes in electronic structures.

At a finite temperature, phonon modes are thermally occupied following Bose-Einstein

80 statistics. Since thermal vibrations are random fluctuations, effect of thermal phonons can be a source of the noise. Unlike the existing techniques for reducing magnetic and electric field noise, suppression of decoherence from thermal lattice vibrations is typically achieved only by lowering the temperature of operation. Specifically, the impact of thermal phonons is irreversible, and fundamentally limits qubit coherence.

Phonons couple to solid-state emitters directly through periodic deformation of the electronic wavefunctions.108 Electron-phonon interactions are responsible for relaxation and decoherence processes in a variety of quantum systems.41,41,42,44,55,85 In particular, for systems with spin-orbit coupling, phonon-mediated processes can necessitate operation below typical liquid helium temperatures (4 K), and the use of more complex cryogenic setups in order to achieve long spin relaxation and coherence times.90,110,127 In contrast, our approach takes advantage of the large strain susceptibility of the orbital, which is also the limiting cause of the spin decoherence. This is counterintuitive, however: we indeed take the feature causing a trou- ble and convert it into a levererage. We use this property to quench the effect of the thermal phonon bath on a single electronic spin qubit without lowering the operating temperature.

4.2 Electronic structure of SiV

Our experiments are performed on the negatively charged silicon-vacancy (SiV) centre in diamond, an emerging platform for photonic quantum networks100 with remarkable optical properties owing to its inversion symmetry.101 This inversion symmetry is also responsible for the particular electronic structure of the SiV, shown in Fig. 4.1a, with similar ground-state

81 a b ￜ4〉 ES-splitting

Δes = 255 GHz 6000 C phonon transitions B ￜ3〉 4000

A B A C 2000 D D PLE count rate [Hz]

optical transitions 0 736.4 736.6 736.8 737 737.2 737.4 Mean ZPL wavelength Mean ZPL ￜ2〉 Wavelength [nm] GS-splitting γup γdown Δgs = 46 GHz ￜ1〉

Figure 4.1: (a) Electronic level structure of the SiV showing the mean zero phonon line (ZPL) wavelength, frequency splittings between orbital branches in the ground state (GS) and excited state (ES) (Δgs and Δes respectively) at zero strain, and the four optical transitions A, B, C, and D. Also shown are single-phonon transitions in the GS and ES manifolds. (b) PLE spectrum of a single SiV center. Each peaks in the spectrum corresponds to optical transitions marked in (a).

(GS) and excited-state (ES) manifolds, each containing two distinct orbital branches.49 Or- bital degeneracy in each manifold is lifted by spin-orbit coupling: |1⟩ , |2⟩ in the GS split by

46 GHz, and |3⟩ , |4⟩ in the ES split by 255 GHz in the absence of strain. Physically, each of the two branches in the GS and ES corresponds to the occupation of a specific E-symmetry orbital by an unpaired hole.49 Due to the inversion symmetry of the defect about the Si atom, the orbital wavefunctions can be classified according to their parity with respect to this inversion center. Thus, the GS configurations correspond to the presence of the unpaired hole in one of two even-parity orbitals |eg+⟩ , |eg−⟩, while the ES configurations have this hole in one of two odd-parity orbitals |eu+⟩ , |eu−⟩. Here the subscripts g, u refer to even and odd parity respectively, and +, − refer to angular momentum lz. This specific level structure gives rise to four distinct optical transitions in the ZPL indicated by A, B, C, D in Fig. 4.1a. These four

82 transitions can be experimentally probed by taking photoluminescence excitation (PLE) spectrum. By sweeping the laser around the orbital transition frequencies, SiV center is excited to a corresponding branch and re-emit photons, whose emission spectrum is consist of both zero phonon line (ZPL) and phonon sideband (PSB). By collecting the part of PSB by using band- pass filter, it is possible to measure the amount of absorption while rejecting photons from pump laser.

4.3 Device description

a b

10 μm 2 μm

Figure 4.2: (a) Scanning electron microscope (SEM) image of a representative diamond NEMS cantilever. Dark regions correspond to diamond, and light regions correspond to metal electrodes. (b) Confocal photoluminescence image of three adjacent cantilevers. The array of bright spots in each cantilever is ﬂuorescence from SiV centers.

The NEMS device used is a monolithic single-crystal diamond cantilever with metal electrodes patterned above and below it , as shown in the scanning electron microscope (SEM)

83 image in Fig. 4.2a. An opening in the top electrode allows optical access to SiV centers inside the diamond cantilever. SiV centers are located in an array as shown in Fig. 4.2b, precisely positioned by focused ion-beam (FIB) implantation of 28Si+ ions.95,112 Upon applying a DC voltage across the electrodes, the cantilever deflects downwards due to electrostatic attrac- tion. This results in controllable static strain directed predominantly along the long axis of the cantilever, oriented along the [110] crystal axis. The strain profile can be visualized in the finite-element-method (FEM) simulation in Fig. 1c. While there are usually four equivalent orientations of the SiV in the diamond crystal, application of strain with our device generates two distinct classes as indicated by the blue and red arrows in Fig. 1c.

4.3.1 Fabrication

The diamond NEMS (nano-electro-mechanical system) device used in this work was fabricated in three steps in the following order: (i) fabrication of bare diamond cantilevers, (ii) creation of silicon vacancy color centers, and (iii) deposition of electrodes. We use commercially available, ⟨100⟩-cut, ultra-high purity, single-crystal diamond (type IIa, nitrogen concentration less than 5 ppb) synthesized by chemical vapor deposition (CVD).

Cantilevers are fabricated in two steps. First, diamond with patterned electron-beam resist is etched vertically with oxygen plasma.46 These vertically-etched structures are then made free-standing by etching the sample at a tilted angle. Specifically, we employ an oxygen- plasma assisted ion-milling process in which the sample is mounted at an angle that is man- ually adjustable within a few degrees of precision. An illustration of this process, and the

84 [001] _ [110] 2 μm [110]

Strain Z

[110]

Figure 4.3: Simulation of the displacement of the cantilever due to the application of a DC voltage of 200 V between the top and bottom electrodes. The component of the strain tensor along the long axis of the cantilever is displayed using the colour scale. Crystal axes of diamond are indicated in relation to the geometry of the cantilever. Arrows on top of the cantilever indicate the highest symmetry axes of four possible SiV orientations, and their colour indicates separation into two distinct classes upon application of strain. SiVs shown by blue arrows are oriented along [111¯ ], [111¯ ] directions, are orthogonal to the cantilever long- axis, and experience strain predominantly in the plane normal to their highest symmetry axis. SiVs shown by red arrows are oriented along [111], [1¯11¯ ] directions, and experience appreciable strain along their highest symmetry axis. Inset shows the molecular structure of a blue-labelled SiV along with its internal axes, when viewed in the plane normal to the [110] axis. resulting suspended structure is schematically shown in Fig. 4.4a.The etching occurs over a period of a few hours, during which the stage is rotated constantly. Further discussion of these techniques can be found in the literature.12,13

After cantilever fabrication, silicon ions (Si+) are implanted at target spots on the cantilevers using a custom focused-ion-beam (FIB) system at Sandia National Labs. The spot

85 a (i) (ii) (iii) c

20μm

b (i) (ii) (iii) (iv)

(v) (vi) (vii) (viii)

Diamond ESPACER PMMA MMA Gold Tantalum

Figure 4.4: (a) Schematic of oxygen-plasma assisted ion-milling process for angled-etching of diamond cantilevers. The ion beam is directed at the diamond sample, with a vertically-etched device pattern. The tilted stage is continuously rotated during the etching process. After the cantilevers are freely standing, the etch- mask is stripped. (b) Fabrication process for the placement of electrodes. First, the coarsely aligned bonding pad is deﬁned with a bi-layer PMMA process followed by gold evaporation. Then the same process is repeated to deﬁne tantalum electrodes near cantilevers, but with better alignment precision. Conductive layer (ESPACER 300Z) on top of the cantilever is helpful for precise alignment. (c) SEM image of the complete chip showing connection between the bonding pad and electrodes on top of the cantilevers. size of the ion-beam on the sample is 40 nm, and is expected to determine the lateral precision of the implantation procedure. The beam energy is chosen to be 75 keV, which is predicted to yield a mean implantation depth of 50 nm with a straggle of 10 nm according to Stopping and

Range of Ions in Matter simulations. Further details of the FIB implantation procedure can be found elsewhere.95,100 After FIB implantation, the sample is subjected to a tri-acid clean

(1:1:1 sulfuric, perchloric, and nitric acids), and a three-step high-temperature high-vacuum annealing procedure18,33 in an alumina tube furnace. The annealing sequence followed com-

86 prises steps at 400◦C (1.5◦C per minute ramp, 8 hour dwell time), 800◦C (0.5◦C per minute ramp, 12 hour dwell time), and 1100◦C (0.5◦C per minute ramp, 2 hour dwell time). During the entire procedure, the pressure is maintained below 5 × 10−7 torr. Annealing generates a small amount of graphite on the diamond surface, which is subsequently etched away by a tri-acid clean. Following this step, we perform a cleaning in piranha solution to ensure a high level of oxygen-termination at the diamond surface. With regards to conversion efficiency, we implant approximately 50 Si+ ions per target spot on the sample, and typically generate 1-3

SiVs at each spot after annealing.

Subsequently, electrode patterns are made by a conventional bi-layer PMMA process followed by metal evaporation. Since the distance between the top surface of the cantilever and the bottom substrate is approximately 4 μm, bi-layer PMMA is spun multiple times until the cantilevers are buried completely. Patterns are written by electron-beam lithography, and metals are evaporated to define the electrodes. Detailed fabrication steps are schematically shown in Fig. 4.4b. Here, the triangle represents the cantilever, and the pedestal to the right of the triangle is the location of the bonding pad for electrical contact. The bi-layer PMMA process is repeated twice - first, to define the bonding pad, and second, to define the electrode pattern near the cantilever. This is because, we use a 200 nm thick gold layer for the bonding pad, but only a 10 nm thick tantalum (Ta) layer for the cantilever electrodes. Fig. 4.4c shows the scheme to connect electrodes on top of the cantilevers to the bonding pad on the diamond pedestal. Electrodes on the substrate below the cantilevers are connected to a second bonding pad (now shown in the figure) that is directly on the surface of the diamond.

87 We now discuss the choice of 10 nm tantalum film for our cantilever electrodes. For five different metals we have tested as a cantilever electrode material (aluminium, chromium, copper, titanium and tantalum), our device always shows a continuous, non-zero leakage-current upon applying voltage. While the exact reasons for this leakage-current are unknown, there have been numerous studies of the surface conductivity of diamond under various conditions.73,120,126 For aluminum, chromium, copper and titanium, this leakage current destroys the electrode when a high voltage (in the few hundred volts range) is applied. The destruc- tion of electrodes appeared to be the result of melting or bursting of the thin metal film, likely caused by Joule heating.15 Tantalum is one of the metals with the highest melting and evaporation points among those available for e-beam evaporation. We find that devices with tantalum electrodes are robust enough to operate at very high applied voltage (∼600 V across an electrode gap of approximately 4 μm, which corresponds to an electric-field of 1.5 MV/cm). The thickness of tantalum is kept below 10 nm in order to avoid thin film-induced stress in the cantilever, which leads to pre-strained SiV centers.

4.3.2 Device design

An important figure of merit for our NEMS device is the maximum achievable strain at the location of SiV. In this section, we discuss two key design aspects that need to be considered towards this goal: (i) ‘pull-in instability’, and (ii) practical limits for high voltage operation.

Pull-in instability is a well-known phenomenon for an electrostatic actuator made with a parallel plate capacitor as shown in Fig. 4.5a. In these devices, voltage is applied to induce

88 an electrostatic force between two plates, where either one or both of them are free to move.

Upon applying a voltage, the capacitor deforms until it reaches equilibrium, when there is a balance between the electrostatic force, and the restoring force exerted by the elasticity of the material. The net force acting on the free top plate in Fig. 4.5a can be modeled as

∂U(x, V) 1 V2 F(x, V) = − = ϵA − kx (4.1) ∂x 2 (d − x)2

where x is the displacement of the plate, U(x, V) is the potential energy and ϵ is the permittivity of the material between the two plates. A is the area of the capacitor, V is the voltage applied, d is the distance between the two plates at 0 V, and k is the spring constant, respectively. By integrating Eq. (4.1), we can calculate U(x, V) at various voltages as shown in Fig.

4.5b. The local minimum in the potential represents a condition of stable equilibrium. As the voltage is increased, the local minimum shifts towards the bottom plate, indicating that the top plate gets displaced downwards, thereby reducing the capacitor gap. When the voltage changes from 3V0 to 4V0 in the Fig. 4.5b, the stable local minimum disappears. This occurs, when the top plate is displaced by about one-third of the initial gap between the two plates, i.e. when x = d/3. At this point, the system reaches a condition in which the top plate snaps down to the bottom plate. Our device is a slight variation of this conceptual model, and hence, the maximum deflection of our cantilever will be limited by pull-in instability (but not at exactly x = d/3).

A 3D finite element method (FEM) calculation can be used to simulate pull-in instability

89 a c

k + V d -

b at SiV zz ε SiV V 0

2 V 0 3 V 0 z 4 V 0 potential energy displacement

Figure 4.5: (a) Illustration of a parallel plate capacitor with one freely movable plate (top), and one ﬁxed plate (bottom). The top electrode can be actuated by applying a voltage. (b) Potential energy of the system in (a) with the different voltages. The stable minimum in the potential disappears, when the system reaches the condition of pull-in instability at a voltage of 4Vo. (c) FEM simulation of the strain-component along the long-axis of the cantilever (most dominant strain-tensor component) near the clamp of the cantilever (inset). Turnaround points in the graph represent pull-in instabilities. accurately for complex structures. Typically, a simulation can be run by setting a particular voltage on the electrode, and solving for the resultant deformation of the structure, thereby arriving at the strain profile inside the cantilever. One can also run the inverse of this procedure. By setting a target displacement of the cantilever tip, the voltage required to achieve this displacement can be calculated. Such an inverse calculation can help arrive at the condition for pull-in instability. Fig. 4.5c shows the results of a simulation run so as to solve this inverse problem. Strain at the SiV location is plotted as a function of the voltage for different cantilever lengths. Turnaround points represent the pull-in instability condition at which both the displacement of the cantilever-tip and the applied voltage reach the maximum value pos-

90 sible before the cantilever snaps down. Fig. 4.5c provides two important conclusions: First, for a given voltage, longer cantilevers provide larger strain, because they have a smaller spring constant. Second, the maximum attainable strain is higher for shorter cantilevers, because they reach the pull-in instability condition at a higher voltage. Therefore, shorter devices are pre- ferred to generate high strain, when arbitrarily high voltage can be applied.

In practice, however, there are mechanisms that limit the maximum possible voltage e.g.

Townsend breakdown, field emission and surface current109,115,126, all of which can be significant depending on experimental conditions. With the fabrication method described in Section

4.3.1, our devices could be operated safely up to voltage as high as 600 V under high vacuum

(∼ 10−7 torr) at cryogenic temperature (4 K). Given that the minimum electrode gap is 4 μm, this condition corresponds to an electric field of approximately 1.5 MV/cm. Experiments described in Section 4.6 are carried out in a helium closed-cycle cryostat with the sample sur- rounded by helium exchange gas at a pressure of 1 mbar. Under these conditions, we observed safe operation up to 500 V. The maximum voltage in this setup is thought to be limited by dielectric breakdown of helium gas.

Considering all the design limitations discussed above, we chose cantilevers of width 1.2-

1.3 μm and length 25-30 μm for the experiments in this work.

4.4 Effect of strain on electronic structure

To understand the effect of strain on the SiV electronic levels, we the employ group theory as done in previous work on point defects. Using the resultant strain Hamiltonian, we explain the

91 experimentally observed strain responses in Figs. 4.6 and 4.7, and extract the susceptibility of the SiV to various strain components.

4.4.1 Strain Hamiltonian of SiV using group theory

It is shown in the literature that the stress tensor transforms as the irreducible representation

49 A1g + Eg, which has even parity about the inversion center of the SiV. Since the ground states of the SiV transform as Eg (even), and the excited states transform as Eu (odd), stress does not couple the ground and excited states with each other to first order. As a result, we can describe the stress-response of the ground and excited state manifolds independently. In particular, Hstrain is identical in form for both manifolds, but will involve different numerical values of strain-response coefficients. Therefore, we drop the subscripts g and u used to refer to the ground and excited states, and simply write the interaction Hamiltonian in the doubly- degenerate basis of E-states {|eX⟩ , |eY⟩}.

We follow the procedure in Section 2.4 for NV center. Minor difference is that the point symmetry group and basis set are different from those of NV center. Again, we write Eq.

(1.21) in their symmetry adapted form (D3d point group for SiV center).

Hstrain = Hstrain + Hstrain + Hstrain A1g Egx Egy (4.2) where Hstrain transforms as the irreducible representation Γ . ΓIR IR

Similarly, operators for each irreducible representations can be calculated in the same was

92 as in NV center, but now with the basis of {|eX⟩ , |eY⟩}.

  1 0 Hstrain =   A1g MA1g   (4.3) 0 1   −1 0 Hstrain   Egx = MEx   (4.4) 0 1   0 1 Hstrain   Egy = MEy   (4.5) 1 0

where components MA1g , MEgx , MEgy are the same as in Eq. (2.7), except with change of subscripts, 1 → g, x → gx and y → gy due to the difference in symmetry group between NV and

SiV centers. Also, the four coefficients A1, A2, B, C are the same with Eq. (2.8), except that

|κ⟩ and |λ⟩ both transforms as Eg here.

The two terms multiplied by A1 and A2 that make up MA1g are stress-modes that transform as A1g, and lead to a common mode energy shift of the {|eX⟩ , |eY⟩} orbitals. On the other

hand, the terms in MEgx and MEgy multiplied by B and C are stress-modes that transform as

Eg. They change the energy-splitting, and mix the orbitals respectively.

Going through the transformation from the crystal system to SiV axis, we get the same result as Eq. (2.10). We repeat the result below with updated subscripts for SiV center.

93 Stress term Effect Stress susceptibility σXX + σYY Common-mode shift (A1 − A2) σZZ Common-mode shift (A1 + 2A2) − σXX σYY Relative shift √(B + C) σZX Relative shift 2(C − 2B) − σXY Mixing √ 2(B + C) σYZ Mixing 2(2B − C)

Table 4.1: Various stress-modes, and their effects on the {|eX⟩ , |eY⟩} orbitals.

− MA1g = (A1 A2)(σXX + σYY) + (A1 + 2A2)σZZ √ − − MEgx = (B + C)(σXX σYY) + 2(2B C)σZX (4.6) √ − − MEgy = 2(B + C)σXY + 2(2B C)σYZ

Table 4.1 describes the effects of various modes of stress on the {|eX⟩ , |eY⟩} orbitals, and the stress-response coefficients (susceptibilities) of these modes.

Again, as was done for NV center, it is possible to express strain Hamiltonian in terms of strain tensor rather than stress tensor. We arrive at the final result of strain susceptibilities and summarize them in the Table 4.2, with the effect of each symmetry adapted, maximally decomposed strain components on the {|eX⟩ , |eY⟩} manifold.

94 Strain term Effect Susceptibility Relation to stress susceptibilities εXX + εYY Common-mode shift t⊥ A1 − A2 = c11A1 − (c12A1 + c44A2) εZZ Common-mode shift t∥ A1 + 2A2 = c11A1 + 2(c12A1 + c44A2) εXX − εYY Relative shift d B + C = (c11 − c12)B + c44C εXY Mixing −2d εZX Relative shift f C − 2B = c44C − (c12 + 2c11)B εYZ Mixing f

Table 4.2: Various strain-modes, and their effects on the {|eX⟩ , |eY⟩} orbitals.

MA1g = t⊥(εXX + εYY) + t∥εZZ

− MEgx = d(εXX εYY) + fεZX (4.7)

− MEgy = 2dεXY + fεYZ

4.4.2 Stress spectroscopy using NEMS device

By applying a specific DC voltage to electrodes fabricated around the cantilever (Fig. 4.2), we

can achieve the desired amount of static strain via deflection of the cantilever. We carry out

optical spectroscopy via PLE shown in Fig. 4.1 at cryogenic temperatures (4 K). Mapping the

response of these transitions as a function of voltage applied to the NEMS device allows us to

study the strain response of the SiV electronic structure.

The diamond samples used in our study are [001] cut on the top surface, and the direction

of the cantilever long-axis is [110]. In general, there are four equivalent orientations of SiVs

- [111], [1¯11¯ ], [111¯ ], [111¯ ] - in a diamond crystal, indicated by the four arrows above the can-

95 tilever in Fig. 4.3. For each SiV-class, this arrow represents the highest symmetry axis or the

Z-axis in its internal frame (see inset of Fig. 4.3). Upon deflection, the cantilever primarily applies uniaxial strain directed along [110]. This breaks the equivalence of the four SiV orientations, and leads to two classes indicated by the blue and red colored arrows in Fig. 4.3. By symmetry, within each class, we expect the same response to strain. The blue SiVs are [111¯ ],

[111¯ ] SiVs, oriented perpendicular to the cantilever long-axis, [110], which happens to be the

Y-axis in the internal frame of these SiVs (see inset of Fig. 4.1b). Thus we expect predominantly uniaxial strain along the internal Y-axis of these SiVs. On the other hand, the red class, which comprises [111], [1¯11¯ ] SiVs is not orthogonal to the cantilever long-axis. The strain- tensor of these SiVs in their internal frame is non-trivial, but unlike the blue SiVs, they are expected to experience a significant strain component along the internal Z-axis. For simplic- ity, we refer to blue SiVs as ‘transverse-orientation’ SiVs, and red SiVs as ‘axial-orientation’

SiVs. This nomenclature is used with the understanding that it is specific to the situation of predominantly [110] uniaxial strain applied with our cantilevers.

Two distinct types of response in optical transitions correlated with SiV orientation are observed with gradually increasing strain as shown in Figs. 4.6 and 4.7. The orientation of the

SiVs in the cantilever is inferred from polarization-dependence of their optical transitions at zero strain.49 Strain tuning of transverse-orientation SiVs shown in Fig. 4.6 leads to an increasing separation between the A and D transitions with relatively small shifts in the B and

C transitions. This behavior has been observed on a previous experiment with an ensemble of

SiVs.107 On the other hand, axial-orientation SiVs show a more complex tuning behavior in

96 16 280V

270V 14 260V

240V 12 220V

10 200V

175V 8 150V Normalized counts 125V 6 100V

4 75V 50V Z Si 2 25V

0V 0 735.5 736 736.5 737 737.5 738 738.5 Wavelength (nm)

Figure 4.6: Tuning of optical transitions of ransverse-orientation SiV (red in Fig. 4.3). Voltage applied to the device is indicated next to each spectrum. which all transitions shift as seen in Fig. 4.7.

To get an intuition of observed behavior, we transform the strain Hamiltonian (Eq. (4.2))

49 from {|eX⟩ , |eY⟩} to {|e+⟩ , |e−⟩} basis.

      ( ) −1  MA ME − iME  MA 0   0 ME − iME  strain strain  1g gx gy   1g   gx gy  H± = Tˆ H Tˆ =   =   +   (4.8)

MEgx + iMEgy MA1g 0 MA1g MEgx + iMEgy 0

97 280V 12

270V

10 260V

250V

8 225V

200V 6

Normalized counts 175V

4 150V

100V

2 Z Si 50V

0V 0 736.2 736.4 736.6 736.8 737 737.2 737.4 737.6 737.8 738 738.2 Wavelength (nm)

Figure 4.7: Tuning of optical transitions of ransverse-orientation SiV (blue in Fig. 4.3). Voltage applied to the device is indicated next to each spectrum. where Tˆ is the transformation given as below.49

  −1 −i   Tˆ =   (4.9) 1 −i

From the functional form shown in Eq. (4.7), it can be noted that matrix with only MA1g

transforms as A1g and matrix with MEgx and MEgy transforms as Eg. As noted in Section 4.2,

98 {|e+⟩ , |e−⟩} forms the basis of SiV’s electronic structure when no perturbation is applied.

Therefore, we can decompose the effect of the strain in Eg and A1g components. Eg strain mixes |e+⟩ and |e−⟩ and A1g strain incurs a common mode shift of them. Effect of these strains are schematically drawn in Fig. 4.8a and b. More rigorous mathematical treatment of the strain Hamiltonian including spin subspace is followed later in Section 4.4.3.

In the context of photon-mediated entanglement of emitters, photons emitted in the C line, the brightest and narrowest linewidth transition are of interest.100 Upon comparing Figs. 4.6 and 4.7, we note that this transition is significantly more responsive for axial-orientation SiVs.

Particularly in Fig. 4.7, we achieve tuning of the C transition wavelength by 0.3 nm (150

GHz), approximately 10 times the typical inhomogeneity in optical transition frequencies of

SiV centers.33 Thus, NEMS-based strain control can be used to deterministically tune multiple on-chip or distant emitters to a set optical wavelength. In particular, integration of this NEMS- based strain-tuning with existing diamond nanophotonic devices can enable scalable on-chip entanglement and a widely tunable single photon source. Besides static tuning of emitters, dynamic control of the voltage applied to the NEMS can be used to counteract slow spectral diffusion, and stabilize optical transition frequencies.74

By decomposing the strain applied by our device into A1g and Eg components, we can confirm the observations on tuning of transverse and axial orientation SiVs. To estimate strain at the SiV location, when a voltage is applied to the device, we run an FEM simulation with a geometry close to t hat of the fabricated device (dimensions are measured from SEM images).

The simulation gives the strain tensor, a 3-by-3 matrix, at any location inside the cantilever.

99 The resultant strain tensor is in the reference frame of the device, and can subsequently be rotated into the internal reference frame of SiV to gain more insight. For an SiV which is at a distance of 2.5 μm from the cantilever-clamp, and 50 nm below the surface, the simulated strain tensor was rotated into the internal reference frame for both transverse and axial orientation emitters, and plotted in Fig. 4.8c and d, respectively. As expected from the geometry in Fig. 4.3, transverse-orientation SiVs predominantly experience εYY, while axial-orientation

SiVs experience both εZZ and εYZ. This simulation is carried out at all applied voltages, and the computed strain tensors are used to fit our experimental data to the group theoretical strain- response model.

4.4.3 Estimation of strain-susceptibility parameters

Without any external perturbation such as strain or external magnetic field, spin-orbit coupling gives the following interaction Hamiltonian within ground and excited state manifolds,

49 expressed in the {|eX⟩ , |eY⟩} ⊗ {|↑⟩ , |↓⟩} basis.

   0 i  SO λSO   H = −   ⊗ σz (4.10) 2 −i 0

Here, λSO is the spin-orbit coupling strength within each manifold (46 GHz for the ground states, and 255 GHz for the excited states). σz is the S = 1/2 Pauli matrix that takes the spin degree of freedom into account. The strain Hamiltonian (Eq. 4.2) after including the spin de-

100 a b

Δes

Eg strain A ωZPL 1g strain

Δgs

c d Z Si Z Si Excited state splitting (GHz) e f 450 650 ε ε ┴ 737.2 || 350 550 Si 737.1 250 Si 450 150 737 350 50 736.9 250 0 0.5 1 1.5 2 0 0.2 0.4 0.6 0.8 1 -4 -4 Ground state splitting (GHz)

Mean ZPL wavelength (nm) A ε ( Eg strain, ε┴ (x 10 ) 1g strain, || x 10 )

Figure 4.8: Normalized strain-tensor components experienced by (a) transverse-orientation SiV (red in Fig. 4.3), and (b) axial orientation SiV (blue in Fig. 4.3) in the SiV co-ordinate frame upon deflection of the cantilever. (c) Variation in orbital splittings within GS (solid green squares) and ES (open blue circles) upon application of Eg-strain. Data points are extracted from the optical spectra in Fig. 2(a). Solid curves are fits to theory in text. (d) Tuning of mean optical wavelength with A1g strain. Data points are extracted from the optical spectra in Fig. 2(b). Solid line is a linear fit as predicted by theory in text. (e) Dominant effect of Eg-strain on the electronic levels of the SiV. (f) Dominant effect of A1g-strain on the electronic levels of the SiV.

101 gree of freedom can be expressed as

 

MA − ME ME  strain  1g gx gy  H =   ⊗ I2 (4.11)

MEgy MA1g + MEgx

Note that identity matrix for the spin in Eq. (4.11) implies that strain affects purely the orbital part of the wavefunction. Combining Eqs. (4.10) and (4.11), we get the following total

Hamiltonian.

   M − M 0 M − iλ /2 0   A1g Egx Egy SO       0 MA − ME 0 ME + iλSO/2  total SO strain  1g gx gy  H = H +H =      M + iλ /2 0 M + M 0   Egy SO A1g Egx    − 0 MEgy iλSO/2 0 MA1g + MEgx (4.12)

Diagonalization of the total Hamiltonian gives the following two distinct eigenvalues within ground and excited state manifolds, respectively. Each the energy eigenstate is doubly degenerate due to the spin degree of freedom.

( √ ) ( √ ) 1 2 2 2 1 2 2 2 E1 = 2MA − λ + 4M + 4M , E2 = 2MA + λ + 4M + 4M 2 1g SO Egx Egy 2 1g SO Egx Egy (4.13)

Eq. (4.13) is valid within both ground and excited state manifolds. Therefore, we can ex-

102 tract following quantities that can be directly measured via spectroscopy.

( ) − ΔZPL = ΔZPL,0 + MA1g,e MA1g,g ( ) ( ) = ΔZPL,0 + t∥,e − t∥,g εZZ + t⊥,e − t⊥,g (εXX + εYY) (4.14) √ ( ) Δ = λ2 + 4 M2 + M2 gs SO,g Egx,g Egy,g √ [ ] [ ] 2 − 2 − 2 = λSO,g + 4 dg(εXX εYY) + fgεYZ + 4 2dgεXY + fgεZX (4.15) √ ( ) Δ = λ2 + 4 M2 + M2 es SO,e Egx,e Egy,e √ 2 − 2 − 2 = λSO,e + 4 [de(εXX εYY) + feεYZ] + 4 [ 2deεXY + feεZX] (4.16)

Here, the subscript g(e) refers to the ground (excited) state manifold. ΔZPL is the mean zero phonon line (ZPL) frequency, and Δgs (Δes) is the ground (excited) state splitting, respectively.

ΔZPL,0 is the mean ZPL frequency, when there is no external strain. From these equations, we

directly observe that strain of A1g type, which determines the MA1g terms, leads to a change in

the mean ZPL frequency. On the other hand, strain of Eg type, which determines MEgx , MEgy terms, leads to a change in the GS and ES splittings.

All three values, ΔZPL,Δgs, and Δes as a function of strain can be directly extracted from the photoluminescence excitation (PLE) spectra shown in Figs. 4.6 and 4.7. Fitting the model

Eqs. (4.14-4.16) to the experimental data, we should be able to estimate the strain-susceptibility parameters. To extract all the values {t⊥, t∥, d, f} for both ground and excited state manifolds,

103 in principle, strain needs to be applied at least in three different directions for a given SiV.

This procedure gives a set of overdetermined equations in these parameters.52 However, the devices in this study can only induce two types of strain profiles as shown in Fig. 4.3b and c. In particular, for a given SiV in either the ‘axial’ or the ‘transverse’ class, the relative ratio between strain-tensor components remains constant, when the voltage applied to the device is swept. This condition makes it difficult to estimate the individual contributions of the t∥ and

t⊥ terms to MA1g , and of the d and f terms to MEgx and MEgy .

To get around this issue, we follow an approximate approach. From Fig. 4.3, we observe that in the case of an axial SiV, εZZ ≫ (εXX + εYY) is always true. Therefore, we can use ( ) the response of the axial SiV in Fig. 4.7 to approximately estimate t∥,e − t∥,g by neglecting

(εXX + εYY) in Eq. (4.14). Fig. 4.8f plots the mean ZPL frequency of the axial SiV in Fig. 4.7 ( ) vs. strain estimated from FEM simulation. The slope of the linear fit yields t∥,e − t∥,g .

( ) t∥,e − t∥,g = −1.7 PHz/strain (4.17)

Likewise, in the case of the transverse SiV in Fig. 4.3, we can conclude that (εXX − εYY) ≫ max{εZX, εYZ}. With this class of SiVs, we can approximately estimate {dg, de} by neglecting

{εZX, εYZ} in Eqs. (4.15,4.16). Fig. 4.8e plots the GS and ES splittings of the transverse SiV in Fig. 4.8c vs. strain estimated from FEM simulation. Fitting yields

dg = 1.3, de = 1.8 PHz/strain (4.18)

104 ( ) Once we extract t∥,e − t∥,g from an axial SiV, we can use this value to further extract ( ) t⊥,e − t⊥,g by fitting Eq. (4.14) to the tuning behavior of the mean ZPL frequency of the transverse SiV. This procedure yields

( ) t⊥,e − t⊥,g = 78 THz/strain (4.19)

( ) ( ) We immediately note that t∥,e − t∥,g is more than an order of magnitude larger than t⊥,e − t⊥,g .

This implies that εZZ tunes the ZPL much more effectively than (εXX + εYY). This can be intuitively explained by examining the spatial profile of the GS and ES orbitals (Table 2.7 of

? Ref. ). Since the GS and ES are even (g) and odd (u) eigenstates of SiV’s D3d point symmetry group respectively, the charge density distributions of the orbitals egX, euX (and egY, euY) are similar in XY plane of nonzero Z, as it can be confirmed from the orbital pictures.49 As a result, we would expect that the common mode energy shift resulting from the strain-mode

εXX + εYY is very similar for the GS and ES manifolds, i.e. t⊥,g ≈ t⊥,e. On the other hand, the energy shift from εZZ is expected to have the opposite signs for the GS and ES manifolds due to the change in wavefunction parity along the Z-axis.

As the last step, we extract the values fg, fe in Eqs. (4.15, 4.16). We observe from Table

4.2 that knowledge of d and B can allow us to determine f. We note that the stress-response

49 coefficients Bg=484 GHz/GPa and Be=630 GHz/GPa are extracted in the literature based on

107 uniaxial stress measurements carried out previously. Combining our estimates of dg and de

105 with this information, we predict

fg = −250, fe = −720 THz/strain (4.20)

We emphasize that the strain-susceptibility values estimated in this section are subject to errors arising from (i) imprecision in SiV depth from the diamond surface (10% straggle from

SRIM calculations, and in practice, higher due to ion-channeling effects), and (ii) due to the fact that the device geometry cannot be replicated exactly in the FEM simulation for strain estimation. In particular, the values f and t⊥ are subject to large errors, since the Eg and A1g responses are mostly dominated by the susceptibilities d and t∥ respectively.

4.5 Controlling electron-phonon processes

In this section, out subject of study is the spin qubit based on the lower branch of the ground- state and how its quantum characteristic change as we control the strain of the SiV center. Ac- cording to the theory given in the Section 4.4, when static strain is applied to the SiV center the ground state splitting can change as a result of mixing between two orbital branches. From the observation that the spin coherence time and the orbital relaxation time are almost the same, it has been speculated that the limiting mechanism of the spin coherence is the electron- phonon process, where resonant thermal phonons are absorbed while making transitions be-

55 tween two ground state orbital branches |eg+⟩ and |eg−⟩.

As shown in the Fig. 4.6, it is possible to control the static strain of the SiV center and

106 hence it can lead to the manipulation of decoherence rate of the spin qubit. Furthermore, we demonstrate that the same physics also leads to the change in the relaxation rate of spin qubit.

4.5.1 Orbital transition rate of electron-phonon process

Due to the nature of the strain interaction (Eq. 4.11), phonons having non-zero Eg-strain components that are resonant with the ground state splitting Δgs can directly drive the orbital transition. Indeed, single-phonon processes are found to be the dominant cause for thermalization of the ground state orbitals at 4 K.55 We follow the approach used in this work to model the orbital thermalization rate due to single-phonon processes. Summing over all acoustic modes of polarizations λ, and wave-vectors k, we can write the upward and downward orbital relaxation rates as

∑ | |2 − γup(Δgs) = 2π nth(ν) g(λ, k) δ(ν Δgs) (4.21) ∑λ,k | |2 − γdown(Δgs) = 2π (nth(ν) + 1) g(λ, k) δ(ν Δgs) (4.22) λ,k

Here g(λ, k) is the single-phonon coupling rate to the mode with polarization λ and wave- vector k, and nth(ν) is the Bose-distribution.

( 1 ) nth(ν) = (4.23) exp hν − 1 kBT

107 Note that the term +1 in the downward rate in Eq. (4.22) physically corresponds to spontaneous emission of a phonon, a process that is independent of the thermal occupation of the modes.

Consider a longitudinal mode with uniaxial zero-point strain εZPF perfectly aligned to drive the orbital transition with susceptibility d. In this case, we would have g(λ, k) = 2πdεZPF.

εZPF itself can be derived by assuming an elastic energy of hν localized in a mode volume V.

1 hν = Eε2 V 2 ZPF

where E is the elastic modulus. This gives

√ 2hν g(λ, k) = 2πd (4.24) EV

√ Regardless of the exact details of the mode profile, the relation g ∝ ν will hold true. In general, the proportionality constant will include a geometric factor, relevant components of the elasticity tensor, and a combination of the susceptibilities d and f, which describe mixing of the |eX⟩ , |eY⟩ orbitals. Neglecting these details and further assuming an isotropic dispersion relation, we define an average single-phonon coupling rate g(λ, k) with

|g(λ, k)|2 = χν (4.25)

Converting Eqs. (4.21) and (4.22) into integrals in 3D k-space, we get

108 ∫ | |2 − 2 MEgy,up(Δgs) = 2π g(λ, k) nth(ν)δ(ν Δgs)4πk dk ∫ 2 = 2π χν · nth(ν)δ(ν − Δgs)ρν dν

Here ρν2 gives the density of states (DOS) for phonons in the bulk, where ρ is a constant that depends on the speed of sound averaged over various modes. Thus we get

3 γup(Δgs) = 2πχρΔgsnth(Δgs) (4.26)

Similarly, we can write

3 γdown(Δgs) = 2πχρΔgs(nth(Δgs) + 1) (4.27)

3 In these expressions, the first term in the product, 2πχρΔgs corresponds to the mean-squared single-phonon coupling rate multiplied with the DOS at the GS splitting Δgs, while the second term corresponds to the thermal occupation of each mode. Fig. 4.9 shows plots of the calculated dependence of upward and downward rates on Δgs at temperature T = 4 K. We observe that the upward rate shows a non-monotonic behavior, approaching its maximum value around hΔgs ∼ kBT. The increasing DOS term dominates in the regime hΔgs < kBT, and causes

≫ MEgy,up to increase. However, when hΔgs kBT, the thermal occupation of the modes be- ( ) haves as n (Δ ) = exp − hΔgs . This exponential roll-off dominates the polynomially th gs kBT

109 increasing DOS, and causes MEgy,up to decrease at higher strain. In contrast, the downward rate monotonically increases with the GS-splitting, because it is dominated by the spontaneous emission rate, which simply scales as the DOS. This behavior is shown in Fig. 4.9, both upward and downward transition rates normalized to their values at the zero-strain. Here, we calculate them with corrected exponent of the frequency in the density of states expression (approximately 0.9 rather than 2), which takes into account the geometric factor and hence gives a better fit for our data in Fig. 4.12a. This is because the device geometry seen by an SiV is far from that of bulk. Phonon-emission rate, )

gs 2

(Δ 10 up γ 100

101 γ down 0 10 (Δ gs ) Phonon-absorption rate, 10-1 0 200 400 600 Ground state splitting (GHz)

Figure 4.9: Change in orbital transition rates γup and γdown as a function of the ground state splitting Δgs.

110 4.5.2 Orbital thermalization measurements

As a first step to demonstrate the control of spin coherence, we directly measure the orbital relaxation rate between two orbital branches of the ground state manifold. We use time resolved pump-probe fluorescence to characterize the phonon processes in the GS. In this method, two consecutive laser pulses resonant with the D transition are used to, first initialize GS orbital population in the lower branch |1⟩, and after a set delay τ, read-out population in the upper branch |2⟩. A schematic of the pulse sequence, and an example of a resulting fluorescence time-trace are shown in Fig. 4.10. By repeating this sequence for steadily increasing pump- probe delay τ, we measure the rate at which the GS population relaxes towards thermal equilibrium due to resonant phonons.

Pump Probe 500 p 2,th τ ￜ3〉 400 p2(τ)

300 D ￜ2〉 γup γ Photon counts 200 down ￜ1〉 p 100 2,opt

0 100 200 300 400 500 Time (ns)

Figure 4.10: Time-resolved ﬂuorescence signal in pump-probe measurement for a delay τ = 50 ns between the two laser pulses. The laser is resonant with the D transition, and optically pumps the GS population into the lower orbital branch |1⟩ over a timescale of few ns. After time τ, the ﬂuorescence signal from the probe pulse has a leading edge determined by the population in the upper orbital branch |2⟩. The decay rates be- | ⟩ | ⟩ tween levels 1 and 2 - γup due to phonon-absorption, and γdown due to phonon-emission - are also shown.

111 4.5.3 Extraction of orbital T1

Figure 4.11: Fluorescence time-traces for various pump-probe delays between τ=5 ns to 70 ns taken at GS- splitting Δgs=46 GHz. x-axis is time in ns, and y-axis is photon counts integrated over multiple iterations of the pulse sequence.

Example data from implementing the pulse sequence in Fig. 4.10 for various pump-probe delays is shown in Fig. 4.11. This data can be interpreted and processed to yield a GS-population thermalization curve as follows. The leading edge of the first fluorescence signal shown in

Fig. 4.10 corresponds to thermal population p2,th in the GS level |2⟩. Upon switching on the pump pulse, this decays to a residual value p2,opt determined by the competition between the optical pumping rate (above saturation, this is simply the decay rate γe from the excited state

| ⟩ 3 ) and the rates γup, γdown. After time delay τ, the leading edge of the probe fluorescence signal corresponds to partially recovered population p2(τ) due to thermalization. We can describe

112 the population recovery in level |2⟩ as

( ) [ ( ) ] − − − (p2(τ) p2,th) = p2(τ) p2,opt exp γup + γdown t (4.28)

( ) In particular, we calculate the normalized change in photon-counts, (p2(τ) − p2,th) / p2(τ) − p2,opt from each measurement in Fig. 4.11, and carry out an exponential fit in Fig. 4.12a to extract ( )

γup + γdown . Repeating this experiment for various values of GS-splitting Δgs, we arrive at

Fig. 4.12b.

a b 0 16 γ + γ data 0.1 Zero strain up down 14 γup+ γdown fit Δgs = 46 GHz 0.2 γdown fit 0.3 12 γup fit 0.4 10 Zero strain 0.5 8 0.6 6 0.7 γup+ γdown = 6.5 ± 0.2 MHz 0.8 4 Thermal relaxation rate (MHz)

Normalized change in photon counts 0 10 20 30 40 50 60 70 2 Pump-probe delay, τ(ns) 50 60 70 80 90 100 110 Ground state splitting, Δgs (GHz)

Figure 4.12: (a) Fluorescence time-trace for τ=50 ns from Fig. 4.11 showing relevant quantities related to the population in level |2⟩. (b) Thermalization curve constructed by extracting the normalized change in photon- counts for various pump-probe delays τ. Solid line is an exponential ﬁt.

4.5.4 Measurement of orbital T1 at various ground state splitting

With our device we can tune the splitting of the orbitals in the GS manifold from 46 GHz to typically up to 500 GHz, and in the best case, up to 1.2 THz. In doing so, we can probe

113 the interaction between SiV and the phonon bath at different frequencies by measuring the thermal relaxation rate of the orbital as described in Section 4.5.3. Measurements are performed in the frequency range Δgs = 46 GHz to 110 GHz where this technique can be ap-

plied. The total relaxation rate is a sum of the rates of phonon absorption, MEgy,up, and emis-

sion, MEgy,down (shown in Fig. 4.1a), which can be individually extracted using the theory described in Section 4.5.1. Over the range of Δgs measured, phonon processes in both directions are observed to accelerate with increasing orbital splitting, thus indicating that the number of acoustic modes resonant with the GS splitting, i.e. the phonon density of states (DOS) at

n this frequency, increases with an expected dependence in Δgs (n depends on the geometry of material seen by resonant phonons). However, if the orbital splitting is increased far above

120 GHz (at temperature T = 4 K) as plotted in Fig. 4.9, the phonon absorption rate (MEgy,up) is theoretically expected to reverse its initial trend. In this regime, the polynomial increase in phonon DOS is outweighed by the exponentially decrease in thermal phonon occupation

∼ − 55 ( exp( hΔgs/kBT)), and consequently MEgy,up is rapidly quenched.

4.6 Spin dynamics with controlled strain environment

In this section, we explore characteristic time of the spin qubit when SiV center is subject to external stress. Since spin dynamics is correlated with the electron-phonon process studied in the Section 4.5, we expect to observe changes in spin dynamics of SiV center, namely the coherence and relaxation time.

When external magnetic field is applied, each ground orbital branch split up due to the Zee-

114 ￜ2↑〉ￜ2↓〉 ν1 ν2 ν1 ν2 Δgs ￜ1↑〉 ω ￜ1↓〉 s decoherence relaxation

Figure 4.13: Transitions relevant to the time scale of spin dynamics. Blue arrows represent transition processes relevant to spin decoherence. Red arrows are transitions relevant to spin relaxation. Note that spin relaxation involve spin-ﬂipping transitions which are much slower than spin-conserving counterparts. The labels |1 ↓⟩, |1 ↑⟩, |2 ↓⟩, |2 ↑⟩ refer to the eigenstates in the high-strain regime. Reduction of both decoherence and relaxation rate comes from the suppression of resonant absorption of a single phonon. man effect giving us an access to spin qubit. At 4K, phonon modes resonant with the ground- state splitting are thermally occupied because kBT > Δgs. These thermal phonons can be absorbed by an SiV, and induce an upward transition. Since thermal phonons arrive randomly they can behave as a noise source that leads to decoherence. Specifically, two possible decoherence processes are depicted in Fig. 4.13. These spin-conserving transitions cause the random phase accumulation of the spin qubit, and destroy the coherence as a result. In contrast, spin relaxation goes through different pathways from the decoherence processes. Its rate is determined by the spin-flipping transitions shown in Fig. 4.13, accompanying a much faster spin-conserving transition.

115 4.6.1 Spin decoherence rate

When applied strain is sufficiently high, e.g. over 300 GHz of ground-state splitting, upper and lower orbital branches can be approximated by pure orbital state |eX⟩ and |eY⟩. Under these circumstances, spin projection is predominantly determined by external magnetic field.

Therefore, spin-conserving transition rates between two branches are approximately the same with the pure orbital transition rate. When applied magnetic field direction is different from

SiV center’s symmetry axis, the electronic levels split into spin sub-levels provide an optically accessible spin qubit as shown in Fig. 4.14a.80,88,92 We use coherent population trapping (CPT) through simultaneous resonant laser excitation of the optical transitions labeled

C1 and C2 to pump the SiV into a dark state, a coherent superposition of the spin sub-levels

|1 ↓⟩, |1 ↑⟩. When the two-photon detuning is scanned, preparation of the dark state results in a fluorescence dip, whose linewidth is determined by the optical driving and spin dephasing rates. At low laser powers, the linewidth is limited by spin dephasing, which is dominated by phonon-mediated transitions within the GS manifold.55 In Fig. 4.14b, as the dark resonance narrows down due to prolonged spin coherence with increasing strain, we reveal a fine structure not visible before. Further measurements in Section 4.7 suggest that the presence of two resonances is due to interaction of the SiV electron spin with a neighbouring spin such as a

13C nuclear spin. This indicates the possibility of achieving a local register of qubits as has been demonstrated with NV centers.16 Fig. 4.14c shows the decreasing linewidths of the CPT resonances with increasing GS orbital splitting, indicating an improved spin coherence time.

116 Magnetic a Strain b Δ eld gs │4↑〉 88 GHz │4↓〉 │4〉 Δ es 303 GHz │3〉 │3↑〉 │3↓〉 352 GHz C1 C2 407 GHz │2↑〉 │2↓〉 │2〉 467 GHz Δgs │1〉 10% │1↑〉 Normalized counts per second │1↓〉 -10 -5 0 5 10 Spin qubit Frequency (MHz) c d

4.0 5.0 Δ gs = 467 GHz 4.0 3.0 3.0

2.0 2.0

1.0 * CPT linewidth (MHz) 1.0 CPT linewidth (MHz) T2 = 0.25±0.02 μs 0.0 300 350 400 450 500 0 200 400 600 800 1000 Δ Ground state splitting, gs (GHz) Laser power (nW)

Figure 4.14: (a) SiV level structure in the presence of strain and external magnetic field. A spin qubit is defined with levels |1 ↓⟩ and |1 ↑⟩ on the lower orbital branch of the GS. This qubit can be polarized, and prepared optically using the Λ-scheme provided by transitions C1 and C2. Phonon transitions within ground- and excited-state manifolds are also indicated. The upward phonon transition (phonon absorption process) can be suppressed at high strain, thereby mitigating the effect of phonons on the coherence of the spin qubit. (b) Coherent population trapping (CPT) spectra probing the spin transition at increasing values of the GS orbital splitting Δgs from top to bottom. Bold solid curves are Lorentzian fits. Optical power is adjusted in each measurement to minimize power-broadening. (c) Linewidth of CPT dips (estimated from Lorentzian fits) as a function of GS orbital splitting Δgs indicating improvement in spin coherence with increasing strain. (d) Power dependence of CPT-linewidth at the highest strain condition (Δgs=467 GHz). Data points are estimated linewidths from CPT measurements, and the solid curve is a linear fit, which reveals linewidth of ± ∗ ± 0.64 0.06 MHz corresponding to T2 = 0.25 0.02μs.

117 Improvement of spin coherence time is expected from the theory because resonant thermal phonons are depleted due to the Boltzmann distribution when the Δgs gets far above kBT. Be- yond a GS splitting of ∼400 GHz, the linewidths saturate at ∼1 MHz. At the highest strain condition, we perform a power dependent CPT measurement to eliminate the contribution of

∗ ± power broadening, and extract a spin coherence time of T2 = 0.25 0.02 μs (compared with

∗ 88,92 ∗ T2 = 40 ns without strain control). This saturation of T2 suggests the mitigation of the pri- mary dephasing source, single-phonon transitions between the GS orbitals, and the emergence of a secondary dephasing mechanism such as slowly varying magnetic fields from naturally

13 ∗ ± abundant (1.1%) C nuclear spins in diamond. We note that our longest T2 = 0.25 0.02 μs is on par with measurements taken on low-strain SiVs by operating at a much lower temperature of 100 mK,110 the conventional approach to suppress phonon-mediated dephasing.

4.6.2 Spin relaxation rate

In contrast to the decoherence case, the spin relaxation process involves spin-flipping transitions. Spin-flip is allowed due to the transverse magnetic field components, Bx and By, because those lead to the mixing of spin states.49 Although direct relaxation between |1 ↑⟩ and

|1 ↓⟩ and off-resonant two-phonon processes are both allowed, their rates are much lower than resonant two-phonon relaxation processes in Fig. 4.13. Detailed analysis of all four relaxation

76 2 processes can be found elsewhere. From the analysis, we get Γ2ph,res = 4 (d1↓→2↑/d) γup, where d1↓→2↑ is the strain susceptibility between two spin sublevels |1 ↓⟩ and |2 ↑⟩ and Γ2ph,res is the total relaxation rate from resonant two phonon absorption shown in Fig. 4.13. This pro-

118 0.45

(MHz) 0.4 1 0.35 0.3 0.25 0.2 0.15 0.1

Spin relaxation rate, 1/T 0.05 200 250 300 350 400 450 Ground state splitting (GHz)

Figure 4.15: Spin relaxation rate change as a function of the ground state splitting Δgs. Relaxation is progressively suppressed as a function of increasing ground state splitting due to reduced single phonon absorption. cess can be intuitively understood as the following. From the orbital relaxation rates shown

≫ in Fig. 4.9, it can be noticed that γdown γup when Δgs is above 200 GHz. For the two consecutive processes, it is natural that slower process approximately determines the overall rate.

Thereby, γup is only showing in the expression of the spin relaxation rate as a ‘bottle-neck’

2 process. 4 (d1↓→2↑/d) factor represents the probablility of flipping the spin. d1↓→2↑/d is the ratio between the transition dipole moments of {|1 ↓⟩ , |2 ↑⟩} and that of pure orbital states { } | ⟩ | ⟩ eg+ , eg− . In short, the slower electron-phonon orbital relaxation of the two, the upward transition rate, further reduced by the small probability of spin-flipping determines the total spin relaxation rate. Change of the spin relaxation rate as a function of ground state splitting is

119 shown in Fig. 4.15.

4.7 Investigation of double-dip CPT signal

1 . 4

1 . 2

1 . 0

) 0 . 8 τ (

( 2 ) 0 . 6 g 0 . 4

0 . 2

0 . 0 - 6 0 - 4 0 - 2 0 0 2 0 4 0 6 0 D eτ l a ( y n s )

− Figure 4.16: Second-order correlation measurement of the SiV centre investigated. The measured data points are plotted as grey dots. A ﬁt based on a three-level model and accounting for timing jitter is plotted as a blue curve.

Here we investigate the origin of two dips shown in Fig. 4.14b. We first rule out the hy- pothesis that the two dips in our CPT measurements originate from two different SiV centres.

The dips are very similar in width and depth, and their frequency-separation remains constant

(4.0 ± 0.1 MHz) over a wide range of applied strain (see Fig. 4b of main text). In the event that this is caused by two SiV centers, they are required to have the same fluorescence intensity, and experience exactly the same strain conditions at any applied voltage, which is very unlikely. In order to categorically establish that we are investigating a single SiV centre, we perform a Hanbury-Brown-Twiss (HBT) experiment on the phonon-sideband fluorescence upon resonant excitation of transition C, and measure the second order correlation function g(2) as shown in Fig. 4.16. The g(2) function is fitted using a three-level model convolved

120 with the Gaussian response of the avalanche photodiodes used, which have a timing jitter of

350 ps. At zero time delay, a clear anti-bunching reaching g(2)(τ = 0) = 0.12 indicates that the measured photons originate from a single emitter.

To gain more insight into the origin of the two CPT dips, we perform CPT at varying orientation of the applied magnetic field, while keeping its magnitude (0.2 T) constant. We work in the high-strain regime at a ground state splitting of 467 GHz. In our results, shown in Fig.

4.17, we observe that the separation between the two CPT dips displays a periodic variation as the magnetic field is rotated.

Given the similarity of the two dips, a very plausible explanation for the double-dip structure is the presence of a proximal spin in the environment of the SiV centre being studied.

Physically, varying the direction of the applied B-field leads to a variation in the quantization axis of the SiV electron-spin (or semi-classically, the orientation of the electron magnetic moment). Likewise, the quantization axis of the proximal spin has its own variation with the

B-field orientation. For instance, if this proximal spin is a nuclear spin, to first order, its orientation simply follows that of the applied B-field. As a result, the dipole-dipole interaction energy of the SiV electron-spin with the neighboring spin varies with B-field orientation, leading to the periodic behavior observed experimentally in Fig. 4.17. Below, we describe a semi- classical approach to model the CPT dip separation as a dipole-dipole interaction.

121 Data 4.5 Dipole-dipole interaction model

3.5 Double dip splitting (MHz) 3 -200 -100 0 100 200 Angle of magnetic field (°)

Figure 4.17: Dependence of CPT dip separation on magnetic ﬁeld orientation. The angle plotted on the x- ◦ axis is measured with respect to the vertical direction on the sample. 0 corresponds to the [001] axis of ◦ diamond, while 90 corresponds to the [110] axis of diamond, along which the cantilever long-axis is aligned. The SiV investigated is a transverse SiV, so its internal Z-axis is either [111¯ ] or [111¯ ]. Error bars correspond to the standard deviation on the CPT dip frequencies estimated from Lorentzian ﬁts.

The Hamiltonian for two dipoles with magnetic moments μ1 and μ2 is given by

Hd-d − μ0 · · − · = (3 (μ1 ˆr) (μ2 ˆr) μ1 μ2) (4.29) 4π |r|3

where μ0 is the vacuum-permeability, r is the vector from one dipole to the other, and ˆr is given by r/ |r|. If the two dipoles are spins described by spin angular momentum S1 and S2,

122 we can write the interaction Hamiltonian in terms of the spin-operators.

d-d μ0γ1γ2 H = − (3 (S1 · ˆr) (S2 · ˆr) − S1 · S2) (4.30) 4π |r|3

where γi is the gyromagnetic ratio of spin i. Semi-classically, we can treat spin angular mo- ( ) ℏ ⟨ ⟩ ⟨ ⟩ ⟨ ⟩ mentum as a vector quantity S = 2 σx , σy , σz , where σx, σy, σz are Pauli spin-matrices.

This vector describes the mean-orientation of the electron magnetic moment. To calculate the

SiV electron-spin orientation under our experimental conditions, the full ground-state Hamil- tonian including spin-orbit coupling, external strain, and magnetic field must be diagonalized.

The effect of the external magnetic field is described by the Zeeman Hamiltonian below writ-

49 ten in the basis {|eX, ↑⟩ , |eX, ↓⟩ , |eY, ↑⟩ , |eY, ↓⟩},

   0 0 iB 0   z       0 0 0 iBz  Zeeman ˆ ˆ   H = qγ LzBz + γ S · B = qγ   + L S L    −   iBz 0 0 0   

0 −iBz 0 0    B B − iB 0 0   z x y       B + iB −B 0 0   x y z  γ   (4.31) S    −   0 0 Bz Bx iBy   

0 0 Bx + iBy −Bz

123 where the first and second terms are from the orbital angular momentum, and the spin angular momentum, respectively. Lˆz and Sˆ are Lz and S normalized by ℏ, respectively. The gyro-

ℏ ℏ magnetic ratios for each term are given by γL = μB/ , γS = 2μB/ , where μB is the Bohr magneton. q is a quenching factor that is commonly observed in solid-state emitters.49

The total Hamiltonian Htotal is obtained by adding the spin-orbit Hamiltonian HSO and strain Hamiltonian Hstrain.

Htotal = HSO + Hstrain + HZeeman (4.32)

   M − M 0 M − iλ /2 0   A1g Egx Egy SO       0 MA − ME 0 ME + iλSO/2  SO strain  1g gx gy  H + H =      M + iλ /2 0 M + M 0   Egy SO A1g Egx    − 0 MEgy iλSO/2 0 MA1g + MEgx (4.33)

λSO, the spin-orbit coupling is 46 GHz for the ground state of the SiV. We diagonalize the total Hamiltonian, Htotal, and calculate expectation values of the Pauli matrices for the lowest two eigenstates, which comprise the SiV spin-qubit under investigation. This gives us the mean orientation of the SiV electron-spin, say S1 for given experimental conditions. To calculate the mean orientation of the proximal spin S2, we assume it to be either a nuclear spin

13 such as C, or an electron spin such as another SiV-center. In the case of a nuclear spin, S2

124 is simply given by the direction of the external magnetic field. Once the quantization axes of the two spins are known, we can fit our data to the calculated value of Hd-d from Eq. (4.30) by using the distance between the spins r as a fit parameter. The result of such a fitting procedure is shown in Fig. 4.17. In the case of a nuclear spin, the distance between the two spins |r| is estimated to be on the order of 1 Å. If the other spin is an electron spin from another SiV centre, it is possible to obtain similar results as in Fig. 4.17. However, in this case, the distance between the spins is on the order of tens of nanometres.

4.8 Conclusion

In conclusion, we use a NEMS device to probe and control the interaction between a single electronic spin and the phonon bath of its solid-state environment. In doing so, we demonstrate six-fold prolongation of spin coherence by suppressing phonon-mediated dephasing as the dominant decoherence mechanism. As a next step, we can further improve the spin coherence by cancelling the effect of slowly-varying non-Markovian noise from the environment110 using dynamical decoupling techniques that are well-studied with other spin systems.16,24,30

Moreover, the emission wavelength tuning provided by our NEMS platform can enable generation of indistinguishable photons from multiple emitters, and hence scalable photonic quantum networks.13,100 Another natural extension of our work is coherent coupling of the SiV spin to phonons in a well-defined mechanical mode, which will enable the use of phonons as quantum resource. In particular, we can combine the large strain susceptibility of the SiV electronic levels76 with mechanical resonators of dimensions close to the phonon wavelength,

125 such as optomechanical crystals11 to obtain orders of magnitude larger spin-phonon interaction strengths compared with previous works3,64,68,77,86, leading to strong spin-phonon coupling. In this regime, one can realize phonon-mediated two-qubit gates96 analogous to those implemented with trapped ions21, and achieve quantum non-linearities required to deterministically generate single phonons and non-classical mechanical states,19,83,93,103 a long sought- after goal since phonons can be used to interface spins with other quantum systems such as superconducting qubits122.

126 A Technical informations

A.1 Chapter 2

A.1.1 Sample preparation

Single crystal electronic grade bulk diamond chips (4mm x 4mm) from Element Six Ltd are implanted with 14N ions at an implantation energy of 75 keV, and a dose of 6 × 1011 /cm2.

This yields an expected depth of 94 ± 19 nm calculated using software from Stopping and

127 Range of Ions in Matter (SRIM). Subsequently, NVs are created by annealing the samples in high vacuum (< 5 × 10−7 torr). The temperature ramp sequence described in18 is followed with a final temperature of 1200oC, which is maintained for 2 hours. After the anneal, the samples are cleaned in a 1:1:1 boiling mixture of sulfuric, nitric and perchloric acids to remove a few nm of graphite generated on the surface from the anneal. Cantilevers are then patterned using e-beam lithography, and etched using our angled etching scheme12. Post-fabrication, we repeat the tri-acid cleaning treatment to partially repair etch-induced damage, and perform a piranha clean to ensure a predominantly oxygen terminated diamond surface (diagnosed by

X-ray photoelectron spectroscopy (XPS)), which is beneficial for NV photostability18,23,25.

A.1.2 Ensemble effects

We address the effect of inhomogeneous coupling strengths in AC strain coupling measurements on an NV ensemble. The width of our confocal laser spot ≈ 560 nm is about forty times smaller than the length of the cantilever (19 μm). Taking into account the roughly linear variation of strain along the length of the cantilever, we expect a ≈ 2% variation in coupling strength within the confocal spot due to lateral distribution of NVs. This is less than the order of the error in the fitted estimate for G. Now, we consider the more significant effect of ion- implantation straggle, which is expected to be ≈ 20% for our chosen NV depth from SRIM.

Upon fitting the experimental spin echo signal to the formula in Eq. 2.17 convolved with a

20% Gaussian straggle in G, we noticed that our estimate for G did not change to within the error bars. We believe that this is because the level of photon shot noise in our measurement

128 (±0.05 error in spin population estimates) does not allow us to ultimately resolve the effect of any inhomogeneity in G across the ensemble.

A.2 Chapter 4

A.2.1 Pump-probe measurement

The pump-probe pulse sequence described above is implemented by pulsing our resonant- excitation laser with a Mach-Zehnder intensity electro-optic modulator (EO Space AZ-AV5-

5-PFA-PFA-737) driven by a digital-delay generator with rise- and fall-times of 2 ns (SRS

DG645). Over the course of the measurements, the operation point of the intensity electro- optic modulator (EOM) is stabilized against long-term drifts with continuous feedback on the

DC-bias voltage. The feedback loop is implemented with a lock-in amplifier (SRS SR830) generating a low-frequency (1 KHz) modulation of the DC-bias voltage. Photon-count pulses from the single-photon-detector are time-tagged on a PicoHarp 300 module triggered by the delay-generator. The laser frequency itself is stabilized by continuous feedback with a wavemeter (High Finesse WS7).

129 B SiV Hamiltonian

B.1 Matlab code for the calculation of SiV’s electronic structure

Following matlab codes can calculate electronic structure of SiV center with all perturbations: electric fields, magnetic fields and strain. Usage of the codes is self-explanatory if one reads comments of the first file carefully. All three files need to be in the same folder to work cor- rectly.

130 B.1.1 ‘SiV_H_Tutorial.m’

1 %% Description of the file: Tuthroial of SiV full Hamiltonian calculation

2 % This file is the tutorial of how SiV full Hamiltonian can be caculated

3 % when strain and magentic field conditions are varied.

4 % This is a wrapper file that runs actual two function files

5 % SiV_Hamiltonian_XYbase and SiV_Hamiltonian_PMbase. All inputs are given

6 % in XY base. SiV_Hamiltonian_PMbase function internally performs

7 % conversion.

8 % There are four secitions.

9 % 1. Strain sweep

10 % 2. B field magnitude sweep with a fixed direction

11 % 3. B field rotation with a fixed magnitude

12 % 4. Comparison of (ex,ey) basis and (e+,e−) calculation results

13 % Author: Joonhee Choi wrote the basic structure of the Hamiltonain and

14 % YoungIk Sohn made further updates for parameter sweeps

15 % Date: 2/16/2017

16 % Version: ver 1.0

18 %% Strain sweep

19 % Define magnetic field condition.

20 % Following definition of B field is just an example, it can be arbitrary.

21 B _ f i e l d = 2; % i n Tesla

131 22 Bx = B _ f i e l d / s q r t ( 3 / 2 ) ;

23 By = 0;

24 Bz = B_field/ s q r t ( 3 ) ;

25 Bvec = [Bx, By, Bz]; % Needs to be prepared in vector form. XYZ frame used

here is SiV’s internal reference frame (z axis is parallel with SiV’s [111]

d i r e c t o i n )

27 % Define sweeping parameter of strain. In fact , defining strain sweep with

28 % ag,ae alone may not be phyiscal. The choice of sweeping parameter is

29 % arbitray and for demonstration only.

30 a g _ l i s t = linspace (0,200,501);

31 bg = 0;

32 ae_list = 3 * ag_list;

33 be = 0;

35 ground_eigenergy_list = zeros ( length (ag_list) ,4); % A l l o c a t e the memory

for the ground state energy eigenvalues.

36 excited_eigenergy_list = zeros ( length (ag_list) ,4); % A l l o c a t e the memory

for the ground state energy eigenvalues.

38 f o r i d x = 1 : length ( a g _ l i s t )

39 ag = ag_list(idx);

40 ae = ae_list(idx);

132 41 Hg_str = [ag, bg; bg, −ag ] ;

42 He_str = [ae, be; be, −ae ] ;

43 [ground_eigvector , ground_eigenergy , excited_eigvector , excited_eigenergy] =

SiV_Hamiltonian_XYbase(Hg_str , He_str , Bvec) ;

44 ground_eigenergy_list(idx ,:) = ground_eigenergy;

45 excited_eigenergy_list(idx ,:) = excited_eigenergy;

46 end ;

48 % Result plotting

49 fh = f i g u r e ( 1 ) ;

50 set ( fh , ’ p o s i t i o n ’ , [100 700 [1200 900]]);

52 % ground state energy levels

53 subplot ( 2 , 1 , 2 ) ; c l a ()

54 p l o t (ag_list , ground_eigenergy_list , ’linewidth ’ , 2)

56 x l a b e l ( ’Strain contribution (GHz) ’ )

57 y l a b e l ( ’Level frequency (GHz) ’ )

58 set ( gca , ’ f o n t s i z e ’ , 15)

59 % set(gca, ’YTick’, [−150 0 150])

60 t i t l e ( ’Ground state ’ )

61 ylim ([−300 300]) ;

133 63 % e xcited state energy levels

64 subplot ( 2 , 1 , 1 ) ; c l a ()

65 p l o t (ag_list , excited_eigenergy_list , ’linewidth ’ , 2)

67 x l a b e l ( ’Strain contribution (GHz) ’ )

68 y l a b e l ( ’Level frequency (GHz) ’ )

69 set ( gca , ’ f o n t s i z e ’ , 15)

70 % set(gca, ’YTick’, [−150 0 150])

71 t i t l e ( ’Excited state ’ )

72 ylim ([−700 700]) ;

74 %%B field magnitude sweep

75 % Define fixed strain condition.

76 % Here we use zero−strain condition but it can be arbitrary.

77 ag = 0; bg = 0;

78 ae = 0; be = 0;

79 Hg_str = [ag, bg; bg, −ag ] ;

80 He_str = [ae, be; be, −ae ] ;

82 % Define sweeping parameter for magnetic B field magnitude.

83 %B field magnitude is increased from zero to 5 Tesla.

84 B _ l i s t = linspace (0 ,5 ,101) ; % magnitude of sweeping B field in Tesla.

134 86 ground_eigenergy_list = zeros ( length (B_list) ,4); % A l l o c a t e the memory

for the ground state energy eigenvalues.

87 excited_eigenergy_list = zeros ( length (B_list) ,4); % A l l o c a t e the memory f o r

the ground state energy eigenvalues.

89 f o r i d x = 1 : length ( B _ l i s t )

90 % This particular example is where B field points [001] direction when SiV

91 % z−axis is [111] direction

92 Bx = B_list(idx)/ s q r t ( 3 / 2 ) ;

93 By = 0;

94 Bz = B_list(idx)/ s q r t ( 3 ) ;

95 Bvec = [Bx, By, Bz]; % Needs to be prepared in vector form. XYZ frame is

SiV’s internal reference frame (z axis is parallel with SiV’s [111]

d i r e c t o i n )

97 [ground_eigvector , ground_eigenergy , excited_eigvector , excited_eigenergy] =

SiV_Hamiltonian_XYbase(Hg_str , He_str , Bvec) ;

98 ground_eigenergy_list(idx ,:) = ground_eigenergy;

99 excited_eigenergy_list(idx ,:) = excited_eigenergy;

100

101

102 end ;

103

135 104 %Result plotting

105 fh = f i g u r e ( 2 ) ;

106 set ( fh , ’ p o s i t i o n ’ , [100 700 [1200 900]]);

107

108 % ground state energy levels

109 subplot ( 2 , 1 , 2 ) ; c l a ()

110 p l o t (B_list , ground_eigenergy_list , ’linewidth ’ , 2)

111

112 x l a b e l ( ’B field amptliude (T)’ );

113 y l a b e l ( ’Level frequency (GHz) ’ );

114 set ( gca , ’ f o n t s i z e ’ , 15)

115 % set(gca, ’YTick’, [−150 0 150])

116 t i t l e ( ’Ground state ’ )

117 ylim ([−100 100]) ;

118

119 % excited state energy levels

120 subplot ( 2 , 1 , 1 ) ; c l a ()

121 p l o t (B_list , excited_eigenergy_list , ’linewidth ’ , 2)

122

123 x l a b e l ( ’B field amptliude (T)’ );

124 y l a b e l ( ’Level frequency (GHz) ’ );

125 set ( gca , ’ f o n t s i z e ’ , 15)

126 % set(gca, ’YTick’, [−150 0 150])

136 127 t i t l e ( ’Excited state ’ )

128 ylim ([−200 200]) ;

129

130 %%B field rotation

131 % Here we rotate magnetic field with fixed magnitude in YZ plane of lab

132 % frame (? = ?/2, ? varies from −?/2 to +?/2). Since lab frame is different

133 % from SiV frame, B field needs to be rotated accordingly.

134

135 % Define strain condition.

136 % Here we use zero−strain condition but it can be arbitrary.

137 ag = 0; bg = 0;

138 ae = 0; be = 0;

139 Hg_str = [ag, bg; bg, −ag ] ;

140 He_str = [ae, be; be, −ae ] ;

141

142 % Define magnetic field rotation parameter

143 % In this particular example, azimuthal angle is fixed and polar angle is

144 % swept from 0 to ?.

145 %B field magnitude is fixed at 2 T

146 B_abs = 2;

147 t h e t a _ r o t = linspace (0,360,101); % Polar angle is rotated from zero to

360? in the lab frame

148 phi_rot = 45; % Fixed at 45? in the lab frame

137 149

150 ground_eigenergy_list = zeros ( length (theta_rot) ,4); % A l l o c a t e the memory f o r

the ground state energy eigenvalues.

151 excited_eigenergy_list = zeros ( length (theta_rot) ,4); % Allocate the memory for

the ground state energy eigenvalues.

152

153 f o r i d x = 1 : length (theta_rot)

154

155 %B field is calculated at each rotation in the lab frame

156 Bx = B_abs * cosd(theta_rot(idx)) * cosd(phi_rot);

157 By = B_abs * cosd(theta_rot(idx)) * sind(phi_rot);

158 Bz = B_abs * sind(theta_rot(idx));

159 Bvec_LabFrame = [Bx, By, Bz]; % Needs to be prepared in vector form. XYZ

frame used here is SiV’s internal reference frame (z axis is parallel

with SiV’s [111] directoin)

160

161

162 % Here we need to express the B field in the SiV frame basis

163 % In order to do that we imagine how Lab frame is rotated to overlap

164 % with SiV lab frame

165 % Operation is the following: first lab frame is rotated by ? around

166 % z−axis of the lab frame. Then this intermedeate frame is rotated by ?

167 % around y−axis of the intermediate frame. Note that this y−axis i s not

138 168 % the same with that of original lab frame. As a result , now two frames

169 % overlap. In this picture , B field vector in lab frame (Bvec_LabFrame)

170 % can be converted to a vector read from SiV frame (Bvec)

171 % SiV z−axis is [111] direction as usual

172 % Note phi/phi_rot and theta/theta_rot are completely different. phi

173 % and theta are for the conversion between lab and SiV frame. phi_rot

174 % and theta_rot gives the direction of B field in lab frame.

175

176 theta = acosd( s q r t ( 1 / 3 ) ) ;

177 phi = 45;

178

179 % R_phi, R_theta sign conventions look different because of the way ?

180 % and ? are defined. ? is an angled in xy−plane and ? is an angle in zx

181 % plane. Therefore they are indeed correct.

182 R_phi = [cosd(phi), sind(phi), 0;

183 − sind(phi), cosd(phi), 0;

184 0 , 0 , 1 ] ;

185

186 R_theta = [ cosd(theta), 0, − sind(theta);

187 0 , 1 , 0;

188 sind(theta), 0, cosd(theta)];

189

139 190 Bvec = R_theta * R_phi * Bvec_LabFrame’; % Note the order of rotation

matrix applied to the magnetic field.

191

192 [ground_eigvector , ground_eigenergy , excited_eigvector , excited_eigenergy] =

SiV_Hamiltonian_XYbase(Hg_str , He_str , Bvec) ;

193 ground_eigenergy_list(idx ,:) = ground_eigenergy;

194 excited_eigenergy_list(idx ,:) = excited_eigenergy;

195

196 end ;

197

198 % Result plotting

199 fh = f i g u r e ( 3 ) ;

200 set ( fh , ’ p o s i t i o n ’ , [100 700 [1200 900]]);

201

202 % ground state energy levels

203 subplot ( 2 , 1 , 2 ) ; c l a ()

204 p l o t (theta_rot , ground_eigenergy_list , ’linewidth ’ , 2)

205

206 x l a b e l ( ’Polar rotation angle (?)’ );

207 y l a b e l ( ’Level frequency (GHz) ’ );

208 set ( gca , ’ f o n t s i z e ’ , 15)

209 % set(gca, ’YTick’, [−150 0 150])

210 t i t l e ( ’Ground state ’ )

140 211 ylim ([−100 100]) ;

212

213 % excited state energy levels

214 subplot ( 2 , 1 , 1 ) ; c l a ()

215 p l o t (theta_rot , excited_eigenergy_list , ’linewidth ’ , 2)

216

217 x l a b e l ( ’Polar rotation angle (?)’ );

218 y l a b e l ( ’Level frequency (GHz) ’ );

219 set ( gca , ’ f o n t s i z e ’ , 15)

220 % set(gca, ’YTick’, [−150 0 150])

221 t i t l e ( ’Excited state ’ )

222 ylim ([−200 200]) ;

223

224

225 %%B field magnitude sweep for xy basis and plus minus basis to compare

226 % This is a sanity check to see if they give the same result.

227 % Define strain condition.

228 % It can be arbitrary.

229 ag = 30; bg = 0;

230 ae = 200; be = 0;

231 Hg_str = [ag, bg; bg, −ag ] ;

232 He_str = [ae, be; be, −ae ] ;

233

141 234 %Define sweeping parameter for magnetic B field magnitude

235 %B field magnitude is increased from zero to 5 Tesla

236 B _ l i s t = linspace (0 ,5 ,101) ; % magnitude of sweeping B field in Tesla

237

238 ground_eigenergy_list_XY = zeros ( length (B_list) ,4); % A l l o c a t e the memory f o r

the ground state energy eigenvalues.

239 excited_eigenergy_list_XY = zeros ( length (B_list) ,4); % Allocate the memory for

the ground state energy eigenvalues.

240

241 ground_eigenergy_list_PM = zeros ( length (B_list) ,4); % A l l o c a t e the memory f o r

the ground state energy eigenvalues.

242 excited_eigenergy_list_PM = zeros ( length (B_list) ,4); % Allocate the memory for

the ground state energy eigenvalues.

243

244

245 f o r i d x = 1 : length ( B _ l i s t )

246 % This particular example is where B field points [001] direction when SiV

247 % z−axis is [111] direction

248 Bx = B_list(idx)/ s q r t ( 3 / 2 ) ;

249 By = 0;

250 Bz = B_list(idx)/ s q r t ( 3 ) ;

251 Bvec = [Bx, By, Bz]; % Needs to be prepared in vector form. XYZ frame is

SiV’s internal reference frame (z axis is parallel with SiV’s [111]

142 d i r e c t o i n )

252

253 [ground_eigvector , ground_eigenergy , excited_eigvector , excited_eigenergy] =

SiV_Hamiltonian_XYbase(Hg_str , He_str , Bvec) ;

254 ground_eigenergy_list_XY(idx ,:) = ground_eigenergy;

255 excited_eigenergy_list_XY(idx ,:) = excited_eigenergy;

256

257 [ground_eigvector , ground_eigenergy , excited_eigvector , excited_eigenergy] =

SiV_Hamiltonian_PMbase(Hg_str , He_str , Bvec) ;

258 ground_eigenergy_list_PM(idx ,:) = ground_eigenergy;

259 excited_eigenergy_list_PM(idx ,:) = excited_eigenergy;

260

261

262 end ;

263

264 % Result plotting for both solutions

265 fh = f i g u r e ( 4 ) ;

266 set ( fh , ’ p o s i t i o n ’ , [100 700 [1200 900]]);

267

268 % ground state energy levels

269 subplot ( 2 , 1 , 2 ) ; c l a ()

270 p l o t (B_list , ground_eigenergy_list_XY , B_list , ground_eigenergy_list_PM , ’ o ’ , ’

l i n e w i d t h ’ , 2) ;

143 271

272 x l a b e l ( ’Strain contribution (GHz) ’ )

273 y l a b e l ( ’Level frequency (GHz) ’ )

274 set ( gca , ’ f o n t s i z e ’ , 15)

275 % set(gca, ’YTick’, [−150 0 150])

276 t i t l e ( ’Ground state ’ )

277 ylim ([−200 200]) ;

278

279 % excited state energy levels

280 subplot ( 2 , 1 , 1 ) ; c l a ()

281 p l o t (B_list , excited_eigenergy_list_XY , B_list , excited_eigenergy_list_PM , ’ o ’ ,

’linewidth ’ , 2) ;

282

283 x l a b e l ( ’Strain contribution (GHz) ’ )

284 y l a b e l ( ’Level frequency (GHz) ’ )

285 set ( gca , ’ f o n t s i z e ’ , 15)

286 % set(gca, ’YTick’, [−150 0 150])

287 t i t l e ( ’Excited state ’ )

288 ylim ([−350 350]) ;

B.1.2 ‘SiV_Hamiltonian_PMbase.m’

This file is written in {|e+⟩ , |e−⟩} basis.

144 1 f u n c t i o n [ground_eigvector , ground_eigenergy , excited_eigvector ,

excited_eigenergy] = SiV_Hamiltonian_PMbase(Hg_str , He_str , Bvec)

2 %%% Simulating the energy level of SiV with interactions %%%

3 %%% File created by Joonhee Choi, 170107 %%%

4 %%% Reference: Thesis by Hepp 2014, Hepp et al , PRL 112, 036405 %%%

6 %%% Revision made by YoungIk Sohn, 170216 %%%

8 %%% manifold of ground state levels %%%

9 % orbital states: e+, e− %%%

10 % electronic spin states: up, down %%%

11 % the ground state basis: (e+,up), (e−,down ) , ( e−,up ) , ( e−,down ) %%%

13 %% SiV parameters %%

14 % Basis conversion matrix between (ex,ey) basis and (e+,e−) basis

15 T=[−1 −1i ;1 −1i ] ;

17 % define spin and orbital operators

18 Id = [1 0; 0 1];

19 Sx= [0 1; 1 0];

20 Sy = [0 −1i ; 1 i 0 ] ;

21 Sz = [1 0; 0 −1];

22 Lz = [0 1 i ; −1i 0 ] ;

145 23

24 % define Jahn−Teller operator

25 gYx = 11; % GHz

26 gYy = 11; % GHz

27 eYx = 20; % GHz

28 eYy = 20; % GHz

29 ground_JT = [gYx gYy; gYy −gYx ] ; % ground state

30 excited_JT = [eYx eYy; eYy −eYx ] ; % excited state

32 % direct product with identity matrix for spin

33 Hg_JT = kron (ground_JT, Id); % ground state

34 He_JT = kron (excited_JT, Id); % excited state

36 % conversion to PM basis

37 Hg_JT=kron (T, Id)*Hg_JT* i n v ( kron ( T , Id ) ) ;

38 He_JT=kron (T, Id)*He_JT* i n v ( kron ( T , Id ) ) ;

40 % define strain operatior

41 % direct product with identity matrix for spin

42 Hg_str = kron (Hg_str, Id);

43 He_str = kron (He_str, Id);

45 % strain Hamiltonian conversion to PM basis

146 46 Hg_str=kron (T,Id)*Hg_str* i n v ( kron ( T , Id ) ) ;

47 He_str=kron (T,Id)*He_str* i n v ( kron ( T , Id ) ) ;

49 % spin−orbit coupling constant

50 lambda_g = 46; % GHz

51 lambda_e = 257; % GHz

53 % spin−o r b i t term

54 Hg_SO = −lambda_g /2 * kron ( Lz , Sz ) ; % ground state

55 He_SO = −lambda_e /2 * kron ( Lz , Sz ) ; % excited state

57 % conversion to PM basis

58 Hg_SO=kron ( T , Id ) *Hg_SO* i n v ( kron ( T , Id ) ) ;

59 He_SO=kron ( T , Id ) *He_SO* i n v ( kron ( T , Id ) ) ;

61 % Zeeman term coefficients

62 f = 0 . 1 ; % quenching factor for orbitals

63 gamma_S = 2 8 . 2 ; % GHz/T, gyromagnetic ratio for spin

64 gamma_L = 0.5 * gamma_S; % GHz/T, gyromagnetic ratio for orbital

66 Bx = Bvec(1); By = Bvec(2); Bz = Bvec(3);

147 68 H_ZL = f * gamma_L/2 * Bz * kron ( Lz , Id ) ;

% Energy from orbital angular momentum

69 H_ZS = gamma_S/2 * ( Bx * kron (Id,Sx) + By * kron (Id,Sy) + Bz * kron ( Id , Sz ) ) %

Energy from spin angular momentum

71 % conversion to PM basis

72 H_ZL = kron (T,Id) * H_ZL * i n v ( kron ( T , Id ) ) ;

73 H_ZS = kron (T,Id) * H_ZS * i n v ( kron ( T , Id ) )

75 % total Hamiltonian

76 Hg_total = Hg_SO + Hg_JT + H_ZL + H_ZS + Hg_str; % Total Hamltonian of

ground state

77 He_total = He_SO + He_JT + H_ZL + H_ZS + He_str; % Total Hamltonian of

excited state

79 [ Pg , Eg ] = eig (Hg_total);

80 [ Pe , Ee ] = eig (He_total);

82 ground_eigvector = Pg;

83 ground_eigenergy = diag (Eg) ;

85 excited_eigvector = Pe;

86 excited_eigenergy = diag (Ee) ;

148 B.1.3 ‘SiV_Hamiltonian_XYbase.m’

This file is written in {|eX⟩ , |eY⟩} basis.

1 f u n c t i o n [ground_eigvector , ground_eigenergy , excited_eigvector ,

excited_eigenergy] = SiV_Hamiltonian_XYbase(Hg_str , He_str , Bvec)

2 %%% Simulating the energy level of SiV with interactions %%%

3 %%% File created by Joonhee Choi, 170107 %%%

4 %%% Reference: Thesis by Hepp 2014, Hepp et al , PRL 112, 036405 %%%

6 %%% Revision made by YoungIk Sohn, 170216 %%%

8 %%% manifold of ground state levels %%%

9 % orbital states: ex, ey %%%

10 % electronic spin states: up, down %%%

11 % the ground state basis: (ex,up), (ex,down), (ey,up), (ey,down) %%%

13 %% SiV parameters %%

15 % define spin and orbital operators

16 Id = [1 0; 0 1];

17 Sx= [0 1; 1 0];

18 Sy = [0 −1i ; 1 i 0 ] ;

19 Sz = [1 0; 0 −1];

149 20 Lz = [0 1 i ; −1i 0 ] ;

22 % define Jahn−Teller operator

23 gYx = 11; % GHz

24 gYy = 11; % GHz

25 eYx = 20; % GHz

26 eYy = 20; % GHz

27 ground_JT = [gYx gYy; gYy −gYx ] ; % ground state

28 excited_JT = [eYx eYy; eYy −eYx ] ; % excited state

30 Hg_JT = kron (ground_JT, Id); % ground state

31 He_JT = kron (excited_JT, Id); % excited state

33 % define strain operatior

34 % direct product with identity matrix for spin

35 Hg_str = kron (Hg_str, Id);

36 He_str = kron (He_str, Id);

39 % spin−orbit coupling constant

40 lambda_g = 46; % GHz

41 lambda_e = 257; % GHz

150 43 % spin−o r b i t term

44 Hg_SO = −lambda_g /2 * kron ( Lz , Sz ) ; % ground state

45 He_SO = −lambda_e /2 * kron ( Lz , Sz ) ; % excited state

48 % Zeeman term coefficients

49 f = 0 . 1 ; % quenching factor for orbitals

50 gamma_S = 2 8 . 2 ; % GHz/T, gyromagnetic ratio for spin

51 gamma_L = 0.5 * gamma_S; % GHz/T, gyromagnetic ratio for orbital

53 % Zeeman term

54 Bx = Bvec(1);

55 By = Bvec(2);

56 Bz = Bvec(3);

58 H_ZL = f * gamma_L/2 * Bz * kron ( Lz , Id ) ;

% Energy from orbital angular momentum

59 H_ZS = gamma_S/2 * ( Bx * kron (Id,Sx) + By * kron (Id,Sy) + Bz * kron ( Id , Sz ) ) ;

% Energy from spin angular momentum

61 % total Hamiltonian

62 Hg_total = Hg_SO + Hg_JT + H_ZL + H_ZS + Hg_str; % Total Hamltonian of

ground state

151 63 He_total = He_SO + He_JT + H_ZL + H_ZS + He_str; % Total Hamltonian of

excited state

65 % eigen value solving

66 [ Pg , Eg ] = eig (Hg_total);

67 [ Pe , Ee ] = eig (He_total);

69 ground_eigvector = Pg;

70 ground_eigenergy = diag (Eg) ;

72 excited_eigvector = Pe;

73 excited_eigenergy = diag (Ee) ;

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