DESIGNING HIGH SENSITIVITY NUCLEAR INERTIAL SENSORS IN DIAMOND A Dissertation Presented to the Faculty of the Graduate School of Cornell University in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy by Johnathan Kuan December 2024 © 2024 Johnathan Kuan ALL RIGHTS RESERVED DESIGNING HIGH SENSITIVITY NUCLEAR INERTIAL SENSORS IN DIAMOND Johnathan Kuan, Ph.D. Cornell University 2024 The nitrogen-vacancy (NV) center in diamond is a mature platform for quantum technologies, consisting of an electron spin that has long coherence times and is sensitive to a variety of external fields at room temperature. These features make NV centers a useful resource for precision metrology and its electron spin has been used to design sensors for magnetic, electric, temperature,and strain fields. The nitrogen nuclear spin that is native to every NV center is also of interest since it presents an opportunity to build highly sensitive sensors. This potential high sensitivity of NV nuclear sensors comes from its small gyromag- netic ratio, which makes it insensitive to magnetic noise, and its long bare coher- ence time on the order of seconds. Developing these nuclear sensors is an active area of research, with multiple demonstrations successfully implementing the nuclear spin as a rotation sensor. However, the sensitivity of these sensors are orders of magnitude worse than conventional gyroscopes such as micorelec- tromechanical systems (MEMS) and ring laser gyroscopes. In this thesis, we will discuss the fundamental limitations behind the poor sensitivity of current nuclear spin sensors, which stems from the strong hyper- fine interaction between the NV center’s electron spin and nuclear spin. This hyperfine interaction limits the coherence time of nuclear spins in diamond to the electron T1. We will also discuss strategies to decouple the nuclear and elec- tron spins from each other, to extend the nuclear spin coherence time. In partic- ular, we focus on strong driving of a magnetically forbidden double-quantum transition. We propose to drive this double-quantum transition using acoustic strain from a MEMS resonator fabricated on diamond and present the devel- opment and performance of a new generation of piezoelectric thin film bulk acoustic resonators on diamond that are designed for this application. Finally we also investigate and design a readout protocol for high sensitivity nuclear spin sensors operating beyond the electron T1 to guide future nuclear sensor development. BIOGRAPHICAL SKETCH Johnathan Kuan spent his childhood in New Jersey, where he developed and interest in math and science. He went on to do his undergraduate work at Columbia College, Columbia University in New York City worked in astro- physics in the VERTIAS collaboration. He received his B.A. in Physics and Mathematics in 2016 and went on to Cornell University immediately after. When he first arrived to Cornell, he was interested in X-ray science and started working at the Cornell High Energy Synchrotron Source (CHESS) under Carl Franck. He ultimately ended up working on quantum information with NV centers under Gregory Fuchs. iii To my parents, for their continual and unconditional love and support. iv ACKNOWLEDGEMENTS I want to begin with thanking my special committee: Daniel Ralph, Erich Mueller, and Gregory Fuchs. I want to thank Greg in particular for his patience and his efforts to guide me through (and after) graduate school. It’s under his steady hand that I came to gain some confidence in myself as an experimental physicist. I also want to acknowledge Carl Franck, who taught me that kindness can go a long way. Outside of Cornell, I would like to thank Sunil Bhave for all his help and good humor. The saying goes that ”it takes a village” and I’ve come to appreciate how true this aphorism is. Without the help and support of the following individuals, I don’t think I would have managed to finish my degree. First I would like to thank my fellow Fuchs Group members, both old and new. In particular, I want to thank Harry Cheung, Michael Chilcote, and Bren- dan McCullian for answering my numerous questions. I want to thank Anthony D’Addario for his help and great company. There is no one else I would rather spend a 14 hour car ride with to transport an AWG to Illinois. I would also like to thank Nicole Guo, Jialun Luo, and Jaehong Choi for the positive atmosphere they helped cultivate around the lab. Outside the Fuchs group, I would like to thank Ozan Erturk for being one of the best collaborators I’ve had the fortune of working with. Outside the lab, but still in Ithaca, I also want to thank Alex Zhou, Peter Ko- ufalis, William Li, Thomas Oseroff, Chad Pennington, and Jai-Kwan Bae who helped make life in Ithaca richer. Outside of Ithaca, I would like to thank my dear friends: Jonathan Ting, Christopher Wang, Gracie Gilbert, Mounika Boda- pati, Royce Zhou, and Kaleel Hatten. Thank you for celebrating the good times and helping me get through the bad. Finally, I want to end by thanking the most v important people to me– my family, who have unconditionally given me their love and support and have always provided me with a home that I can always return to. vi TABLE OF CONTENTS Biographical Sketch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii Dedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi 1 Introduction 1 2 Rotation Sensing with Nitrogen-Vacancy Centers in Diamond 4 2.1 Chapter Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.3 NV Center Electronic Structure . . . . . . . . . . . . . . . . . . . . 6 2.4 NV Center Hyperfine Structure . . . . . . . . . . . . . . . . . . . . 8 2.5 Nuclear Spin Readout . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.6 NV Center Magnetometry . . . . . . . . . . . . . . . . . . . . . . . 14 2.7 Sensitivity of an NV Magnetometer . . . . . . . . . . . . . . . . . . 17 2.8 Rotation Sensing with Spins . . . . . . . . . . . . . . . . . . . . . . 19 2.9 Sensitivity of a NV Nuclear Gyroscope . . . . . . . . . . . . . . . . 21 2.10 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 3 Continuous Dynamical Decoupling of the Nuclear Spin from the NV Center Electron 24 3.1 Chapter Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 3.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 3.3 Semi-classical Model of Nuclear Spin Decoherence . . . . . . . . . 25 3.4 Dynamical Decoupling for the Nuclear Spin . . . . . . . . . . . . . 28 3.5 NV Center Spin-Strain Interaction . . . . . . . . . . . . . . . . . . 31 3.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 4 Thin-Film Bulk Acoustic Resonators on Diamond for Continuous Dy- namical Decoupling 37 4.1 Chapter Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 4.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 4.3 Diamond Gyroscopes with FBAR Concept . . . . . . . . . . . . . . 38 4.4 FBAR on Diamond Fabrication . . . . . . . . . . . . . . . . . . . . 40 4.5 Acoustically Driven Rabi Oscillations . . . . . . . . . . . . . . . . 42 4.6 First Generation FBAR Device . . . . . . . . . . . . . . . . . . . . . 44 4.7 Second Generation FBAR Device . . . . . . . . . . . . . . . . . . . 46 4.7.1 50 Ω Impedance Matching . . . . . . . . . . . . . . . . . . . 46 4.7.2 Device Packaging . . . . . . . . . . . . . . . . . . . . . . . . 49 4.7.3 Spin Measurements . . . . . . . . . . . . . . . . . . . . . . 50 vii 4.8 Conclusion and Outlook . . . . . . . . . . . . . . . . . . . . . . . . 52 4.9 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 5 Optical Pumping Dynamics of the NV Center Electron and Nuclear Spins 54 5.1 Chapter Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 5.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 5.3 The Lindblad Master Equation . . . . . . . . . . . . . . . . . . . . 55 5.4 7 Level Rate Model of the NV Center . . . . . . . . . . . . . . . . . 58 5.5 21 Level Rate Model of the NV Center . . . . . . . . . . . . . . . . 63 5.6 Excited State Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . 64 5.7 Nuclear Polarization . . . . . . . . . . . . . . . . . . . . . . . . . . 67 5.8 Field Dependent Contrast of NV Center Spin States . . . . . . . . 71 5.9 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 6 Optical Readout of Coherent Nuclear Spins Beyond the Electron T1 74 6.1 Chapter Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 6.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 6.3 Readout Protocol past the Electron T1 . . . . . . . . . . . . . . . . 76 6.3.1 Visiblity and the Sensitivity of an NV gyroscope . . . . . . 76 6.3.2 Physical Considerations for Readout Sensitivity . . . . . . 77 6.3.3 Thermal Electron State Preparation . . . . . . . . . . . . . 80 6.3.4 Artificial Thermal Electron State Fidelity . . . . . . . . . . 81 6.3.5 Simulated Rotation Sensing Measurement . . . . . . . . . 84 6.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 6.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 6.5.1 Analysis of Longitudinal Ramsey Results . . . . . . . . . . 93 6.5.2 Qualitative Model of the Phase Susceptibility . . . . . . . . 101 6.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 6.7 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 6.8 Supplementary Information . . . . . . . . . . . . . . . . . . . . . . 109 6.8.1 Nuclear Polarization from 200-800 G . . . . . . . . . . . . . 109 7 Conclusion 111 8 Appendix 113 8.1 Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 8.1.1 Confocal Microscopy Setup . . . . . . . . . . . . . . . . . . 113 8.1.2 Microwave Electronics and RF Switch Network . . . . . . 116 8.2 Quantum State Tomography . . . . . . . . . . . . . . . . . . . . . . 120 8.2.1 Spin-1/2 Case . . . . . . . . . . . . . . . . . . . . . . . . . . 121 8.2.2 Spin-1 Case . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 8.3 Quantum Process Tomography . . . . . . . . . . . . . . . . . . . . 126 8.3.1 General Theory . . . . . . . . . . . . . . . . . . . . . . . . . 126 viii 8.3.2 Process Tomography of One Qubit Processes . . . . . . . . 128 8.4 AlN Deposition and Processing . . . . . . . . . . . . . . . . . . . . 132 8.4.1 Sputtering and Characterization . . . . . . . . . . . . . . . 132 8.4.2 Etching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 Bibliography 137 ix LIST OF TABLES 3.1 Spin-stress coupling coefficients calculated and measured in the literature. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 5.1 Rates used for the 7 level model for the optical cycle of the NV center’s electron spin. GS and ES denote the ground state and excited state manifolds respectively. . . . . . . . . . . . . . . . . . 60 x LIST OF FIGURES 2.1 a) The NV center in the diamond. b) NV center spectrum at room temperature showing the ZPL (637 nm) and phonon sideband. c) Single NV center resolved in a confocal microscope. . . . . . . . . 5 2.2 Optical cycle of the NV center electron, which leads to polariza- tion to mS = 0 and spin-dependent photoluminescence. . . . . . . 7 2.3 a) (14N) Hyperfine structure of the NV center. b) Hyperfine tran- sitions in ODMR at 200 G. A radio-frequency pulse is applied to increase the visibility of the three hyperfine transitions. . . . . . . 10 2.4 a) NV center dispersion with varying external field. b) Zoomed in view of the GSLAC between |−1,+1⟩ and |0, 0⟩ . . . . . . . . . . 11 2.5 CNOT pulse applied to create nuclear spin-dependent photo- luminescence. mI = 0 appears ”bright” and mI = +1 appears ”dark” after the mapping. . . . . . . . . . . . . . . . . . . . . . . . 13 2.6 Ramsey sequence used for NV magnetometry and rotation sens- ing. The spheres below show the NV center electron state plotted on the Bloch sphere. . . . . . . . . . . . . . . . . . . . . . . . . . . 15 3.1 a) Nuclear spin precession with no hyperfine coupling. b) Nu- clear spin precession slowing down after T1 relaxation to mS = −1. c) Nuclear spin precession speeding up after T1 relaxation to mS = +1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 3.2 a) Dynamical decoupling of the nuclear spin from the electron spin by using repeated π pulses on the electron spin. b) Theoret- ical enhancement over bare T ∗2,n with no dynamical decoupling. The dotted lines mark where the enhancement is a factor of 10. . 29 3.3 Coordinate system for spin-strain Hamiltonian. Lower case let- ters denote the NV frame and upper case letters denote the cubic crystal frame. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 4.1 Schematic of the FBAR on diamond gyroscope concept. An AlN piezoelectric transducer launches standing acoustic waves into the diamond. An NV ensemble is placed at an anti-node (red and blue) of the wave to implement the decoupling protocol. A MW antenna is fabricated on the opposite side (potentially off the diamond chip) for magnetic spin control. . . . . . . . . . . . . 39 4.2 AlN FBAR on diamond process flow. The lithography and etch- ing is done at Purdue and the AlN deposition is done at Cornell. Figure Credit: Ozan Erturk. . . . . . . . . . . . . . . . . . . . . . . 41 4.3 a) Driving the DQ transition on resonance. b) Pulse sequence for acoustically-driven Rabi oscillations. . . . . . . . . . . . . . . . . 43 xi 4.4 a) SEM image of first generation FBAR device with an antenna for MW spin control. b) COMSOL calcuation of the acoustic mode generated by the FBAR. c) Probe measurements of the FBAR electromechanical resonance measured at ∼2.55 GHz. The yellow region between the dotted black lines indicates the range of drive frequencies that the FBAR performance was character- ized via spin measurements. Figure is based on images and data from Ref [1]. Probe measurements were provided by Noah Op- pondo and Ozan Ertutk. . . . . . . . . . . . . . . . . . . . . . . . . 44 4.5 a) Acoustically driven Rabi oscillations across ∼ FBAR reso- nance. b) FFT of acoustically driven Rabi oscillations with drive frequency. c) Trace of the Rabi oscillation at the FBAR resonance. Figure is based on images and data from Ref [1]. . . . . . . . . . . 45 4.6 Impedance of the first generation FBAR device at the elecrome- chanical resonance. The impedance is around 700 Ω at 2.55 GHz. 47 4.7 a) Optical and b) SEM image of the second generation FBAR de- vice. c) Probe measurement of the electromechanical response and impedance of the second generation FBAR device at around 1.93 GHz. Optical and SEM images were provided by Ozan Erturk. 48 4.8 a) Image of sapphire chip with a Pt antenna fabricated for mi- crowave spin control. b) Image of the diamond bonded to the sapphire. c) Alignment marks used to align the sapphire and diamond chips to each other. d) Image of the antenna aligned to the FBAR. The red circle denotes the location of the antenna beneath the diamond. Images were provided by Ozan Erturk. . . 50 4.9 a) Rabi oscillations of the DQ transition at 1.92 GHz. The max- imium measured Rabi frequency is about 7 MHz. b) Fitted Rabi frequencies plotted against the drive power into the device. Past 27 dBm, the device performance falls off the observed trend (solid black line). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 5.1 7 level model for the optical cycling of the NV center’s elec- tron spin. The green arrows denote transitions due to laser ex- citation. The red arrows denote transitions due to spontaneous emission. The blue arrows denote transitions associated with the non-radiative intersystem crossing. . . . . . . . . . . . . . . . . . 59 5.2 a) Calculated mS = 0 occupation using the 7 level model for opti- cally pumping a thermal electron spin state at optical saturation. b) Corresponding electron polarization to mS = 0 . . . . . . . . . 60 5.3 a) Spin dependent PL emission from mS = 0 and mS = −1 calcu- lated from 7 level model. The yellow shade area is the contrast that we can experimentally observe. . . . . . . . . . . . . . . . . . 62 xii 5.4 a) ES spin flip-flops that are mediated by the transverse hyper- fine coupling A⊥ near the ESLAC. b) ESLACs opened by the transverse hyperfine coupling. The solid lines denote the ES- LAC between |0, 0⟩ and |−1,+1⟩ and the dotted lines denote the ESLAC between |0,−1⟩ and |−1, 0⟩. . . . . . . . . . . . . . . . . . . 64 5.5 Average spin flip-flop probability pavg between |0, 0⟩ and |−1,+1⟩ from 200-800 G. The black dotted line denotes the location of the ESLAC. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 5.6 Pathways to polarize nuclear spin to mI = +1 for Group I, II, and III. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 5.7 a) Occupation in mI = +1 after optical pumping at saturation. b) Corresponding polarization in mI+ = 1 . . . . . . . . . . . . . . . 70 5.8 Calculated PL emission from |0,+1⟩ and |−1,−1⟩ from the 21 level model at a) 200 G b) 500 G. The yellow region is the observed contrast. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 6.1 a) Artificial electron thermal state preparation. A pair of MW pulses is used to equalize the electron spin populations in all sublevels. A 10 µs wait (greater than T ∗2,e ∼ 600 ns) eliminates any coherences between electron spin sublevels. b) Quantum state tomography of the artificial thermal state at 400 . The image on the left shows the density matrix for a true thermal state and the image on the right shows the measurement. The fidelity is ap- proximately 98%. c) Pulse sequence for the optical repump of the longitudinal component of the nuclear spin coupled to the arti- ficial electron thermal state (simulating rotating sensing beyond the electron T1). d) Pulse sequence for the optical repump of the transverse component of the nuclear spin coupled to the artifi- cial electron thermal state. e) Pulse sequence for control mea- surement. The nuclear spin is coupled to a pure mS = 0 state, which represents the visibility obtained in the long time limit of the electron T1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 6.2 a) Quantum state tomography (QST) for checking artificial ther- mal state. After the thermal state is prepared, a set MW pulses is used to measure the expectation values of the Gell-mann spin observables λi necessary for measuring the qutrit state. b) QST pulses used to diagonalize the spin operators for S z measurement. 83 6.3 Calibration pulse sequences of PL emission from mS = 0,±1. The adiabatic passage pulses are applied after the 20 µs green exci- tation to transfer the NV center electron spins from mS = 0 to mS = ±1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 xiii 6.4 Representative measurements of the visibility and initial phase ϕ of the Ramsey fringe at 400 G for a) the optically repumped lon- gitudinal Ramsey sequence [Figure 6.1(c)] and b) the optically repumped transverse Ramsey sequence [Figure 6.1(d)]. There are only significant increases for the visibility for the longitudi- nal Ramsey. The transverse Ramsey shows a linear dependence between the initial phase ϕ and the repump time. . . . . . . . . . 85 6.5 Summary of data collected from 200-800 G. a) Visibility of the CNOT readout for a state initialized to |0, 0⟩ (green). Maximum visibility of Ramsey sequences with no repump is shown in blue, which demonstrates the contribution of the contrast enhance- ment mechanism. b) Maximum recovered visibility for the lon- gitudinal Ramsey (blue) compared to the maximum possible vis- ibility (red). The maximum occurs at 500 G, where the ES spin flip-flops are the strongest. c) Maximum recovered visibility for the transverse Ramsey (blue) compared to the maximum possi- ble visibility (red). The maximum also occurs at 500 G, but is worse compared to the longitudinal Ramsey. d) Phase suscepti- bility χϕ for the transverse Ramsey from 200-800G. The antisym- metric trend about 500 G suggests that the physical origin behind the phase susceptibility come from the ES dynamics. . . . . . . . 89 6.6 Plot of the NV ensemble photoluminescence as a function of lase excitation power. The black dotted line indicates the saturation power extracted from the fit, which is used in the Lindblad cal- culations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 6.7 a) Schematic of simulated quantum process tomography. The ex- perimental process consists of the nuclear spin undergoing opti- cal pumping, followed by measurement of the nuclear spin pop- ulations. The ideal process consists of the identity process on the nuclear spin, followed by measurement of the nuclear spin populations. b) Plot of the nuclear spin fidelity (Eqn 6.17) from 200-800 G for times up to 2 µs. The fidelity is the worst near the ESLAC at approximately 500 G and for long pump times due to the accumulation of many ES spin flip-flops. . . . . . . . . . . . . 97 6.8 Calculated contrasts of all 9 basis states using the Lindblad model. The contrasts are calculated by normalizing to the PL emission from |0,+1⟩ [Eq 6.4]. The collection time and power are chosen to match our experiment. Different states have an enhancement or loss in contrast at 500 G due to the ES spin flip- flops. The states |0,+1⟩ and |+1,+1⟩ are not affected by the ES spin flip-flops, making the contrasts independent of field. . . . . 99 xiv 6.9 Lindblad calculation showing the occupations in the nuclear spin sublevels when the state a) ρtherm 3 ⊗ |+1⟩ ⟨+1| and b) ρtherm 3 ⊗ |0⟩ ⟨0| are optically pumped. Both cases show negligible leakage into mI = −1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 6.10 Comparison between the visibility of the longitudinal Ramsey over repump time and the calculated occupation in |0, 0⟩ for a) 400G and b) 500 G after optically repumping the state ρtherm e ⊗ |0⟩ ⟨0|. The maximum of the occupation coincides with the max visibility, which supports |0, 0⟩ being the state that is dominating the visibility. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 6.11 Plot of the calculated phase susceptibility from the four level qualitative model from the ES Hamiltonian. The model shows the antisymmetric dependence about the ESLAC. The excited state lifetime used for the calculation is 10 ns. . . . . . . . . . . . 102 6.12 Nuclear polarization into mI = +1 at the fields used in this work. The polarization is calculated according to Eqn. 6.39, where 0% corresponds to an unpolarized ensemble and 100% corresponds to a perfectly polarized ensemble. . . . . . . . . . . . . . . . . . . 110 8.1 Schematic of confocal microscope setup used for data collection. Green indicates the excitation path and red indicates the collec- tion path. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 8.2 a) Schematic of MW and RF lines used for magnetic and acous- tic spin control. b) RF switch network used for routing single photon counts. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 8.3 a) AlN chuck used to sputter on small diamond pieces. b) Di- amond membrane mounted in chuck. c) AlN chuck in the de- position chamber of the OEM Endeavor M1 tool at the Cornell Nanoscale Facility. d) XRD measurement of AlN deposited on a Si wafer and a 3 mm by 3 mm CVD diamond piece. . . . . . . . . 133 8.4 a) Patterned etch window after the hard bake. b) Etch window with the exposed bottom Pt layer (white). The undercut at the edge of the etch window is visible. c) Etch depth versus etch time with TMAH developer (MIF726). . . . . . . . . . . . . . . . . 135 xv CHAPTER 1 INTRODUCTION The nitrogen-vacancy (NV) center in diamond is an ”atom-like” solid state de- fect that has properties that are of interest for many quantum information appli- cations. At cryogenic temperatures, the narrow optical linewidth of single NV centers make them a candidate platform generating indistinguishable photons for building quantum networks. At room temperature, the NV center main- tains excellent quantum mechanical properties, such as long spin coherence times (T1 ≈ 10 ms, T2 ≈ 100 µs), which make it a good resource for precision metrology. The electron spin of the NV center has been extensively studied and has been used to sense magnetic fields [2, 3, 4, 5, 6, 7], electric fields [8, 9, 10], and temperature in a multitude of settings [11, 12, 13]. These include sensing in biological systems [14, 15], high pressure systems [16], and exotic magnetic materials and textures [17]. Another potential application of the NV center is towards building iner- tial sensors, in particular rotation sensors such as gyroscopes. Since quantum mechanical spins are forms of angular momentum, they are sensitive to ex- ternal rotations of their environment. In the literature, both the electron spin and nitrogen nuclear spin of the NV center has been used to build sold-state spin-based gyroscopes [18, 19, 20]. In particular, the nuclear spin is of interest since it has a lower gyromagnetic ratio (14N– 0.300 kHz/G) compared to the electron spin (2.8 MHz/G), making it less susceptible to magnetic fluctuations that can harm the sensitivity of the sensor [21]. In addition, the phase coher- ence (T ∗2) of isolated nuclear spins can be very long– on the order of seconds [22, 23, 24]. These factors make nuclear spin-based gyroscopes potentially more 1 sensitive than conventional gyroscopes, which operate using microelectrome- chanical systems (MEMS) to measure the Coriolis force or interferometers that measure the optical Sagnac effect. Because of this potential, there are various demonstrations using the native nitrogen nuclear spin as a rotation sensor. However these sensors have sensitiv- ities that are orders of magnitude worse than current state of the art gyroscopes. The main culprit for this is a fundamental limit on the phase coherence of nu- clear spins in diamond. Specifically, the nitrogen nuclear spin has a strong hy- perfine interaction (∼2.1 MHz) with the NV center’s electron spin. This causes the nuclear spin’s phase coherence to be highly dependent on the electron spin state. When the electron spin undergoes spin-lattice relaxation (T1 processes), the phase coherence is lost limiting the phase coherence time (T ∗2) to the elec- tron T1. This thesis is focused on strategies to decouple the nuclear spin from the electron spin to build the next generation of NV center based gyroscopes. If these strategies are successful, these new sensors will require strategies to read out the nuclear spin past the electron T1, which is a regime current NV center sensors do not operate in. The layout of the thesis is as follows. In Chapter 2, we discuss the physics of the NV center spin (electron and nuclear spins) and its application to rotation sensing. In Chapter 3, we discuss the coupling between NV centers and lattice strain and the theory behind continuous dynamical decoupling (CDD) of the nuclear spin from the electron spin. In Chapter 4, we present work on fabricat- ing Aluminum Nitride thin-film bulk acoustic resonators (FBARS) on diamond to realize CDD. In Chapter 5, we discuss the NV center’s optical dynamics and its effect on the nuclear spin. Finally in Chapter 6, we propose a method for nu- 2 clear readout beyond the electron T1. We test this protocol for nuclear readout past the electron T1 and discuss its impact on future nuclear spin gyroscopes. 3 CHAPTER 2 ROTATION SENSING WITH NITROGEN-VACANCY CENTERS IN DIAMOND 2.1 Chapter Abstract We introduce the optical and spin properties of the nitrogen-vacancy (NV) cen- ter at room-temperature and discuss the the physics behind its Hamiltonian. We discuss the hyperfine interaction between an NV center and its native nitrogen nuclear spin, which allows for optical readout of the nuclear spin. Finally, we review the basic principles behind magnetometry with NV centers and relate this to rotation sensing. We identify the key parameters that are important to consider for the sensitivity of NV gyroscopes. 2.2 Introduction The negatively-charged nitrogen-vacancy (NV) center in diamond is a point de- fect that consists of a substitutional nitrogen adjacent to a lattice vacancy [Figure 2.1(a)]. The structure of the defect allows for discrete electronic states to exist in the band gap of diamond (∼5.5 eV) which are well separated from the states in the conduction and valance band. This makes the NV center an ”atom-like” system, which can be interrogated optically. For this thesis, we will focus on the properties of the NV center at room temperature [25]. Upon non-resonant excitation (commonly with 532 nm light), the NV center emits photoluminescence (PL) consisting of a zero phonon line 4 a) b) c) ZPL (637 nm) Phonon Sideband Figure 2.1: a) The NV center in the diamond. b) NV center spectrum at room temperature showing the ZPL (637 nm) and phonon side- band. c) Single NV center resolved in a confocal microscope. (ZPL) at 637 nm and a broad phonon sideband [Figure 2.1(b)], which contains the majority of the emission at room temperature. Using confocal microscopy techniques, single NV centers can be resolved and probed in diamond [Figure 2.1(c)]. For this thesis, we will work with ensembles of NV centers since this is the setting that most NV center sensors operate at to maximize their sensitivity. NV centers can naturally be found in diamond with high nitrogen concentra- tions. The NV centers used for this thesis come from diamond grown through chemical-vapor deposition (CVD). These diamonds are optical grade with ni- trogen introduced as a dopant to accelerate the growth. The optical grade dia- monds have nitrogen concentrations that are on the order of parts per million (ppm). The high nitrogen concentrations allow these diamonds to naturally host ensembles of NV centers. A metric for the quality of the diamond for quantum applications is the spin coherence the available NV centers. The spin coherence of qubits sets the timescale that gate operations and sensing must operate (i.e. shorter than the smallest coherence time). The inhomogeneous phase coherence T ∗2 largely de- 5 pends on the interaction of the NV center spin with other paramagnetic spins in the diamond lattice (e.g. P1 centers, other NV centers). For developing quan- tum sensors, we want the qubit to maximize the spin coherence to improve its sensitivity. One way to approach this is from a materials perspective, where we engineer the growth of CVD diamond to minimize the amount of paramag- netic impurities (such as nitrogen) that are introduced into the reactor. Diamond grown using this process is considered electronic grade and has nitrogen con- centrations on the order of parts per billion (ppb). Usually these diamonds do not host native NV centers. This requires ion implantation and annealing of the electronic grade diamond to create NV centers [26, 27, 28]. Growing isotopi- cally pure diamond (12C) removes another source of paramagnetic spins (13C). Exceptionally long coherence times (T2 ∼1 s) have been observed for single NV centers at room temperature for isotopically pure diamond [29, 30]. Alternatively, diamond can be grown through delta-doping, where nitrogen is introduced for a short fraction of the growth [31, 32, 33]. After subsequent annealing, the NV centers are well confined to a plane at a specific depth in the diamond. This provides an additional advantage over ion implantation since NV center ensembles can be engineered at a desired depth, rather than a spread of depths that is dictated of the kinetics of ion implantation. 2.3 NV Center Electronic Structure The electronic structure consists of a triplet ground state (3A2) and triplet excited state (3E) (Figure 2.2) [34]. The emitted PL from the NV center is dependent on the spin projection of the electron spin. The spin-dependence of the PL is 6 3A2 3E Orbital Spin Metastable States |+1〉 |−1〉 | 0 〉 |+1〉 |−1〉 | 0 〉 Strong Weak Non-radiative Absorb Emit Figure 2.2: Optical cycle of the NV center electron, which leads to polar- ization to mS = 0 and spin-dependent photoluminescence. due to the presence of a non-radiative intersystem crossing (ISC) that couples strongly to the mS = ±1 projections in the excited state, but weakly to the mS = 0 projection. This non-radiative ISC consists of a group of singlet states, which we will parameterize in this thesis as a single singlet state. As a consequence of this ISC, photon emission from mS = 0 will be higher on average than mS = ±1, making mS = 0 the ”bright” state and mS = ±1 the ”dark” state. In addition, since the ISC preferentially depopulates spins in mS = ±1, the electron spin can be polarized to mS = 0 after multiple optical cycles. We will go into detail about this optical cycling in Chapter 5 when we discuss optical readout of nuclear spins beyond the electron T1. One prominent application of the NV center is employing its electron spin as a quantum sensor. This is possible because a variety of external fields (e.g. mag- netic, electric, thermal) shift the spin-resonance frequencies of the NV center electron spin, which can be detected through magnetic resonance techniques. The NV center ground state has the canonical spin-1 defect Hamiltonian: Hgs = DgsS 2 z + γeB · S (2.1) 7 where Dgs is the zero-field splitting in the ground state (2.87 GHz), γe is the electron gyromagnetic ratio (2.8 MHz/G), B is the external magnetic field, and S = (S x, S y, S x) are the spin-1 operators [25]. There are other fields that the spin couples to such as electric fields (through the Stark effect). However, for this chapter, we will focus on magnetic fields. The NV center excited state shares a Hamiltonian of the same form: Hes = DesS 2 z + γeB · S (2.2) where Des is the zero field splitting in the excited state (1.42 GHz) [35, 36]. In quantum sensing applications, we will work with the electron spin in the ground state. However, the spin structure of the excited state will be important in subsequent chapters when we discuss the optical excitation and relaxation of the electron spin and its impact on the readout of a nuclear sensor. 2.4 NV Center Hyperfine Structure The NV center electron spin can interact with other spins in diamond, for exam- ple nuclear spins (e.g. 13C), that give rise to hyperfine structure. In particular we will focus on the interaction between the electron spin of the NV center and the native nitrogen nuclear spin that make up the defect. The two isotopes of nitrogen that can be found naturally are 14N (spin-1) and 15N (spin-1/2). We will focus on 14N since it is the most abundant isotope, but much of the general physics will carry over to 15N. The nuclear spin interacts with the electron spin through the hyperfine in- 8 teraction. For the ground state, this is described by the Hamiltonian: HGS h f = PI2 z + γnB · I + A∥S zIz + A⊥(S xIx + S yIy) (2.3) where P is the quadrupolar moment of 14N (-4.96 MHz), γn is the nuclear gy- romagnetic ratio (0.300 kHz/G), I are the spin-1 operators for the nuclear spin, A∥ is the longitudinal hyperfine coupling (2.14 MHz) and A⊥ is the transverse hyperfine coupling (2.70 MHz) [25, 37, 38]. The excited state has an analogous hyperfine Hamiltonian: HES h f = PI2 z + γnB · I +C∥S zIz +C⊥(S xIx + S yIy) (2.4) where C∥ is the longitudinal hyperfine coupling (−40 MHz) and C⊥ is the trans- verse hyperfine coupling (−23 MHz) [25, 39]. The ground state hyperfine struc- ture can be observed in optically-detected magnetic resonance (ODMR) mea- surements using small microwave fields commensurate with the hyperfine cou- pling [Figure 2.3(b)]. One of the consequences of the hyperfine interaction is the presence of level anti-crossings (LAC) in the NV center’s dispersion with field. These LACs occur at fields where the energy of the mS = −1 state is nominally equal to the mS = 0 state (Figure 2.4) [40]. For 14N, the transverse hyperfine coupling opens two LACs. One LAC corresponds to the crossings between |0, 0⟩ and |−1,+1⟩ and the other to the crossing between |0,−1⟩ and |−1, 0⟩. For the excited state level anti-crossing (ESLAC), these crossings occur at approximately 500 G [41, 36]. For the ground state level anti-crossing (GSLAC), these crossings occur at ap- proximately 1000 G [42]. To understand the physics behind these LACs, we will focus on the LAC corresponding to the |0, 0⟩ and |−1,+1⟩ crossing in the excited state. This LAC 9 a) b) |+1〉 |−1〉 | 0 〉 |+1〉 |−1〉 | 0 〉 |+1〉 |−1〉 | 0 〉 |+1〉 |−1〉 | 0 〉 Electron Spin 14N Spin Figure 2.3: a) (14N) Hyperfine structure of the NV center. b) Hyperfine transitions in ODMR at 200 G. A radio-frequency pulse is ap- plied to increase the visibility of the three hyperfine transitions. 10 ms = −1 ms = 0 ms = +1 GSLAC GSLAC|−1, +1〉 |0, 0〉 a) b) Figure 2.4: a) NV center dispersion with varying external field. b) Zoomed in view of the GSLAC between |−1,+1⟩ and |0, 0⟩ is significant because it introduces dynamics between the electron and nuclear spins of the NV center that will have consequences for reading out the nuclear spin, which is discussed in the latter half of this thesis. In the subspace of these two states, the Hamiltonian reduces to H =  0 A⊥ A⊥ ω−1,+1  (2.5) with ω−1,+1 = DES − γeB + γN B + P − A∥. (2.6) At low and high fields far from the ESLAC (much farther than A⊥/γe ∼20 G), the eigenstates of the Hamiltonian are well approximated by |0, 0⟩ and |−1,+1⟩. In the vicinity of the ESLAC (within A⊥/γe), the eigenstates are described by superpositions of |0, 0⟩ and |−1,+1⟩, with equal superpositions when ω−1,+1 = 0. The fact that the eigenstates are spin-mixed states between the electron and nuclear spins allows for consequential dynamics to occur in the excited state. For instance, if the NV center spin is in the state |0, 0⟩ after optical excitation, be- cause this is not an eigenstate of the system, this state will precess over time and 11 oscillate between |0, 0⟩ and |−1,+1⟩. This allows for spin flip-flop processes be- tween the electron and nuclear spin of the NV center near the LACs. These spin flip-flop processes are important because in combination with optical pumping, they allow for polarization of the NV center into |0,+1⟩, which will have conse- quences for nuclear spin readout beyond the NV center electron T1. For the work presented in this thesis, the external fields applied to the NV center will be up to 800 G. These fields lie well outside of the range of the of the GSLAC at 1000 G since the spin flip-flops are only significant within C⊥/γe ≈ 1 G. In this range of fields, the eigenstates will be well defined by the Zeeman basis |mS ,mI⟩. Therefore, we will neglect the effect of C⊥ in the ground state. 2.5 Nuclear Spin Readout The nuclear spin provides another degree of freedom to store information. This has been used to employ nuclear spins that are hyperfine coupled to NV center electron spins to create nuclear spin registers [43, 44, 45] and to help enhance conventional optical readout of the electron spin [46, 47]. Because of the poten- tial applications of the nuclear spin, it is important to be able to determine its state to extract the relevant information. Conventionally, nuclear spin readout takes advantage of the hyperfine cou- pling between the nuclear and electron spins to create nuclear spin-dependent PL. To see this, we will try to read out the information in the {+1, 0} qubit of 14N. By the end of sensing protocol or gate application (for a time shorter than the electron T1), the NV center is in the state a |0,+1⟩ + b |0, 0⟩. To create nuclear spin-dependent PL, we apply a selective microwave pulse (i.e. resonant with 12 |+1〉 |−1〉 | 0 〉 |+1〉 |−1〉 | 0 〉 |+1〉 | 0 〉 CNOT Nuclear SpinElectron Spin "Bright" "Dark" Figure 2.5: CNOT pulse applied to create nuclear spin-dependent photo- luminescence. mI = 0 appears ”bright” and mI = +1 appears ”dark” after the mapping. the |0,+1⟩ and |+1,+1⟩ transition, which we will refer to as a CNOT pulse (Fig- ure 2.5). This pulse will cause nuclear spins in mI = +1, which were originally hyperfine coupled to mS = 0, to be hyperfine coupled to the mS = +1 electron spins. Upon optical illumination, the mI = +1 spins will appear dark and the mI = 0 spins will appear bright. Therefore by observing the PL emission, we will be able to deduce the relative populations of nuclear spins in mI = +1 and mI = 0. In this thesis, we will use the term CNOT pulse to refer to any selective mi- crowave pulse that creates nuclear spin-dependent PL. For example, we could have used the CNOT pulse corresponding to the |0, 0⟩ and |−1, 0⟩ transition. This would have created a similat effect as the previous CNOT pulse, except mI = +1 will appear bright because it is hyperfine coupled to mS = 0 and mI = 0 would appear dark. 13 2.6 NV Center Magnetometry Rotation sensing with quantum mechanical spins is analogous to magnetic field sensing, so we will first detail the basic principles of NV magnetometry [2]. We will focus on DC magnetometry, though rotation sensing with spin echo tech- niques (AC magnetometry) have been demonstrated with the electron spin [18]. The central idea behind magnetometry is that the Zeeman effect will cause shifts in the spin resonance frequencies of the spin. Specifically, if the applied mag- netic field is along the NV axis, the spin transition frequency will depend lin- early on the applied field by γeBz where Bz is the magnitude of the applied field. This shift can be observed in optically detected magnetic resonance (ODMR) or pulsed measurements (i.e. Ramsey interferometry) to measure the magnetic field. We will focus on the latter as this will have the higher sensitivity. The magnetometer will consist an ensemble of NV center spins in the pres- ence of an unknown magnetic field Bz that we want to measure. First, we will consider a single NV center. We apply a magnetic field B0 with a known magni- tude to lift the degeneracy of the mS = ±1 states. We also apply a transverse field B1 cos(ω0t + ϕ)x̂ to drive spin transitions between |0⟩ and |+1⟩. We will assume that B1 is significantly larger than Bz so that this drive is effectively resonant with the transition. To simplify the discussion, we will focus on the case where our spin qubit consists of the |mS = 0⟩ and and |mS = +1⟩ states. To measure Bz, we will use the Ramsey interferometry sequence (Figure 2.6). First, we initialize the spin to |0⟩ (I). We then turn on the transverse field to apply a π/2 pulse (with ϕ = 0), which rotates the spin qubit to the equator of the Bloch sphere (II). After this, we have some free precession time T where the 14 Initialize Read Free Precession (Sensing) |+1〉 |0〉 |+1〉 |0〉 |+1〉 |0〉 |+1〉 |0〉 (I) (II) (III) (IV) Figure 2.6: Ramsey sequence used for NV magnetometry and rotation sensing. The spheres below show the NV center electron state plotted on the Bloch sphere. spin interacts with Bz (III). Finally we apply a second π/2 pulse, which is phase shifted by ϕ0 and read out the spin state (IV). We will work in the rotating frame of the microwave drive, which is rotating at ω0. In the absence of the transverse field (III), the Hamiltonian in the laboratory frame is H = ω0 + γeBz 0 0 0  = 1 2 (ω0 + γeBz)I + 1 2 ω0 + γeBz 0 0 −(ω0 + γeBz)  (2.7) where ω0 = D + γeBz. (2.8) We will neglect the first term in the Hamiltonian since this is a constant energy offset which will only contribute a global phase. After transforming to the rotat- ing frame using the unitary transformation URF = eiω0tσz/2, where σz is the Pauli operator for the z component of the spin, the Hamiltonian is H = 1 2 γeBz 0 0 −γeBz  = 1 2 γeBzσz. (2.9) 15 Therefore the time evolution operator for (III) is given by U3 = e −iγeBzt 2 0 0 e iγeBzt 2  . (2.10) When the transverse field is on (II and IV), the Hamiltonian is given in the laboratory frame as H = 1 2  ω0 + γeBz √ 2γeB1 cos (ω0t + ϕ) √ 2γeB1 cos (ω0t + ϕ) −(ω0 + γeBz)  . (2.11) We will define Ω = 2 √ 2γeB1, which corresponds to the Rabi oscillation fre- quency induced by the transverse field. After transforming to the rotating frame and apply the rotating wave approximation (RWA), the Hamiltonian is H = 1 2  0 Ωeiϕ Ωe−iϕ 0  = 1 2 (Ω cos ϕ)σx − 1 2 (Ω sin ϕ)σy (2.12) where σx and σy are the x and y Pauli operators. Applying a π/2 pulse means that the pulse length tp is chosen such that Ωtp = π 2 . The operator corresponding to applying a π/2 pulse in (II) is U2 =  1 √ 2 − i √ 2 − i √ 2 1 √ 2  (2.13) and in (IV) U4 =  1 √ 2 − i √ 2 eiϕ0 − i √ 2 e−iϕ0 1 √ 2  . (2.14) Therefore the evolution of the state due to the entire pulse sequence is described by the operator U = U4U3U2. (2.15) 16 To determine the signal we expect from our magnetometer, we follow the time evolution of the NV center spin. At (I), the ensemble is initialized to |ψ(0)⟩ = |0⟩. After all the pulses are applied, the final state, up to a global phase, is |ψ(t)⟩ = cos ( γeBzt − ϕ0 2 ) |+1⟩ + e−iϕ0 sin ( γeBzt − ϕ0 2 ) |0⟩ . (2.16) Therefore the final occupations in mS = 0 and mS = +1 are p0 = 1 2 − 1 2 cos (γeBzT − ϕ0) (2.17) p+1 = 1 2 + 1 2 cos (γeBzT − ϕ0). (2.18) If the NV ensemble interacts with a spin bath through dipolar interactions [48, 49, 50], then the populations in mS = 0 and mS = −1 will decay on a timescale corresponding to the inhomogeneous phase coherence time T ∗2 p0 = 1 2 − 1 2 e−T/T ∗2 cos (γeBzT − ϕ0) (2.19) p+1 = 1 2 + 1 2 e−T/T ∗2 cos (γeBzT − ϕ0). (2.20) Therefore, the magnetic field Bz can be measured by measuring the oscillation frequency of the Ramsey fringe γeBz. We will assume, for the rest of this chapter, that we are working in a regime where T is much smaller than T ∗2 so we can use Equations 2.17 and 2.18 as the occupations. 2.7 Sensitivity of an NV Magnetometer A key metric for the performance of the magnetometer is its sensitivity. This sensitivity has been derived and characterized in a variety of previous works [2, 4]. We will briefly summarize the arguments behind this derivation to allow us to motivate an analogous expression for the sensitivity of an NV gyroscope. 17 To measure the magnetic field Bz, we will perform a differential measure- ment where we compare the intensity change in the Ramsey fringe during a short amount of time τ. The simplest choice of points would be when T = 0 and T = τ. To introduce shot noise into our model, we will assume that |0⟩ will optically excite into a coherent state |α0⟩where the average number of pho- tons emitted in a collection time ts is n0 = |α0| 2. The state |+1⟩ will optically excite into a coherent state |α+1⟩ where the average number of photons emitted in the same collection time is n+1 = |α+1| 2. Because the spin states are orthogo- nal to each other, we will assume these coherent states are also orthogonal (i.e. ⟨α0|α+1⟩ = ⟨α+1|α0⟩ = 0). Thus, the final spin state of the NV center will emit into the state |Ψ(t)⟩ = cos ( γeBzt − ϕ0 2 ) |α+1⟩ − e−iϕ0 sin ( γeBzt − ϕ0 2 ) |α0⟩ (2.21) For a given time T , the expectation value for the number of photons emitted is ⟨N(T )⟩ = navg[1 − V cos (γeBzT − ϕ0)] (2.22) with the average number of photons navg and the visibility V defined as navg = n0 + n+1 2 (2.23) V = n0 − n+1 n0 + n+1 (2.24) The change in intensity ∆N is then given by ∆N = N(τ) − N(0) = ∂N ∂T ∣∣∣∣∣ T=0 τ (2.25) This gives Bz = ∆N γenavgVτ sin ϕ0 (2.26) The smallest detectable field δBz corresponds to when ∆N is equal to the uncertainty in the change in photon counts σN . The sensitivity η is then given 18 by η = δBz √ τ = σN γenavgV √ τ (2.27) where we have chosen ϕ0 = π/2 to minimize η. The uncertainty is calculated using the final photon state, which gives σN = navgV √ 1 + 1 V2navg ≈ √ navg (2.28) The approximation in the previous expression is because V2navg ≪ 1 for typical NV center measurements. This gives an expression for the shot-noise limited sensitivity of the NV magnetometer for an ensemble of N spins η ≈ 1 γeV √ Nnavgτ (2.29) In practice, τ is chosen to be less than the phase coherence time T ∗2 of the ensem- ble, which limits η to η ≈ 1 γeV √ NnavgT ∗2 (2.30) From this expression, we see that the sensitivity is limited by three key param- eter: the choice of spin (gyromagnetic ratio), the visibility of the Ramsey fringe, and the phase coherence time T ∗2 . 2.8 Rotation Sensing with Spins Spins are quanta of angular momentum and are therefore sensitive to the rota- tion of their environment. We will find that rotations appear to spins as a ficti- tious magnetic field, which allows us to use Ramsey interferometry to measure them. Our rotation sensor will consist of a spin-1/2 sitting in a solid, which is rotating at angular velocity ω. An antenna is fabricated on the structure to 19 provide magnetic control of the spin. We apply in a magnetic field B0ẑ to lift the degeneracy of the spin states, which will will label using the computational basis. In this basis, |0⟩ will correspond to m = +1/2 and |1⟩ will correspond to m = −1/2. This applied field gives us a spin transition of ω0 = γB0, where γ is the gyromagnetic ratio of the spin. In a stationary frame, we would apply a linearly polarized transverse field B1 cos(ω0t+ϕ)x̂ with the antenna to resonantly drive the spin transition. The field can be decomposed into two circularly polarized fields, one of which will drop out of the calculation due to the RWA. To simplify our calculation, we will work instead with the counterclockwise rotating field B1 cos(ω0t+ϕ) ˆx + B1 sin(ω0t + ϕ)ŷ. Because our antenna is rotating with the sensor, the transverse field that is ap- plied in the laboratory frame will be B1 cos[ω0 +ω)t + ϕ] ˆx + B1 sin[(ω0 + ω)t + ϕ]ŷ. We will follow the same Ramsey protocol as before. First we initialize the spin to |0⟩ (I). Then turn on the transverse field to apply a π/2 pulse (with ϕ = 0), which rotates the spin qubit to the equator of the Bloch sphere (II). After this, we have some free precession time T . Finally we apply a second π/2 pulse, which is phase shifted by ϕ0 and read out the spin state (IV). The difference now is that given external rotation, we need to work in frame rotating at ω0 + ω. The Hamiltonian in (II) and (IV) in the laboratory frame is H = 1 2  ω0 Ωe−i(ω0+ω)t Ωei(ω0+ω)t −ω0  (2.31) where Ω = γB1 is Rabi oscillation frequency corresponding to the applied trans- verse field. We transform to the rotation frame with the unitary transformation 20 URF = ei(ω0+ω)t. The effective Hamiltonian is then H = 1 2  −ω Ωe−iϕ0 Ωeiϕ0 ω  ≈ 1 2  0 Ωe−iϕ0 Ωeiϕ0 0  . (2.32) This is the same as eqn 2.12, where we assume that ω ≪ Ω. This means the operators describing the π/2 pulses are given as eqn 2.13 and 2.14 as before. In the rotating frame, the Hamiltonian during the free precession time (III) is H = 1 2 −ω 0 0 ω  = −1 2 ωσz. (2.33) Therefore the populations in |1⟩ and |1⟩ can be calculated by substituting −ω for γeBz in eqns 2.17 and 2.18. This gives p0 = 1 2 − 1 2 cos (ωT + ϕ0) (2.34) p1 = 1 2 + 1 2 cos (ωT + ϕ0). (2.35) Therefore the observed Ramsey fringes will oscillate by the rotation rate ω, anal- ogous to the magnetometry result. 2.9 Sensitivity of a NV Nuclear Gyroscope In the previous section, we calculated the Ramsey fringes for a psuedospin-1/2 rotation sensor. In this thesis, we will choose our effective two level system as the nitrogen nuclear spin of the NV center. The hyperfine interaction outlined previously allows us to use selective pulses to map the spin-dependent PL of the electron spin to the nuclear spins. Therefore we can define average PL navg and visibility V analogously to the NV center magnetometer. Our expected signal 21 for the measured photon counts is then ⟨N(T )⟩ = navg[1 − V cos (ωT + ϕ0)]. (2.36) Our rotation sensor will also operate by comparing the change in intensity ∆N between T = 0 and a short time T = τ. Therefore the shot-noise limited sensitivity of our gyroscope is η ≈ 1 V √ Nnavgτ . (2.37) This places a limit on the sensitivity that is determined by T ∗2,n of the nuclear spin η ≈ 1 V √ NnavgT ∗2,n . (2.38) A key feature here is that there is no dependence on the gyromagnetic ratio, making both electron spin and nuclear spins equally sensitive to rotations if they have the same phase coherence times. However, due to the higher gyromagnetic ratio of the electron spin, it is more sensitive to magnetic fluctuations than the nuclear spin, which causes T ∗2,e to be much smaller than T ∗2,n. For isolated nuclear spins T ∗2,n can be on ther order of seconds. This makes the nuclear spin of the NV center a better potential platform to construct a solid-state spin gyroscope than its electronic spin. However, as described in Chapter 3, T ∗2,n will be limited to the electron T1 due to the hyperfine interaction between the nuclear and electron spins of the NV center. 2.10 Conclusion In this chapter, we have discussed the electronic and hyperfine structure of the NV center at room temperature. We also discussed how rotation sensing with 22 NV centers is analogous to magnetometry, which allows us to use Ramsey in- terferometry to measure rotations. This allowed us to derive an expression for the sensitivity of a nuclear NV gyroscope, which depends on two key parame- ters: the visibility V and the phase coherence time T ∗2,n, which is limited to the electron T1. The rest of this thesis will concern how to maximize the sensitivity of an NV gyroscope with respect to these two parameters. In Chapters 3 and 4, we will discuss strategies to decouple the nuclear and electron spins from each other, potentially extending the T ∗2,n beyond the electron T1. In Chapter 5 and 6, we discuss optical readout of the nuclear spin beyond the electron T1, which will impact the V . 23 CHAPTER 3 CONTINUOUS DYNAMICAL DECOUPLING OF THE NUCLEAR SPIN FROM THE NV CENTER ELECTRON 3.1 Chapter Abstract We discuss a limitation on the phase coherence T ∗2,n of the native nitrogen nu- clear spin in diamond, which comes from the strong hyperfine interaction be- tween the nuclear spin and the NV center electron spin. We will introduce a semi-classical model to describe the coherence of nuclear spins in the presence of the NV center electron undergoing spin-lattice relaxation. Then we will dis- cuss a strategy to enhancing the phase coherence of a nuclear spin in diamond by decoupling it from the electron spin using continuous dynamic decoupling (CDD). Finally, we discuss the interaction between lattice strain and the NV center electron spin, which will provide us with an avenue to realize CDD. 3.2 Introduction A key parameter that determines that sensitivity of a nuclear-spin-based gyro- scope is the phase coherence time T ∗2,n of the nuclear spin. While T ∗2,n of isolated nuclear spins can be exceptionally long (on the order of seconds), it turns out that T ∗2,n for nuclear spins in diamond are orders of magnitude shorter (on the order of milliseconds) [51, 52, 22, 23, 24]. This physical origin of this poor phase coherence comes from the strong hyperfine interaction between the nuclear and electron spins in the NV center ground state, which will be a source of noise that 24 will induce decoherence. Mitigating the effect of decoherence in quantum sensors and quantum gates is a fundamental problem in quantum information and is often dealt with by using dynamical decoupling protocols (e.g. spin echo, CPMG) [53, 54, 55]. The goal of these protocols is to cancel out the effect the effect of the noise source us- ing control pulses on the qubit, allowing for longer phase coherence times. The ultimate goal will be to develop a protocol analogous to dynamical decoupling to cancel out the effect of the hyperfine interaction on the nuclear spin sensor. 3.3 Semi-classical Model of Nuclear Spin Decoherence We use a semi-classical model of the nuclear and electron spin to describe the decoherence of the nuclear spin. This model is used in [56] and [57] to model the effect of a noisy fluctuator on the coherence of a qubit. In this section, we will go over the conclusions of these studies to help motivate dynamical decoupling protocols to extend the phase coherence time T ∗2,n of the nuclear spin. In the model, the nuclear spin will be the system of interest and the electron spin will be considered as part of the environment whose fluctuations are driv- ing the decoherence. We will treat the nuclear spin quantum mechanically, but the electron spin as a classical fluctuator that is telegraphing between discrete states. This telegraphing occurs with a characteristic transition rate γ deter- mined by the spin-lattice relaxation time T1. The hyperfine interaction between the nuclear and electron spins of the NV 25 center is described by the Hamiltonian Hh f = S · C · I ≈ C∥S zIz (3.1) where S and I are the electron and nuclear spin operators respectively and C is the ground state hyperfine tensor. We use the secular approximation to neglect terms proportional to S x and S y since the effect of these terms are suppressed due to the large splitting (on the order of Dgs) between the electron spin states. We can draw an analogy between the hyperfine term and the Zeeman term γnBzIz, which describes the dynamics of the nuclear spin in a magnetic field. We can interpret the hyperfine interaction as an effective magnetic field gener- ated by the electron that is dependent on its spin state. For example, mS = ±1 generates an effective field of ±C∥ and mS = 0 generates no field. The decoherence of the nuclear spin is qualitative described as follows (Fig- ure 1). The nuclear spin undergoes precession due to the external rotation it is measuring ω0. Initially, the electron spin is in mS = 0, which means that there is no effective field that can affect the precession. However, when the electron undergo a T1 process, then it will change the effective field the nuclear spin experiences. The precession velocity v will speed up or slow down by C∥ de- pending on if the electron spin telegraphs to mS = +1 or mS = −1 respectively. Therefore the phase of the nuclear spin can be considered as a random walker whose precession velocity depends on the state of the electron spin. In the strong fluctuator regime where v/γ ≫ 1, one T1 jump of the electron spin is enough to decohere the nuclear spin. The nuclear spin of the NV center operates in this regime since v = C∥ = 2 MHz and γ ≈ 1/T1 ≈ 1 kHz. This will cause the phase coherence time T ∗2 to be limited to T ∗2,n = 3 2 T1,e. (3.2) 26 Figure 3.1: a) Nuclear spin precession with no hyperfine coupling. b) Nuclear spin precession slowing down after T1 relaxation to mS = −1. c) Nuclear spin precession speeding up after T1 relax- ation to mS = +1. This sets the fundamental limit on the nuclear spin coherence in the absence of any dynamical decoupling protocol [56]. 27 3.4 Dynamical Decoupling for the Nuclear Spin The nuclear spin decoherence is caused by telegraphing of the electron spin to states mS = ±1 with effective fields ±C∥. These states have opposite but equal magnitude coupling to the nuclear spin, so in principle if the electron state is driven between mS = ±1 on a timescale faster than than C∥, then the effective field experienced by the nuclear spin approaches zero. Then, the nuclear spin is effectively decoupled from the electron spin, allow us to extend the phase co- herence limit set by T1,e. The question is how fast of a timescale does the electron spin need to be driven in order to experimentally observe this enhancement in T ∗2,n. To get an estimate of the driving strength, we will look at dynamical decou- pling protocols investigated in Ref [56] and Ref [57]. These protocols consist of a series of π pulses on the fluctuator or the qubit to average over the coupling between the fluctuator and the qubit. This is shown schematically in Fig. We will model the electron spin as a two level system for our estimate, where the two states are the mI = ±1 states. We make this simplification since there is no simple analytic solution for the full three model fluctuator [56]. In dynamical decoupling, the interpulse delay τ sets the timescale of the noise that is canceled out (i.e. noise with frequency components smaller than 1/τ do not affect the coherence).The π pulses used for dynamical decoupling are also limited by the interpulse delay τ. This sets a restriction on the Rabi frequency of the pulse Ω ≲ 1/τ. For a 2LS, the coherence time of the nuclear spin in the presence of a fluctuator is given by [56] 1 T ∗2,n = γ − 1 τ ln γ sin(Wτ) + √ v2 − γ2 cos2(Wτ) W  (3.3) 28 ... ... |+1〉 |-1〉 |+1〉 |-1〉 |+1〉 |-1〉 Electron Spin Dynamical Decoupling a) b) Figure 3.2: a) Dynamical decoupling of the nuclear spin from the electron spin by using repeated π pulses on the electron spin. b) The- oretical enhancement over bare T ∗2,n with no dynamical decou- pling. The dotted lines mark where the enhancement is a factor of 10. 29 where W = √ v2 − γ2. For our application, γ ≈ 1/T1,e and v = C∥. In the limit where τ approaches ∞ (i.e. no dynamical decoupling is done), the expression reduces to T2,n∗ ≈ T1,e as expected. In the limit in which τ ap- proaches 0, the expression reduces to T ∗2,n → ∞, which is a signature of complete decoupling between the qubit and the fluctuator. Realistically, T ∗2,n cannot be ex- tended to an arbitrarily long time as dipole-dipole interactions with other spins start to become the dominant source of dephasing. The result of this model is plotted in Figure 3.2. To observe an order-of-magnitude enhancement of the coherence time would require Ω ∼ 1/τ ≈ 10 MHz. The description we have of dynamical decoupling is based of a pulsed scheme with interpulse delays τ. Experimentally, it could be difficult to im- plement very small interpulse delays (e.g. due to long rise and fall times of the pulses), so instead of dynamical decoupling through multiple π pulses, the drive could be kept continually on with Ω ≈ 10 MHz. This would be an exam- ple of continuous dynamics decoupling [58, 59, 60, 61]. In practice, this will be the most likely implementation of the decoupling we will implement, since we aim to use piezoelectric resonators to drive the double quantum transition. The rise and fall times times will be limited by the ring-up and ring-down times of the resonator. This in turn will we determined by the quality factor Q of the resonator. In the previous estimate, we implicitly assumed that the decoupling was strong enough to drive two hyperfine transitions equally well. That is, while the nuclear spin was in a superposition state on the equator of the Bloch sphere, we drove the transitions |−1,+1⟩ ↔ |+1,+1⟩ and |−1, 0⟩ ↔ |+1, 0⟩. These transi- tions are detuned from each other by ∼ C∥ which makes them comparable to the 30 estimated driving field. Therefore if we drive one transition at Ω ≈ 10 MHz, the other transition will experience a lower drive because it is off resonance. This issue could be solved in a couple ways. For example, the drive can have a large bandwidth (i.e. coming from a low Q resonator) so that the two transitions can achieve comparable decoupling. Another method would be to have narrow bandwidth drive (i.e. coming from a high Q resonator) and drive both transitions simultaneously. There some potential issues with this solution. First, destructive interference between the two drives will affect the amount of protection provided to each hyperfine transition. Additionally, if the drive is large compared to the detuning, then higher order effects such as the Bloch- Siegart shift will start to influence the dynamics because of the breakdown of the RWA. 3.5 NV Center Spin-Strain Interaction In the previous section, we discussed that strong driving the double quantum (DQ) transition of the NV center’s electron spin could potentially allow for the nuclear spin to be decoupled from the electron spin, thereby enhancing its phase coherence past the electron T1. The question is then how to implement this driv- ing experimentally, as this transition is cannot be driven with magnetic fields. The most direct way is to find some other field that can access this transition. Lattice strain in diamond is able to access this transition directly. AC strain has been demonstrated to drive double quantum [62, 59] and single quantum transitions [63] coherently In general, strain interacts with with spins through 31 their quadrupolar moment. The quadrupolar Hamiltonian is given by HQ = eQ 6I(2I − 1) ∑ i, j Vi j [3 2 (IiI j + I jIi) − δi jI2 ] (3.4) where e is the elementary charge, Vi j is the electric field gradient tensor, Q is the spectroscopic nuclear quadrupole moment, I corresponds to the spin operators, and the indicies run over the Cartesian axes x, y, and z [64, 65, 66]. The electric field gradient Vi j is dependent on the position of external charges in the lattice, which can be introduced by strain. To first order in the strain tensor ϵ, electric field gradient is given by Vi j = ∑ lm S jklmϵlm. (3.5) The fourth-rank tensor S jklm describes the coupling between the electric field gradient and the strain tensor [66]. In cases of symmetry, the number of inde- pendent parameters that determine the S-tensor decreases. For the diamond NV center, which has C3v symmetry, the number of independent parameters in the S-tensor is six [66, 67]. Following the notation of [68], [67], and [69], the strain-stress Hamiltonian is given by Hσ = Hσ0 + Hσ1 + Hσ2 (3.6) Hσ0 =MzS 2 z (3.7) Hσ1 = Nx{S x, S z} +Ny{S y, S z} (3.8) Hσ2 =Mx(S 2 x − S 2 y) +Ny{S x, S y} (3.9) where the number in the subscript denotes the type of transition strain can drive (e.g. subscript σ2 being a DQ transition). The prefactors to the spin operators 32 Figure 3.3: Coordinate system for spin-strain Hamiltonian. Lower case let- ters denote the NV frame and upper case letters denote the cu- bic crystal frame. are given by Mz = a1(σXX + σYY + σZZ) + a2(σYZ + σZX + σXY) (3.10) Nx = d(2σZZ − σXX − σYY) + e(2σXY − σYZ − σZX) (3.11) Ny = √ 3[d(σXX − σYY) + e(σYZ − σZX)] (3.12) Mx = b(σZZ − σXX − σYY) + c(2σXY − σYZ − σZX) (3.13) My = √ 3[b(σXX − σYY) + c(σYZ − σZX)] (3.14) where σi j denote the components of the stress tensor and the set {a1, a2, b, c, d, e} are the six parameters that set the coupling. The value of the parameters cal- culated through density functional theory (DFT) calculations and experimental measurements are given in Table 4.1. For this Hamiltonian, we will define the X, Y , and Z axes to be the axes corresponding to the cubic unit cell and x, y, z to be the axes of the NV frame where z is along the symmetry axis. 33 Table 3.1: Spin-stress coupling coefficients calculated and measured in the literature. Reference a1 a2 b c d e Udvarhelyi et al. (DFT) [67] 2.66 ± 0.07 −2.51 ± 0.06 −1.94 ± 0.02 2.83 ± 0.03 −0.12 ± 0.01 0.66 ± 0.01 Barson et al. (Exp.) [69] −11.7 ± 3.2 6.5 ± 3.2 7.1 ± 0.8 −5.4 ± 0.8 - - Barfuss et al. (Exp.) [68] 4.86 ± 0..2 −3.7 ± 0.2 −2.3 ± 0.3 3.5 ± 0.3 - - Chen et al. (Exp.) [63] - - - - √ 2(0.5 ± 0.2)b - 34 To implement the decoupling protocol, we will use the stress generated by a bulk acoustic resonator to drive the Hσ2 term in the Hamiltonian. For the devices that are discussed in this thesis, the resonators will drive acoustic waves along [001], which makes b the parameter that describes the coupling between the acoustic wave and the NV center electron spin. 3.6 Conclusion In this chapter, we discussed the fundamental limitation to the nuclear spin co- herence T ∗2,n due to the hyperfine interaction between the electron and nuclear spins. We physically motivated the idea of decoupling these two spins from each other and made an estimate of the drive strength for dynamical decou- pling. We will attempt to implement this protocol by taking advantage of the spin-strain coupling in diamond, which can be used to drive the DQ transition. Our next goal will be to realize devices that will enable this driving in diamond. Decoupling the electron spin and the nuclear spin is not the only step that is necessary to improve nuclear phase coherence. To observe the effect of the decoupling, the hyperfine interaction needs to be the dominant source of de- coherence. Another major source of decoherence is the thermal dependence of various parameters in the Hamiltonian that can shift the Larmor frequency of the nuclear qubit. For 14N, this includes the temperature dependence of the quadrupolar moment, which can vary in the laser focus [70, 71]. There has been progress in protecting the qubit from these thermal fluctuations [72]. Further work needs to be done first to identify and evaluate other sources of decoher- ence that limit the phase coherence before the improvement from the decou- 35 pling can be observed. 36 CHAPTER 4 THIN-FILM BULK ACOUSTIC RESONATORS ON DIAMOND FOR CONTINUOUS DYNAMICAL DECOUPLING 4.1 Chapter Abstract Continuous dynamical decoupling of the nuclear spin from the NV center’s electronic spin is theoretically possible with strong driving of the electron spin’s double quantum (DQ) transition. This transition can not be directly driven mag- netically, but can be driven using lattice strain. We will discuss the fabrication of aluminium nitride (AlN) thin film bulk acoustic resonators (FBAR) on diamond, which are used to drive acoustic waves on diamond. We also directly charac- terize the strain generated by measuring Rabi oscillations of the DQ transition driven by these FBAR devices. Finally, we outline future steps and outlooks for next generation FBAR devices on diamond for continuous dynamical decou- pling. 4.2 Introduction In the previous chapter, we discussed how to theoretically decouple the nuclear spin from its hyperfine interaction with the NV center’s electron spin. This can be achieved by strong driving (∼ 10 MHz) of the double quantum (DQ) transition of the electron spin. The DQ transition, which has a change in the spin projection of ∆mS = ±2, cannot be directly addressed with magnetic fields, which can only drive single quantum (∆mS = ±1) transitions. Lattice strain 37 can directly drive the DQ transition, so this motivates us to design devices on diamond that can generate the necessary ac lattice strain. Previous works have demonstrated using microelectromechanical (MEMS) systems to drive spin defects in diamond and other defect systems. These include high overtone bulk acoustic resonators (HBARs) and surface acoustic wave resonators (SAWs) [62, 59, 73, 63, 74]. We will focus on developing film bulk acoustic resonators on diamond (FBARs), using aluminum nitride (AlN) as the piezoelectric material. The ba- sic premise is to generate large strain by confining the energy in the acoustic mode generate by the FBAR device into a small mode volume. In addition, we will work the fundamental mode or low harmonics. We will detail the fabrica- tion process flow and the measurements used to characterize the performance of these devices. 4.3 Diamond Gyroscopes with FBAR Concept The goal of this device development is to integrate FBARs with nuclear spin- based gyroscopes in diamond. Nuclear spin-based gyroscopes, using Ramsey interferometry, have already been demonstrated in the literature [71, 20]. We will describe the overall concept in integrate FBAR devices, which will be used to test continuous dynamical decoupling, with current concepts of nuclear spin- based gyroscopes in diamond. This will provide context that will inform the design and fabrication of these devices. A schematic of a potential nuclear spin-based gyroscope with FBAR integra- tion is shown in Figure 4.1. The substrate will be a thin membrane of diamond 38 Figure 4.1: Schematic of the FBAR on diamond gyroscope concept. An AlN piezoelectric transducer launches standing acoustic waves into the diamond. An NV ensemble is placed at an anti-node (red and blue) of the wave to implement the decoupling proto- col. A MW antenna is fabricated on the opposite side (poten- tially off the diamond chip) for magnetic spin control. (∼ 10 µm). The FBAR device, which is fabricated on the top side of the dia- mond, consists of an AlN transducer. This transducer is a film stack of AlN that is between two platinum (Pt) electrodes. At these Pt electrodes, AC voltage is applied to launch acoustic waves into the diamond substrate. Magnetic control of the spins is achieved with a Pt antenna fabricated on the opposite side, ei- ther on the diamond or off chip on another substrate (e.g. sapphire). The laser is focused through the bottom side of the diamond (through the antenna) to initialize and read out the spins for the Ramsey protocol. The FBAR on diamond device will act as a Fabry-Perot cavity for the acoustic waves, which will result in standing waves in the diamond. The anti-nodes of the standing wave will have the maximum strain (tensile or compressive). Ideally, there are NV centers in the diamond only at the anti-nodes to maximize the coupling between the acoustic wave and the NV center electron spin. This 39 can be achieved, for example, through ion implantation with nitrogen where the implantation energy sets the depth of the NV ensemble in the diamond. Even with ion implantation, there will be some straggle of NV centers in the depth, as ion implantation is a probabilistic process. Another method of introducing an NV center ensemble in a specific depth of the diamond is to introduce nitrogen into the chemical vapor deposition (CVD) growth for a limited amount of time (delta-doping) during the deposition. The depth of the NV center ensemble will be determined by how far into the growth the nitrogen is introduced. For the diamonds used for the devices in this chapter, we do not use ion im- planted or delta-doped samples. Rather, as a proof of principle, we fabricate on optical grade diamond from Element Six. For these types of diamond, nitrogen is introduced through the growth to enhance the deposition rate, which gives diamonds with ∼1 ppm nitrogen content. This naturally gives rise to a native NV center ensemble throughout the entire depth of the diamond. Because of this, our measured signal from the NV center ensemble will be a convolution of the acoustic wave and the depth profile of the NV centers at the focus. At specific depths of the acoustic wave where there is a large inhomogeneity in the strain at the focus, this will lead to an apparent damping of the signal. How- ever, this will not have a dramatic effect on the measured oscillation frequency, which is the parameter of interest. 4.4 FBAR on Diamond Fabrication We will describe the fabrication process flow for the FBAR devices presented in this chapter. We begin with a 50 µm membrane of diamond from Element 40 a) b) c) d) e) f) g) h) i) Diamond Quartz Ni Pt AlN Diamond etch Figure 4.2: AlN FBAR on diamond process flow. The lithography and etching is done at Purdue and the AlN deposition is done at Cornell. Figure Credit: Ozan Erturk. Six. First a quartz mask is applied to the backside of the diamond and the di- amond is etched in the exposed region to a thickness of approximately 20 µm. A nickel mask is then applied to expose a small circular region to further etch down the diamond to approximately 12 µm. The quartz and nickel masks are removed. Then on the front side, aligned to the etched region, another nickel mask is placed where the exposed region defines the FBAR shape. After the etch, the FBAR shape is well-defined but is still anchored to the diamond sub- strate. The nickel mask is stripped again and the bottom Pt electrodes of the FBAR are patterned. Subsequently, approximately 1.5 µm of AlN is sputtered on the frontside. The top Pt electrode is then patterned to complete the FBAR. 41 To release the device, a blanket etch is done on the backside until the device is released. A schematic of the process flow is shown is Figure 4.2. This process flow is roughly used for all the devices we tested. The AlN deposition was performed at Cornell and the rest of the process flow was per- formed at Purdue. There are small differences between some of the devices however. In particular, the first generation device, which will be discussed, was accidentally released during the etch step that defined the FBAR shape because it is difficult to accurately judge the membrane thickness. The second genera- tion device we discuss does follow this complete process flow. 4.5 Acoustically Driven Rabi Oscillations To extract the strain produced by the FBAR device, we measure the Rabi oscilla- tions of the electron spin that are driven acoustically. These Rabi oscillations are resonant with the electron spin’s DQ transition (mS = −1 to mS = +1). To begin, we measure the electromechanical resonance of interest using a vector network analyzer (VNA). The clock of this VNA is synced with the clock of the signal generators. From this data, the resonance frequency is extracted. The external magnetic field applied to the NV axis is adjusted so that the DQ transition matches the electromechanical resonance [Figure 4.3(a)]. To check that the resonance condition is met, we measure pulsed ODMR of the mS = 0 to mS = −1 transition and of the pulsed ODMR of the mS = 0 to mS = +1 transition. The difference between these two transitions is 2γeB, which should match the FBAR resonance. 42 532 nm MW (f0) FBAR (fmech) ff0 Dgs fmech = 2γeB ms = 0 → ms = −1 ms = 0 → ms = +1 t L − t AP AP Initalize Read a) b) Figure 4.3: a) Driving the DQ transition on resonance. b) Pulse sequence for acoustically-driven Rabi oscillations. After the external field is adjusted, the Rabi oscillations are measured using the pulse sequence in Figure 4.3(b) [59]. First the NV centers are initialized to |0⟩ with 532 nm illumination. Then an adiabatic passage (AP) pulse is used move the NV center spins to |−1⟩. After this, the FBAR is turned on for a variable amount of time t to drive the DQ transition. Another AP pulse is then applied to move spins in |−1⟩ to |0⟩. The FBAR then is turned on again for a time L − t, where L is the length of the longest duration data point, to keep the thermal load on the device constant for all measurements. Finally 532 nm illumination is used to read out the population of spins in |0⟩, thereby measuring the number of spins left in |+1⟩. Note that after the second AP pulse, the spins we measure in the readout are already in mS = 0, so this acoustic pulse has no effect on out measurement. 43 50 μm MW Antenna AlN FBAR a) [001] S tr a in [ a rb . u n its ]12 μm FBAR (Pt/AlN/Pt) Diamond b) c) Figure 4.4: a) SEM image of first generation FBAR device with an antenna for MW spin control. b) COMSOL calcuation of the acoustic mode generated by the FBAR. c) Probe measurements of the FBAR electromechanical resonance measured at ∼2.55 GHz. The yellow region between the dotted black lines indicates the range of drive frequencies that the FBAR performance was characterized via spin measurements. Figure is based on im- ages and data from Ref [1]. Probe measurements were pro- vided by Noah Oppondo and Ozan Ertutk. 4.6 First Generation FBAR Device The first generation device consisted of an AlN FBAR fabricated on a released pentagonal membrane of diamond of approximately 10 µm in thickness and approximately 20 µm in size [Figure 4.4(a)] [?]. The AlN tranducer consists of 1.5 µm of AlN fabricated between two Pt electrodes. A loop antenna is fabri- cated around the entire device for magnetic control of the NV center spins. A finite-element calculation (COMSOL) of the mode profile is shown in [Figure 4.4(b)], showing approximately two antinodes in the diamond substrate. The electromechanical response is shown in Figure 4.4(c). We focus on the 44 a) b) c) d) Figure 4.5: a) Acoustically driven Rabi oscillations across ∼ FBAR reso- nance. b) FFT of acoustically driven Rabi oscillations with drive frequency. c) Trace of the Rabi oscillation at the FBAR resonance. Figure is based on images and data from Ref [1]. electromechanical mode at around 2.55 GHz. There are two key metrics that indirectly characterize the strain generated by our device. The first metric is The quality factor Q, which characterizes the degree to which energy is lost in the FBAR (e.g. through dissipating energy through the FBAR’s anchor). This is extracted from a Q circle fit to a modified Butterworth-van Dyke model of the FBAR [75]. The second metric is the electromechanical coupling k2 t which de- scribes the percentage of electrical energy stored in the AlN FBAR is converted to acoustic energy. To achieve high ac strain, we aim for devices with high Q and k2 t . For the first generation device, the Q extracted from the fit is approximately 300 and k2 t is 0.3%. 45 To directly measure the ac strain, we measure Rabi oscillations of the DQ transition driven by the FBAR from 2.53 GHz to 2.57 GHz (shaded yellow in Figure 4.4(c)). At each drive frequency, the external magnetic field is adjusted so that the drive is resonant with the DQ transition. The Rabi frequency, ex- tracted from a FFT, over these range of fields is shown in Fig and shows that the resonance occurs at 2.551 GHz. Given that the linewidth of the resonance is approximately 15 MHz, this corresponds to a Q of approximately 175. Since the Rabi oscillations directly probe the strain in the diamond, this suggests that there is some systematic error in the probe measurements, which indirectly ex- tract the Q via an electromechanical model. With 30 dBm of input power (∼ 1 W), we measure a maximum Rabi frequency of 3.74 MHz when driving at the electromechanical resonance. This is not large enough to observe decoupling between the nuclear and electron spins based on our estimate (∼10 MHz). 4.7 Second Generation FBAR Device 4.7.1 50 Ω Impedance Matching With our current design, we are able to drive the DQ transition with a Rabi fre- quency of 3.74 MHz at 30 dBm of power at the port. The main reason why such a relatively large power is needed is the fact that our FBAR is poorly impedance matched (∼700 Ω) to our 50 Ω microwave line (Figure 4.6). This leads to ma- jority of the power being reflected from the device back to its source with only a small fraction of the power actually exciting the acoustic mode. Therefore a straightforward way to potentially improve our device is to better matched its 46 Figure 4.6: Impedance of the first generation FBAR device at the ele- cromechanical resonance. The impedance is around 700 Ω at 2.55 GHz. impedance at the electromechanical resonance to 50 Ω. Because the FBAR is essentially a parallel plate capacitor with AlN as the dielectric, the impedance will be inversely proportional to the area of the de- vice. Therefore, with a sufficiently large area device, the FBAR will be able to absorb more of the incident MW power to help it generate more strain. How- ever, increasing the area of the device also increases the mode volume of the acoustic mode, which tends to decrease the maximum achievable strain as the total energy stored in the acoustic mode is now spread through a larger volume. The goal is to balance these two effects to maximize the strain generated by the second generation device. 47 a) b) c) 50 Ω Figure 4.7: a) Optical and b) SEM image of the second generation FBAR device. c) Probe measurement of the electromechanical re- sponse and impedance of the second generation FBAR device at around 1.93 GHz. Optical and SEM images were provided by Ozan Erturk. Images of the second generation FBAR device is presented in Figure 4.7(a- b). The largest dimension of our device ∼200 µm. Its total area is approximately a factor of three larger than the first generation FBAR. The electromechanical response (S11) is shown in Figure 4.7(c). The mode used for the measurement is the second harmonic at approximately 1.93 GHz. The impedance is better matched to 50 Ω, with a impedance of approximately 200 Ω. The Q of the elec- tromechanical resonance is approximately 550 with a k2 t of 0.9%. We can estimate the improvement in the Rabi frequency from driving this mode by looking at the ratio √ Q2k2 t2 Q1k2 t1 Γ1 Γ2 ∼ 3 (4.1) 48 where Γ is the reflection coefficient calculated from the device impedance. The subscripts 1 and 2 denote whether the quantity is associated with the first gen- eration or second generation device. The basic intuition behind this quantity is that the first ratio describes how much more energy is stored in the acoustic mode and the second ratio describes how much much more energy is actually used by the device. The square root of the product of these two ratio is taken because the energy density stored in the acoustic mode is proportional to the square of its amplitude. Therefore, using the parameters extracted from our VNA measurements for both devices, we expect that the second generation de- vice is able to achieve a Rabi frequency of around 9 to 12 MHz, which is in range of our estimate for the necessary driving to observe dynamical decoupling. 4.7.2 Device Packaging Because of the increased size of the the FBAR device, it is not practical to cre- ate an antenna that encircles the FBAR like in the first generation devices and keep it impedance matched to 50 Ω. This is because the inductive component of the antenna’s impedance scales to the radius r as r ln r, so a factor of 10 in- crease in size would correspond to a approximately a factor of 20 increase in the impedance. To circumvent this problem, we fabricate 30 µm loop antennas off chip on a sapphire substrate. The layout of the chip is shown in Fig. The chip features alignment marks that coincide with reference marks on the diamond. These are used to align the FBAR on the diamond to an antenna on the sapphire by hand. When aligned with the antenna, the pads for the antenna are accessible for wire 49 a) b) c) d) Figure 4.8: a) Image of sapphire chip with a Pt antenna fabricated for mi- crowave spin control. b) Image of the diamond bonded to the sapphire. c) Alignment marks used to align the sapphire and diamond chips to each other. d) Image of the antenna aligned to the FBAR. The red circle denotes the location of the antenna beneath the diamond. Images were provided by Ozan Erturk. bonding to the sample mount. 4.7.3 Spin Measurements To compare the strain generated by the device, we measure acoustically driven Rabi oscillations. The oscillations are driven at different powers as shown in 50 ΩR = 6.63(6) MHz ΩR = 3.98(3) MHz ΩR = 2.00(2) MHz 28.5 dBm a) b) 10.9 dBm 16.8 dBm 22.6 dBm ΩR = 0.96(2) MHz Figure 4.9: a) Rabi oscillations of the DQ transition at 1.92 GHz. The max- imium measured Rabi frequency is about 7 MHz. b) Fitted Rabi frequencies plotted against the drive power into the de- vice. Past 27 dBm, the device performance falls off the ob- served trend (solid black line). Fig. Compared to the first generation device, which achieved 3 MHz Rabi fre- quency at approximately 30 dBm of power, the second generation device is able to achieve the same perforamce at approximately 20 dBm of input power. The Rabi frequency driven by the device scales linearly with the applied voltage (which scales with power as √ P) as expected. However, the trend becomes non- linear at higher powers (greater than 27 dBm) and falls off greatly with higher applied power. We hypothesize that the nonlinear behavior at applied powers above 27 dBm is most likely due to thermal heating that is damaging the AlN and/or the Pt films of the device. Because the FBAR is fabricated on a released diamond struc- ture, the only way for device to dissipate heat is through the anchor. As a result, the FBAR device can potentially heat up enough to damage the AlN and/or Pt 51 films. This damage can change the acoustic properties of AlN, which will shift the location of the resonances. As a result, after applying large powers to the FBAR on resonance initially, the drive frequency will be off resonant over time ad the resonance shifts away. One important thing to note is that given the linear trend of the Rabi fre- quency with applied voltage in Fig, driving at approximately 10 MHz is realis- tically achievable. At 27 dBm of input power, we are able to achieve Rabi fre- quencies of approximately 7 MHz. If we are able to mitigate the thermal dam- age up to 30 dBm of power, then we can reach approximately 10 MHz drive, which would be sufficient to test our proposed decoupling protocol. This value is also consistent with our estimated device performance from our VNA mea- surements. 4.8 Conclusion and Outlook We developed AlN FBAR on diamond devices to implement the decoupling of the nuclear spin from the electron spin with strong driving of the electron spin’s DQ transition. The first generation device was able to reach 3 MHz of Rabi at approximately 30 dBm of input power. The second generation device actually only reaches a maximum of 6 MHz at 27 dBm of input power and its performance deteriorates most likely due to thermal damage to the AlN and Pt films. Further improvements to the device will focus on mitigating thermal damage to the film. This could mean investigating the AlN growth parameters to get higher quality films or changing the anchor size to increase heat dissipa- tion, while keeping the performance at a comparable level. Additionally, higher 52 quality films could lead to higher k2 t . Another avenue that we can approach this problem from is to make a semi- confocal FBAR device. This device geometry has been explored with ZnO HBARs on diamond [73]. The main idea would be to fabricate a solid immer- sion lens (SIL) on a thin membrane of diamond. The SIL will not only help to collect the PL from the NV center ensemble, but also to act as an acoustic mirror to confine the acoustic mode in a limited volume, like in the FBAR design. The benefit of this design is that it is not released and so should dissipate heat better than the FBAR. 4.9 Acknowledgements We acknowledge Ozan Erturk, Noah Oppondo, and Sunil Bhave for the FBAR fabrication, probe measurements, and electromechanical analysis at Purdue University. We acknowledge Anthony D’Addario and Brendan McCullian for assistance in collecting and analyzing spin measurement data. This work was supported by the DARPA DRINQS program (Cooperative Agreement No. D18AC00024). Device fabrication at Cornell was done at the Cornell Nanoscale Facility, a member of the National Nanotechnology Coordinated Infrastructure (NNCI), which is supported by the National Science Foundation (Grant No. NNCI-2025233) and the Cornell Center for Materials Research Shared Facili- ties, which is supported through the NSF MRSEC program (Grant No. DMR- 1719875). 53 CHAPTER 5 OPTICAL PUMPING DYNAMICS OF THE NV CENTER ELECTRON AND NUCLEAR SPINS 5.1 Chapter Abstract We focus on the optical dynamics of the NV Center at room temperature. First we will describe the Lindblad master equation model and focus on the dynam- ics of the electron spin to provide a quantitative description of the NV center’s spin-dependent photoluminescence and optical polarization. Then we will ex- tend this model to include the hyperfine interaction with 14N and describe field- dependent excited state dynamics. Finally, we will look at some consequences of the excited state dynamics, which include nuclear polarization through opti- cal pumping and field-dependent contrast of NV center spins. These processes will be important when we investigate optical readout of nuclear spins beyond the electron T1. 5.2 Introduction In the previous chapters, we focused on the idea of decoupling the nuclear and electron spins from each other to extend the phase coherence T ∗2,n. Now we will focus on how to read out the nuclear spin to extract the phase information from the Ramsey interferometry sequence. A key feature that makes the NV center a promising platform for quantum sensing is its electron spin-dependent pho- toluminescence (PL) under nonresonant excitation (e.g. 532 nm). With a CNOT 54 pulse, we can map electron spin-dependent PL to nuclear spin-dependent PL. However, this CNOT pulse operates best when the NV electron spin is in mS = 0. If we use decoupling to extend the T ∗2,n past the electron T1, then this is not true since the electron spin will be driven to thermal equilibrium. In the next chapter, we will discuss a readout scheme for this scenario that involves an optical repump pulse to regain electron polarization to increase the effectiveness of the CNOT readout. To describe the effect of this optical repump pulse, we need a quantitative model of the optical pumping process. This will allow us to look at field-dependent dynamics between the electron and nuclear spins in the excited state. The goal is to understand the effect of these dynamics on the sensitivity of our nuclear sensor, namely the visibility of the Ramsey fringes. 5.3 The Lindblad Master Equation We will describe the dynamics of the optical cycling of the NV center electron and native nuclear spin under optical illumination with the Lindblad master equation formalism[76, 77, 78, 79]. In this formalism, the NV center is treated as an open system that is weakly coupled to the environment (e.g. the diamond lattice, spin bath) in an unspecified manner. We make some key assumptions when we use the Lindblad master equation. • The correlation time τC between the system and bath is much smaller than the timescale of the relaxation τr of the system induced by the environ- ment. This has two consequences. First, we can assume the the state of the environment ρE and the state of the system ρS are uncorrelated at t = 0 55 and remain uncorrelated for all time. That is, we can always express the total state of the system and environment as ρS ⊗ ρE. The second is that it allows us to use the Markov approximation, where τc is so short that it allows us to neglect the prior history of the state when considering its time evolution. • We work within the Rotation Wave Approximation (RWA) where we as- sume that the system is only affected by resonant couplings between the system and the environment. Under these assumptions, the time evolution of the state will be described by the master equation dρ dt = −i[H, ρ] + ∑ i Γi ( LiρL†i − 1 2 {L†i Li, ρ} ) (5.1) where Li are the jump operators describing the relaxation of the system and Γi are the corresponding rates. The first commutator corresponds to the Louiville- von Nuemann equation, which describes unitary evolution of the the system. The second part is the Lindbladian which describes the non-unitary dynamics due to the interaction of our system with the environment. The jump operators Li are phenomenological operators we impose on the system to model various relaxation processes that we believe are occurring in the system. It is possible to derive every operator Li if we knew the exact form of every coupling between the system and the environment, but we do not usually have this knowledge, hence the phenomenological nature of this approach. In general the jump operators we use fall into two different groups. The first group of jump operators describes incoherent transitions between different states of the system. For the NV center, this will include the non- 56 resonant laser excitation; the transitions between between the ground, ex- cited, and metastable states; and T1 processes. These operators take the form Li = |ψ1⟩ ⟨ψ2| where |ψ1⟩ and |ψ2⟩ are the final and initial states of the relaxation respectively. The simplest way is to see the effect of this jump operator is to consider the relaxation of a spin-1/2 in a magnetic field. H = 1 2 γBzσz (5.2) We will ignore the time evolution from this Hamiltonian (spin precession in the field) by moving to the rotating frame of the spin (ω0 = γBz) to focus on the relaxation. We can describe a T1 process from |↑⟩ to |↓⟩ as the lowering operator L = σ− = |↓⟩ ⟨↑|. Then the master equation is dρ dt = Γ(σ−ρσ+ − 1 2 {σ+σ−, ρ}) (5.3) whereσ+ = σ†− is the raising operator. This gives the partial differential equation d dt ρ00 ρ01 ρ10 ρ11  = −Γρ00 − Γ 2ρ01 −Γ2ρ10 Γρ00  (5.4) which has the solution ρ(t) = ρ00(0)e−Γt ρ01(0)e− Γ 2 t ρ10(0)e− Γ 2 t 1 − ρ00(0)e−Γt  . (5.5) Therefore this operator describes population transfer from |↑⟩ to |↓⟩. This popu- lation transfer comes with a exponential decay in coherence (ρ01 and ρ10) since every T1 process is a T2 process. The next group of operators describes T2 processes (i.e. dephasing in the ground and excited states). These operators are of the form |ψ⟩ ⟨ψ|. To see its 57 effect, we will look at the the spin-1/2 again, where we will model the jump operator as L0 = |0⟩ ⟨0| = σ+σ−. The master equation is then dρ dt = Γ(σ+σ−ρσ−σ+ − 1 2 {σ+σ−, ρ}. (5.6) This gives the differential equation d dt ρ00 ρ01 ρ10 ρ11  =  0 −Γ2ρ01 −Γ2ρ10 0  . (5.7) which has the solution ρ(t) =  ρ00(0) ρ01(0)e− Γ 2 t ρ10(0)e− Γ 2 t ρ11(0)  . (5.8) Therefore, the population of |↑⟩ and |↓⟩ do not change over time, but their coher- ences (ρ01 and ρ10) will damp exponentially at a rate Γ/2. If we include the other possible operator L1 = |↓⟩ ⟨↓| = σ−σ+, then the coherences would damp at a rate Γ. 5.4 7 Level Rate Model of the NV Center We will first begin by describing the dynamics of the electron spin at zero field. At this field, the dynamics of the nuclear spin can be neglected as the large zero- field splitting DES suppresses spin-mixing caused by the transverse hyperfine coupling A⊥. The Hilbert space we will use will consist of three states from the ground state (GS), described by the Hamiltonian: Hgs = DgsS 2 z + γeBz§z (5.9) where DGS corresponds to the zero-field splitting in the ground state (2.87 GHz), γe corresponds to the electron gyromagnetic ratio (2.8 MHz/G), and Bz is the 58 |ms = -1⟩ ES GS SingletΓ0 Γ1 Γ2 Γ3 Γ4 Γ5 Γ6 Γ7 |ms = 0⟩ |ms = +1⟩ |ms = 0⟩ |ms = -1⟩ |ms = +1⟩ Figure 5.1: 7 level model for the optical cycling of the NV center’s electron spin. The green arrows denote transitions due to laser exci- tation. The red arrows denote transitions due to spontaneous emission. The blue arrows denote transitions associated with the non-radiative intersystem crossing. external applied field (Bz = 0). Another three states will come from the excited state (ES), which is described by a Hamiltonian of the same form: HES = DES S 2 z + γeBz§z (5.10) where DES corresponds to the zero-field splitting in the excited state (1.42 GHz) The final state will be the singlet state of the ISC. This 7 level model is depicted in Figure 5.1 and the corresponding rates used are presented in Table 5.1. The are a variety of rate models in the literature [80, 81, 82, 83, 84]. The rates used in this thesis are based off of a 5-level fit done in [84]. Transitions between the GS and ES are spin-preserving. The excitation rate from the GS and ES for all calculations in this chapter is set at optical saturation unless otherwise specified. Saturation will correspond to an excitation rate that is equal to the spontaneous emission rate. 59 Table 5.1: Rates used for the 7 level model for the optical cycle of the NV center’s electron spin. GS and ES denote the ground state and excited state manifolds respectively. Transition/Decoherence Rate [Linear Units] Excitation (GS to ES) [MHz] Γ0 67.4 Spontaneous Emission [MHz] Γ1 67.4 ES to singlet (mS = ±1) [MHz] Γ2, Γ3 91.6 ES to singlet (mS = 0) [MHz] Γ4 9.9 Singlet to GS (mS = ±1) [MHz] Γ5 , Γ6 1.06 Singlet to GS (mS = 0) [MHz] Γ7 4.83 Electron T1 (GS) 1/TGS 1 1/10 ms Electron T2 (GS) 1/TGS 2 1/100 µs Electron T1 (ES) 1/T ES 1 1/1 ms Electron T2 (ES) 1/T ES 2 1/10 ns a) b) Figure 5.2: a) Calculated mS = 0 occupation using the 7 level model for optically pumping a thermal electron spin state at optical satu- ration. b) Corresponding electron polarization to mS = 0 60 We will use this rate model to describe optical polarization to mS = 0 through optical pumping. This occurs because this optical cycle on average depletes spins from mS = ±1. This depletion comes from two specific sets of transitions: the transitions from the ES into the ISC’s singlet and the transition from the ISC’s singlet to the GS. The transition rate to the singlet state of the ISC from the ES is spin dependent, where the transition rates for the mS = ±1 (91.6 MHz) state is 10 times higher than that of the mS = 0 state (9.9 MHz). This transition dominates the optical polarization process since it allows for the electron spin to undergo relaxation that does not preserve its spin projection. Because the rate is higher for mS = ±1 by an order of magnitude compared to mS = 0, this makes the rate of spins exiting mS = ±1 and into mS = 0 significantly higher than the rate for the inverse process. Compounding this effect is the transition from the ISC’s singlet to GS. The transition rate into mS = 0 (4.83 MHz) is 5 times larger than the transition rate into mS = ±1 (1.06 MHz). The optical polarization calculated from the Lindblad calculation is shown in Figure 5.2. The initial state used for the calculation is a mixed state corre- sponding to the electron spin at thermal equilibrium: ρtherm = 1 3 I =  1/3 0 0 0 1/3 0 0 0 1/3  . (5.11) The polarization as a function of optical pump time is defined as: P = 1 − 3 2 ( p−1 + p+1 p0 + p−1 + p+1 ) (5.12) where P = 0 corresponds to a completely unpolarized electron spin (i.e. at thermal equilibrium) and P = 1 represents complete polarization into mS = 0. At saturation, the polarization reaches its steady state value of 0.8 at approximately 1 µs. 61 ms = −1 ms = 0 Figure 5.3: a) Spin dependent PL emission from mS = 0 and mS = −1 calcu- lated from 7 level model. The yellow shade area is the contrast that we can experimentally observe. Another key consequence of this rate model is the spin-dependent PL be- tween mS = 0 and mS = ±1. PL is the result of spontaneous emission when the electron spin relaxes from the ES to the GS (6.75 MHz) which corresponds to an ES lifetime of ∼10 ns. The spin-dependence of the emission is again because of the presence of the ISC, which competes with the radiative relaxation. Be- cause the transition rate into the ISC is an order of magnitude larger for mS = ±1 than for mS = 0. this makes the mS = ± states significantly more likely to relax through the ISC. Thus the total emission from the mS = ±1 is on average lower than the emission from mS = 0. The spin-dependence of the PL calculated from the rate model is shown in Figure 5.3, where the PL emission is plotted as a function of pumping time, as well as the contrast between mS = 0 and mS = ±1. 62 5.5 21 Level Rate Model of the NV Center We will include the nuclear spin states of 14N into our rate model to look at the effect the optical pumping dynamics have on the nuclear spin. 14N is spin-1, which splits each state in the 7 level model into three states due to the hyperfine interaction, labeled by the nuclear spin projection. The GS hyperfine Hamilto- nian is: Hh f GS = PI2 z + γnBzIz + A∥S zIz (5.13) where γn is the gyromagnetic ratio of the nuclear sopin (0.300 kHz/G), P is the quadrupolar moment of the nuclear spin (−4.95 MHz) and A∥ is the longitudinal hyperfine coupling ( 2.1 MHz) [25, 37, 38]. We neglect the transverse hyperfine coupling A⊥ since we will focus on fields up to 800 G in this thesis. The ES hyperfine Hamiltonian has a similar form Hh f ES = PI2 z + γnBzIz +C∥S zIz +C⊥(S xIx + S yIy) (5.14) where C∥ is the longitudinal hyperfine coupling in the ES (−40 MHz) and C⊥ is the longitudinal transverse coupling in the ES (−23 MHz) [25, 39]. The trans- verse hyperfine coupling here is not neglected because it opens a LAC at ap- proximately 500 G. The 21 level model shares the same transition rates and jump operators as the 7 level model (Table 5.1). This is because we assume that every transition preserves the nuclear spin projection. We do not include the effect of decoher- ence on the nuclear spin because the T1 and T2 of the nuclear spins are orders of magnitude longer than the timescale of a single optical cycle. 63 |+1,0⟩ |+1,-1⟩ |+1,+1⟩ |0,0⟩ |0,-1⟩ |0,+1⟩ |-1,0⟩ |-1,-1⟩ |-1,+1⟩ A⟂ A⟂ ES GS |+1,0⟩ |+1,-1⟩ |+1,+1⟩ |0,0⟩ |0,-1⟩ |0,+1⟩ |-1,0⟩ |-1,-1⟩ |-1,+1⟩ Key Radiative relaxation (Bright) Non-radiative relaxation (Dark) Excite a) b) |-1,0⟩ |0,-1⟩ |-1,+1⟩ |0,0⟩ Figure 5.4: a) ES spin flip-flops that are mediated by the transverse hyper- fine coupling A⊥ near the ESLAC. b) ESLACs opened by the transverse hyperfine coupling. The solid lines denote the ES- LAC between |0, 0⟩ and |−1,+1⟩ and the dotted lines denote the ESLAC between |0,−1⟩ and |−1, 0⟩. 5.6 Excited State Dynamics While the transitions in our 21 level model are nuclear spin preserving, the ES Hamiltonian does allow changes to the nuclear spin projection (Figure 5.4a). These changes are accompanied by changes in the electron spin projection to obey energy conservation. We will refer to these changes in the electron and nuclear spin as spin flip-flop processes. There are two spin flip-flop that are available in the excited state correspond- ing to two LACs in the ES. These LACs are shown in Figure 5.4b. The first LAC involves a gap that is opened between the |0, 0⟩ and |−1,+1⟩ states. The Hamil- 64 tonian in this subspace is H =  0 A⊥ 2 A⊥ 2 ω−1,+1  (5.15) with ω−1,+1 = DES − γeBz + P + γnBz − A∥. (5.16) The second LAC is opened between the |0,−1⟩ and |−1, 0⟩ states. The Hamilto- nian takes a similar form H = ω0,−1 A⊥ 2 A⊥ 2 ω−1,0  (5.17) with ω0,−1 = P − γnBz (5.18) ω−1,0 = DES − γeBz. (5.19) For the rest of this section we fill focus on the |0, 0⟩ and |−1,+1⟩ states, but the general physics carries over to the other LAC. The Hamiltonian can be simpli- fied by subtracting out a global energy shift, which gives H = 1 2 −ω0 A⊥ A⊥ ω0  (5.20) with ω0 = ω−1,+1 2 . When the splitting between the states ω0 is comparable to the transverse hy- perfine coupling, the Zeeman states |mS ,mI⟩ are no longer good approximate eigenstates of the system [40, 35, 85]. The eigenstates instead are superposi- tions of the electron and nuclear Zeeman states. This Hamiltonian is the same as the canonical Rabi Hamiltonian, which describes Rabi oscillations in a two level system. Therefore, if the system begins in the state |0, 0⟩, the state of the 65 |0,0⟩ |-1,+1⟩ A⟂ pavg Figure 5.5: Average spin flip-flop probability pavg between |0, 0⟩ and |−1,+1⟩ from 200-800 G. The black dotted line denotes the lo- cation of the ESLAC. system will precess |0, 0⟩ between |−1,+1⟩. The probability p(t) of a spin flip-flop occurring is p(t) = A2 ⊥ A2 ⊥ + ω 2 0 sin2  √ A2 ⊥ + ω 2 0 2 t  (5.21) Therefore the precession will occur at a frequency Ω = √ A2 ⊥ + ω 2 0 with a maxi- mum amplitude that decreases as the field moves away from the LAC. The average spin flip-flop probability pavg will be determined by the lifetime of the state in the excited state, which is limited by the spontaneous emission rate to approximately 1/Γ1 ≈ 10 ns. Assuming that exiting the ES is a Poisson process, the average spin flip-flop probability will be pavg = ∫ ∞ 0 Γ1e−Γ1t p(t)dt = A2 ⊥ 2(Γ2 1 + A2 ⊥ + ω 2 0) (5.22) which corresponds to a Lorentzian centered around the LAC (ω0 = 0) with a 66 full width at half maximum (FWHM) of 2 √ Γ2 1 + A2 ⊥. The average probability as a function of external field is plotted in Figure 5.5. The maximum spin flip probability occurs at the LAC and decreases to zero away from it. 5.7 Nuclear Polarization The optical dynamics of the electron and nuclear spins are due to the effect of the transitions described in the 21 level model (non-unitary evolution) and the ES spin flip-flops (unitary evolution). The most important consequence of these dynamics is optical polarization of the nuclear spin near the ESLAC. This feature is convenient because it allows protocols to initialize both the electron and nuclear spin simultaneously to a known state with a single optical pulse. The optical polarization process of nuclear spins (e.g/ 13C, 14N, 15N) has been studied extensively in prior works with single and ensemble NV centers [44, 35, 85, 86, 87] and has been used to extract the hyperfine coupling between the nuclear and electron spins [39]. Under optical illumination, the nuclear spin of 14N will polarize to mI = +1. Therefore using an sufficiently long and intense optical pulse, the NV center can easily be initialized to the state |0,+1⟩. This occurs because of every possible spin state, there is a pathway to |0,+1⟩ through the combination of the optical cycling of the electron and the ES spin flip-flops. The probability that the spin goes through this pathway is small, but nonzero. Therefore, after many optical cycles, an excess of spins will end up in |0,+1⟩. We will qualitatively describe the polarization pathways for all the states. These are shown schematically in Figure 5.6. We will categorize these pathways 67 into three groups. Group I will consist of states do not require any ES spin flip- flops to polarize the nuclear spin. Group II will consist of states that only require one ES spin flip-flop to polarize the electron spin. Group III will consist of states that require two spin-flip flops to polarize the nuclear spin. The pathway these different groups take to polarize to |0,+1⟩ is as follows: • The states in Group I are |−1,+1⟩ and |+1,+1⟩, and |0,+1⟩ which will just have the electron spin polarize to mS = 0 through the optical cycle. • The states in Group II are |0, 0⟩, |−1, 0⟩, and |+1, 0⟩. Through the optical cycle, these states will polarize the electron spin and end in |0, 0⟩. This state then undergoes a spin flip-flop in the ES to the state |−1,+1⟩. This state finally ends in |0,+1⟩ after the electron spin is polarized by another optical cycle. • The states in Group III are |−1,−1⟩, |0,−1⟩, and |+1,−1⟩. These states will polarize the electron spin to |0,−1⟩, which then undergoes a ES spin flip- flop to the state |−1, 0⟩. Then this state will undergo the pathway for Group II to end in |0,+1⟩ 68 |+1,0⟩ |0,0⟩ |-1,0⟩ |+1,+1⟩ |0,+1⟩ |-1,+1⟩ Key Radiative relaxation (Bright) Non-radiative relaxation (Dark) ES spin flip-flop |0,+1⟩ |0,0⟩ |-1,+1⟩ |+1,-1⟩ |0,-1⟩ |-1,-1⟩ |0,-1⟩ |-1,0⟩ Group I polarization Group II polarization |0,0⟩ |0,-1⟩ Laser excitation Group I Polarization Group II Polarization Group III Polarization Figure 5.6: Pathways to polarize nuclear spin to mI = +1 for Group I, II, and III. 69 a) b) Figure 5.7: a) Occupation in mI = +1 after optical pumping at saturation. b) Corresponding polarization in mI+ = 1 Figure 5.7 shows the result of a Lindblad calculation that shows the amount of polarization in mI = +1 over pumping times at the 500 G and 200 G. In this calculation, the NV center starts in a state corresponding to thermal equilibrium of the nuclear and the electron spins. The system is allowed to relax for 1 µs be- fore the populations in each nuclear projection is extracted. In general, better polarization is obtained closer to the ESLAC due to the high average spin flip- flop probability. In addition polarization improves with higher optical power and long pulses. These conditions maximize the average spin flip-flop proba- bility and provide the spin with multiple opportunities to enter a pathway to polarize to |0,+1⟩. Relatively large nuclear polarization is achievable for fields relatively far from the ESLAC (i.e. greater that C⊥/γe ∼ 10 G). For example, it is possible to achieve at least 50% polarization at fields as far as 300 G away from the ES- LAC (Fig). This is because while the average spin flip probability is low at these fields (less than 1%), it is still sufficient enough to flip a majority of nuclear spin spins. The trade-off is that it will take many more optical cycles (much longer 70 pumping times) to achieve nuclear polarization as the electron and nuclear spin will require many more opportunities to relax to |0,+1⟩. 5.8 Field Dependent Contrast of NV Center Spin States Because the ES spin flip-flops can change both the electron and nuclear spin projections, they will have an effect on the PL emission from each state |mS ,mI⟩. The PL emission from each state will depend on the external applied field, since this affects the average probability of a spin flip-flop occurring. At low fields (e.g. 10 G), the average spin flip-flop probability in the ES is negligible, which allows the state |0, 0⟩ have spontaneous emission be the primary pathway the NV center spin relaxes. Close to the ESLAC, the average spin flip-flop proba- bility approaches its maximum, making it significant more likely that |0, 0⟩ will flip-flop into the |−1,+1⟩ state. Therefore |0, 0⟩ has a non-negligible probabil- ity to relax through the ISC. This causes the average PL emission of |0, 0⟩ lower near the ESLAC. This causes an increase in contrast compared to the NV cen- ter’s ”bright” state, which can be used to increase the signal to noise ratio (SNR) of optical readout near the ESLAC [37]. The reduction in PL emission for a state |mS ,mI⟩ is proportional to the the number of times the spin non-radiatively relaxes before polarizing to |0, 1⟩. For the state |0, 0⟩, as outlined above, the state enters the ISC twice before polariza- tion, which reduce its emission by a factor of two (increase in the SNR by √ 2 at optical saturation. The state with the largest difference in PL emission close to the ESLAC is the state |−1,−1⟩. This state enter the ISC crossing three times before polarizing, leading to a factor of three reduction in PL (increase in SNR 71 200 G 500 G a) b) |0,+1⟩ |-1,-1⟩ |0,+1⟩ |-1,-1⟩ Figure 5.8: Calculated PL emission from |0,+1⟩ and |−1,−1⟩ from the 21 level model at a) 200 G b) 500 G. The yellow region is the ob- served contrast. by √ 3). The calculated PL emission from these to states at low field (10 G) and near the ESLAC (500 G) is shown in Figure 5.8. As discussed in Chapter 1, an important parameter that determines the sen- sitivity of a gyroscope is the visibility V of the Ramsey fringes. In terms of the brightest (Imax) and lowest (Imin) intensities of the Ramsey fringe, the visibility V is defined as V = Imax − Imin Imax + Imin . (5.23) Therefore the ES spin flip-flops will impact the visibility by changing Imax and/or Imin depending on how the states corresponding to Imax and Imin relax at different fields. This makes the choice of external field an important factor in optimizing the sensitivity of the NV gyroscope. 72 5.9 Conclusion We described the optical pumping dynamics of the NV electron and nuclear spins using a 21 level rate model and the Lindblad master equation formalism. Important optical properties of the NV center, such as spin-dependent PL and optical polarization to mS = 0 are quantitatively shown using this model. The 21 level rate model will be of key importance when we discuss the physics behind the optical readout protocol for nuclear spins beyond electron T1 in Chapter 6. Finally, we discuss two important consequences of the 21 level rate model, which includes optical polarization of the NV center into |0,+1⟩ and field de- pendent PL emission from spin states |mS ,mI⟩. 73 CHAPTER 6 OPTICAL READOUT OF COHERENT NUCLEAR SPINS BEYOND THE ELECTRON T1 6.1 Chapter Abstract While dynamically decoupling the native nitrogen spin of the NV center from the proximal electron spin can theoretically extend the phase coherence time T ∗2,n beyond the electron T1, this prospect comes with an additional challenge– read- ing out the information stored in the nuclear spin in this new sensing regime. In this chapter, we will discuss the central issue behind nuclear spin readout beyond the electron T1, which involves the ineffectiveness of the conventional CNOT readout on a thermalized elecron spin. We will discuss a readout proto- col that addresses this issues, which involves optically repumping the NV center before the CNOT issue. Finally, we will evaluate the effect this readout scheme has on the sensitivity of a theoretical nuclear based NV gyroscope, which will be the result of the dynamics of the NV center’s excited state discussed in the previous chapter. 6.2 Introduction Rotation sensing with NV centers involves detecting small frequency shifts in the spin resonances using Ramsey interferometry. These shifts are mapped to changes of the population in the nuclear spin sublevels, which can be mapped to an observable PL contrast through the hyperfine interaction between the elec- 74 tron and nuclear spins. This hyperfine interaction, which allows us to perform traditional CNOT readout, turns out to be a major limitation to the phase co- herence time T ∗2,n, which drastically harms the sensitivity of current NV center gyroscopes. Using dynamical decoupling techniques described in Chapter 3, we are in principle able to extend T ∗2,n past the electron T1 if the hyperfine inter- action is the dominant decoherence channel. However, observing this dynamical decoupling requires being able to read- out the nuclear spin beyond the electron T1, which is a new regime that NV center based sensors do not operate in. In this regime, T1 processes drive the electron spin to a thermal mixed state (i.e. equal spin populations in all spin sublevels), while the nuclear spin remains coherent. The conventional CNOT readout takes advantage of the fact that for sensing times less than the electron T1, the electron spin is in the mS = 0 state. A selective microwave pulse (e.g. resonant with |0, 0⟩ to |−1, 0⟩) maps a single nuclear spin projection (mI = 0) to an optically dark electron spin projection (mS = −1). This allows us to create nu- clear spin-dependent PL, which we can use to extract out the relevant sensing information from our nuclear qubit. In our theoretical sensing scenario, where the electron and nuclear spin are successfully decoupled, the electron spin has thermalized. This means that nu- clear spin information is not just stored in mS = 0, but is stored equally in all three electron spin projections. As a result, after using the selective CNOT pulse, the system is unchanged, which prevents us from creating additional nuclear spin-dependent PL. Thus, there is a need to develop protocols for reading out the nuclear spin beyond the electron T1 to enable us to actually observe the dy- namical decoupling. 75 In this chapter, we develop and test an optical readout protocol for nuclear spin sensing beyond the electron T1, which consists of using an optical repump pulse to reset some of the electron spins back into the mS = 0 before applying the conventional CNOT readout. We evaluate the effect that this readout protocol has on the sensitivity of nuclear based NV center sensors. If it turns out that the readout protocol severely limits the sensitivity (i.e. by more than a order of magnitude), then this makes the prospect of making competitive NV center gyroscopes poor. 6.3 Readout Protocol past the Electron T1 6.3.1 Visiblity and the Sensitivity of an NV gyroscope The shot-noise limited sensitivity of an NV center gyroscope, as shown in Chap- ter 2, is given by ηshot Ramsey ≈ 1 V √ NT ∗2,n (6.1) where V is the Ramsey fringe visibility and N is the ensemble size. In Chapter 2, we defined the Ramsey fringe visiblity using the definition defined in literature [2, 4] Vlit = Imax − Imin Imax + Imin (6.2) where Imax and Imin are the maximum and minimum intensities of the Ramsey fringe. In this chapter we will use an alternative definition of the visibility V = Cmax −Cmin (6.3) 76 where Cmax and Cmin are the relative contrast of the maximum and minimum of the Ramsey fringe to the emission from a reference state |0,+1⟩ (I0,+1). The contrast of a point i in the Ramsey fringe is given by Ci = Ii − I0,+1 I0,+1 (6.4) where Ii is the emission from the point on the Ramsey fringe. This definition for visibility will be easier to use when we compare to our Lindblad simulations to our measurements since it allows us to look at relative PL intensities, rather than absolute values. These two definitions for the visibility are essentially equivalent. They are algebraically related by the expression Vlit = V[2 − (Cmax +Cmin)]. (6.5) For typical measurements with ensembles of NV centers, Cmax +Cmin ≪ 1 so Vlit ≈ 2V. (6.6) Therefore our definition of visibility is proportional to the literature definition, which means we can use our definition to evaluate relative changes in the sen- sitivity due to our readout protocol. 6.3.2 Physical Considerations for Readout Sensitivity Our readout protocol involves using an optical repump pulse to regain some electron polarization after sensing before the conventional CNOT readout. Therefore the main physics that will determine the impact of the optical re- pump pulse on the sensitivity will be the dynamics in the excited state (ES) 77 [39, 25, 80, 82, 84]. The ES Hamiltonian is HES = DesS 2 z + γeBz + PI2 z + γnBz + A∥S zIz (6.7) + A⊥(S xIx + S y + Iy) where Des is the zero field splitting in the excited state (1.42 GHz), γe is the elec- tron spin gyromagnetic ratio (2.8 MHz/G), γn is the nuclear spin gyromagnetic ratio (0.300 kHz/G), P is the quadrupolar moment of the nuclear spin (−4.96 MHz), and A∥ and A⊥ are the longitudinal and transverse hyperfine couplings (−40 MHz and −23 MHz respectively) [25, 39]. The transverse hyperfine coupling in the excited state opens a level anti- crossing (ESLAC) at approximately 500 G [40, 36, 35, 85]. This strong transverse hyperfine coupling allows for fast electron-nuclear spin flip-flops before the or- bital relaxation of the NV center. One consequence of these spin flip-flops is optically polarization of the nuclear spin to mI = +1. While this is useful in robustly initializing the nuclear spin qubit, this property is detrimental to read- out because polarization essentially erases the information stored in the nuclear spin qubit. Therefore, there is a trade-off between generating electron polariza- tion for the conventional CNOT readout and preserving the information from sensing stored in the nuclear spin. In the previous chapter, we saw that nuclear polarization was not the only physical consequence of these ES spin flip-flops. The spin flip-flops can cause the contrast of certain spin states (relative to the PL emission from |0,+1⟩) to in- crease near the ESLAC. For these spin states (e.g. |−1,+1⟩), the spin flip-flops can prevent the electron from relaxing radiatively for multiple optical cycles. This increase in contrast will help increase visibility of the Ramsey fringe, thereby improving the readout sensitivity. 78 1.23 rad π 2 |0⟩e→|-1⟩e |0⟩e→|+1⟩e ... ...Wait 10 μs a) 0 ms -1 +1 b) Thermal state density matrix Measured density matrix 400 G c) 532 nm RF MW Thermal State Prep CNOT (0.8 MHz) Initialize (20 μs) Readout (350 ns) π 2( ) X π 2( ) X,θ |0,+1⟩ → |0,0⟩ |0,0⟩ → |-1,0⟩ Wait 10 μs 532 nm RF MW Thermal State Prep CNOT (0.8 MHz) Initialize (20 μs) Readout (350 ns) π 2( ) X π 2( ) X,θ |0,+1⟩ → |0,0⟩ |0,0⟩ → |-1,0⟩ Wait 10 μs |0,+1⟩ → |0,0⟩ Longitudinal Ramsey (with thermal electron state, T > T1 simulated) Transverse Ramsey (with thermal electron state, t > T1 simulated) 532 nm RF MW CNOT (0.8 MHz) Initialize (20 μs) Readout (350 ns) π 2( ) X π 2( ) X,θ |0,+1⟩ → |0,0⟩ |0,0⟩ → |-1,0⟩ Control Ramsey (with ms=0 pure state, t < T1) d) e) Repump (vary) Repump (vary) T2,e * dephasing Figure 6.1: a) Artificial electron thermal state preparation. A pair of MW pulses is used to equalize the electron spin populations in all sublevels. A 10 µs wait (greater than T ∗2,e ∼ 600 ns) eliminates any coherences between electron spin sublevels. b) Quantum state tomography of the artificial thermal state at 400 . The image on the left shows the density matrix for a true thermal state and the image on the right shows the measurement. The fidelity is approximately 98%. c) Pulse sequence for the opti- cal repump of the longitudinal component of the nuclear spin coupled to the artificial electron thermal state (simulating ro- tating sensing beyond the electron T1). d) Pulse sequence for the optical repump of the transverse component of the nuclear spin coupled to the artificial electron thermal state. e) Pulse se- quence for control measurement. The nuclear spin is coupled to a pure mS = 0 state, which represents the visibility obtained in the long time limit of the electron T1. 79 To evaluate the overall effect of these two competing consequences of the ES spin flip-flops on our optical repump readout protocol, we simulate a rotation sensing measurement beyond the electron T1 and evaluate the net effect on the visibility of the Ramsey fringe. For our measurements, we use an ensemble of NV centers in an optical grade diamond from Element Six (approximately 100 defects/µm3). To minimize the effect of ensemble inhomogeneity on our measurements, we excite the NV ensemble at an optical power of 1.6 mW, which is approximately a factor of 10 below optical saturation. 6.3.3 Thermal Electron State Preparation A central component behind developing a readout protocol for the nuclear spin beyond the electron T1 is being able to simulate the final electron-nuclear spin state from the dynamically decoupled rotation sensing protocol. To accomplish this, we couple coherent nuclear spin states to artificial thermal electron spin states states by mapping the state |0⟩ ⟨0| ⊗ ρn to ρtherm e ⊗ ρn. We artificially prepare a thermal electron state using the following pulse se- quence. First, while the electron is polarized to mS = 0, we prepare the nuclear spin state (i.e. with control pulses on the nuclear spin). Then, we artificially thermalize the electron spin with the microwave pulse sequence shown in Fig- ure 6.1(a). The first pulse corresponds to a 1.23 radian rotation between the mS = 0 and mS = −1 electron spin projections. This moves 1/3 of the electron spins from mS = 0 to mS = −1. The second pulse is a π/2 pulse that moves half of the spins remaining in mS = 0 into the mS = +1 state. The net result is that the electron spin population is equalized in all three electron spin projections. To 80 minimize the impact that the electron thermal state preparation has on the nu- clear spin, the pulses are significantly harder than C⊥ ∼ 2.1 MHz (a Rabi field of ∼ 12 MHz). Finally, to eliminate any coherences between the electron spin pro- jections, we have a 10 µs wait, which is much longer than the T ∗2,e of the electron spin (around 600 ns). In our electron thermal state preparation, we ideally use a 1.23 rad pulse followed by a π/2 pulse. However, the B1 field in the confocal volume in in- homogeneous. As a consequence, these pulses do not rotate the electron spin enough for some NV centers. To correct for this, we measure the expectation value of S z from our artificial thermal state. Then we adjust the rotations for both pulses until the measured expectation value is zero. 6.3.4 Artificial Thermal Electron State Fidelity To assess the fidelity of our artificial thermal state preparation, we perform quantum state tomography to measure the electron spin-1 density matrix [88, 89]. To measure the density matrix, we have to measure eight independent spin observables {λt} (Gell-Mann matrices). The specific details and background for state tomography can be found in Appendix 8.2. After our measurement, we compare our artificial state to an ideal thermal electron state. Our procedure is robust with a fidelity of approximately 98% [Figure 6.1(b)]. The pulse sequences for measuring each observable is given in Figure 6.2. When we perform state tomography, we measure the NV center’s PL emis- sion to determine the relative electron spin populations for measuring the ob- servables. However, extracting the electron spin populations is complicated 81 by the fact that the contrast of different electron states (i.e. |0,+1⟩, |−1, 0⟩, and |−1,+1⟩) depend on the applied field due to the presence of the ES spin flip-flops. Therefore, we use the following procedure to account for this effect. To extract the spin populations in mS = +1 and mS = −1, we first use calibra- tion pulse sequences that measure the contrasts of |+1,+1⟩ and |−1,+1⟩ relative to |0,+1⟩ (Figure 6.3). We denote C+ and C− as the contrasts of |+1,+1⟩ and |−1,+1⟩ from the calibration sequences and P0, P+, and P− as the spin popula- tions in mS = 0, mS = +1, and mS = −1 respectively. For our artificial thermal state, we measure three contrasts R0, R+, and R− that are proportional the mS = 0, mS = +1, and mS = −1 populations. This relationship is described by the matrix equation R = CP R0 R− R+  =  0 C− C+ C− 0 C+ C+ C− 0   P0 P− P+  . (6.8) We can invert the matrix C and extract the electron spin populations. The measured qutrit from this procedure may be an unphysical state (i.e. have negative eigenvalues and/or non-unit trace) due to experimental noise. To estimate a physical state, we use a numerical estimation procedure, similar to maximum-likelihood estimation, which uses our initial unphysical state as an initial guess for the least squares optimization problem detailed in Ref[90]. To estimate the measurement error for the individual elements of the qutrit state, we use statistical bootstrapping of our data to find the distribution of physi- cal density matrices that can be estimated from our noisy measurements. The estimated measurement error from bootstrapping is approximately 1%. Addi- 82 532 nm MW Thermal State Prep Adiabatic Passage Initialize (20 μs) Readout (350 ns) |0⟩ → |-1⟩ for ms=±1 readout only Wait 10 μs Quantum state tomography (QST) of artificial thermal electron state QST Pulses a) b) QST Pulses |0⟩ → |+1⟩ π 2( ) -Y λ1 λ2 π 2( ) X λ3 λ4 π 2( ) -X πX λ5 |0⟩ → |-1⟩ |0⟩ → |+1⟩ π 2( ) -Y λ6 λ7 π 2( ) X λ8 ( ) π 2( ) -Y πX Figure 6.2: a) Quantum state tomography (QST) for checking artificial thermal state. After the thermal state is prepared, a set MW pulses is used to measure the expectation values of the Gell- mann spin observables λi necessary for measuring the qutrit state. b) QST pulses used to diagonalize the spin operators for S z measurement. 83 MW Adiabatic Passage Initialize (20 μs) Readout (350 ns) for ms=±1 only ( ) 532 nm Figure 6.3: Calibration pulse sequences of PL emission from mS = 0,±1. The adiabatic passage pulses are applied after the 20 µs green excitation to transfer the NV center electron spins from mS = 0 to mS = ±1. tionally, there can be systematic errors caused by experimental imperfections such as imperfect pulse fidelity and ensemble inhomogeneity. Given that we use a wait much longer than T ∗2,e after our MW pulses for thermal state prepa- ration, we are confident that any coherences should be negligible making the systematic error on the order of the magnitude of coherences of the estimated state. 6.3.5 Simulated Rotation Sensing Measurement Now that we are able to simulate the states from a rotation sensing measurement beyond the electron T1, we look at the effect of the optical repump pulse on the nuclear spin [Figure 6.1(c-d)]. We will look at the effect of the repump on both the longitudinal component of the nuclear spin (along z on the Bloch sphere) and on the transverse component of the nuclear spin (on the xy plane of the Bloch sphere). By looking at both components, we get a complete picture of how the nuclear spin information is lost during the repump. 84 a) Optically repumped Ramsey (400 G, Longitudinal) b) Optically repumped Ramsey (400 G, Transverse) Figure 6.4: Representative measurements of the visibility and initial phase ϕ of the Ramsey fringe at 400 G for a) the optically repumped longitudinal Ramsey sequence [Figure 6.1(c)] and b) the op- tically repumped transverse Ramsey sequence [Figure 6.1(d)]. There are only significant increases for the visibility for the lon- gitudinal Ramsey. The transverse Ramsey shows a linear de- pendence between the initial phase ϕ and the repump time. The pulse sequences to look at both the longitudinal and transverse compo- nents are shown in Figure 6.1(c-d). The key difference between the sequences is when the optical repump pulse is applied. For the longitudinal component of the nuclear spin [Figure 6.1(c)], we first illuminate the NV centers for 20 µs with a 532 nm laser. This initializes the electron spin and nuclear spin to |0,+1⟩. Next, a pair of π/2 pulses, resonant with the |0,+1⟩ to |0, 0⟩ transition, is applied, with the second pulse shifted by an additional phase θ relative to the first pulse. We then perform our artificial thermal state preparation as described in the pre- vious section. After this, we apply the optical repump pulse and perform the convention CNOT readout. The CNOT pulse we use is resonant with the |0, 0⟩ 85 to |−1, 0⟩ transition. To read out the spin populations we use a 350 ns optical pulse and measure the PL. For the transverse component of the nuclear spin [Figure 6.1(d)], we use a similar protocol. The main difference is that the artificial thermal electron state is created in between the two π/2 pulses on the nuclear spin. For the longitudi- nal component, we apply the repump pulse after the accumulated phase from sensing is mapped back to z on the Bloch sphere by the second π/2 pulse. How- ever, for the transverse component, the optical repump pulse is applied while the nuclear spin is on still on the equator of the Bloch sphere during the Ramsey sequence, prior to the second π/2 pulse. Finally, as a control measurement we perform a conventional Ramsey protocol, which does not include the thermal electron state preparation or the optical repump pulse [Figure 2(e)]. The pur- pose behind this control measurement is to simulate the maximum theoretical signal we can achieve if we could effectively ”turn off” electron T1 processes (i.e. in the long time limit of the electron T1). The CNOT pulse we use needs to be selective with respect to the hyperfine split nuclear states. This is done using a pulse that has a Rabi field of ∼0.8 MHz which will, in the worst case, move about 13% of the population in |0,+1⟩ to |−1,+1⟩. We cannot make the CNOT pulse more selective by reducing the Rabi field because of ensemble inhomogeneity (i.e. T ∗2 ∼600 ns), which causes the contrast of our CNOT readout to decrease with Rabi field. This ensemble in- homogeneity can come from multiple sources such as differences in the B1 field in the ensemble. In addition to the ensemble inhomogeneity, there is also in- homogeneous broadening of the electron spin transition, which can be caused by dipolar interactions between the NV center ensemble with the surrounding 86 spin bath (e.g. P1 centers, 13C spins, and other NV centers). In general, inhomogeneous broadening is a fundamental problem for NV center sensing applications. Because the sensitivity scales by √ N where N is the ensemble size, we naturally we want to work with a large number of nuclear spins. Nuclear spins have a small gyromagnetic ratio, which allows us to work with a dense ensemble without significant inhomogeneous broadening caused by interactions between nuclear spins. However, the nuclear spin is not the only spin that is important in our system as we are using the optical properties of the NV center electron spin for readout. Therefore, if we increase the density of nu- clear spins, we will also increase the density of electron spins. These electron spins consist of the NV electronic spins that is associated with each nuclear spin and other paramgagnetic defects such as P1 centers. Because the electron spin has a gyromagnetic ratio that is orders of magnitude larger than the nuclear spin, this √ N increase in sensitivity will come at the cost of inhomogenoeus broadening due to interactions between electron spins [91, 49, 50], which will decrease the contrast of the CNOT readout. Thus, it is important to characterize the impact these inhomogeneities have on the our readout protocol in represen- tative NV center ensembles (e.g. NV centers in optical grade diamond). For both the longitudinal and transverse components of the nuclear spin, we measure Ramsey fringes corresponding to the simulated rotation sensing measurement. The PL of each data point is normalized to the PL emission from the |0,+1⟩ as described in Eqn 6.4. These Ramsey fringes, S (θ), are fitted to S (θ) = V 2 cos (θ + ϕ) + B (6.9) where B is the background, and θ is the relative angle between the π/2 pulses in radians. The parameter ϕ is related to the acquired phase from our sensing 87 protocol, so it is important to track if has a dependence on our repump proce- dure. The goal of our measurements is to look at V over different repump times to track the impact the repump has on the readout sensitivity (Eqn 6.1). 6.4 Results For both components of the nuclear spin, the data roughly share the same quali- tative features from 200-800 G. We present the result of the measurements taken at 400 G as a representative example. For the longitudinal spin component [Fig- ure 6.4(a)], the visibility starts at a minimum due to the fact that the nuclear spin is coupled to the artificial thermal state. Then as the optical repump pulse is ap- plied, the visibility increases until it reaches a maximum (∼ 1 µs). The amount of recovery in the visibility depends on the field because it depends on competition between the gain in electron polarization with the loss in nuclear spin fidelity from nuclear spin re-polarization. Past this point, the visibility decreases as the nuclear polarization process dominates. We also find that the initial phase ϕ from our fit is independent of repump time. Even though we only excite our NV ensemble with optical intensity that is a factor of 10 below optical saturation, our results should scale predictably with optical power (assuming we are sufficiently below saturation). Since the visi- bility depends only on the number of ES spin flip-flops that occur during the optical repump pulse, we expect that the maximum recovered visibility should be independent of excitation power. The repump time required to obtain this maximum visibility will change. This should be inversely proportional to the optical power, since increasing the laser excitation rate also increases the aver- 88 CNOT With thermal state (No repump) a) With pure ms = 0 state (t < T1) With thermal state (Repump) Longitudinal Ramsey b) Transverse Ramsey c) With thermal state (Repump) With pure ms = 0 state (t < T1) d) Figure 6.5: Summary of data collected from 200-800 G. a) Visibility of the CNOT readout for a state initialized to |0, 0⟩ (green). Maxi- mum visibility of Ramsey sequences with no repump is shown in blue, which demonstrates the contribution of the contrast enhancement mechanism. b) Maximum recovered visibility for the longitudinal Ramsey (blue) compared to the maximum possible visibility (red). The maximum occurs at 500 G, where the ES spin flip-flops are the strongest. c) Maximum recov- ered visibility for the transverse Ramsey (blue) compared to the maximum possible visibility (red). The maximum also oc- curs at 500 G, but is worse compared to the longitudinal Ram- sey. d) Phase susceptibility χϕ for the transverse Ramsey from 200-800G. The antisymmetric trend about 500 G suggests that the physical origin behind the phase susceptibility come from the ES dynamics. 89 age number of spin flip-flops that occur, keeping the pulse length constant. In addition, the behavior of ϕ is not unexpected considering the context of the longitudinal Ramsey measurement. Because the relevant phase information is mapped back to z on the Bloch sphere before the repump, any effective field induced by the repump will not have any effect on it. One way to view this is through the semi-classical picture we presented previously in Chapter 2. The nuclear polarization process is caused by the ES spin flip-flops which can be viewed as an effective T1 process on the electron spin. This effective T1 process changes the effective field acting on the nuclear spin, which causes changes to its precession on the Bloch sphere. However, because the relevant phase infor- mation from sensing is mapped back to z prior to the repump, any additional change in the phase on the transverse plane will have no effect on the measure- ment. For the visibility, the transverse Ramsey [Figure 6.4(b)] shares the same trend as the longitudinal Ramsey. The main difference is that the maximum recovered visibility is dramatically smaller. We can explain this using the previous semi- classical picture. Since the ES spin flip-flops change the effective field by acting as an effective electron T1 process, this will cause the transverse component of the spin, whose information has not been mapped back to z yet, to undergo a random walk. Therefore, the optical repump acts as an effective T2 process on the nuclear spin by averaging over this random walk over multiple optical cycles, which leads to the loss in visibility in the Ramsey fringes. In comparison, the longitudinal component is insensitive to this random walk, which enables the optical repump to recover some visibility of the Ramsey fringe. For the transverse component, the initial Ramsey phase ϕ extracted from the 90 fit increases linearly with repump time. Thereofre, even though the phase un- dergoes a random walk, this stochastic process is biased in one direction, which appears as a slope in this plot. We define the slope of this linear dependence as a phase susceptibility, χϕ. When normalized to the laser excitation power, χϕ represents the average phase gain in one optical cycle from the transverse component’s random walk in the excited state. We also look at the CNOT’s effectiveness without the repump [Figure 6.5(a)]. For fields far from the ESLAC, the visibility is low because the is no electron polarization for the CNOT pulse to create nuclear spin-dependent PL. However, at 500 G, even though there is no electron polarization to generate contrast using the CNOT, we observe a relatively high visibility (∼2.5%). This is because at near the ESLAC (∼500 G), the ES spin flip-flops cause nuclear spins in mI = 0 to relax through the dark ISC during optical readout more times on average than the spins in mI = +1, creating contrast even when the electron is in a thermal mixed state. We also plot the visibility of the CNOT pulse, which is extracted from a Rabi oscillation of the NV center from |0, 0⟩ to |−1, 0⟩. The visibility of the CNOT pulse dips to its minimum near the ESLAC, which is what we expect theoretically. The visibility generated by the CNOT pulse is the difference between the contrasts of |−1, 0⟩ and |0, 0⟩. Since |−1, 0⟩ reaches its minimum at the ESLAC and |0, 0⟩ reaches its maximum, the difference will be lowest near the ESLAC (Figure 6.8). We also see that the CNOT visibility falls off as the field approaches 200 and 800 G, which we attribute to poorer nuclear initialization to |0,+1⟩. Now, we will look at the effect of the repump across 200-800 G. We plot the maximum visibility obtained at each field and compare then to the correspond- 91 ing control measurements. The control measurements (with the electron spins polarized in mS = 0) represent the maximum visibility that our readout proto- col can achieve if we could ”turn off” electron T1 processes (i.e. the limit where T1 approaches infinitely long times). We see that the highest recovered visibil- ity for the longitudinal and transverse components occur near the ESLAC. The maximum recovered visibility for the longitudinal component is only about a factor of two below the control [Figure 6.5(b)]. For the transverse component it is about a factor of four [Figure 6.5(c)]. Compared to the recovered visibility at low field (i.e. 200 G), the maximum recovered visibility is about a factor of six higher for the longitudinal component and about a factor of four for the trans- verse component. Our measurements suggest that the sensitivity gain from the PL contrast enhancement mechanism is comparable to the sensitivity loss from the nuclear polarization process. One aspect of our data that might be surprising is the fact that the maximum recovered visibility is so low far from ESLSC (e.g. less than 1% at 200 G). For a single NV center, we would expect that the visibility of the control measure- ment would reach its maximum away from the ESLAC and reach its minimum near it. This is because all the spins in |0, 0⟩ should be moved to |−1, 0⟩, so we expect the visibility to follow the same trend at the contrast of |−1, 0⟩ with field. We attribute this opposite trend in our measurements to the inhomogeneous broadening of the electron spin transition, which harms the contrast generated by the CNOT pulse. As a result, after the CNOT is pulse applied, the major- ity of spins remain in |0, 0⟩, which has a significantly lower contrast than |−1, 0⟩ compared to |0,+1⟩ Finally, we look at the phase susceptibility of the transverse component 92 across 200-800 G [Figure 6.5(d)]. The general trend with respect to field ap- pears to be anti-symmetric about 500 G. This provides strong evidence that the physical origin behind the phase susceptibility is due to the ES spin-flip flops at the ESLAC. In combination with the poor recovery in the visibility, optically repumping the transverse component is not ideal for recovering nuclear spin information beyond the electron T1. 6.5 Discussion 6.5.1 Analysis of Longitudinal Ramsey Results From our measurements, we find that the maximum visibility occurs at the ES- LAC for both the longitudinal and transverse components of the nuclear spin. This suggests that gain in visibility from the PL contrast enhancement is compa- rable to the loss from the nuclear polarization process. This makes the ESLAC to be the best field to operate our repump protocol, despite the fact that this is where the ES spin flip-flops are most significant. The trend that we observe, where the visibility is highest near the ESLAC, is consistent with the idea that the CNOT pulse is being limited by inhomoge- neous broadening, suggesting the |0, 0⟩ is a key contributor to the visibility trend we observe. There are other electron-nuclear spin states, however, that the NV center and the nuclear spin cycle through during the optical pumping process. Therefore it is possible that these other states (e.g. |±1,+1⟩) are contributing sig- nificantly as well. To determine if this is feasible, we use the 21 level Lindblad model we introduced in Chapter 5. 93 Figure 6.6: Plot of the NV ensemble photoluminescence as a function of lase excitation power. The black dotted line indicates the satu- ration power extracted from the fit, which is used in the Lind- blad calculations. The goal is to summarize the central principles behind the observed trend and potentially extend these results to other experimental contexts (i.e. different optical powers) and to identify any potential improvements that can be made to the readout protocol. In this section, we will focus on the physics behind the longitudinal component rather than the transverse, since repumping the latter is a significantly worse for readout. However, we will present a qualitative model in the next section that explains the observed linear trend in the phase susceptibility. A key parameter for our Lindblad calculations is the optical excitation rate of the NV center by the 532 nm laser. To estimate this parameter, we measure the emitted PL from our NV center ensemble at different incident optical powers 94 (Figure 6.6). We fit the curve to I(P) = I0 1 + P/Psat + B (6.10) where I0 corresponds to the saturation PL intensity, B is the background inten- sity, and Psat is the saturation power [92]. The saturation power from the fit, which corresponds to when the laser excitation rate is comparable to the spon- taneous emission rate (67.4 MHz), is 18 ± 3 mW. To estimate our experimental excitation rate, we scale the spontaneous emission rate by the ratio between the optical power used in the measurements (1.6 mW) and the saturation power measured from the fit. To quantify the loss of information from the nuclear polarization process, we use our Lindblad model to look at the nuclear fidelity after different repump times across 200-800 G. Specifically, we simulate quantum process tomography of a psuedospin-1/2 consisting of the {0, +1} nuclear spin qubit [76, 41, 90, 93]. We treat the dynamics of the electron spin caused by optical pumping as an external source of dephasing with respect to the nuclear spin. We calculate the process matrix for our experimental process and compare it to an ideal process where the nuclear spin populations are perfectly preserved [Figure 6.7(a)]. Our ideal process, written in the operator-sum representation is εideal(ρ) = P↑ρP↑ + P↓ρP↓ (6.11) = 1 2 ρ + 1 2 σzρσz (6.12) where εideal(ρ) describes the evolution of the nuclear spin from the ideal process and P↑ and P↓ are the projection operators P↑ = |+1⟩ ⟨+1| (6.13) P↓ = |0⟩ ⟨0| . (6.14) 95 To find the process matrix, we calculate the reduced density matrices of the nuclear spin by evolving the initial states ρ1 = ρ therm e ⊗ 1 0 0 0  ρ2 = ρ therm e ⊗ 0 1 0 0  ρ3 = ρ therm e ⊗ 0 0 1 0  ρ4 = ρ therm e ⊗ 0 0 0 1  (6.15) with ρtherm e =  1 3 0 0 0 1 3 0 0 0 1 3  . (6.16) We write the above initial states using a reduced Hilbert space consisting the ground state of the electron spin and the {0, +1} nuclear spin qubit. However, we use full 21-dimensional states as initial inputs for the 21 level model, where elements outside of this reduced Hilbert space are set to zero. The system is then allowed to relax for 1 µs before tracing over the electron spin to calculate the reduced density matrix of the nuclear spin. We calculate density matrices for different pumping times up to 2 µs. After we calculate the density matrices at a given pump time, we can deter- mine the process matrix χexp. The nuclear fidelity for each pump time is calcu- lated as F = Tr[ √ χidealχexp]. (6.17) 96 a) ρideal Output ρn(0) Input Optical Repump ρn(t) Measure ρexp P0=|0〉〈0|^ P+1 ρn(t)P+1 +P0 ρ n(t)P0 ^ ^ ^ ^ P+1=|+1〉〈+1| ^ 𝛘exp ρn(0) Identity ρn(0) Measure P+1 ρn(0)P+1 +P0 ρ n(0)P0 ^ ^ ^ ^ 𝛘ideal Experiment Ideal b) Figure 6.7: a) Schematic of simulated quantum process tomography. The experimental process consists of the nuclear spin undergoing optical pumping, followed by measurement of the nuclear spin populations. The ideal process consists of the identity process on the nuclear spin, followed by measurement of the nuclear spin populations. b) Plot of the nuclear spin fidelity (Eqn 6.17) from 200-800 G for times up to 2 µs. The fidelity is the worst near the ESLAC at approximately 500 G and for long pump times due to the accumulation of many ES spin flip-flops. 97 This fidelity is a metric for how well the nuclear spin information is preserved, where F = 1 corresponds to the populations in both spin projections being per- fectly preserved and F = 0.5 corresponds to a complete loss of nuclear spin information. A simple interpretation of the fidelity is as the square-root proba- bility that the experimental process will give the same result as the ideal process. Figure 6.7(b) shows the calculated nuclear fidelity from 200-800 G for pump times up to 2 µs. In general, the fidelity gets worse closer to the ESLAC and with longer pump times. This is expected since, under these conditions, the ES spin flip-flops occur with high probability, which is the dominant source of decoher- ence. However, even at the ESLAC, we can find pumping times (less than 500 ns) where the nuclear spin fidelity can be relatively large (i.e. F ≥ 0.8). It is in this regime where the PL contrast enhancement mechanism is comparable with the nuclear spin polarization process. Next, we look at the contribution of the PL contrast enhancement to the vis- ibility of the Ramsey fringe. As discussed in Chapter 5, the amount of enhance- ment depends on the number of times each electron-nuclear spin state enters the ISC during readout due to the ES spin flip-flops. We calculate the contrast for each state, relative to the PL emission from |0,+1⟩ (Figure 6.8). For these calculations, we initialize the NV center to a pure state of each electron-nuclear basis state. A significant component to the observed trend in the visibility is the contrast enhancement of |0, 0⟩, which is due to the CNOT pulse being limited by inho- mogeneous broadening. Another state that follows a similar trend in contrast as |0, 0⟩ is |0,−1⟩ (Figure 6.8). However, we rule out |0,−1⟩ a major contributor to the visibility since our nuclear qubit only involves mI = 0 and mI = +1. While 98 |-1,0⟩ |-1,-1⟩ |-1,+1⟩ |+1,0⟩ |+1,-1⟩ |+1,+1⟩ |0,+1⟩|0,-1⟩|0,0⟩ Contrast of basis states (normalized to PL|0,+1⟩): 350 ns collection, 10x below saturation Figure 6.8: Calculated contrasts of all 9 basis states using the Lindblad model. The contrasts are calculated by normalizing to the PL emission from |0,+1⟩ [Eq 6.4]. The collection time and power are chosen to match our experiment. Different states have an enhancement or loss in contrast at 500 G due to the ES spin flip- flops. The states |0,+1⟩ and |+1,+1⟩ are not affected by the ES spin flip-flops, making the contrasts independent of field. 99 mI = +1 mI = +1 mI = -1 mI = -1 mI = +1 mI = 0 ρ(0) = ρe therm⊗|+1⟩⟨+1| ρ(0) = ρe therm⊗|0⟩⟨0| a) b) Figure 6.9: Lindblad calculation showing the occupations in the nuclear spin sublevels when the state a) ρtherm 3 ⊗ |+1⟩ ⟨+1| and b) ρtherm 3 ⊗ |0⟩ ⟨0| are optically pumped. Both cases show negligible leak- age into mI = −1. spins can enter mI = −1 during the optical repump, our Lindblad calculations show that a negligible number of spins do this (Figure 6.9). This suggests that the origin of the contrast enhancement, which is competing with the nuclear polarization process, comes predominantly from the spins in |0, 0⟩. One observation is that the full width at half maximum (FWHM) of the re- pumped state’s visibility is wider than the FWHM of the contrast of |0, 0⟩ calcu- lated in the Lindblad calculation. We attribute this to the fact that the amount of inhomogenous broadening limiting the CNOT pulse varies with field. Specifi- cally, near the ESLAC, inhomogenous broadening due to coupling between NV centers and P1 centers will be significant [94]. However, away from the ES- 100 LAC, this becomes less dominant, which decreases the amount of inhomoge- neous broadening at these fields. The CNOT pulse is therefore able to move spins more efficiently from |0, 0⟩ to |−1, 0⟩ away from the ESLAC, which leads to a larger observed visibility compared to |0, 0⟩ at these fields. This causes the apparent broadening of the repump state’s visibility. The fact that the contrast enhancement from |0, 0⟩ is the cause for the ob- served maximum visibility near the ESLAC can be used to predict the best op- tical repump pulse at a given set of conditions. The observed visibility of the Ramsey fringe is determined by spin population differences between two opti- cally pumped initial states ρe therm ⊗ |+1⟩ ⟨+1| and ρe therm ⊗ |0⟩ ⟨0|. The |0, 0⟩ state’s contrast enhancement will have the largest effect when most of the spins are in |0, 0⟩ after optically pumping ρe therm ⊗ |0⟩ ⟨0|. This supported by our 21 level Lind- blad model, where we calculate the populations in |0, 0⟩ of the repumped state as a function of time for 400 G and 500 G. The maximum visibility coincides with the maximum population in |0, 0⟩ for both these fields [Figure 6.10]. This obser- vation provides a general rule for extending our results to other experimental conditions (i.e. different optical powers). 6.5.2 Qualitative Model of the Phase Susceptibility To develop a qualitative model for the phase susceptibility, we will focus on the excited state and ignore the other states in the optical pumping cycle. We will focus on the subspace consisting of the states |0,+1⟩, |0, 0⟩, |−1, 0⟩, and |−1,+1⟩. 101 a) b) 400 G 500 G Figure 6.10: Comparison between the visibility of the longitudinal Ram- sey over repump time and the calculated occupation in |0, 0⟩ for a) 400G and b) 500 G after optically repumping the state ρtherm e ⊗ |0⟩ ⟨0|. The maximum of the occupation coincides with the max visibility, which supports |0, 0⟩ being the state that is dominating the visibility. Figure 6.11: Plot of the calculated phase susceptibility from the four level qualitative model from the ES Hamiltonian. The model shows the antisymmetric dependence about the ESLAC. The excited state lifetime used for the calculation is 10 ns. 102 The Hamiltonian in matrix form is HES =  ω0 0 0 0 0 0 A⊥ 2 0 0 A⊥ 2 ωe + ω0 + A∥ 0 0 0 0 ωe  (6.18) where ω0 = P + γnBz (6.19) ωe = DES − γeBz (6.20) are the Larmor frequencies for the nuclear and electron spin respectively. We subtract out a global energy shift of ω 2 , where ω = ω0 + ωe + A∥ HES = 1 2  2ω0 − ω 0 0 0 0 −ω A⊥ 0 0 A⊥ ω 0 0 0 0 2ωe − ω  . (6.21) We then move to the rotating frame of the nuclear spin (RF1) with the unitary transformation U1. U1 =  eiω0t/2 0 0 0 0 e−iω0t/2 0 0 0 0 eiω0t/2 0 0 0 0 e−iω0t/2  . (6.22) which leads to the following effective Hamiltonian HRF 1 . HRF 1 = 1 2  −ωe − A∥ 0 0 0 0 −ωe − A∥ A⊥e−iω0t 0 0 A⊥eiω0t ωe + A∥ 0 0 0 0 ωe − A∥  . (6.23) 103 From HRF 1 , we see that in the rotating frame of the nuclear spin, the trans- verse hyperfine coupling appears as a time-varying effective magnetic field field which drives Rabi oscillations (i.e the ES spin flip-flops). We will calculate the phase susceptibility in this frame, since our measurements are done in the ro- tating frame of our RF pulses. To eliminate the time dependence in the |0, 0⟩ and |−1,+1⟩ subspace, we ap- ply another unitary transformation U2. U2 =  1 0 0 0 0 eiω0t/2 0 0 0 0 e−iω0t/2 0 0 0 0 1  . (6.24) This gives a new effective Hamiltonian HRF 2 . HRF 2 = 1 2  −ωe − A∥ 0 0 0 0 −ω A⊥ 0 0 A⊥ ω 0 0 0 0 ωe − A∥  . (6.25) We can calculate the time-evolution operator for the nuclear spin by exponenti- ation. exp ( −iHRF 2 t ) =  ei(ωe+A∥)t/2 0 0 0 0 α −iβ 0 0 −iβ α∗ 0 0 0 0 e−i(ωe−A∥)t/2  (6.26) 104 where α = cos ( Ωt 2 ) + i ω Ω sin ( Ωt 2 ) (6.27) β = A⊥ Ω sin ( Ωt 2 ) (6.28) Ω = √ A2 ⊥ + ω 2. (6.29) In the original rotating frame (RF1), the time evolution operator is U(t) = U†2 exp ( −iHRF 2 t ) U2. (6.30) In this frame (RF1), the initial state is ρtherm e,n (0) = ρtherm e (0) ⊗ ρn(0) = 1 2 1 0 0 1   ⊗ 1 2  1 e−iΦ0 eiΦ0 1   (6.31) where the nuclear spin in on the equator of its Bloch sphere at some angle Φ0 relative to x and the electron spin is in a thermal mixed state. We then ap- ply U(t) to our initial state and look at the coherence of the matrix element ⟨0,+1|ρtherm e,n (t)|0, 0⟩. ⟨0,+1|ρtherm e,n (t)|0, 0⟩ = α∗ exp [ −i(Φ0 − ωe + A∥ 2 t) ] . (6.32) The complex phase Φ(t) of α∗ is determined by the relation tanΦ(t) = ω Ω tan ( Ωt 2 ) (6.33) so the total phase ϕ in the matrix element is ϕ = Φ0 + Φ(t) − ωe + A∥ 2 t. (6.34) We can calculate the time derivative of Φ(t) by implicit differentiation ∂Φ(t) ∂t = 1 2 Ω2ω Ω2 cos2(Ωt 2 ) + ω2 sin2(Ωt 2 ) . (6.35) 105 Finally, the phase precession velocity is ∂ϕ ∂t = 1 2  Ω2ω Ω2 cos2(Ωt 2 ) + ω2 sin2(Ωt 2 ) − ωe − A∥  . (6.36) To get the average phase, ⟨∆ϕ⟩, gained in a single excitation, we assume an average ES lifetime, T , of 10 ns. ⟨∆ϕ⟩ = ∫ ∞ 0 ∫ t 0 1 T e− t T ∂ϕ ∂t′ dt′ dt. (6.37) The phase susceptibility χϕ is given by χϕ = ⟨∆ϕ⟩ T . (6.38) We show the result of this qualitative model in Figure 6.11. The key aspect is that it captures the antisymmetric behavior of the phase susceptibility about the ESLAC. This model is not exact because we have not included the complete Lind- blad dynamics involving the other 17 available states. For example, the electron polarization process will also create an effective field through the hyperfine cou- pling in the ground state as this process does not conserve the electron spin pro- jection. This results in additional phase precession of the transverse component of the nuclear spin. However, since the electron polarization process is essen- tially independent of the field, we expect that this process would appear as an constant offset in the susceptibility. In addition, to work with a tractable Hamil- tonian, we ignored the |0,−1⟩ to |−1, 0⟩ spin flip-flop, which could contribute to the susceptibility. This model also predicts the linear relationship between the phase suscepti- bility with respect to time. We can show that as long as the coherences in the electron spin are small (i.e the electron spin remains sufficiently mixed during 106 the optical pumping), the phase precession velocity is independent of the elec- tron spin polarization. This results in a time-independent phase susceptibility, which is what we observe in our measurements [Figure 6.4(b)]. 6.6 Conclusion In summary, we investigate a readout protocol for coherent nuclear spins be- yond the NV electron T1, which involves repumping the mixed electron ther- mal state prior to a CNOT readout. We test this protocol on a simulated rotation sensing measurement beyond the NV electron T1. We find that the sensitivity of our readout protocol is best near the ESLAC at around 500 G, despite the fact that this is where the ES spin flip-flops are the most significant. There are various ways to improve our optical repump protocol. One method is to work with higher fidelity microwave pulses for the CNOT read- out (i.e. working with microwave structures that minimize sources of ensemble inhomogeneity) and to minimize sources of inhomogeneous broadening of the electron spin transition. A metric for these is the T ∗2,e coherence time for the NV center’s electron spin. This requires further work on determining NV cen- ter ensemble densities that balance the sensitivity loss from a shorter T ∗2,e (lim- ited by dipolar interactions between electron spins) with the √ N enhancement in sensitivity due to the increase in the ensemble size. This would require di- amond growth/processing engineering to minimize the influence of the spin bath, which is the largest contributor to the inhomogeneous broadening. We can also improve the sensitivity of by changing our choice of CNOT read- out. For example, using a CNOT readout from |0, 0⟩ to |+1, 0⟩ would reduce the 107 sensitivity loss since both states have contrast enhancement near the ESLAC. We did not use this specific CNOT readout in our experiment, since this makes it difficult to separate the effects of the CNOT readout and contrast enhancement. Inertial sensors, such as gyroscopes, can in principle be built out of any spin, including other solid-state defect systems. Our artificial thermal electron state preparation protocol can be applied to test readout protocols for defects with native nuclear spins that strongly hyperfine coupled to nearby defect spins. Ex- amples of these are the Group-IV vacancy defects in diamond [95, 96]. For NV center gyroscopes, the most relevant context to extend our technique to is to- wards 15N nuclear spins. Unlike 14N, this isotope has no quadrupolar moment as it is spin-1/2. The quadrupolar moment of 14N has a temperature depen- dence, which allows laser heating of the diamond to be an additional source of decoherence [71, 97]. 6.7 Acknowledgements We thank Brendan McCullian, Anthony D’Addario, Ozan Erturk, and Sunil Bhave for helpful insights and discussions. This work was supported by the DOE Office of Science (National Quantum Information Science Research Centers) and the DARPA DRINQS program (Cooperative Agreement No. D18AC00024). Device fabrication was done at the Cornell Nanoscale Facility, a member of the National Nanotechnology Coordinated Infrastructure (NNCI), which is supported by the National Science Foundation (Grant No. NNCI- 2025233) and the Cornell Center for Materials Research Shared Facilities, which is supported through the NSF MRSEC program (Grant No. DMR-1719875). 108 6.8 Supplementary Information 6.8.1 Nuclear Polarization from 200-800 G In our measurements, we use the ESLAC to polarize the nuclear spin to mI = +1. While this is a straightforward method for nuclear polarization, it is less effec- tive for fields farther away from the ESLAC [86, 87, 85]. In addition, polarization depends on the excitation power and initialization time. Imperfect polarization of the nuclear spin contributes to various features in our data, such as the loss in CNOT visibility at fields away from the ESLAC. To get a qualitative understanding the impact that imperfect nuclear polar- ization has in our measurements, we measure the nuclear spin populations with pulsed ODMR measurements of the electron spin at each field. The measured ODMR is fit to a sum of three Lorentzians, where we constrain the hyperfine splitting between the transitions. We obtain the intensity of each hyperfine tran- sition (I0, I+1, and I−1) by integrating the corresponding Lorentzians. We define the polarization as P = 1 − 3 2 ( I0 + I−1 I0 + I−1 + I+1 ) (6.39) where P = 1 corresponds to perfect polarization in mI = +1 and P = 0 corre- sponds to the nuclear spin being completely unpolarized. We show the nuclear spin polarization from 200-800 G in Figure 6.12. Far away from the ESLAC (at 200 and 800 G), the polarization is approximately 50%. We obtain the maximum polarization near the ESLAC at around 85%. Away from the ESLAC, the nuclear polarization produced by a rate model (e.g. Ref [80, 81, 82, 83, 84]) has a strong dependence on the branching ratio in 109 Figure 6.12: Nuclear polarization into mI = +1 at the fields used in this work. The polarization is calculated according to Eqn. 6.39, where 0% corresponds to an unpolarized ensemble and 100% corresponds to a perfectly polarized ensemble. the ES (Γ2,3/Γ4) and the GS (Γ7/Γ5,6) since the average spin flip probability in the ES is very low (Figure 5.5). This does not affect the qualitative features of the trends presented in this chapter (e.g. Figure 6.8), but makes direct quantitative comparisons between our Lindblad calculations and data difficult due to the discrepancy in the nuclear polarization present at these fields (Lindblad (200 G) – ∼12%, Exp (200 G) – ∼50%). Therefore, we restrict our quantitative compar- ison (Figure 6.10) between our Lindblad simulation and data to fields close to the ESLAC where there is good agreement in the amount nuclear polarization between our simulation and measurements (Lindblad (400 G) – ∼77%, Exp (400 G) – 86%). 110 CHAPTER 7 CONCLUSION In summary, nuclear spin gyroscopes made from NV center electron spins are an emerging technology that will require several innovations to achieve sen- sitivities that are competitive with conventional gyroscopes (MEMS and ring gyros). We have discussed in this thesis a potential way to overcome a fun- damental limitation– decoherence induced by the telegraphing of the strongly hyperfine coupled electron spin – to the sensitivity of an nuclear based NV cen- ter gyroscope which involves use pulses to drive the double quantum transition of the NV center’s electron spin to achieve dynamical decoupling of the electron and nuclear spins. This, theoretically, can enhance the phase coherence time T ∗2,n past the electron T1, which will improve the sensitivity proportional to 1/ √ T ∗2,n. We presented a MEMS device geometry and fabrication that can achieve this driving. Additionally, we created and tested a protocol to read out these nuclear sen- sors beyond the electron T1. This protocol involves using an optical repump pulse to regain some electron polarization of the NV center’s electron spin be- fore applying a CNOT pulse for optical readout. This readout protocol affects the sensitivity as it affects the visibility of the observed Ransey fringes. The key lesson from this experiment is that sensitivity loss from repumping the electron spin is not extreme enough to preclude the prospect of creating a nuclear based NY center gyroscope and can be in principle be compensated a large enough enhancement in T ∗2,n. The next step to advance this technology will involve experimentally observ- ing the enhancement from the MEMS driving in an NV gyroscope. Currently 111 this is not possible because decoherence inducted by the electron-nuclear hy- perfine interaction is not the dominant decoherence channel. The dominant decoherence channel is in fact due to thermal fluctuations which causes shifts in the quadrupolar coupling of the nuclear spin. There has been work to overcome this and other sources of decoherence through dynamical decoupling pulse se- quences, which allow T ∗2,n to begin to approach the electron T1 [72, 98]. This additional dynamical decoupling will need to be integrated with the MEMS platform we introduced in this thesis. One final avenue to observe decoupling using our MEMS device is to en- gineer our system so that the electron T1 is the dominant form of nuclear spin decoherence in our system (i.e. outcompetes or is competitive with thermally induced decoherence). The simplest way to achieve this would be to use a mag- netic material to decrease the electron T1. This would come with the challenge of identifying the correct magnetic material and integrating it with the FBAR device. Alternatively, as in [56], we can artificially engineer T1 processes with a series of π pulses on the electron spin’s single quantum transition. Further work would be necessary to determine the conditions that would allow this to successfully mimic a naturally induced T1 process. 112 CHAPTER 8 APPENDIX 8.1 Experimental Setup The measurement setup used for the experiments with NV centers in this thesis consists of three parts: a home-built confocal microscope to excite and collect PL from NV centers, a microwave (MW)/ radio-frequency (RF) line for mag- netic and acoustic control of the NV center’s electron and nuclear spins, and a RF switch network to route photon counts for data collection. All instruments are controlled using a LabView Virtual Instrument (VI) while the waveforms outputted are written using a Python script. 8.1.1 Confocal Microscopy Setup A comprehensive schematic of the confocal microscope setup is presented in Figure 8.1. We will outline its key components. A 532 nm laser is used to excite the NV center. The laser mode is first focused into an acoustic-optic modulator (AOM), which allows for fast on/off switching of the laser intensity with is done using TTL logic. The TTL of the AOM’s driver is controlled by a digital delay generator, which allows us to create optical pulses with rise times of approxi- mately 40 ns. This laser mode passes through a 550 nm dichoric mirror, which reflects the green excitation and transmits the red PL. This dichoric is essential for separating the excitation of the defect from the collection of the PL. After the dichoric, the excitation laser is reflected off a fast steering mirror (FSM) and then through a 4f correlator before it is focused by a microscope objective onto 113 the diamond. The FSM and 4f correlator allows us to raster the excitation across the diamond to create images of the PL from the NV centers. The microscope objective sits of a piezoelectric stage, which allows us to focus into the diamond at different depths. The PL collection begins at the NV centers where the light is collected by the microscope objective and travels back to the dichoric. The dichoric allows the red PL emission from the phonon sideband of the NV center. An additional 675 nm low pass filter is introduced after the dichoric to suppress any spurious green excitation light. The PL is then focused through a 30 µm pinhole, which acts as a spatial filter to restrict the PL collection to the objective focus. After the pinhole, the PL is then focused into a multimode fiber where it is routed to a sin- gle photon avalanche photodiode (APD). This APD saturates at count rates of approximately 15 MHz so prior to the pinhole, we have a neutral density filter to cut down counts as needed for NV ensemble measurements to prevent over- saturation of the detector. Following the APD, the photon counts are converted to voltage pulses that are routed through an RF switch network. 114 Figure 8.1: Schematic of confocal microscope setup used for data collec- tion. Green indicates the excitation path and red indicates the collection path. 115 8.1.2 Microwave Electronics and RF Switch Network In our experiments, we often work with sequences of optical and magnetic pulses to coherently control the electron and nuclear spins of the NV center. We will classify these pulses into two groups. Microwave (MW) pulses are pulses whose carrier frequency range from approximately 1 to 2 GHz. Radio- frequency (RF) pulses are pulses whose carrier frequencies range from approx- imately 300 kHz to 10 MHz. The pulse sequences require precise temporal con- trol over the output of various electronics, which is achieved by using an Tek- tronixs Arbitrary Waveform Generator (AWG7122C). The 10 MHz reference of the AWG is used as the master clock for the timing of the pulse sequences. The AWG has two modes of operation. In the first (normal operation), the wave- forms produced by the AWG, sampled up to 12 GS/s, are output from the ana- log channels (Ch1 analog and Ch2 analog). In the second (interleaved opera- tion), the outputs of the two analog outputs are interleaved into a single output to create waveforms sampled up to 24 GS/s. In addition, there are two marker outputs (Mrk1 and Mrk2) per channel. There are two separate configurations for this setup [Figure 8.2(a)]. Each configuration can be switched using series of mechanical switches which are controlled via a TTL signal produced by a programmable power source. The first configuration is used for the measurements for mechanically driving spins using bulk acoustic resonators described in Chapter 4. The second configuration is used for nuclear spin manipulation and testing a readout protocol for nuclear spins past the electron T1 described in Chapter 6. In the first configuration, an signal generator (SRS SG386) other than the AWG is used to produce MW pulses to manipulate the electron spin. The two 116 analog outputs of the AWG are connected to the SG386 to generate the MW pulses through IQ modulation. The AWG’s marker outputs are used to trigger other electronics during the pulse sequence. Ch 2 Mrk2 is used to amplitude modulate the output of a separate signal generator (SG384) that is used to drive the bulk acoustic resonators. Channel 2 Mrk1 is used as a trigger for the digital delay generator (DDG). The DDG (SRS DG645) is used to create TTL signals for modulating the laser intensity using the acoustic optical modulator (AOM). In addition, two DDG delays are used for routing the photon counts from the single photon avalanche photodiode (SPAPD) through an RF switch network. The RF switch network [Figure 8.2(b)] consists of two solid state RF switches that are controlled using TTL signals from the DDG (Delay EF and Delay GH). When both TTL signals are low, the pulses from the SPAPD are routed to the PFI0 channel of the DAQ. This channel is used to count photon counts for non- pulsed measurements (e.g. 2D scans of the diamond). If only Delay GH is high, then the photon counts are routed to the PFI1 channel, which is the normaliza- tion (norm) counting window. This counting window is typically a 1 µs window at the end of the optical initialization of the NV electronic spin. If only Delay EF is high, then the photon counts are routed to the signal counting window, which corresponds to the readout of the counts after all the spin manipulation is finished. Typically the quantity we plot in our data is derived the ratio of the signal to norm counts. Looking at this ratio allows us to reject laser noise whose characteristic frequencies are smaller than the repetition rate of our pulse sequence (i.e. acting as a high pass noise filter). The second configuration shares much of the same structure as the first con- figuration [Figure 8.2(a)]. The main difference is that the MW pulses used for 117 electron spin control are generated directly by the AWG’s interleave output. This configuration is important for sequences where control of the phase of the MW pulses are important (e.g. quantum state tomography). In addition, Ch1 Mrk1 is used to trigger an arbitrary function generator (AFG3102C). The AFG has two channels. The first channel is used to synthesize RF pulses for nuclear spin control. The second channel is used to provide the TTL signal to modu- late the laser intensity with the AOM. For both configurations, the MW pulses and RF pulses are combined using a frequency diplexer and are routed to the antenna fabricated on our diamond substrate. 118 AWG7122C Interleave Ch1 Analog Ch1 Mrk1 Ch1 Mrk2 Ch2 Analog Ch2 Mrk1 Ch2 Mrk2 Input 1 Input 2 TTL Mechanical Switch 50 Ω Term Output AFG3102 Ch1 Ch2 Ext Trig DG645 Ext Trig Delay AB Delay CD Delay EF Delay GH SG386 I (Mod) Q (Mod) Output SG384 I (Mod) Output Input 1 Input 2 TTL Mechanical Switch 50 Ω Term Output Frequency Diplexer RF IN MW IN Output Diamond Sample To Antenna To FBAR Acoustic Optical Modulator (ISOMET) RF Switch Network Power Supply Out 1 Out 2 InputTTL RF Switch Output 2 Output 1 Single Photon APD Output PFI0 InputTTL RF Switch Output 2 Output 1 PFI1 (Norm) PFI2 (Signal) DG645 EF DG645 GH a) b) Figure 8.2: a) Schematic of MW and RF lines used for magnetic and acous- tic spin control. b) RF switch network used for routing single photon counts. 119 8.2 Quantum State Tomography A fundamental problem in quantum information is developing quantum gates that robustly initialize and control qubit states. It is important to quantify how well these gates perform, which requires the experimenter know the initial state of the qubit before the gate operation and the final state of the qubit after the operation. The goal of quantum state tomography is to infer the this informa- tion experimentally. This will allow us to compare the final state after the gate operation to the ideal output that we are targeting, which in principle will give us some insight into any physical mechanisms in the gate that may be produc- ing errors. In the context of this thesis, the gate that is studied is the gate that corresponds to generating an thermal electron state ρtherm from the initial state |0⟩. Because the goal of state tomography is to characterize the performance of quantum gates, this requires us to quantify how close our output state is to the target state. The (psuedo-)metric we use is the fidelity F [76]. For states ρ0 and ρ1, this is defined as F(ρ0, ρ1) = Tr [√ ρ1/2 1 ρ0ρ 1/2 1 ] (8.1) Note that the definition of fidelity is symmetric with respect to ρ0 and ρ1, that is F(ρ0, ρ1) = F(ρ1, ρ0). The square of the fidelity F2 can be thought roughly as the overlap between ρ0 and ρ1 (i.e. the probability that the two states give the same measurable result). 120 8.2.1 Spin-1/2 Case We will first introduce state tomography through a spin-1/2 system [76]. In general, a spin-1/2 density matrix can be parameterized as ρ = 1 3 I + a1σx + a2σy + a3σy (8.2) where I is the identity operator and σx, σy, and σz are the Pauli operators. We will refer to this set of four operators as the Pauli basis. This representation ensures that ρ is unit trace and is semi-positive definite. The goal is to determine the coefficients a1, a2, and a3. Note that the elements in the Pauli basis σi follow the orthogonality condition Tr[σiσ j] = 2δi j (8.3) where δi j is the Kronecker delta. Using this condition, we find that a1 = 1 2 ⟨σx⟩ = ⟨S x⟩ (8.4) a2 = 1 2 ⟨σy⟩ = ⟨S y⟩ (8.5) a3 = 1 2 ⟨σz⟩ = ⟨S z⟩ (8.6) where ⟨σi⟩ and ⟨S i⟩ are the corresponding expectation values. Thus we see that measuring the state ρ only requires the measurement of the expectation values of the spin components. Usually, we are restricted to measurements along Z since we work in the eigenbasis of S z. This makes measuring ⟨S z⟩ straightforward, but creates addi- tional work to determine ⟨S x⟩ and ⟨S y⟩. We need to rotate the corresponding spin component to be along z to measure it. This requires the use of MW pulses 121 that are resonant with the transition between the two eigenstates. This implic- itly means that we are measuring the spin in the rotating frame, that is rotating at the Larmor frequency of the spin. In the rotating frame, a rotation R by θ about an axis on the xy plane of the Bloch sphere is represented as Rϕ(θ) =  cos ( θ 2 ) −ie−iϕ sin ( θ 2 ) −ieiϕ sin ( θ 2 ) cos ( θ 2 )  (8.7) where ϕ is the angle between the rotation axis and X on the xy plane. We will use the convention that X corresponds to ϕ = 0, Y corresponds to ϕ = π/2, −X corresponds to ϕ = π, and −Y corresponds to ϕ = 3π/2. With this, we see that we rotate ⟨S x⟩ to the Z axis of the Bloch sphere using a MW pulse that corresponds to a π/2 rotation about −Y . Similarly for ⟨S y⟩, we use a MW pulse corresponding to a π/2 rotation about X. 8.2.2 Spin-1 Case Now we expand our previous discussion to spin-1 density matrices, which we will refer to as a qutrit state. Theoretical descriptions of spin-1 and higher spin tomography can be found in [88, 89], which we summarize in this section. In- tuitively spin-1 state tomography should be similar to the spin-1/2 case since we can ”decompose” the spin-1 space into three different qubit states ({0,+1}, {0,-1}, and {+1.-1}). Note that we are not making the statement that the qutrit state can be written as three independent Bloch spheres, since the individual subspaces are constrained by the unit trace and semi-positive definite proper- ties of the complete qutrit density matrix. However this still gives us a useful 122 guide to visualize what is happening in the qutrit case. Just like in the spin-1/2 case, we parameterize our qutrit state as ρ = 1 3 I + 8∑ i=1 aiλi (8.8) where I is the identity operator and λi are trace-less spin operators known as the Gell-Mann matrices. These are given by λ1 =  0 1 0 1 0 0 0 0 0  λ2 =  0 −i 0 i 0 0 0 0 0  λ3 =  1 0 0 0 −1 0 0 0 0  λ4 =  0 0 1 0 0 0 1 0 0  λ5 =  0 0 −i 0 0 0 i 0 0  λ6 =  0 0 0 0 0 1 0 1 0  λ7 =  0 0 0 0 0 −i 0 i 0  λ8 = 1 √ 3  1 0 0 0 1 0 0 0 −2  (8.9) This parametrization of the qutrit state guarantees that the state has unit trace and is semi-positive definite. These operators follow the orthogonality condi- tion Tr[λiλ j] = 3δi j (8.10) where δi j is the Kronecker delta. From this, we find ai = 1 3 ⟨λi⟩ (8.11) 123 Therefore the problem of finding the qutrit state depends on measuring eight different expectation values λi, analogous to the spin-1/2 case. In general for a spin-n system, the density matrix can be parameterized using the identity I and the generators of SU(n) (group of special unitary n × n matrices). Measuring the expectation values λi requires us to rotate the spin compo- nents such that λi is diagonal after the rotation. To accomplish this, we use MW pulses to manipulate spin transitions between m = 0 and m = ±1. This implicitly means that we are working in the interaction frame, where the individual qubit subspaces are represented by rotating frames rotating at their respective Larmor frequencies. Therefore rotations in the {+1,0} subspace are represented by R+ϕ(θ) =  cos ( θ 2 ) −ieiϕ sin ( θ 2 ) 0 −ie−iϕ sin ( θ 2 ) cos ( θ 2 ) 0 0 0 1  (8.12) and rotations in the {-1,0} subspace are represented by R−ϕ(θ) =  1 0 0 0 cos ( θ 2 ) −ieiϕ sin ( θ 2 ) 0 −ie−iϕ sin ( θ 2 ) cos ( θ 2 )  (8.13) These rotations depend on the rotating wave approximation (RWA), which make them valid only in the regime where the spin transitions are detuned from each other by an amount that is significantly larger than the Rabi oscillation frequency of the applied field. As an example, we show the rotations needed for λ4. We first apply the 124 rotation R−X(π) (i.e. a π pulse in the {-1, 0} subspace). This gives R−X(π)λ4R−†X (π) =  1 0 0 0 0 −i 0 −i 0   0 0 1 0 0 0 1 0 0   1 0 0 0 0 i 0 i 0  (8.14) =  0 i 0 −i 0 0 0 0 0  (8.15) = −λ2 (8.16) Next we apply a rotation corresponding to R+ −X(π2 ) (i.e. a π/2 rotation about −X). This gives R+−X ( π 2 ) (−λ2)R+† −X ( π 2 ) =  1 √ 2 i √ 2 0 i √ 2 1 √ 2 0 0 0 0   0 i 0 −i 0 0 0 0 0   1 √ 2 − i √ 2 0 − i √ 2 1 √ 2 0 0 0 0  (8.17) = λ3 (8.18) Now that the operator is diagonal, we can measure the expectation value (i.e. through measuring the occupations in m = +1, 0,−1). One important detail from the above example is that the X direction for the {+1, 0} subspace does not coincide with the X direction for the {0, -1} subspace. This is because the two qubit subspaces have different Larmor frequencies so their respective coordinate systems are rotating at different rates relative to the laboratory frame. Therefore, it is important to be choose phases for the MW pulses that are consistent with each rotating frame. The rotations for all eight operators are listed in Chapter 6 [Figure 6.2(b)]. 125 8.3 Quantum Process Tomography Quantum process tomography is a technique to quantitatively characterize the performance of quantum processes such as quantum gates. It is useful because it can provide a metric for how closely gates perform to ideal performance and can help identify and evaluate different sources of decoherence. The basic idea behind process tomography is the same as state tomography, except we are de- termining a process matrix χ that describes the quantum process, rather than a density matrix ρ. A complete treatment of quantum process tomography can be found in Ref [76]. We will describe the general procedure first and then go over the specific application, which is used this thesis. 8.3.1 General Theory In process tomography, we treat our quantum process as a black box, where details about the internal dynamics are not necessarily known. For example, we can perform tomography on a CNOT gate without known how the CNOT gate is physically implemented. We will denote the quantum process as ε(ρ). Since ε(ρ) describes evolution of quantum states, it satisfies the following properties: • ε(ρ) is a trace-preserving superoperator: Tr[ρ] = Tr[ε(ρ)] • ε(ρ) is a linear superoperator of ρ. That is ε(cρ) = cε(ρ) (8.19) ε(ρ1 + ρ2) = ε(ρ1) + ε(ρ2) (8.20) where c is a complex scalar. 126 ε(ρ) has a representation called the Kraus representation. There exist operators Ki, called the Kraus operators, such that the quantum process can be expressed as ε(ρ) = ∑ i KiρK†i (8.21) where the Kraus operators satisfy a completeness relation∑ i K†i Ki = I (8.22) The Kraus operators can be expressed in terms of an operator basis {Ei} (e.g. the Pauli basis). That is, Ki = ∑ j ei jE j (8.23) Then ε(ρ) can be written as ε(ρ) = ∑ m,n eme∗nEmρE†n = ∑ m,n χmnE†mρEn (8.24) where χmn is the process matrix, which contains all the information about the quantum process. Determining χmn is the central problem behind process to- mography. To experimentally characterize a quantum process, we need to put in known states and look at the output (which requires state tomography). Let {ρi} be a linearly-independent basis for the density matrix states. Then the output state can be expressed as ε(ρ j) = ∑ k λ jkρk (8.25) We can express the action of the operator basis on the a basis element ρi as Emρ jE†n = ∑ k βmn jk ρk (8.26) Combining these gives ε(ρ j) = ∑ k ∑ mn χmnβ mn jk ρk = ∑ k λ jkρk (8.27) 127 From which we conclude ∑ mn χmnβ mn jk = λ jk (8.28) Let κ be the generalized inverse of β satisfying βmn jk = ∑ st,xy βmn st κ st xyβ xy jk (8.29) Then the solution to Eq 8.28 is χmn = ∑ jk κmn jk λ jk (8.30) Therefore, by experimentally measuring λ and calculating β, we can determine the process matrix χ. 8.3.2 Process Tomography of One Qubit Processes As a concrete example, we will apply the general theory in the previous section on a one qubit gate. This procedure is what is used in the process tomogra- phy calculation in Chapter 6. As a simple example, we will calculate the pro- cess matrix χ associated with the Hadamard gate. In the computational basis, a Hadamard gate A has the following matrix representation A = 1 √ 2 1 1 1 −1  (8.31) We pick our density matrix basis to be the states ρ1 = 1 0 0 0  ρ2 = 0 0 1 0  ρ3 = 0 1 0 0  ρ4 = 0 0 0 1  (8.32) which follows the orthogonality condition Tr[ρiρ † j] = δi j (8.33) 128 After applying the Hadamard gate, the output states are ε(ρ1) = 1 2 1 1 1 1  ε(ρ2) = 1 2  1 1 −1 −1  ε(ρ3) = 1 2 1 −1 1 −1  ε(ρ4) = 1 2  1 −1 −1 1  (8.34) The matrix λ can be determined by measuring the output states of the Hadamard gate. Using the orthogonality of the density matrix basis, we have λ jk = Tr[ε(ρ j)ρ † k] (8.35) This gives λ = 1 2  1 1 1 1 1 −1 1 −1 1 1 −1 −1 1 −1 −1 1  (8.36) The next step is to calculate β. For this, our operator basis will be the Pauli basis σ0 = 1 0 0 1  σ1 = 0 1 1 0  σ2 = 0 −i i 0  σ3 = 1 0 0 −1  (8.37) To calculate β, we use the orthogonality condition again βmn jk = Tr[σmρ jσnρ † k] (8.38) 129 This gives a 16 by 16 matrix β =  1 0 0 1 0 0 0 0 0 0 0 0 1 0 0 1 0 1 i 0 0 0 0 0 0 0 0 0 0 1 i 0 0 0 0 0 1 0 0 1 −i 0 0 −i 0 0 0 0 0 0 0 0 0 1 i 0 0 −i 1 0 0 0 0 0 0 1 −i 0 0 0 0 0 0 0 0 0 0 1 −i 0 1 0 0 −1 0 0 0 0 0 0 0 0 1 0 0 −1 0 0 0 0 0 1 −i 0 0 −i −1 0 0 0 0 0 0 0 0 0 1 0 0 −1 −i 0 0 i 0 0 0 0 0 0 0 0 1 0 0 1 i 0 0 i 0 0 0 0 0 0 0 0 0 1 i 0 0 i −1 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 −1 0 0 −1 0 1 i 0 0 0 0 0 0 0 0 0 0 −1 −i 0 0 0 0 0 0 1 −i 0 0 i 1 0 0 0 0 0 0 0 0 0 1 0 0 −1 i 0 0 −i 0 0 0 0 0 1 −i 0 0 0 0 0 0 0 0 0 0 −1 i 0 1 0 0 −1 0 0 0 0 0 0 0 0 −1 0 0 1  (8.39) This matrix representation for β can be used whenever the Pauli basis is used for process tomography. The columns are labeled by mn and the rows are labeled by jk. The inverse of this matrix, κ, exists and be calculated computationally. To get a matrix equation that we can solve for χ, we can define the vector- ization of λ. To give an example of the vectorization process, for a 2 by 2 matrix M, M = a b c d  (8.40) 130 we will define its vectorization to be M⃗ =  a b c d  (8.41) which is equivalent to concatenating the rows of M into one large row vector and taking the transpose. Then to find χ, we just need to evaluate the matrix equation χ⃗ = κλ⃗ (8.42) Finally, we can obtain χ by undoing the vectorization. For the Hadamard gate, this gives χHadamard = 1 2  0 0 0 0 0 1 0 1 0 0 0 0 0 1 0 1  (8.43) This means that the Hadamard gate can be expressed as the following operator sum in the Pauli basis ε(ρ) = 1 2 (σxρσx + σxρσz + σzρσx + σzρσz) (8.44) To find the Kraus operators Ki, we can diagonalize and find the eigenvectors of χ, which will give us the Kraus representation. In this example, there is only one eigenvector with a nonzero eigenvalue (λ = 1) K = 1 √ 2 (σx + σz) = A (8.45) Unsurprisingly, the Kraus operator of a unitary one qubit gate is itself as the state transforms as AρA† after applying the gate, which is already in the Kraus 131 representation form. ε(ρ) = AρA† (8.46) = 1 2 (σx + σz)ρ(σx + σz) (8.47) 8.4 AlN Deposition and Processing We will outline the deposition and processing (i.e. etching) of the AlN films used to fabricate the FBAR devices used in this thesis. The standard operat- ing procedures used at the Cornell Nanoscale Facility (CNF) are outlined in the subsequent sections. 8.4.1 Sputtering and Characterization The AlN films in this thesis were sputtered using the OEM Endeavor M1 at CNF. This sputter system uses a reactive sputter with an aluminum target to create polycrystalline films. The films that are deposited in this work are ap- proximately 1.5 µm thick. This tool is designed for depositing films on Si wafers, which are the typical substrates for MEMS devices. However, we work with 3 mm by 3 mm diamond pieces from Element Six, which is not compatible with the tool. To deposit AlN onto these small diamond pieces we used an anodized alu- minum chuck that can hold small pieces up to 5 mm by 5 mm. We are able to deposit onto three 3 mm by 3 mm samples simultaneously with our chuck. Af- ter deposition, the backside of the chuck (which faces the target) will be covered 132 a) b) c) d) AlN (0002) Pt (111) Si Diamond Figure 8.3: a) AlN chuck used to sputter on small diamond pieces. b) Di- amond membrane mounted in chuck. c) AlN chuck in the de- position chamber of the OEM Endeavor M1 tool at the Cornell Nanoscale Facility. d) XRD measurement of AlN deposited on a Si wafer and a 3 mm by 3 mm CVD diamond piece. in AlN. To prevent excess AlN from accumulating and flaking off our chuck onto the target between depositions, the chuck is sandblasted between deposi- tions as necessary. Additionally, the deposition process heats up the aluminum significantly, which causes warping over time. To correct for this, the chuck is hammered back down to a flat state when the warp is significant. The standard operating procedure is as follows. The diamond sample is 133 cleaned in acetone and IPA. After this is done, the chuck is cleaned in IPA along with a spacer 3 by 3 mm Si pieces. An Si witness pieces is also cleaned as well. Following this the diamond is placed into the chuck followed by the Si spacer. A 3” wafer then covers the chuck so that there are no exposed gaps during the deposition.After the deposition, we check the orientation of the AlN film by performing a θ − 2θ X-ray diffraction scan. We look for the diffraction peak that corresponds to the (0002) orientation to confirm that the film is appropriate for fabrication. A diffraction measurement showing the correct AlN alignment is shown in Figure 8.4. Future improvements to this deposition will involve two aspects. The first is optimizing the recipe for diamond growth. The current recipes for AlN and the Ti/Pt layer that the piezeoelectric is deposited on is developed by the Lal group at Cornell in collaboration with CNF and is primarily used for 4” and larger Si wafers. Due to the differences in the thermal expansion coefficients between the two substrates (Diamond (500 K): ∼ 2 × 10−6), Si (500 K): ∼ 4 × 10−6, a recipe that produces stress-free films on Si does not necessarily create a stress-free film on diamond [99, 100]. The second improvement will come from improving the consistency of the film across depositions. Currently there is a wide variation in film performance (i.e. some devices are not well modeled by a mBVD or Mason model of the MEMS transducer). Again this could come from temperature variations of the diamond between different depositions as each diamond chip may be thermally anchored to the chuck differently depending on its placement and/or the cleanliness of the chuck [Figure 8.3(b)]. 134 a) b) TMAH (MIF 726) c) Figure 8.4: a) Patterned etch window after the hard bake. b) Etch window with the exposed bottom Pt layer (white). The undercut at the edge of the etch window is visible. c) Etch depth versus etch time with TMAH developer (MIF726). 8.4.2 Etching For our devices, AlN needs to be etched in order to expose the bottom electrode for wire bonding. In general, AlN etching is accomplished by introducing an etchant to the film to oxide the aluminum in the film [101]. At Purdue, this is accomplished using a recipe that involves hot phosphoric acid. At Cornell, we use TMAH/KOH based developers as the etchant. KOH developers have a 135 drastically faster etch rate compared to TMAH so this is only used if fine control over the etch is not needed (e.g. to stripping off the entire film). For etching a window to expose the bottom electrode of bulk acoustic resonators, we use TMAH based developers. The standard operating procedure for etching is as follows. An etch window is defined lithographically with photoresist (Shipley SC1827) that is baked at 115oC for 1 minute. 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