QUANTUM RELAXOMETRIC SPECTROSCOPY USING THE NITROGEN-VACANCY CENTRE IN DIAMOND

Transcription

1 QUANTUM RELAXOMETRIC SPECTROSCOPY USING THE NITROGEN-VACANCY CENTRE IN DIAMOND By James David Antony Wood SUBMITTED IN TOTAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY SCHOOL OF PHYSICS, THE UNIVERSITY OF MELBOURNE AUSTRALIA FEBRUARY 2018

2 DECLARATION OF ORIGINALITY This is to certify that 1. The thesis comprises only their original work towards the degree of doctor of philosophy except where indicated in the statement of contributions, 2. due acknowledgement has been made in the text to all other material used, 3. the thesis is fewer than 10,000 words in length, exclusive of tables, maps, bibliographies and appendices. I authorize the Head of the School of Physics to make or have made a copy of this thesis to any person judged to have an acceptable reason for access to the information, i.e., for research, study or instruction.

3 Abstract This thesis investigates the use of the Nitrogen Vacancy (NV) centre in diamond as a novel sensor for magnetic resonance spectroscopy on the nanoscale. The need for novel nanoscale sensors of nuclear spins, for the application of nuclear magnetic resonance (NMR), and electron spins, for the application electron paramagnetic resonance (EPR), has been well established in recent decades. The nanoscale resolution, magnetic sensitivity, room temperature operation and optical initialisation and readout of the NV centre has led to it being of intense interest for this purpose. The biological compatibility of diamond strengthens the utility of the NV centre for interesting EPR and NMR applications. Existing techniques for EPR and NMR with the NV centre have focused on dynamically controlling the NV spin, and at times the environmental target spin, in order to match the frequency selectivity of the NV centre to the environmental spin species. However this introduces both physical and technical complexities on the measurement which can limit the applicability to certain, highly interesting applications. While microwave control of the NV centre looks a promising technology in the highly controlled lab environment for targetting applications such as protein structure determination, the 3

4 requirement for control of both the static magnetic field and the strength of the microwave driving field are significant difficulties for more complex environments, such as potential in-vitro or in-vivo applications. This thesis is a detailed investigation of a new method of nanoscale EPR and NMR with the NV centre where we apply a precise static magnetic field to tune the NV into resonance with the environmental spin species. We can then measure this resonance via the reduced axial relaxation time, T 1. We investigate the theoretical possibility and then conduct an experimental demonstration of both EPR and NMR in this way at the nanoscale with a single NV centre. Initially, in Chapter 2, we consider the theoretical application of T 1 spectroscopy both to a classical noise signal and also, with a fully quantum treatment, to a single environmental spin. In addition, we detail the experimental setup used subsequently for implementing T 1 spectroscopy. As an initial investigation, in Chapter 3, we map the spectrum of the substitutional Nitrogen defect (P1 centre) within diamond and compare this method to the current state-of-the-art techniques for nanoscale EPR with the NV centre. Furthermore we map the nuclear transitions of the P1 centres and demonstrate how the nuclear spin signal is enhanced by the strongly interacting electron spin of the P1 centre, thus providing a method for hyperfineassisted NMR. After demonstrating our technique on an electronic spin system within the diamond lattice, we look to extend the technique to general nuclear spins external to the diamond surface in Chapter 4. This entails an extensive investigation of the NV centre s behaviour near the ground state level anticrossing (GSLAC) during which we demonstrate all-optical nanoscale AC 4

5 magnetometry down to sub MHz level. Subsequently, we are able to demonstrate all-optical nanoscale NMR on proton spins external to the diamond surface and show that this technique achieves similar sensitivity to current techniques while removing the requirement for complex microwave pulsing of the NV centre. Finally, in Chapter 5, we look at how the ability to directly tune the NV into resonance with environmental spins allows for hyperpolarisation of the surrounding spin bath. Via the same mechanism used for the magnetic resonance spectroscopy studies, we show that the 13 C nuclear spins within the diamond lattice can be polarised with high efficiency via this direct crossrelaxation. This work shows that coupling the NV centre to environmental spins simply via a static magnetic field is possible, opening up a new avenue for future sensing with the Nitrogen-Vacancy centre in diamond. 5

6 Acknowledgements Through my time at the University of Melbourne I have had the good fortune of working with some truly fantastic people. Firstly I would like to acknowledge the guidance and support offered to me by my supervisors, Professor Lloyd Hollenberg and Dr. David Simpson over the course of my thesis. While I am continually amazed by the depth and breadth of Lloyd s scientific knowledge, it is the generosity and patience that he showed towards me that I will recall most fondly. He deserves all of his success due to the care and support he shows those who work with him as much as due to his extensive scientific achievements. David s positive outlook made the day to day work within the group thoroughly enjoyable and his advice was always helpful and constructive. I will be forever grateful for his willingness to help an experimental novice understand the operation of an NV confocal setup. My colleagues within the Hollenberg group deserve special praise for assisting me in so many ways and for making the group a fun and productive place to be. I m lucky enough to have become good friends with many of them. Jean-Philippe Tetienne s arrival in the group really spurred on this project and his passion and enthusiasm for physics is infectious. A particular thanks 6

7 to him for putting me in touch with his PhD supervisor, Vincent Jacques, which allowed me to begin my current European adventure. I am similarly indebted to Liam Hall for providing significant guidance in understanding the theory of T 1 spectroscopy. In addition he gave me valuable life advice by teaching me that tractor mechanic, semi-professional motorbike rider and theoretical physicist are not mutually exclusive careers. Alastair Stacey deserves a lot of credit for education me on the experimental side of this research and assisting in my understanding of diamond-based technologies. David Broadway joining the group similarly helped the project along with his incredible work rate and positive outlook, even if he tried to make our lunch time inhumanely early. Although not directly involved in my PhD project, Charles Hill was always an incredibly helpful, generous person to be able to discuss physics with. I m also greatly indebted to him for co-supervising me in my Masters project where he gave me a thorough grounding in NV physics. Viktor Perunicic was always a pleasure to be around and we had many informative discussions. Perhaps more than anybody in our group, Patricia Gigliuto deserves immense credit for helping me through my PhD. I d hate to think what would ve happened to our group without her assistance in our interactions with the real world. In addition, she has become a dear friend who is always there to talk to in the good times or the bad. On top of these people I would like to acknowledge a number of people who I have worked with and have provided numerous informative discussions: Marcus Doherty, Nikolai Dontschuk, Liam McGuinness, Brant Gibson, Fedor Jelezko and Vincent Jacques. It has been a privilege to get to know and work 7

8 with all of these people. I wouldn t be the person I am today without your input. The final group that deserve huge thanks are my family. To Mum and Dad, thank you for all your love and support through my PhD and indeed throughout my life. To my sister Sarah, thank you for leading by example with your diligence and being such a great sounding board for me from childhood to now. To Eryn and Mike, thank you for welcoming me so generously into your family and offering me your advice on so many aspects of my work and life. Finally and most importantly to Leah. You are a constant source of amazement with your generousity, care and love. For putting up with and loving me for all my quirks and oddities I cannot thank you enough. This would never have been possible without you. 8

9 Statement of Contributions In writing this thesis I have endeavoured to present a coherent, complete story on the development of T 1 spectroscopy and hence there are parts that are not exclusively my work. Throughout the thesis I explicitly state the contributions of others to the work contained within this thesis through appropriate referencing. However I will now state the contributions of my colleagues within the Hollenberg group as their assistance can t be summarised from references to published work alone. The work by Dr. Liam Hall in Ref. [1] is the starting point of T 1 spectroscopy with the NV centre and therefore the theory section of Chapter 2 contains significant overlap with this. In addition, Liam contributed significantly to the calculation of the P1 resonances in Chapter 3, particularly the semi-classical derivation developed in Section 3.3. The development of the experimental setup used throughout this work was greatly assisted by Dr. David Simpson and Dr. Alastair Stacey. Dr. Jean-Philippe Tetienne and Mr. David Broadway joined the T 1 spectroscopy project within the Hollenberg group subsequent to the initial P1-EPR measurements (Sec 3.2). From here on the work was done among a team with the major aim to acheive T 1 -NMR. In particular, the measure- 9

10 ments conducted on the NV centre near the ground state level anti-crossing (GSLAC) presented in Section 4.3, were performed predominantly by David and are presented in Ref. [2]. These measurements are included in detail to give a full picture of the steps taken towards nanoscale T 1 -NMR. In addition, the XY8 measurements (Fig. 4.19), used for means of depth determination were taken by Jean-Phillipe. The work on hyperpolarisation of nuclear spins via this T 1 resonance is briefly discussed in Chapter 5 and is presented in Ref. [3]. While the understanding that this hyperpolarisation mechanism was preventing our attempted sensing of 13 C and very preliminary data showing this was made at the end of my time in the Hollenberg group, the detailed study of this process was done by David and Jean-Phillipe, with theory from Dr. Liam Hall. Indeed the data of 13 C polarisation, presented in Fig. 5.1, was taken by David Broadway. This work is presented in Ref. [3]. Finally, the measurements on environmentally mediated resonance (EMR), presented in the Appendix, were taken by Mr. Scott Lillie and are detailed in Ref. [4]. Dr. Tetienne and I initially observed the behaviour and then concluded as to the mechanism we were witnessing however it was Scott who undertook the detailed study of EMR. 10

11 List of Publications 1. J. D. A. Wood, D. A. Broadway, L. T. Hall, A. Stacey, D. A. Simpson, J.-P. Tetienne, and L. C. L. Hollenberg. Wide-band nanoscale magnetic resonance spectroscopy using quantum relaxation of a single spin in diamond. Physical Review B 94, , D. A. Broadway, J. D. A. Wood, L. T. Hall, A. Stacey, M. Markham, D. A. Simpson, J-P. Tetienne, and L. C. L. Hollenberg, Anticrossing Spin Dynamics of Diamond Nitrogen-Vacancy Centers and All-Optical Low-Frequency Magnetometry. Physical Review Applied 6, , S. E. Lillie, D. A. Broadway, J. D. A. Wood, D. A. Simpson, A. Stacey, J.-P. Tetienne, and L. C. L. Hollenberg. Environmentally Mediated Coherent Control of a Spin Qubit in Diamond. Physical Review Letters 118, , J. D. A. Wood, J.-P. Tetienne, D. A. Broadway, L. T. Hall, D. A. Simpson, A. Stacey, and L. C. L. Hollenberg. Microwave-free nuclear magnetic resonance at molecular scales. Nature Communications 8, 15950,

12 5. D. A. Broadway, J.-P. Tetienne, A. Stacey, J. D. A. Wood, D. A. Simpson, L. T. Hall, and L. C. L. Hollenberg. Quantum probe hyperpolarisation of molecular nuclear spins. arxiv: ,

13 Contents 1 Introduction Layout of this Thesis Magnetic Resonance Spectroscopy at the Nanoscale Traditional EPR and NMR Techniques A Spin-Based Nanoscale Magnetometer The Nitrogen Vacancy Centre in Diamond The NV Centre as a Spin Qubit Spin Hamiltonian of the NV Centre Ground State Magnetometry with the NV Centre DC Magnetic Field Sensing Microwave Control for Improved Magnetometry Environmental Spin Sensing and Spectroscopy Decoherence Decoherence Spin Sensing with the NV centre From Detection to Spectroscopy: Current Techniques for Magnetic Resonance Spectroscopy Double Resonance: DEER and ENDOR

14 1.5.5 Spectral Filtering via High Order Pulse Sequences T 1 Spectroscopy Quantisation as an Environmental Filter Potential Improvements from Relaxation Spectroscopy Theory of T 1 Spectroscopy The Spin Relaxation Curve from a Classical Field The Spin Relaxation Curve from a Single Spin Single Spin Detection Sensitivity Linewidth of the Spin Relaxation Experimental Setup and Method Confocal Microscope Magnetic Field Alignment Nuclear Spin Polarisation of the NV Centre Experimental Procedure T 1 Spectroscopy of the P1 Centre in Diamond The P1 Centre in Diamond P1 Hamiltonian P1 Centre Energy Levels Determining the Frequency Spectrum of the P1 Centre Evolution Electronic Paramagnetic Resonance on the P1 Centre in Diamond EPR Spectrum of the P1 Centre Single Quantum EPR Transitions

15 3.3.2 Double Quantum Transitions Comparison with Double Electron Electron Resonance Hyperfine-Assisted Nuclear Magnetic Resonance Theory on the competing interaction pathways Measurement of the P1 Nuclear Spin Spectrum On the suppressed hyperfine-enhanced NMR transitions Evaluation of the P1 Hamiltonian Parameters Summary Nuclear Magnetic Resonance via T 1 Relaxometry Introduction Theory of NMR via T 1 Spectroscopy Interaction with a Single Nuclear Spin NV state-mixing at the GSLAC Detecting an ensemble of nuclear spins Sensing Volume Feasibility Study towards T 1 -NMR Optically Detected Magnetic Resonance at the GSLAC T 1 Decay Features at the GSLAC caused by NV level crossings AC Magnetometry via T 1 Relaxation Demonstration of NMR via T 1 Relaxometry Proton NMR using a 15 NV Proton NMR using a 14 NV Sensitivity of T 1 -NMR and Comparison to T 2 -based Methods

16 5 Nuclear Spin Hyperpolarisation via Direct Cross-Relaxation Detection and Hyperpolarisation of 13 C Spins Conclusion and Outlook 182 Appendix A Environmentally Mediated Spin Manipulation

17 List of Figures 1.1 Optically Detected Magnetic Resonance for DC Magnetometry The Structure, Energy Levels and Spin Readout Mechanism of the NV Centre Evolution of the Spin State During a Ramsey Pulse Sequence Represented on the Bloch Sphere Magnetometry Pulsing Schemes with the NV Centre Schematic and Angular Dependence of a Double Electron- Electron Resonance Measurement Schematic and Angular Dependence of an XY8-N Measurement Schematic of a T 1 Magnetic Resonance Spectroscopy Experiment Model Used to Describe Relaxation Within the NV Electronic Ground State Schematic of the Experimental Setup for T 1 Spectroscopy and NV Fluorescence as a Function of Applied Magnetic Field An NV Photo-Luminescence Measurement as a Function of Magnet Position in the x,y plane for the Purpose of Magnetif Field Alignment

18 2.5 Pulsed ODMR on a Single NV Centre Demonstrating Nuclear Spin Polarisation T 1 -EPR Measurements of P1 Centres in Diamond as Measured by a Single NV Centre Spin Probe Comparison of T 1 -EPR Measurements to Double Electron Electron Resonance T 1 Measurements of the Nuclear Transitions of P1 Centres with a Single NV Centre Spin Probe The On-Resonance Decay Rate of the NV Centre Produced by a Single Proton as a Function of Separation Distance and Angle The On-Resonance Decay Rate of the NV Centre Produced by a Single Proton at Various Positions on a Diamond Surface Energy Levels of the NV Across the GSLAC and the Theoretical Spectrum Produced by a Single 13 C and 1 H The Induced Decay Rate of the NV Centre Produced by a Semi-Infinite Layer of Protons External to a Diamond Surface The Sensing Volume of an NV Centre Spin Probing Nuclei External to the Diamond Optically Detected Magnetic Resonance of the NV Centre Across the GSLAC The Energy Levels and State Crossings of the 14 NV Across the GSLAC

19 4.8 T 1 Spectra Across the GSLAC for 14 NV in Various Spin Baths Showing the Impact on the Crossing Features The Energy Levels and State Crossings of the 15 NV Across the GSLAC T 1 Decay From a Single Proton Compared to the Decay from the Intrinsic Crossing Feature of the 15 NV Measurements Demonstrating All-Optical AC Magnetometry Via T 1 Relaxometry Experimental Setup for NMR on an Organic Sample External to the Diamond Substrate ODMR Spectra Across the GSLAC and Relaxometry Across the Crossing Feature Measured on a 15 NV ODMR Measurement and Calculated Energy Levels Across the GSLAC for a 15 NV Strongly Coupled to an Environmental 13 C T 1 -NMR Measurements of Protons using a Single 15 NV Spin Probe T 1 -NMR Measurements of Protons using a Single 14 NV Spin Probe Comparative Spectra on the Same 14 NV in Air and then Exposed to Oil Model Used To Describe Relaxation Within the NV Centre Ground State when Magnetic Noise is Stronger than the Phonon Background

20 4.19 Comparison of Microwave-Free T 1 -NMR to the XY8-N Measurement Technique T 1 Spectroscopy on the 13 C Bath Showing the Impact of Hyperpolarisation on the Spectral Features A.1 Schematic of Environmentally Mediated Magnetic Resonance. 208 A.2 ODMR on a Single 15 NV at ω and the Pulse Sequence Used for Environmentally Mediated Resonance A.3 Experimental and Theoretical Maps of EMR Driving as a Function of Driving Field Strength and Frequency A.4 Rabi Oscillations, Ramsey Curve and Spin-Echo Curve with Control of the NV Achieved via EMR

21 List of Tables 1.1 Parameters of the NV Hamiltonian The Spectral Components of the Evolution of a P1 Centre Theoretical and Experimental EPR Transition Frequencies of P1 Centres in Diamond as Measured with a Single NV Spin Probe Summary of the Theoretical and Measured T 1 -NMR Resonances of P1 Centres in Diamond

22 Chapter 1 Introduction The quest to see nature in ever increasing detail has a long and proud history in physics. While understanding the various interaction mechanisms that allow us to acheive improved sensing can in itself be of deep interest, often the most important applications of such technology are in diverse and distant fields. For example it was in Roman times that the ability of curved glass to magnify an object became apparent. Yet even now, such as with the work in this thesis, we are utilising this ability to control and focus light to investigate novel atomic-level systems and their quantum behaviour. Recently the field of quantum sensing has opened new methods for viewing the natural world. It is this quest, to improve our ability to sense and image the world around us, that this thesis further explores. One of the great advancements of novel sensing technology in the 20th century was the ability to detect and image both electron and nuclear spin species. Electron paramagnetic resonance (EPR) gave us the ability to detect the presence and density of unpaired electron spins while nuclear magnetic 22

23 resonance (NMR) gave similar insights into nuclei and their distribution in samples of interest. By being able to differentiate spin species according to their gyromagnetic ratio, and image the spatial distribution of these spins, significant advances have been made in fields ranging from materials science to healthcare. Perhaps the most widely known application, magnetic resonance imaging (MRI) for medical purposes has led to vital improvements such as early detection of tumours, dynamic imaging of neuronal processes and identification of damaged tissue. While these are just a small subset of the advances made by current technologies, the ability to sense at the nanoscale would open up a range of other applications of great interest. Imaging dynamic processes on the sub-cellular scale or determining the structure of single molecules, such as proteins that are unable to be crystalysed, could solve some of the greatest outstanding questions in the health sciences. These require sensing at a scale beyond the scope of current techniques and, with this in mind, there has been a lot of work in recent times to develop novel probes that go beyond the resolution limitations of traditional magnetic resonance detection methods. The work within this thesis investigates a novel method for doing just that. In particular, we have considered the Nitrogen-Vacancy (NV) centre in diamond, and investigated a technique involving finely tuning an external magnetic field in order to reach a direct cross-relaxation point for nanoscale magnetic resonance spectroscopy. Through this investigation we contrast this new technique with existing methods to determine the potential technological applicability of this new scheme. 23

24 1.1 Layout of this Thesis The work in this thesis is set out as follows. Firstly, in this introduction, we motivate the research and look at the need to move beyond traditional EPR and NMR techniques for nanoscale implementation. We introduce the NV centre as a nanoscale magnetometer based on its electron spin and detail some of the relevant previous sensing work with the NV. In particular, we introduce and discuss different methods for detecting and spectroscopically differentiating environmental spin species with the NV centre. In Chapter 2 we introduce in detail the technique of T 1 spectroscopy with the NV centre. We calculate the expected relaxation curve of the NV centre when interacting with both a classical noise signal and, with a fully quantum treatment, a single spin. We consider both the on-resonance decay, as a measure of the sensitivity of this technique, as well as the spectral resolution to determine the ability of this technique to differentiate spin species. We then detail the experimental setup used for these experiments in T 1 spectroscopy. In particular we look at the fine magnetic field control required for T 1 spectrocopy as well as summarising the measurement scheme employed for the work throughout this thesis. To finish, we motivate the reasons for developing new non-invasive measurement techniques with the NV centre. This looks at general weaknesses of current techniques that could be alleviated by T 1 spectroscopy as well as looking at where the techniques could be used in a complementary fashion. Chapter 3 covers the initial experimental investigation of the spectrum of the P1 centre in diamond. This was an ideal test system for T 1 spectroscopy 24

25 due to its rich hyperfine structure and its natural presence within type Ib diamond. We demonstrate EPR spectroscopy on a small number of P1 centres surrounding a single NV, thus moving this T 1 technique to the nanoscale for the first time. In addition to the EPR measurements we also demonstrate detection of the nuclear spectrum of the P1 centre. This is achieved via hyperfine-assisted NMR, where the hyperfine coupling between the P1 centre s nuclear spin and electron spin significantly enhances the measured signal. In Chapter 4, we investigate shifting our technique beyond the diamond substrate and to pure nuclear spin signals where there is no proximal electron spin to enhance the coupling to the NV. First, we undertake a detailed investigation of the energy levels and behaviour of the NV centre near its ground state level anticrossing (GSLAC), where the NMR spectra occur. With this understanding, we are able to experimentally detect protons within an organic sample external to the diamond, thus demonstrating nanoscale T 1 - based NMR for the first time, and showing that the NV centre in diamond can be brought directly into resonance with nuclear spins external to the diamond by applying a precise static magnetic field. Finally, in Chapter 5 we discuss the ability of this technique to be used to hyperpolarise nuclear spins. In particular, we look at 13 C within the diamond lattice and show how choosing whether to interleave our measurement with π- pulses allows for either polarisation build-up within the spin bath or sensing of the spin bath. 25

26 1.2 Magnetic Resonance Spectroscopy at the Nanoscale Traditional EPR and NMR Techniques All spin-resonance detection techniques rely on detecting the varying gyromagnetic ratio, γ, of a particular electronic or nuclear spin species. The gyromagnetic ratio defines the rate at which the energy levels of a spin system shift with applied magnetic field, a process first identified by Pieter Zeeman and hence known as Zeeman splitting [5]. By measuring these spin transitions as a function of magnetic field, we are ableit is possible to differentiate between unpaired electron spins of particular chemical species or the spins of different nuclei. This can extend even to detecting chemical shifts, the small shift in the gyromagnetic ratio of a particular nuclei due to interactions with the electrons of surrounding atoms. The measurement of chemical shifts allows the chemical makeup of a sample to be probed [6]. EPR is a method for detecting unpaired electrons and is achieved by measuring the energy required to resonantly excite an electron spin to a higher energy state. Under the application of a background magnetic field, an electron spin s energy levels will be Zeeman split depending on their spin projection, m s, such that the energy of these spin levels is E = m s g e µ B B, where g e is the electron spin s g-factor (g e 2 for a free electron), µ B is the Bohr magneton and B is the applied magnetic field. The energy gap corresponding to a unitary change in the electron spin projection ( m s = 1) can be tuned via adjusting this magnetic field. When light of a particular frequency, f, is shone upon the sample being investigated, the light will be 26

27 absorbed only if the resonance equation, gµ B B = hf, is satisfied where g is now the general g-factor of the electron spins being probed. By determining the field at which unpaired electrons within the sample absorb these photons the g-factor of these electrons can be determined. In addition, the relative density of these electrons can be determined via the strength of the photon absorption. NMR is an analogous method for detecting nuclear spin species. Due to the low gyromagnetic ratio, and hence the low energy splitting of nuclear spins, the absorption method used for EPR would require magnetic fields of unfeasible strength. Instead, resonant microwaves are used to probe the nuclei. A microwave pulse scheme is applied in order to manipulate the spin projection of any nuclear spins on-resonance with the applied microwave field. The manipulation of the nuclear spins produces a changing magnetic field which induces a current within a detection coil and thus detection is achieved through magnetic induction. While these measurement methods have led to enormous technological advances, the lack of signal compared to the noise of these detection methods puts a lower bound on the number of spins able to be detected. Currently, the lower-bound spatial resolution for induction-based NMR is still only 10 µm, many orders of magnitude above the single spin level. However, there are many interesting applications that could be investigated via sensing either nuclear or electron spins at, or close to, the single spin level. These include molecular structure determination [7, 8], detection of ion flow through ion channels [9], investigation of single-molecule magnets [10] and many more. 27

28 To be able to investigate this regime a fundamentally new magnetic resonance probe is required to move to the nanoscale regime. One potential method would be to directly probe the magnetic field produced by a single spin. Achieving single spin resolution and sensitivity through a nanoscale magnetometer would naturally require a probe that is highly sensitive to magnetic fields and highly localised in order to bring the probe close to the sample of interest. This is important both for spatial resolution, which is dependent on probe-to-sample standoff distance, as well as sensitivity since the magnetic field produced by a single spin or single magnetic dipole scales with 1 r A Spin-Based Nanoscale Magnetometer A single electronic spin would represent a probe that would allow for the interrogation of environmental magnetic fields or environmental spins with nanoscale resolution and open the possibility of moving beyond the µm scale limitations of current magnetic resonance methods. In addition, the interaction of an electron with environmental spins, via either the dipole-dipole or exchange interaction, provides a method for spin-based detection. Of course the realisation of a robust electronic spin probe requires a very specific type of spin system. Indeed, if we are to detect environmental spins via their dipole-dipole or exchange interaction with the probe spin, we need a spin probe where we can measure the impact of such environmental spins on the probe. Hence it is not surprising that our minimum requirements have a strong overlap with the DiVincenzo criteria for quantum computing [11]. The minimum necessary requirements are a probe that is: 28

29 1. Chemically and physically stable. 2. Able to have its spin state initialised. 3. Able to have its spin state controlled. 4. Able to have its spin state measured. 5. Has coherence properties allowing highly-sensitive detection. Chernobrod and Berman [12] were the first to propose using the spin sublevels of a single atomic system for DC magnetometry on the nanoscale, where the spin state could be detected optically. This would allow the measurement of a DC magnetic field via Optically Detected Magnetic Resonance (ODMR). The idea of ODMR is to monitor the probability of driving a particular transition by the change in photoluminescence (PL) this causes. Figure 1.1a) shows the PL from an optically active quantum system under the application of a microwave driving field. In this case there exists a transition from a bright initial state to a comparatively darker final state, shown by the dip in PL at the resonance frequency. If a driving field frequency is chosen to be at the point of maximum gradient in the transition with respect to the DC background field (slightly detuned from resonance), there is the optimal sensitivity to DC magnetic fields. A DC magnetic field in one direction (Figure 1.1b) will cause a lower measured PL from the quantum system as the driving field comes more into resonance with the transition. Conversely, a DC magnetic shift in the opposite direction (Figure 1.1b), would shift the transition away from the driving field frequence, causing less probability of this transition occurring and leading to a higher measured PL. In this way, 29

30 ODMR can be used to give the static magnetic field directly by the measured PL. (a) B env = 0 PL Contrast (b) B env > 0 (c) B env < 0 Low PL High PL MW frequency Figure 1.1: Optically detected magnetic resonance (ODMR) for DC magnetometry. (a) A driving field frequency is chosen to be at the maximum gradient of photoluminescence (PL) with respect to the background magnetic field, PL. (b) and (c) An external magnetic field will shift the transition B resonance due to Zeeman splitting, leading to a (b) lower or (c) higher PL. One proposed implementation of ODMR was to use a spin-1 quantum dot at liquid helium temperatures where the linewidth of the transition would be significantly smaller than the shift induced by a single electron at a distance from the probe of < 10 nm [12]. By mounting the probe on an AFM tip, it would be possible to scan the probe across a sample to within these distances. 30

31 This would allow the spatial mapping of unpaired electron spin, potentially to the single spin level. While this was an interesting first proposal for a relatively simple implementation of quantum sensing using an electronic spin probe, the requirement for the quantum dot to be kept at cryogenic temperatures would significantly limit the potential applications. A probe operable at room temperature would open up a whole range of interesting applications. 1.3 The Nitrogen Vacancy Centre in Diamond The Nitrogen-Vacancy centre in diamond [13] has been the platform for an extensive amount of work on nanoscale magnetometry [14]. The high Debye temperature of diamond means that the diamond lattice behaves as a lowtemperature system even under ambient conditions leading to the NV centre having long coherence times even at room temperature [15]. Thus we do not have the issues caused by thermal broadening of the relevant transitions seen in many other quantum systems such as quantum dots [16] and oter donor systems like P in Si [17]. In addition, the ability to control the depth of NV centres to below 10 nm from the diamond surface has opened huge possibilities for nanoscale sensing beyond the material substrate [18, 19]. The biocompatibility of diamond is another important factor allowing for comparatively non-invasive biological sensing and imaging [20, 21, 22]. 31

32 (a) (b) 3 E m s = +1 m s = -1 m s = 0 Weak Strong Singlet States (c) 3 A m s = +1 m s = -1 m s = 0 Similar Probability Figure 1.2: (a) The NV Centre within the diamond lattice consisting of a nearest neighbour substitutional nitrogen (red) and vacancy. Image reproduced from Ref. [23]. (b) The energy levels of the NV centre showing the relevant transitions causing optical initialisation and readout of the electron spin under green illumination. (c) An example of collected red PL during a laser pulse from the bright (red curve) and dark (green curve) spin states as well as the contrast. The NV reaches equilibrium after 500 ns. Image reproduced from Ref. [24] The NV Centre as a Spin Qubit The NV centre is a point-like defect within the diamond lattice. It consists of a substitutional nitrogen atom combined with a nearest neighbor vacancy as shown in Fig. 1.2a) which is thermodynamically stable. This means that 32

33 upon high-temparature annealing, allowing diffusion of the vacancies within the diamond, NV centres will be produced and the system satisfies the first condition for a spin probe of being chemically stable. The diamond lattice renders it physically stable under all but the most extreme conditions. The nearest-neighbour configuration leads to a single unpaired electron. This neutral charge state, referred to as NV 0, exhibits the behaviour of a paramagnetic spin- 1 defect. Unfortunately, the ground state of the 2 NV0 has very broad EPR signals and thus is not utilised as a probe for quantum magnetometry. The negative charge state, or NV, is a spin-1 paramagnetic point defect. Due to its significantly narrowed spectral features, it is this system that is used for quantum sensing and magnetometry applications. The NV is generally referred to as simply the NV centre and this is the convention that will be used throughout this thesis. Although charge-state control of the NV centre has received attention in recent times [25, 26], the experiments considered in this thesis are taken in the regime where the ionisation of NV is low and thus we will only consider the NV centre in its negative charge state. The ground state of the NV centre is a 3 A triplet electronic state with 0 and ±1 states separated by a crystal-field splitting of 2.88 GHz, where m s represents the magnetic spin projection of the system [13]. It is the spin in the ground state that is utilised for magnetometry and thus is the major focus of this thesis. The Hamiltonian will be introduced in detail in Sec First we will discuss the important optical properties that allow readout and initialisation of this electron spin ground state. 33

34 The energy levels of the ground, excited and singlet states of the NV centre as well as the relevent transitions are shown in Fig. 1.2b). Under the application of a 532 nm laser, all 3 spin states of the NV centre are pumped via non-resonant optical excitation into the excited ( 3 E) state. This spin conserving transition is shown by the green arrows. From this excited state there are two decay pathways available. Firstly, the NV can decay via the emission of a red photon back to its initial ground state spin. Once again this is a spin-conserving transision. Additionally, there is a decay pathway where no red photon is emitted via a pair of intermediate singlet states. Being singlet states, the transition out of these does not conserve the initial electronic spin state of the NV. The decay back to 0 and ±1 has been measured to be of the same order [27]. Importantly however, the transition rate from the excited state to these intermediate states is strongly spin dependent, with the transition from the ±1 states approximately an order of magnitude faster than from the 0 state [28]. The strong spin dependence of the transition into the singlet state has two important implications. Firstly, under continual optical excitation the NV will accumulate population in the 0 state leading to spin initialisation. Due to the lower energy intermediate singlet state being a comparatively longlived metastable state with lifetime of order 100 ns, the NV centre population reaches equilibrium in less than 1 µs. In addition to spin polarisation, under the green laser pulse the spin state can also be read out. Due to the significantly lower probability of decaying via the singlet states, the 0 state has a higher fluorescence than the ±1 state. Fig. 1.2c) shows the fluorescence from the respective states during 34

35 a typical laser readout pulse as well as the measured contrast. Integrating the fluorescence over the first 500ns of the laser pulse leads to a contrast of around 30% between 0 and ±1. After this time the fluorescence from the two states equilibrate in the same steady state with strong spin polarisation in 0. In this way, the NV is able to have its spin states initialised and measured, thus satisfying those important requirements for a spin-based magnetometer. It is possible at temperatures below 10 K to conduct on-resonance excitation of the spin levels in order to give single shot readout of the NV centre (ie 100% contrast between the spin levels) [29]. However the broadening of the excitations means at room temperature this is not possible. In addition, it is possible to transfer the electronic spin state to a proximal nuclear spin and improve readout fidelity via a repetitive readout scheme. However the fidelity of this method improves at higher magnetic fields and thus is usually limited to fields > 0.2 T [30]. Since this thesis considers sensing at room temperature and relatively weak magnetic fields of less than 0.15 T we use the non-resonant mechanism discussed above in our experiments. Our theory calculations use values of readout contrast typical for this method Spin Hamiltonian of the NV Centre Ground State Having briefly discussed the initialisation and readout mechanisms for the NV centre spin state, we now detail the ground state of the NV centre. It is the physics of this ground state that will be investigated in detail throughout this thesis to acheive nanoscale magnetic resonance spectroscopy. In addition to the electron spin-1, the NV centre contains a nitrogen 35

36 nuclear spin which can be either the 14 N or 15 N isotope. The ground state Hamiltonian can be written as: H NV = S. D. S + S. A. I + I. Q. I hγ NV B. S hγn B. I (1.1) where S is the NV electron spin-1 operators, I is the nitrogen nuclear spin operators, D is the crystal-field tensor, A is the NV hyperfine tensor, Q is the nuclear quadrupole tensor, B is the applied external magnetic field and γ NV and γ N are the gyromagnetic ratios of the NV electron spin and nitrogen nuclear spin respectively. This can be simplified by going to the secular approximation. By defining the z-axis as the crystal-field quantisation axis, the magnetic couplings to the x and y projections of the NV electron spin only contribute to second order and thus can be ignored in our measurements which include fine alignment of the magnetic field along the z-axis. This is because the NV quantisation along z is by far the dominant energy term and thus no environmental coupling can cause the NV to flip along this quantisation axis. This approximation gives the NV Hamiltonian as: H NV /h = DS 2 z +A S z I z +A (S x I x + S y I y )+QI 2 z γ NV B z S z γ N B z I z. (1.2) Here we have expressed the Hamiltonian in frequency and the symbol γ refers to gyromagnetic ratios expressed in units of Hz/T, that is γ = The only term coupling different spin projections that has been kept is the perpendicular hyperfine component, A. This term becomes important at low NV quantisations near the ground state level anticrossing which will be discussed in Chapter 4. γ 2π. 36

37 The different isotopes of nitrogen found in creating NV centres have different spin configurations. The 14 N isotope, a spin-1 system, has a natural abundance of 99.64% and thus is the dominant species of NV centres in samples without isotopic purification. The 15 N isotope, a spin- 1 system, has a 2 natural abundance of 0.36%. However, implanting with isotopically purified 15 N is possible and will create NV centres of this isotope. This is often used to differentiate shallow NV centres created by ion implantation from native NV centres existing throughout the bulk sample. Throughout this thesis, I will refer to NV centres containing a 14 N as 14 NV and similarly 15 NV for those containing a 15 N. The parameters of Eq. 1.2 for each isotope are shown in Table 1.1. It should be noted that the nuclear quadrupole moment comes from a non-spherical distribution of the nuclear spins and thus requires a nuclear spin of at least 1. Hence there is no nuclear quadrupole for the 15 NV due to it being a nuclear spin The ground state level anticrossing of the NV centre is where the 0 and 1 states reach the same energy and cross. This occurs at a field of 1024G. While the aligned magnetic field is at a value far from the GSLAC, the non-axial hyperfine, A, can be discarded and both the nuclear quadrupole moment and the axial hyperfine, A, act simply as static shifts on the energy levels of the NV electron spin. However, as the NV centre approaches the GSLAC, leading to the energy difference between the 0 and 1 states being less than 10 MHz (comparable with A ), this non-axial hyperfine acts to mix the spin states of the NV and thus this term cannot be discarded. This regime is important for the nuclear spin sensing discussed in detail in Chapter 4. 37

38 Parameter Symbol 14 N Isotope 15 N Isotope Electron Spin S Spin-1 Spin-1 Nuclear Spin I Spin-1 Spin- 1 2 Zero-field Splitting D 2.88 GHz 2.88 GHz Axial Hyperfine A 2.14 MHz 3.03 MHz Non-axial Hyperfine A 2.7 MHz 3.65 MHz Quadrupole Moment Q MHz NA Elec. Gyro. Ratio γ NV GHzT GHzT 1 Nuc. Gyro. Ratio γ N MHzT MHzT 1 Table 1.1: Parameters of the NV Hamiltonian (Eq. 1.2) for both isotopes of nitrogen. 1.4 Magnetometry with the NV Centre All sensing with the electronic spin of the NV centre is acheived by some form of initialise-evolve-measure protocol. As discussed, both the initialisation and measurement are acheived via optical excitation. By applying different control schemes during the evolution of the state it is possible to ensure the final measurement of the system is sensitive to a particular coupling within the NV Hamiltonian. Via manipulating the NV centre in a way that makes the final measurement sensitive to a change in the Zeeman splitting, it is possible to undertake magnetometry [14, 31, 32, 33, 34, 35, 36]. Similarly, by implementing a different series of controls, it is possible to have the final state strongly dependent on the crystal field splitting, D, of the NV electron spin which allows for thermometry due to the dependance of D on thermal 38

39 expansion and contraction of the diamond lattice [37, 38]. Electrometry, the sensing of electric fields is also possible [39]. In this section we detail some such schemes that are discussed throughout this work DC Magnetic Field Sensing Gruber et al. in 1997 demonstrated optically detected magnetic resonance (ODMR) on NV centres in diamond [40]. However, the application of ODMR to magnetic field sensing was initally proposed by Chernobrod and Berman [12] with this approach being reformulated specifically for the NV centre in 2008 [32, 41]. As previously discussed, ODMR involves measuring the resonance energies of a quantum system by applying a driving field and optically detecting when that driving field hits a resonance causing a spin transition. In the case of the NV centre, the 532 nm laser is applied in order to initialise the system into the 0 state. A microwave driving field is then applied to the system and a laser pulse reads out the final spin state. By sweeping this driving field across a range of frequencies it is possible to detect when the driving field matches the 0 ±1 transitions due to a reduction in final fluorescence. Conveniently, the final laser pulse also acts to reinitialise the system. ODMR can be done in the pulsed mode, as is described here, where the laser pulse and the microwave driving are applied at different intervals. Alternatively ODMR can be done in continuous mode where both laser and driving fields are applied continuously throughout the measurement. In continuous mode it is the steady state fluorescence that is probed. Throughout the experimental section of this thesis we used pulsed rather 39

40 than continuous ODMR in order to ascertain the NV transition frequencies and hence the applied magnetic field. This allowed for low power microwave pulsing in order to not break the rotating wave approximation for low energy splittings near the GSLAC without having to accordingly adjust the laser power in order to maintain optimal fluorescence contrast. In addition, pulsed ODMR removes laser broadening and the slow driving strength strongly reduces microwave broadening to allow an accurate measurement of the static background field [42]. Although ODMR is a relatively simple method for DC magnetometry, its development has allowed for many important sensing applications to be investigated. For example, DC magnetic sensing through ODMR can be coupled with scanning capability when the NV centre is placed at the end of an atomic force microscope (AFM) tip, either via deposition of a nanodiamond or through fabrication of an all-diamond scanning probe [43]. This leads to high sensitivity magnetometry with spatial resolution on the order of 10 nm which has been used to image many interesting magnetic systems. These include imaging domain walls in ferromagnets [44], pinning sites in superconductors [45] and many other complex magnetic materials at greater sensitivity and spatial resolution than previously possible. When an ensemble of NV centres shallow within the diamond are utilised in widefield imaging mode, imaging of current flow in graphene [46] and of hardrive data bits has been demonstrated [47]. 40

41 1.4.2 Microwave Control for Improved Magnetometry Beyond ODMR, more complex pulsing schemes may be implemented in order to improve magnetometry sensitivity and tune the frequency of magnetic fields that the NV centre is sensitive to. These require the ability to manipulate the state of the NV centre which is normally achieved via the application of an on-resonance microwave driving field in order to drive Rabi oscillations. When a static magnetic field is applied, the energy levels of the ±1 electronic spin states of the NV centre shift at a rate of ±2.8 MHz/G. Applying such a field breaks the degeneracy between the +1 and 1 states, allowing a microwave driving field to be applied on-resonance between the 0 state and either the +1 or 1 states. This limits the population to 2 levels of the 3-level spin-1 system and thus the system approximates a 2- level spin qubit. From hereon the notation ±1 will refer to the full 3-level system while the state 1 will signify one of the ±1 states which has been chosen as the second level of the effective qubit. Under the application of this near-resonant microwave driving field, detuned from resonance by δ, the probability of finding the system having transitioned to the 1 state, after an evolution time τ is simply given by the Rabi formula [( Ω ) ] cos P 1 (τ) = Ω2 MW + MW δ2 τ Ω 2 MW +, (1.3) δ2 2 where Ω MW is the frequency of on-resonance Rabi oscillations which is defined by the strength of the microwave driving field and δ is the detuning from resonance. At resonance, with δ = 0, this simply become sinusoidal oscillations with frequency Ω MW. When detuned from resonance, the frequency of these Rabi oscillations increases and the contrast decreases (ie the 41

42 population is never fully transferred from 0 to 1 ). By taking on-resonance Rabi oscillations it is possible to ascertain the time to drive the NV population from its initial 0 state to the 1 state. A pulse of this length, which fully transfers population, is a π-pulse. A pulse which transfers half the population from 0 to 1 is a π 2 -pulse. For DC magnetometry the Ramsey pulse sequence is used. The evolution of the qubit state during a Ramsey pulse sequence is shown in Fig 1.3 through the use of the Bloch sphere. This is a convenient representation of the state of a two-level quantum system. Any two-level quantum state, ψ = c c 1 1, can be represented on the Bloch sphere by rewriting the complex constants c 0 and c 1 in terms of θ and φ, thus rewriting the state as ( ) ( ) θ θ ψ = cos 0 + e iφ sin 1. (1.4) 2 2 We can then represent all pure quantum states of the two-level system on the surface of a sphere where θ is the polar angle and φ is the azimuthal angle. Generally the state is then drawn as a vector from the origin to the surface of a unit radius sphere. The basis states 0 and 1 then correspond to the north and south pole of the sphere respectively. Figure 1.3a) shows the state of the NV after a green laser pulse, initialised into 0. Pulse-based magnetometry schemes with the NV centre involve applying a π -pulse to the NV centre immediately after the laser initialisation pulse, 2 thus putting the NV centre into an equal superposition of 0 and 1. This is shown in Fig. 1.3b). This means that over the evolution time, τ, any environmental magnetic field oriented along the quantisation axis of the NV centre will cause a phase accumulation, ϕ, to the NV state. The state after this phase accumulation is shown in Fig. 1.3c). This simplifies magnetom- 42

43 (a) (b) (c) (e) (di) No Phase Shift (dii) 90 o Phase Shift Figure 1.3: A Ramsey pulse sequence for DC magnetometry represented on the Bloch sphere. a) The state initialised by a green laser pulse into 0. b) A π -pulse puts the state into an equal superposition of 0 and 1. c) After 2 a time of free evolution the state gains a phase ϕ due to a static background field. d) A final π -pulse transfers the phase ϕ into a measurable change in 2 population. The final pulse can be done about the same axis as the initial pulse (di) or around the complementary axis with a 90 o phase shift (dii). e) The measured signal after the final pulse. The 90 o phase-shifted final pulse (green) gives a maximum gradient in signal with respect to accumulated phase. etry with the NV centre into a phase estimation problem since this phase accumulation is directly proportional to the strength of the magnetic field (for a DC magnetic field along the NV-quantisation axis, ϕ = B z γ NV τ). A final π -pulse transfers this accumulated phase into a measurable pop- 2 43

44 ulation difference between the basis states. This final pulse can be applied along the same axis as the initial pulse (see Fig. 1.3di). However by phaseshifting the driving field by 90 o for the final π -pulse (see Fig. 1.3dii), the 2 measurement is most sensitive to small phase accumulations as well as being sensitive to the sign of the phase accumulation. This can be seen in Fig. 1.3e) where the change in population with respect to accumulated field is at a maximum for the state after the final pulse around the complementary axis. With only this initial and final π -pulse, this is a Ramsey sequence where 2 the final phase, ϕ, is proportional to the DC magnetic field strength during the evolution time. This effectively transforms the frequency-domain measurement of ODMR into a time-domain measurement. The minimum detectable magnetic field via an NV measurement can be defined as where the signal to noise goes to 1. This can be expressed as B min = S. (1.5) S B Here S is the measured signal (in this instance the number of photons collected from the NV centre), S is the uncertainty on that measured signal and S B is the change in signal as a function of magnetic field. By defining T 2 to be the NV coherence time, this minimum detectable field for a Ramsey measurement, given 1 s of total measurement time, is then B min h gµ B C. (1.6) T2 where C 1 is a parameter describing the efficiency of the detection mechanism with C going to 1 for ideal, single-shot readout. Improved sensitivity 44

45 comes from extending the NV coherence time T2. Taylor et al. [32] showed the NV centre could achieve sensitivities below 1 µthz 1 2 via implementation of a Ramsey pulse sequence. Purely in terms of sensitivity, this is many orders of magnitude from the state of the art SQUID magnetometers which can achieve sensitivities down to 10 fthz 1 2. Other high-sensitivity magnetometers such as vapor cells and single trapped ions can achieve pt sensitivity. However, the room temperature operation of the NV centre makes it applicable to a vast range of problems unable to be interrogated in other ways. Additionally, the atomic size and ability to engineer NV centres shallow within the diamond allows for significantly smaller stand-off distances to magnetic samples. This leads to significantly stronger signals from the same samples as well as vastly improved spatial resolution making a single NV a vital magnetic sensing tool despite the sensitivity not approaching that achieved by other methods. By alterring the Ramsey sequence to include π-pulses within the evolution time of the measurement, the NV sensitivity can be tuned to particular environmental frequencies. For example, adding a single π-pulse at the middle of the evolution time makes this a Hahn-echo sequence. This and the Ramsey sequence are shown in Fig The π-pulse acts to flip the state of the NV centre and hence any phase accumulation due to a static field will cancel before and after the π-pulse. Importantly, this rephasing of the static magnetic field also cancels DC dephasing components from the spin bath and hence extends the coherence time of the NV centre from T2 out to the longer T 2. This sequence is optimally sensitive to magnetic fields where a single period of the field occurs over the total evolution time with the field being 45

46 Laser MW pulses Ramsey Hahn Echo Final Pulse 90 o phase shifted Figure 1.4: Magnetometry Pulsing schemes with the NV centre in diamond. An initial and final laser pulse are used to initialise and readout the NV spin state. The second row shows a Ramsey magnetometry sequence, sensitive to DC magnetic fields. Adding a π-pulse at the centre of the evolution time creates the Hahn-echo sequence, sensitive to AC magnetic fields. For magnetometry with optimal sensitivity, the final π -pulse must be phase-shifted by 2 90 with respect to the initial pulse. Additional π-pulses can be added during the evolution time to tune the NV response to a particular environmental frequency. of opposite sign in the first and second half of the evolution. Since the sensitivity improves with the square-root of the coherence time, this extension of the coherence time leads to a better sensitivity for measuring these AC fields than for the previous DC case. More complex pulsing schemes such as Uhrig [48, 49], CPMG [50] and XY-8 [51] incorporate more π-pulses within the evolution time in order to further extend the coherence time and to tune the sensitivity to a particular environmental frequency [52]. In this way, AC magnetic field sensitivity can be achieved below 10 nthz 1 2 with a single NV 46

47 centre [15]. 1.5 Environmental Spin Sensing and Spectroscopy Up until now we have simply considered the case of detecting fully coherent environmental magnetic fields with sensitivities quoted for the optimal case of the field being in phase with the applied pulse sequence. In reality, AC sensing of interesting environments often requires sensing environmental fluctuations which either are not coherent over many repetitions of the measurement scheme, or where we do not have enough information to lock our detection scheme in phase with the environmental fluctuations. Thus a method which can sense the strength of these environmental fluctuations must be considered Decoherence Much discussion of quantum mechanics considers fully coherent quantum states. In reality no quantum system can ever be completely shielded from its environment. The interaction of a quantum system with its environment causes the pure state depiction to be modified in experimental realisations. Rather than a closed quantum system, we have an open system interacting with a bath that is generally far too complex to fully mathematically solve. This interaction between our quantum system and the environment leads to a removal of information from the quantum system and non-unitary evolution. 47

48 This loss of information, about either the phase or population of our quantum state, is known as decoherence and leads to the quantum system transitioning from a fully quantum state upon initialisation towards a classical or mixed state. Decoherence provides a significant problem for quantum technologies. In order to benefit from the potential advantages of quantum processes, such as in quantum computation, we require our quantum information to remain in such a state in a robust manner while operations are performed. Furthermore, we must be able to apply the quantum protocols needed within this timeframe. Hence two important figures of merit for any system for quantum information processing applications are the time to apply a particular gate and the time that the state remains coherent. Indeed, much of the work in advancing quantum technologies has been aimed at reducing the effects of decoherence on the systems being used by better shielding the systems from their environment or controlling them in a manner to make them less sensitive to environmental fluctuations. For a general qubit, decoherence processes can be split into two forms, dephasing and relaxation. Relaxation describes the process by which the relative populations of the qubit s basis states decay to a thermal mixture. As we are always considering room temperature systems in this work, where the energy splitting is far smaller than thermal excitations, that is k B T E, the thermal population of the NV electron spin will be an equal probability in each of the three basis states. Thus, for an NV centre initialised in 0, the relaxation time, T 1, is the characteristic time to decay to an equal mixture of 0, +1 and 1. This may seem to contradict the fact that these states are 48

49 eigenstates of our qubit Hamiltonian and therefore should remain constant over time. In reality, every quantum system has some environmental impact not considered in the Hamiltonian which can cause a transition away from these eigenstates. In the case of an NV centre, at room temperature and off-resonance from any magnetic bath fluctuations, this is the phonon bath which causes this relaxation on a timescale of a few ms [53]. Importantly, due to a quantised energy splitting between our basis states, the relaxation time is sensitive specifically to noise on-resonance with the transition energy. This is a fact we will exploit throughout this thesis. Dephasing is the other form of decoherence that we will consider. This is the time taken for a qubit to lose phase coherence. That is, the characteristic time for a coherent superposition state (eg: 1 2 [ ]) to dephase into a mixed state where we have no knowledge of the relative phase. This dephasing time is referred to as T 2. For a spin qubit, due to all superpositions being energetically degenerate, a magnetic field along the quantisation axis of the qubit will cause a phase rotation. When taking many repetitions of the same experiment, any randomness in this magnetic field between subsequent repetitions will cause the final states to be evolved with different final phases. In the measurement process we average over these states which leads to a loss of phase information. It is possible to apply pulse sequences to rephase some of this magnetic noise and thus extend the coherence time of the state. T 2 refers to the natural dephasing time without any outside control applied to rephase this information. T 2 is the dephasing time under the application of a particular pulsing sequence. We will keep to this convention throughout this thesis. 49

50 1.5.2 Decoherence Spin Sensing with the NV centre Quantum sensing is the act of utilising the probe s quantum properties in order to determine information about its environment. In this context, rather than being a problem needing to be mitigated against, decoherence can provide a valuable tool for sensing [49, 54]. Rather than protecting our quantum system from the environment, decoherence sensing aims to expose our system to the environment in a controlled manner and interrogate the environment based on the decoherence rate measured. If we can understand how particular aspects of the environment affect our system s coherence, we can then measure the decoherence of the systems as a direct probe for those environments. Cole and Hollenberg developed the idea for using the decoherence of a quantum probe, such as an NV centre, on a scanning AFM tip in order to interrogate magnetic fluctuations within a sample [54]. One major advantage of such a technique is the ability to detect randomly fluctuating fields about a zero mean [49]. Such a signal would be critical in biological sensing since many biological processes produce such fluctuations that would be insensitive to Ramsey magnetometry. Similarly, the inability to lock environmental fluctuations with the phase of an AC sensitive experiment would lead to no net accumulation of phase here either. By detecting the decoherence rate rather than the phase accumulation directly, it allows information to be inferred about the relative strength of the environmental fluctuations. For the NV centre in diamond, T 1 has been found to be limited by phonon processes under low magnetic field and at temperatures above 100 K [53]. This regime gives standard T 1 times of 5 ms at room temperature. At 50

51 low temperatures these can extend out to seconds as phonon processes are frozen out. Importantly, at magnetic fields where the NV transitions are offresonance from other magnetic environmental noise, this phonon based T 1 is very similar for most samples. That is, apart from extreme cases, and as we will show, specific resonant magnetic fields, the T 1 time is generally similar between diamonds and does not depend strongly on the isotopic purification or presence of defects within the diamond such as substitutional nitrogen. Due to the phonon processes being dependent on the vibrational modes of the diamond lattice, the presence and variation of magnetic spin impurities has little impact on T 1. The exception to this, which this thesis focuses on, is at precise magnetic fields where the NV comes into resonance with environmental spins, where this resonance process can then govern the T 1 decay. In contrast, the dephasing rate of NV centres in diamond at room temperature is highly sample, and NV, dependent. This is because it is the magnetic noise that defines T 2 rather than phononic behaviour. For a highly impure sample with a large density of nitrogen impurities( 10 ppm), T2 times are on the order of 100ns and even upon rephasing, T 2 times are as low as a few microseconds. In contrast, high purity diamond grown by chemical vapour deposition (CVD) will produce NV centres with T2 values of a few µs and T 2 of around 100 µs. Finally, in isotopically purified diamond this T 2 time can be extended out to a few milliseconds even at room temperature [15] as the magnetic impurities within the diamond are further reduced. In this instance it is approaching the limit set by the phonon bath, T 2 2T 1 [55, 56]. Also within these samples, because the accumulated phase is changed by 51

52 the strength of magnetic noise, the dephasing times can vary significantly between different NV centres. The specific environment, in particular the spatial distribution of spin defects around a particular NV will define the dephasing rate [57]. In short, decoherence sensing with the NV centre is acheived by measuring a change in the decoherence time of the NV centre when it is exposed to a changed magnetic environment. We will look at this in the context of sensing of environmental spins. Dephasing and, for specific resonant fields, relaxation are caused by interactions between the NV and the environmental spin bath. This opens up the possibility for sensing unpaired electron spins on the nanoscale, even down to the single spin level. Due to the highly localised wave-function of the NV centre s electron spin [58], exchange coupling to environment spin species can be ignored. Therefore the NV centre s interaction with environmental spins is mediated via the magnetic dipole-dipole interaction. The Hamiltonian of the magnetic dipole-dipole interaction between two spins is: H int = µ [ 0 γ 1 γ 2 h 2 3 ( ) ( ] S1 r S2 r) S 4πr 3 r 2 1 S 2 (1.7) where γ 1,2 are the gyromagnetic ratio of each spin, S 1,2 are the spin operators of each spin, r is the separation vector between the two spins with magnitude r, µ 0 is the vacuum permeability and h is Planck s constant. Importantly, the strength of this interaction means that detecting small numbers of electron spins external to the diamond is very feasible with NV centers within 20 nm of the surface. At a separation of 20 nm and the case of a free electron spin interacting with an NV centre, the pre-factor of Eqn. 1.7 is 6.5 khz. This is on the same order as dephasing rates for NV centres 52

53 with long coherence times and an order of magnitude faster than the room temperature phonon relaxation rate. This allows for electron spin detection beyond the background decoherence processes at the sensitivity of a single electron spin at an achievable separation distance. The NV centre has been used in a number of different studies to sense unpaired electron spins by sensing changes in either T 1 or T 2. T 1 detection was used for sensing Gadolinium spins (Gd 3+ ) with the presence of the unpaired environmental spins leading to a characteristic drop in T 1. These were measured both in bulk diamond [59] and in nanodiamonds [60]. In addition, detection of a Gd 3+ labelled lipid bilayer demonstrated sensitivity to less than 5 spin labels [61]. However the intrinsic relaxation rate of the Gd 3+ complexes (dominated by its own photon relaxation mechanisms) results in a very broad spectrum which overlaps with the NV centre even at small magnetic fields. As we will see in this thesis, there are many examples of electron spins that do not have such a broad spectrum and hence, the presence of these electron spins does not cause any change in the relaxation without specifically tuning the magnetic field to bring the species into resonance with the NV [1, 62]. In addition to T 1 based measurements, T 2 has also been used as a detection mechanism for electron spins [63, 64], described in more detail below From Detection to Spectroscopy: Current Techniques for Magnetic Resonance Spectroscopy While sensing the presence of unpaired electron spins is an important first step, spectroscopic differentiation of spin species is a vital piece of the puzzle. 53

54 Rather than simply detecting the presence of spins via additional environmental fluctuations, characterising them by gyromagnetic ratio would allow analysis of those spin species present within the environment. To do this requires the ability to filter the environment in order to eliminate those species away from the resonances that you plan to interrogate. In recent years there have been two main methods of environmental filtering using the NV centre in diamond for the purpose of spectroscopic differentiation of spin species. In both cases, the signature of an additional environmental species is a reduction in T Double Resonance: DEER and ENDOR (a) π/2 π π/2 NV τ/2 τ/2 π target (c) (111) 0 (b) T 2 0 δ 180 Figure 1.5: (a) The pulse sequence of a double resonance experiment. (b) Schematic of the measured T 2 as a function of detuning of the target driving field, δ. (c) The angular dependence of double resonance experiments with a maximum at θ = 0 and the interaction strength dropping to zero at the magic angle. The first of these techniques for filtering the environment are double resonance methods. These are split into two forms, double electron-electron 54

55 resonance (DEER) and electron-nuclear double resonance (ENDOR) [63, 65]. The pulse sequence applied for these techniques are the same. DEER refers to utilising such a technique to detect electron spins while ENDOR refers to the detection of nuclear spins. I will refer to this technique as DEER since it is applied far more regularly to the sensing of electron spins although it should be noted that this analysis applies equally to ENDOR in the limited situations where detection of nuclear spins is possible in this way. DEER involves applying a pulse sequence to the NV centre that is sensitive to AC noise. At the same time, a second microwave field is applied to flip the target spin at the correct moment so that the field produced by the target spin at the NV centre is an AC field. A Hahn-echo sequence is applied to the NV centre however the final pulse is now not phase shifted with respect to the initial pulse. A second microwave pulse is applied to the target spin species at the same point as the NV centre undergoes a π-pulse. The pulsing sequence for a double resonance experiment is shown in Fig 1.5a). If the frequency of this second driving field matches the transition frequency of an environmental target spin this will cause a pulse on the environmental spin, changing its magnetic dipole. This pulse on the environmental spin alters the strength of the effective magnetic field that the spin is producing at the NV centre, leading to the NV centre experiencing an increased AC field across the entire evolution time. The length of the pulse on the environmental spin can be optimised to give a π-pulse leading to a reversal of the magnetic field for the second half of the evolution and thus a maximal AC field produced at the NV centre. While this at first appears as a standard measurement of an AC field, 55

56 because the initial state of the target spin is not able to be controlled, there will be no average phase acquired by the NV centre. Instead it will cause an increase in the variance of the phase evolution which causes a reduction in T 2. Since this reduction in T 2 only occurs when the applied pulse hits resonance with the environmental spin, by sweeping the frequency of the applied pulse, the entire spectrum is able to be interrogated. A schematic of the T 2 of the NV centre, as a function of δ, the detuning of the second driving field from the environmental spin s resonance, is shown in Fig 1.5b). Only when coming into resonance does the T 2 of the NV change. Importantly, the strength of the increase in T 2 is directly related to the interaction strength of the environmental spin with the NV. DEER interrogates the ZZ term in the dipole-dipole interaction. This leads to an interaction strength with the environmental spin that depends on the polar angle of separation, θ, of [3 cos 2 (θ) 1] 2. This angular dependence is shown in Fig 1.5c) by the blue curve where the radial distance represents the interaction strength at a given polar angle. It should be noted that ( 1 the interaction drops to zero at the magic angle of θ = arccos 3 ). This angle also corresponds to the angle between the (100) surface and the (111) orientation of the NV quantisation axis. As a result, if we have an NV beneath a (100) surface, the most common and simplest growth surface for diamond, a target spin as close as possible to the NV centre external to the diamond will have no ZZ interaction with the NV centre and cannot be measured via DEER. However, spins shifted slightly from this position will couple to the NV and this coupling has been used for the detection of electronic spins both internal [66, 67] and external [68] to the diamond. It 56

57 has also been used for imaging a single electron spin [69]. In addition to magnetic resonance spectroscopy on electron spins, double resonance has been demonstrated on hydrogen nuclear spins [63]. However, the requirement of pulsing the target spins makes the application to nuclear spins difficult due to the high RF field amplitudes required to pulse a nuclear spin in a sufficiently fast manner due to their low gyromagnetic ratios. The demonstration by Mamin et al. [63] was achieved with comparatively deep NV centres in isotopically purified diamond in order to have a long background T 2, giving longer time in which to pulse the target protons. For any NMR sensing application, due to the 1 r 3 dependence of the dipole-dipole interaction, near-surface NVs are essential. Currently, production of nearsurface NVs with sufficiently-long coherence times has proven to be a major challenge, limiting the utility of such an approach for nuclear spin detection. Due to the weak gyromagnetic ratio of nuclear spins, requiring a longer microwave driving time than electron spins, and the strong reduction in T 2 of the NV seen close to a diamond surface, ENDOR is not feasible for shallow NV centres and thus is of limited utility for nuclear spin sensing Spectral Filtering via High Order Pulse Sequences The second major technique for filtering the environment is through highorder pulsing schemes such as XY8 [70] or CPMG [50]. These higher order pulsing schemes lead to a narrow filter function dependent on the delay time between the π pulses. Figure 1.6a) shows the pulse sequence for XY8-N where the central 8 pulses are repeated N times. This leads to the NV coherence time having a narrow filter function where it is sensitive to noise in a narrow 57

58 (a) X Y ( τ )N (b) (c) 1.0 (111) Figure 1.6: (a) The XY8-N pulse sequence. The 8 π pulses are repeated N times. Pulses in blue and red must be around orthogonal axes (eg: X and Y). (b) Filter function of an XY8-N pulse sequence. Via adjusting τ it is possible to interrogate different environmental frequencies. Image reproduced from ref [51]. (c) The angular dependency of the dipole-dipole interaction interrogated by the XY8 pulse sequence. spectrum around f = 1 2τ π-pulse as shown in Figure 1.6b). where τ is the delay time between each successive Both XY8 and CPMG interrogate the ZX and ZY terms of the dipoledipole interaction. This can be considered as a flip of the target spins (X or Y operators) causing a phase rotation on the NV (Z). These terms have an angular dependence of [cos (θ) sin (θ)] 2 where θ is once again the polar angle of separation. This angular dependency is plotted in Figure 1.6c). Because of the dominant filter frequency depending on the delay time between pulses (f = 1 ), and the requirement for short pulses compared to 2τ the evolution time of the NV, it is only feasible to interrogate noise 10 MHz 58

59 in strength. This means that spectroscopically mapping electron spins is not possible with this technique. However, the major strength of this technique is for NMR spectroscopy. This was also used as a technique for the first demonstration of external proton spin detection [51] but since then it has also been used to spectroscopically differentiate between various nuclear species [71], estimate the depth of a single NV centre [72] and show NMR on various targets such as a single protein [73]. This technique is the current standard for nano-nmr with the NV centre and is being investigated for nanoscale MRI with applications such as protein structure determination in mind [7]. However both current spectral mapping techniques, double resonance and higher order pulse sequences, have limitations. In the following chapter we discuss some of these limitations and consider how T 1 spectroscopy may open another avenue to attack interesting applications. 59

60 Chapter 2 T 1 Spectroscopy Having introduced the NV centre and the current methods for spectroscopic mapping of environmental spin species using the NV centre, we now turn our attention to the major investigation of this thesis; the application of T 1 spectroscopy for magnetic resonance spectroscopy at the nanoscale. We start by introducing the idea of T 1 spectroscopy and what potential advantages there could be by using T 1 spectroscopy [1] rather than T 2 -based techniques. We follow on from this discussion with a detailed theoretical analysis of the technique including the expected decay from a classical signal as well as a fully quantum treatment of the problem. Finally, we detail the experimental setup that was used for the measurements in the following chapters. 2.1 Quantisation as an Environmental Filter The relaxation time, or T 1, of the NV centre is the characteristic time taken to relax away from the initial 0 state. While the NV Hamiltonian depends 60

61 on many aspects of the environment (magnetic field, electric field, strain, temperature, etc), the quantisation of the NV centre means that the relaxation rate is only sensitive to excitations matching a transition away from the NV s initial state. This leaves it insensitive to low-frequency fluctuations of the environment which allows us to consider only those environmental contributions that could exist at the comparably high frequency being probed. At low magnetic fields, the NV centre s 2.88 GHz zero-field splitting is orders of magnitude higher than any environmental magnetic or electric field fluctuations and therefore the relaxation is governed by the strength of the phonon bath[53]. In diamond, due to the high Debye temperature, this is on the order of a few ms at room temperature and extends significantly at low temperatures, out to many seconds, due to the gradual freezing out of the phonon bath. T 1 spectroscopy was developed and initially demonstrated by Hall et al. [1]. This initial demonstration was on a large ensemble of NV centres and our work extends this technique to the single spin domain, thus reaching the nanoscale desired. As discussed in the previous chapter, spectroscopic mapping of a magnetic environment requires both a magnetometer (in this case our NV centre) and a filter of the environment. While previous NVbased magnetic resonance spectroscopy has filtered the magnetic environment through some form of pulse sequence on the NV centre (eg: DEER or XY8), the T 1 decay rate is already sensitive only to noise at the energy of the 0 +1 and 0 1 transitions. This gives us a natural filter of the environment. T 1 spectroscopy uses this filter to detect additional onresonance magnetic fluctuations causing a transition away from the 0 state 61

62 and thus decreasing the relaxation time, T 1. The environmental frequencies being interrogated by the NV centre can be controlled by a static magnetic field and hence the magnetic environment can be filtered and quantitatively mapped. Figure 2.1 shows a schematic of the process involved in T 1 spectroscopy. Fig. 2.1a) shows the energy levels of the NV centre as a function of aligned magnetic field strength with the 0 and 1 states crossing at the GSLAC at 1024 G. Fig. 2.1b) shows the energy levels of sample environmental spins, in this case an electron spin and a nuclear spin each with no zero-field splitting. As we increase the magnetic field from zero, the environmental spin s levels split apart while the transition between the 0 and 1 states of the NV centre reduces. At some point, the spins will come into resonance, where the environmental spin can give a single excitation to the NV centre, driving the NV away from its initial state. The resonance points, and a schematic of the subsequent drop in T 1 when on resonance is shown in Fig. 2.1c). For a spin with a lower gyromagnetic ratio than the NV, such as the nuclear spin, there will be a pair of resonance points before and after the GSLAC. The general, all-optical T 1 sequence, which measures the population being transferred away from the initial 0 state of the NV is shown in Fig. 2.1d). The signal is taken to be the ratio of the fluorescence over the first 300 ns of the laser pulse, I S, where the spin contrast is highest, normalised to a reference measurement at the end of the laser pulse, I r. In general a 3 µs laser pulse was used and the reference measuremetns, I r, was an integration over all of the laser pulse after steady state fluorescence was reached in order to minimize the error incorporated through this normalisation. Thus I r ranged 62

63 (a) (d) Init. τ Read (b) I s I r (e) (c) Wait Time, τ (ms) Figure 2.1: (a) The energy levels of the NV centre as a function of aligned magnetic field. (b) The energy levels of two example environmental spins as a function of magnetic field with a high gyromagnetic ratio (electron spin) and low gyromagnetic ratio (nuclear spin). (c) Schematic of T 1 as a function of applied magnetic field. When the NV comes into resonance with an environmental spin, the T 1 decreases due to cross relaxation. (d) An all-optical T 1 sequence with signal (I s ) and reference (I r ) fluorescence windows. (e) A schematic T 1 curve on (blue) and off (black) resonance, showing the NV population remaining in the initial 0 state. from 500 ns to 1.5 µs throughout this thesis. Fig. 2.1e) shows a schematic for an on-resonsnace (blue curve) and off-resonance (black curve) T 1 measured 63

64 with this sequence. The curves are calculated from Eqn. 2.4 with Γ 1,r = 1 khz and Γ 1,ph = 0.2 khz. T 1 spectroscopy as described here is analogous to an inversion of ODMR that was discussed in Chapter 1. In ODMR, the energy of the 0 ±1 transitions, and hence the strength of the static magnetic field, can be determined via sweeping a driving field and measuring a transition away from the 0 state via a reduction in fluorescence. This is in effect measuring a decrease in T 1. Conversely, in T 1 spectroscpy, the background magnetic field is swept in order to determine the driving frequencies of the environmental spectrum via a decrease in T 1. It should be noted that ODMR generally involves a strong driving field where the microwave driving rate (Ω MW ) is far greater than the dephasing rate ( 1 ) and therefore the spin state oscillates T2 coherently between 0 and ±1. In contrast, T 1 spectroscopy is used to detect environmental resonances where the rate at which the NV is driven by on-resonance environmental magnetic fluctuations is much less than the dephasing rate. Therefore we observe over-damped decay rather than the coherent driving of ODMR [1]. The influence of cross-relaxation on the NV centre T 1 was first observed by Jarmola et al. [53]. While they saw increased relaxation rates near resonance points, the large ensembles of NVs they were using and the few data points meant there was no investigation of the techniques spectroscopic capabilities. Early studies of the NV centre s flourescence as a function of field enabled mapping of some of the NV centre resonances with P1 defects within the diamond [62, 74]. However the reliance on steady-state fluorescence for these measurements limited the sensitivity to strong interactions. 64

65 In contrast, spectroscopy with the NV centre using full T 1 mapping was first proposed and investigated by Hall et al. [1]. In addition to outlining the theoretical proposal, they conducted a first experimental demonstration by mapping the spectrum of P1 centres in diamond detected with an ensemble of NV centres. By including wait times, τ, between the laser pulses to allow T 1 decay rather than simply relying on the steady-state fluorescence of the NV under laser illumination, interrogation of far weaker signals was possible. In addition, they were able to deconvolve the measured spectrum from the NV filter function in order to extract the full spectrum of the P1 centres interrogated by an ensemble of NV centres. In addition to this demonstration of EPR, T 1 spectroscopy has been used to probe spin waves in a ferromagnetic microdisc [75]. This thesis aims to investigate the limits and utility of T 1 spectroscopy. In particular, whether T 1 spectroscopy can be achieved at the nanoscale on a single NV centre, on nuclear spins species as well as the previously shown electron spin measurements, and whether T 1 spectroscopy allows for improvements on current nanoscale spectroscopic techniques. It is this final question we consider now Potential Improvements from Relaxation Spectroscopy As discussed in Chapter 1 there exist two other current techniques for filtering the environmental spectrum acting on the NV centre in order to implement nanoscale spin spectroscopy; double resonance techniques such as DEER and high-order pulse sequences such as XY8. The reason for investigating T 1 65

66 spectroscopy in more detail is to see whether we can overcome some of the inherent limitations in these techniques. As previously mentioned, the utility of double resonance techniques is realistically limited to electron spin measurements while higher-order pulse sequences can only be used to investigate comparatively low frequency noise, limiting them to the nuclear spin regime. In contrast, one potential advantage of T 1 spectroscopy is that it would be able to be implemented across a wide range of frequencies simply by tuning the magnetic field. Furthermore, extending the interrogation time from T 2 out to the longer T 1 suggested the potential for improved sensitivity from the dependence of sensitivity on interrogation time discussed previously. We will cover this comparison in Section Another issue for higher order pulsing schemes is that the filter function contains harmonics away from the central resonance point. This leads to the potential for a signal to be misinterpreted as being from one nuclear species when it is in fact a harmonic of a second spin species. One particular example of such is the overlap of the 3rd 13 C harmonic with the central feature of 1 H [76]. While the use of correlation spectrocopy can mitigate this effect, these spurious resonances do not exist in the case of T 1 spectroscopy. However the major possible improvement of T 1 over these dephasingbased techniques is the ability to completely remove the need for microwave pulsing. The requirement for pulsing the target environmental spins in double resonance techniques is a limitation for two reasons. Firstly, as mentioned, this limits the application to nuclear spins for most applications. In addition, as is shown by the Rabi formula (Eq. 1.3), a strong driving field will cause 66

67 the target electron spin to be driven across a wider range of frequencies than theoretically possible thus limiting the spectral resolution of the technique. T 1 spectroscopy does not require any pulsing of the target spin and thus would remove this requirement. Since T 1 is an all-optical measurement, and we can sweep our environmental filter by adjusting a static magnetic field, there is no need for microwave fields at all when implementing this scheme on a single NV. While this is not a major difficulty for measurements in the controlled lab environment, attempting to achieve nanoscale EPR or NMR within living cells or within an organism would require high power and high precision pulsing if using current techniques. The ability to drive a strong pulse at the NV centre without affecting surrounding tissue would be a significant technological challenge and hence the ability to remove pulsing altogether could be of great use for these applications. It should be noted that the initial demonstration of T 1 spectroscopy [1] didn t use an all-optical technique. This was due to the measurement being conducted on an ensemble NV sample. Since there are four possible orientations of the NV quantisation axis within the diamond lattice, a π-pulse was required to cancel the impact of the NV centres not aligned to the background magnetic field axis. Thus the measurements in Chapter 3 represent the first demonstration of microwave-free spectroscopy with the NV centre. 2.2 Theory of T 1 Spectroscopy We start our investigation of T 1 spectroscopy by modeling the decay from two possible noise sources; a classical on-resonance magnetic field and then 67

68 a single environmental spin The Spin Relaxation Curve from a Classical Field Our first attempt at modeling the T 1 process is through considering an onresonance classical magnetic field and calculating the change in the NV relaxation curve. To do this, we treat the phonon decay as a common 2-way transition rate k ph linking all 3 spin states of the NV. We model the classical field as an on-resonance noise source, that we aim to detect, linking the m s = 0 and m s = 1 states with a rate k r. This ignores coherent evolution of the NV centre and assumes that upon averaging over many realisations of the measurement, any coherence is averaged out. A schematic of this model is shown in Fig. 2.2(a). The rate equations governing this decay are dn 0 dτ = dn 1 (n 1 + n +1 ) k ph 2n 0 k ph + (n 1 n 0 ) k r = (n +1 + n 0 ) k ph 2n 1 k ph (n 1 n 0 ) k r (2.1) dτ dn +1 = (n 1 + n 0 ) k ph 2n +1 k ph dτ where n 0,+1, 1 are the populations in the m s = 0, 1 and +1 states respectively. Solving the rate equations together with the closed-system condition n 0 (τ) + n +1 (τ) + n 1 (τ) = 1 yields the populations as a function of the 68

69 (a) (b) I 0 I τ (ms) Figure 2.2: a) Model used to describe the population dynamics within the NV electronic ground state in response to a classical noise source. The transition rate k ph accounts for the phonon-induced relaxation while k r corresponds to a classical noise source linking the 0 1 transition. b) Plot of NV intensity against evolution time for various on-resonance decay rates (Γ 1, ph is the phonon relaxation rate). evolution time τ as n 0 (τ) = [ ] 1 3 n +1(0) e 3k phτ [n 0(0) n 1 (0)] e 3k phτ 2k rτ n 1 (τ) = [ ] n +1(0) e 3k phτ (2.2) 1 2 [n 0(0) n 1 (0)] e 3k phτ 2k rτ n +1 (τ) = 1 [ ] n +1(0) e 3kphτ. The PL intensity at the start of the readout laser pulse, I s (τ), can be ex- 69

70 pressed as I s (τ) = I 0 n 0 (τ) + I 1 [n +1 (τ) + n 1 (τ)] = I 1 + (I 0 I 1 )n 0 (τ) (2.3) where I 0 and I 1 < I 0 are the PL rates associated with spin states 0 and ± 1. Here we assume that the fluorescence intensity of the dark state is independent of which m s = ±1 state is populated. Inserting Eqs.(2.2) into Eq. (2.3), the relaxation curve can be written as [ I s (τ) = I 1 + C ( e Γ 1,ph τ + 3e )] (Γ 1,ph+Γ 1,r )τ (2.4) 4 where we introduced the phonon decay rate Γ 1,ph = 3k ph and the resonance decay rate Γ 1,r = 2k r. The coefficients I and C are given by I = I 0 + 2I 1 3 C = 1 α 1 + 2α [3n 0(0) 1] (2.5) with α = I 1 /I under typical experimental conditions. In obtaining Eq. (2.4) we assumed that the initial populations are such that n +1 (0) = n 1 (0), that is, the initialisation pulse affects the populations of ± 1 in the same way, as it is generally accepted [27, 77]. Fig 2.2(b) shows a selection of decay curves for different on-resonance decay rates, Γ 1,r. The phonon decay rate in each point, Γ 1,ph, is taken to be 200 Hz giving a background T 1 of 5ms [53]. Plotted in black is the offresonance decay curve. What should be noted is that this produces a double exponential where the pair of states coupled by the resonant signal equilibrate on a timescale of (Γ 1,r + Γ 1,ph ) 1, contributing to 3 4 all 3 spin states equilibrate over the longer timescale of Γ 1 1,ph. 70 of the total decay while

71 2.2.2 The Spin Relaxation Curve from a Single Spin We now consider the microscopic description of the NV interacting with environmental spins rather than a simple classical on-resonance signal. For magnetic-resonance detection of spins, we have the NV interacting with quantum objects in the environment. Here we consider the simple case of a single spin-1/2 as the target, which is treated using a fully quantum mechanical approach. We seek here to calculate the population dynamics of a system composed of the NV spin and a single spin-1/2 target with gyromagnetic ratio γ t. The full Hamiltonian of the coupled NV-target system is H = H NV + H t + H int (2.6) where H NV is the Hamiltonian of the NV spin, H t is that of the target spin and H int is the magnetic dipole-dipole interaction between the two spins. For clarity, we use the superscript (p) to refer to the spin of the probe NV centre while we use the subscript (t) to refer to the target environmental spin whose spectrum we wish to measure. We restrict the NV spin to the {m (p) s = 0, m (p) s = 1} subset of the three level system to consider the case where the target spin causes a coupling between these two states. In the case of coupling to the m (p) s = +1 state the calculation could be repeated with that state included in the reduced spin matrices instead. We also neglect the hyperfine interaction with the NV centre s nitrogen nuclear spin for this calculation. The magnetic field of strength B is applied along the NV centre s symmetry axis, which defines the z direction. The Hamiltonian of the NV 71

72 and target spins are therefore simply ( ) H NV /h = D NV S (p) 2 z γnv BS z (p) (2.7) H t /h = γ t BI (t) z (2.8) where I (t) denotes the spin operators of the target spin which can be electronic or nuclear. The dipole-dipole interaction is the same as Eqn. 1.7 with γ 1 = γ NV (2.9) S 1 = S (p) (2.10) γ 2 = γ t (2.11) S 2 = I (t). (2.12) In the { 0, +1/2, 0, 1/2, 1, +1/2, 1, 1/2 } basis using the notation m (p) s H h =, m (t) I, the total Hamiltonian is γtb H int,31 h H int,41 h γ tb 2 H int,32 h H int,42 h H int,13 h H int,23 h D NV + γ NV B γtb 2 + H int,33 h H int,43 h H int,14 h H int,24 h H int,34 h D NV + γ NV B+ γtb 2 + H int,44 h (2.13) where {H int,ij } are the matrix elements of H int. Denoting {H ij } as the matrix elements of H, the two resonances occur when H 11 = H 44 and H 22 = H 33 since these transitions correspond to a flip of both the target and NV spin state. This yields γ tb 2 = D NV + γ NV B + γ tb 2 + H int,44 h γ t B 2 = D NV + γ NV B γ tb 2 + H (2.14) int,33 h. 72

73 These two resonances occur before (B < 1024 G) and after (B > 1024 G) the GSLAC if γ t < 0, respectively, and after and before the GSLAC if γ t > 0. Note that for weak dipolar coupling the terms H int,33 and H int,44 in Eqs. (2.14) can be neglected. This leads to solutions for the resonance fields of: B (±) res = D NV γ NV ± γ t (2.15) Now that the resonance fields have been calculated, we must evaluate the time evolution of the NV centre in order to ascertain the strength of the interaction and hence the ability of such a method to detect environmental spin species. In order to account for decoherence of the spin state, we utilise the Lindblad equation with dephasing at the rate of Γ (p) 2 for the NV spin and Γ (t) 2 for the target environmental spin. We can then write the rate of change of the density matrix ρ(t) as ρ (t) = ī [Hρ (t) ρ (t) H] h [ + 2Γ (p) 2 S z (p) ρ (t) S z (p) Γ (t) 2 [ I z (t) ρ (t) I z (t) 1 2 ρ (t) S(p) z S z (p) ρ (t) I(t) z I z (t) 1 2 S(p) z 1 2 I(t) z S (p) z I z (t) ρ (t) ] ρ (t) ] (2.16) When on resonance, only those terms in the resonant states need to be considered as those off-resonance will not decay beyond the background phonon rate which we have neglected here. Hence we obtain a system of first order differential equations involving only ρ 11 (t), ρ 14 (t), ρ 41 (t) and ρ 44 (t) for the resonance where H 11 = H 44, and a system involving only ρ 22 (t), ρ 23 (t), ρ 32 (t) and ρ 33 (t) for the resonance where H 22 = H 33. From these, and assuming the NV is fully initialised while the target spin starts in an arbitrary mixture, i.e. 73

74 ρ 11 (0) + ρ 22 (0) = 1, we can generate a single third order differential equation for each resonance, d 3 ρ AA (t) dt 3 d 2 ρ AA (t) = 2Γ 2 [ ] ω 2 dt 2 int,ab + Γ 2 dρ AA (t) 2 dt Γ 2 ω 2 int,abρ AA (t) + Γ 2 2 ω2 int,abρ AA (0) (2.17) where we introduced the effective interaction strength ω int,ab = 2 H int,ab, the h total dephasing rate Γ 2 = Γ (p) 2 + Γ (t) 2 and A, B = 1, 4 or 2, 3 depending on which resonance is being interrogated. With the initial conditions we find the solution ρ AA (t) = ρ AA(0) 2 + dρ AA (0) = 0 dt d 2 ρ AA (0) = ω2 int,ab ρ dt 2 AA (0) 2 + ρ [ AA(0) e Γ 2 2 t cosh 2 Γ 2 Γ 2 2 4ω 2 int,ab sinh ( t ) Γ 2 2 4ωint,AB 2 2 ( t int,ab) Γ 2 2 4ω 2. 2 (2.18) (2.19) In the regime of weak dephasing ( hγ 2 H int,ab and therefore Γ 2 ω int,ab ), this equation simplifies to: ρ AA (t) = ρ AA(0) 2 = ρ AA(0) 2 + ρ AA(0) e Γ 2 t 2 [cosh (itω int,ab )] 2 + ρ AA(0) e Γ 2 t 2 [cos (tω int,ab )]. 2 (2.20) These are oscillations corresponding to a coherent energy transfer back and forth between the two spins. These would be expected in the case of strong interaction compared to dephasing and are commonly referred to as spin flipflops. In most practical cases we will not have engineered the environment 74

75 to contain only a single spin strongly interacting with the NV centre. Since dephasing is due to the random field produced by the surrounding spin bath, when we do not engineer the spin bath in that regime, the coupling strength is much weaker than the dephasing ( hγ 2 H int,ab and therefore Γ 2 ω int,ab ). In this weak coupling regime, Eq. (2.19) simplifies into ρ AA (t) ρ AA(0) 2 + ρ AA(0) 2 e ω 2 int,ab t Γ 2 (2.21) from which we identify the relaxation rate Γ 1,r = ω2 int,ab Γ 2 = 4 Γ 2 ( Hint,AB h ) 2. (2.22) By taking the relevant terms in the dipole-dipole interraction for the two resonances H 11 = H 44 (labelled + ) and H 22 = H 33 (labelled - ), this gives Γ + 1,r = 1 Γ 2 ( µ0 γ NV γ t h 2 2 Γ 1,r = 1 Γ 2 ( µ0 γ NV γ t h 2 2 ) 2 ( 3 sin 2 θ r 3 ) 2 ) 2 ( ) 3 sin 2 2 θ 2. r 3 (2.23) For an electronic target spin ( γ t = γ e γ NV ), there is only one resonance at B r G because the second resonance corresponds to a field Br =. We note that, as written, the resonance decay rate Γ 1,r does not depend on the initial state of the target spin, i.e. on the value ρ AA (0). However, the latter affects the relative contrast of the decay as measured via the PL. Indeed, the PL intensity can be expressed using Eq. (2.3) as I s (τ) = I 1 + (I 0 I 1 ) [ρ 11 (τ) + ρ 22 (τ)] = I 0 + I 0 C [ e Γ1,rτ 1 ] (2.24) 75

76 where the contrast is given by C = I 0 I 1 2I 0 ρ AA (0). (2.25) The contrast is maximum if the target spin is initialised in the resonant state (ρ AA (0) = 1), and is null if the target spin is initialised in the other (non-resonant) state (ρ AA (0) = 0). In general, a target spin at room temperature is initially in a mixed state (ρ AA (0) = 1/2) since the thermal energy greatly exceeds the Zeeman energy, i.e. k B T hγ t B. However, in the presence of a large number of target spins, the probability of finding at least one target spin on resonance with the NV is close to unity, which implies that the contrast is maximum, i.e. C = (I 0 I 1 )/2I 0. The effective relaxation rate, Γ 1,eff, is then simply a sum of the relaxation rates induced by each on-resonance target spin. Under this assumption, Eq. (2.24) is found to match Eq. (2.4) when one sets Γ 1,ph = 0 (no phonon relaxation) and n 0 (0) = 1 (NV electron spin fully initialised in m (p) s = 0). Eq. (2.4) is therefore a generalisation of Eq. (2.24) which includes phonon relaxation and non-perfect NV initialisation, and is valid in the presence of a large number of identical target spins, each with similar interaction strengths with the probe NV Single Spin Detection Sensitivity Having modeled the on-resonance decay rate, this now gives us the ability to take a meaningful measure of the sensitivity of T 1 spectroscopy. We take the measure of sensitivity as the time taken to detect a single spin, that is, to acquire more signal than noise. Here we derive an expression for this time. 76

77 For T 1 spectroscopy, the measurement sequence consists of a 3-µs laser pulse followed by a wait time τ assumed to be much longer than 3 µs. The useful fluorescence I s (τ) is obtained by counting the photons within a readout time t ro = 300 ns. As a result, the PL signal is acquired only for a fraction t ro /τ of the total experiment time, T tot. Therefore total number of photons detected can be expressed as t ro N (Γ 1,r, τ) = T tot τ I s(τ) (2.26) [ ( t ro = T tot τ R 1 C + Ce Γ 1,phτ )] 4 e Γ 1,rτ where R is the photon count rate under continuous laser excitation, Γ 1,ph is the intrinsic phonon relaxation rate, Γ 1,r is the relaxation rate induced by the on-resonance target spins, and C is the contrast of the T 1 relaxation curve. The change in the number of photons caused by the presence of Γ 1,r is N signal (τ) = N (0, τ) N (Γ 1,r, τ) = 3RT tott ro C e ( Γ 1,phτ 1 e ) Γ 1,rτ. (2.27) 4τ The photon shot noise associated with the measurement is N noise (τ) = N (Γ 1,r, τ) RTtot t ro τ (2.28) where we used the approximation C 1. Taking a series solution for a small signal regime (Γ 1,r Γ 1,ph ) and only keeping the first order terms in Γ 1,r gives the signal to noise ratio as SNR T1 (τ) = N signal(τ) N noise (τ) = RT tot t ro 3C 4 e Γ 1,phτ Γ 1,r τ. (2.29) 77

78 To find the optimal probe evolution time we solve where the derivative equals zero. That is dsnr T1 (τ) dτ = = 0 RTtot t ro τ 3C 8 Γ 1,re Γ 1,phτ (2Γ 1,ph τ 1) (2.30) giving an optimal probe time of τ = 1/(2Γ 1,ph ). Thus the signal to noise ratio at this optimal time becomes SNR max = RT tot t ro 3C 4 e 1/2 Γ 1,r 1 2Γ 1,ph. (2.31) We define the spin as detected when the SNR reaches 1. Now we can solve for the total measurement time to give: T tot = 2eΓ 1,ph Rt ro 16 9C 2 (Γ 1,r ) 2. (2.32) If we consider an NV centre interacting with a weak on-resonance signal of Γ 1,r = 75 Hz and standard values for an NV centre of R = 200 khz, Γ 1,ph = 200 Hz, C = 0.25 corresponding to an NV with 33% contrast between 0 and ±1 states interacting with a single unpolarised spin, t ro = 300 ns and T 2 = 1 µs we find a total experiment time required for detecting this signal of 91 seconds. This corresponds to a single electron a distance of 43 nm away, a single proton a distance of 5 nm away or a single 13 C a distance of 3.1 nm away assuming an optimal polar angle of separation in each case. This shows that T 1 spectroscopy has the sensitivity to detect even single nuclear spins beyond the diamond substrate considering NV centres can be engineered within these distances of the surface. 78

79 2.2.4 Linewidth of the Spin Relaxation Beyond detection sensitivity, the spectral resolution is an important aspect of any spectroscopic technique. From this quantum mechanical treatment we can also calculate the spectral linewidth of such a T 1 resonance feature and hence consider what features could be differentiated. To do this we consider the Hamiltonian from Eqn but with the near-resonant energy levels now detuned by a frequency δ. So the energy levels from Eqn now become γ tb 2 + 2π hδ = D NV + γ NV B + γ tb 2 + H int,44 h γ t B 2 + 2π hδ = D NV + γ NV B γ tb 2 + H (2.33) int,33 h. This leads to a more general third-order differential equation d 3 ρ AA (t) dt 3 = 2Γ 2 d 2 ρ AA (t) dt 2 [ ω 2 int,ab + Γ π 2 δ 2] dρ AA (t) dt Γ 2 ωint,abρ 2 AA (t) + Γ (2.34) 2 2 ω2 int,abρ AA (0). We now follow the method of Hall et al. [1] by making the substitution T = Γ 1,r t. Doing this leads to a re-scaled differential equation d 3 ρ AA (T ) = 2 Γ2 2 d 2 ρ AA (T ) dt 3 ωint,ab 2 dt 2 Γ2 2 2ω 4 int,ab ( 2Γ π 2 δ 2 + 2ω 2 int,ab) dρ AA (T ) dt (2.35) Γ4 2 ρ ωint,ab 4 AA (T ) + Γ4 2 ρ 2ωint,AB 4 AA (0). Since the interaction strength is always much smaller than the dephasing (ω int,ab Γ 2 ) we can take an expansion in ω int,ab Γ 2 second order. Doing this gives the equation dρ AA (T ) dt and retain only terms to = ρ AA (0) Γ 2 2 2Γ 2 2ρ AA (T ). (2.36) 2 (Γ π 2 δ 2 ) 79

80 Solving this and rescaling back to t gives the population as a function of time as ρ AA (t) = ρ AA(0) 2 [ ( + ρ Γ AA(0) Γ 1,r t 2 2 Γ e π2 δ )]. 2 (2.37) 2 where Γ 1,r is the on-resonance relaxation rate from Eqn This means that the relaxation rate is simply scaled by a Lorentzian about the resonance point. This Lorentzian has a linewidth of Γ 2 2π. So the linewidth of T 1 spectroscopy scales with the total dephasing of the system and, as we are not applying any rephasing pulses, this dephasing rate is 1. In contrast, the T2 linewidth of CPMG or XY-8 based detection depends on the rephased decoherence rate 1 T 2. In this instance, T 1 spectroscopy will lead to significantly broader features than these T 2 spectroscopic techniques. Importantly, in recent times it has been shown that the spectral resolution of CPMG and XY-8 based techniques can be improved by correlation spectroscopy [78]. Indeed it has now been shown that the spectral resolution can be limited only by the coherence of the field (or spin) being sensed via the implementation of Fourier spectroscopy [79, 80]. These advances have allowed for nanoscale-nmr to differentiate chemical shifts, a vital step towards protein structure determination [81]. Given these advances, it would be unlikely that T 1 spectroscopy can match the spectral resolution of these T 2 -based methods for the application of NMR. However the simplification of the measurement technique and the removal of the requirement for microwave driving motivates T 1 spectroscopy as an interesting area of research for magnetic resonance applications. Furthermore, current EPR techniques require driving of the target electronic spins and this can lead to power broadening (and heating), especially in uncontrolled noisy environments where the tar- 80

81 get electronic spins have shorter coherence times. Thus, it is possible that T 1 spectroscopy could match or even better current T 2 techniques for nanoscale EPR. As Eq shows, the estimate of the gyromagnetic ratio of an environmental spin species depends on the NV zero-field splitting which can be significantly impacted by the presence of strain. Therefore, the zero-field splitting must be independently measured for the particular NV probe being used. In addition, the temperature should be well controlled due to the thermal shift of the zero-field splitting ( 74.2 khz/k). 2.3 Experimental Setup and Method Confocal Microscope As the NV centre is initialised and read out optically, a confocal microscope is the standard apparatus used. The confocal allows us to both address the NV with the green laser light and collect the emitted red fluorescence. The NV centre is initialised into the 0 state under the application of green light. The source used for this is a 532nm laser (Laser Quantum Gem 532). The laser beam is focused and double-passed through an acousto-optic modulator (AOM) in order to give a high extinction ratio (> 10 6 ) between the laser on and laser off states. In this configuration the AOM switching time is 10 ns. This is essential for pulsing the laser to control readout and initialisation timing, as well as ensuring the free evolution of the system is done with no green light exciting the NV from the ground state. After the AOM, the light is passed through a single-mode fibre and colli- 81

82 mated. This light is then reflected off a dichroic beam-splitter and directed to the back of the microscope objective lens (Olympus UPlanSApo 100x, NA = 1.4 Oil). This objective lens is then mounted on an XYZ scanning stage (PI P Nanocube) in order to precisely control the position on the diamond sample at which the laser is applied. The red flourescence from the NV centre is collected by the objective lens and filtered from the excitation light via the dichroic beam splitter and a band-pass filter. Finally this light is coupled in to a multi-mode fibre connected to a single photon avalanche photodiode or APD (Excelitas SPCM- AQRH-14-FC) where the fluorescence is measured. The APD output was sent to a time resolved photon counting card (Fastcomtec MCS6A) to allow integration of the signal and reference photons. The diamond sample was mounted on a PCB board with a wire soldered across the surface of the diamond sample. The PCB board contained a conductor with SMA connector to deliver microwave driving fields through this wire. This was necessary for ODMR to determine the magnetic field strength. While this represents a standard NV confocal setup, the specific addition for this thesis was of a Neodymium magnet mounted on an XYZ scanning stage (PI-M515) with 100 nm minimum increment. This allowed for precise control of the magnetic field alignment and strength at the NV Magnetic Field Alignment We required the ability to control the magnetic field strength. In order to do this we initially mounted the magnet in such a way that both the NV 82

83 (a) APD y Dichroic mirror PL z XYZ stage Laser Obj. (b) Figure 2.3: (a) Schematic of the experimental setup with the magnet tilted by 54.7 o in order to be aligned with the NV quantisation axis. (b) Plot of the NV fluorescence as a function of background magnetic field strength for different angles of misalignment reproduced from Ref. [82]. This shows the excited state level anticrossing (B 512 G) and the ground state level anticrossing (B 1024 G) axis and the magnetic field was approximately radial to the magnet position. Due to the diamond samples having surfaces oriented in the [100] direction, this required the magnet to be tilted by 54.7 o from the horizontal. Fig. 2.3a) shows the experimental setup with the magnet aligned away from the horizontal in order to approximately align with one of the possible NV quantisation axes. 83

84 Ignoring the nuclear spin, the NV centre Hamiltonian, from Eq. 1.1, can be written as H NV /h = D NV Sz 2 γ e (B z S z + B x S x ) (2.38) where B z is a magnetic field along the NV quantisation axis and we have redefined the x coordinate to be aligned with the magnetic field perpendicular to the NV quantisation axis. Solving the energy levels with a series expansion about B x = 0 gives [ ( ) 2 ( ) ] 3 γ e B x γ e B x E = (D NV γ e B z ) 1 + O. (2.39) D NV γ e B z D NV γ e B z Since the energy shift is only second order in the strength of the non-axial field, B x, a small misalignment in the applied field should not greatly affect the transition energies and thus not greatly impact the measurement. In reality though, all of the experimental measurements within this thesis are taken at points where fine alignment is vital. This calculation assumes that the dominant term is the NV ground state quantisation, D NV γ e B z. This however is not always the case. In general nuclear spin measurements are done at low NV quantisations near the ground state level anticrossing (GSLAC). At this point, the NV quantisation drops to a level comparable to the energy of a slightly misaligned magnetic field. Not only would any non-axial field produce an energy shift but, more importantly, it would lead to strong mixing between the NV spin states. This strong mixing means the optical cycling of the NV centre does not conserve the electronic spin state and therefore does not initialize the NV into 0. So in order to be able to utilise our NV as a sensor we must have a well aligned magnetic field. 84

85 Coincidentally, the resonance points with electron spins occur near the excited state level anticrossing (ESLAC). Due to the reduced zero-field splitting in the excited state, the ESLAC occurs at 510 G. While the state mixing is present in the excited state rather than the ground state, our initialisation and measurement rely on the spin preserving transitions to the excited state. With a high degree of spin mixing in the excited state once again the NV would be unable to be initialised and read out so we need to align the field precisely. Importantly, the impact of misaligning the magnetic field at these points is a drop in photo-luminescence produced by the NV centre. The impact of this misalignment was studied by Epstein et al. [82] and a plot from this paper showing this effect of a misalignment in the magnetic field on the photoluminescence at both the GSLAC and ESLAC is shown in Fig. 2.3b). This shows that at full misalignment there is a > 30% drop in fluorescence. At a misalignment of 0.6 o there is a flourescence drop from 400 to 600 G with a maximum decrease of 12% around the ESLAC. Around the GSLAC the decrease in fluorescence is more pronounced with a drop of 30%. This shows we require alignment of significantly better than 0.6 o in order to avoid state mixing for electronic and nuclear spin detection. In order to align the magnetic field we define the z-axis as being oriented along the NV quatisation axis as shown in Fig. 2.3a). The alignment protocol involved stepping the magnet in a grid in the x,y plane. At each point, the NV photo-luminescence was measured. An example of such a scan on our system near the magnetic field required for P1 nuclear spin detection is shown in Fig 2.4 where the magnet is scanned in a 7x7 grid with a range of 40 µm 85

86 y (μm) PL (counts/ms) x (μm) Figure 2.4: An NV photo-luminescence measurement as a function of magnet position in the x,y plane. The scan is taken at 1000 G near the resonance between the NV centre and the P1 nuclear transitions presented in Chapter 3. in both x and y. After mapping out this grid, a Gaussian is fitted to this and the magnet moved to the point of maximum photo-luminescence Nuclear Spin Polarisation of the NV Centre The experimental mesurements undertaken in Chapters 3 and 4 are conducted at magnetic fields near the NV centre s excited state level anticrossing (ESLAC) in the case of the EPR measurements or near the ground state level anticrossing (GSLAC) in the case of NMR measurements. This meant that the nuclear spin of the NV centre was polarised during the initialisation procedure. This is a well known and previously studied phenomenon [83] however here I briefly summarise this polarisation mechanism. 86

87 Section introduced the ground state Hamiltonian of the NV centre as well as the mechanism with which the NV centre s electron spin is optically polarised for the purpose of both initialisation and readout. This process requires pumping the NV centre to its excited state followed by spin-dependent decay via a dark shelving state. At room temperature, the excited state Hamiltonian of the NV centre is also a triplet whose Hamiltonian can be written in the same form as the ground state Hamiltonian in Eqn 1.2 with the values [13]: D e NV A e A e = 1.42 GHz 60 MHz 60 MHz This lower zero-field splitting leads to the excited state level anticrossing (ESLAC) occurring at a magnetic field of 510 G. At this point, the NV electron m s = 1 and m s = 0 states become close to degenerate and the non-diagonal terms in the Hamiltonian cause spin mixing. Thus a misaligned magnetic field leads to a loss of fluorescence from the NV since it contributes a non-diagonal term to the Hamiltonian. In addition to a misaligned magnetic field, there is always the non-axial hyperfine (A ) that is a non-diagonal term which causes spin mixing. This hyperfine term couples the m (p) s, m (p) I = 0, 0 and 1, +1 states as well as the 0, 1 and 1, 0 states. As the NV electron spin is polarised via laser excitation through the excited state, in the time the system is in the excited state the system will undergo evolution which will cause transitions of the form 0, m (p) I 1, m (p) I + 1. Upon the many cycles of an initialisation 87

88 pulse, this leads to optical pumping into the 0, +1 state. That is, we achieve polarisation of the nuclear spin into m I = +1 as well as the electron spin into m s = 0. Here we have explained the process for a 14 NV. The process is identical for a 15 NV except the nuclear spin is polarised into m I = PL (a. u.) B = 7 G PL (a. u.) B = 488 G PL (a. u.) B = 1012 G Microwave frequency (MHz) Figure 2.5: Low microwave power pulsed-odmr conducted on a single NV centre within a type Ib diamond (see Ch. 3) at various magnetic fields. The top plot shows the ODMR at low field with the 3 NV hyperfine transitions clear. The second and third plots show only a single transition due to the nuclear spin polarisation under optical pumping at these fields. The second plot is at a field of B = 488 G, near the electron spin transitions, while the third plot is at B = 1012G, near the NV ground state level anti-crossing. Fig. 2.5 shows pulsed ODMR plots undertaken on a single NV centre at 3 separate magnetic fields. The first plot, at B = 7 G, corresponds to the situation where neither the NV electron spin in the excited state or ground state are near crossings. This leads to the hyperfine coupling causing no spin 88

89 mixing meaning the nuclear spin is not polarised. Hence the ODMR has 3 clear transitions corresponding to the 3 nuclear spin states. The second plot, conducted at B = 488G, is near to the crossing within the excited state. This induces polarisation of the NV nuclear spin and hence we see only a single NV transition. The broad nature of this leads to the NV nuclear spin being polarised across all of the NV-P1 ESR and NMR resonances that are discussed in Chapter 3. This simplifies the resonances of the NV down to a single transition rather than the 3 transitions corresponding to the three nuclear spin states thus significantly narrowing our spectral response. Furthermore this leads to a significantly increased contrast when interacting with a narrow frequency signal due to the NV filter function not being split into 3 equally strong transitions 2 MHz apart. Because of this nuclear spin polarisation, the only NV transition corresponding to an electron spin flip is 0, +1 1, +1 with an energy: E NV = D NV (γ e + γ N ) B z A. (2.40) This is the transition for the NV centre that is used to calculate all resonant fields in measurements of the P1 centre in Chapter 3. For the nuclear spin measurements in Chapter 4, spin mixing within the ground state alters this transition strength. In addition to nuclear spin polarisation at the ESLAC, there is also polarisation at the ground state level anti-crossing (GSLAC). The third plot in Fig. 2.5, conducted at 1012 G is close to the GSLAC of the NV centre. Due to the lower hyperfine coupling within the ground state, the polarisation occurs over a narrower range than the polarisation due to the excited state 89

90 crossing. However polarisation is still visible. This range will be important for NMR measurements in Chapter Experimental Procedure The process for completing a T 1 spectroscopic scan requires that the magnet is stepped along the z direction to vary the magnetic field strength B. The range is calculated in order to cross the resonance field points, B res, for the targets of interest. For each magnet z position, three operations are run consecutively. First, the magnet is scanned in the transverse xy plane in order to fine tune the alignment of the field based on the PL intensity as discussed above. Second, an ODMR spectrum of the NV centre is recorded in order to determine the NV transition frequency, ω NV, and infer the magnetic field strength B. To this end, the PL intensity is measured while sweeping the microwave frequency across the 0 1 resonance. To avoid power broadening [42], 300-ns laser pulses are interleaved with 1-µs microwave pulses corresponding to a π-flip of the NV spin on resonance. The spectrum is fitted with a Lorentzian function to obtain ω NV. The field strength B is deduced from the analytic form of the NV transition energy from Eqn Third, a sequence of 3 µs laser pulses separated by different wait times, τ, are applied. The sequence is repeated typically 10 6 times while the timeresolved PL is integrated. The resulting PL trace is then analysed to extract the quantities I S (τ) and I r (τ). Most results in Chapters 3 and 4 are presented in single-point T 1 mode. That is, a single τ point optimising the sensitivity of the measurement is 90

91 chosen. A reduction in T 1 gives rise to an increase in the decay rate which results in a drop in PL at that magnetic field point for the given τ. Having introduced the method of T 1 -based magnetic resonance spectroscopy, theoretically analysed the spectral response and detailed the experimental setup, we next look at implementing T 1 spectroscopy on a test spin target within the diamond. 91

92 Chapter 3 T 1 Spectroscopy of the P1 Centre in Diamond Much of the work in this chapter has been peer-reviewed and published in Ref. [84]. Testing the feasibility and applicability of T 1 spectroscopy required a test system to implement the measurement on. As discussed, the idea for T 1 spectroscopy was developed by Hall et al. [1] where they initially demonstrated T 1 spectroscopy for an ensemble sample of NV centres by detecting the spectrum of the P1 centre in diamond. The P1 centre is an ideal test target due to its unpaired electron spin and complex hyperfine structure. Additionally, any method for producing NV centres will also produce these substitutional nitrogen defects, allowing for easy interrogation without having to contend with the additional experimental complexities involved in mapping spins external to the diamond. Because of this ease of use, numerous studies have been undertaken on the P1 centre[66, 67, 85]. 92

93 In this chapter we perform the first single NV spectroscopy of immediately proximal P1 centres, extending the previous work on T 1 spectroscopy to the single NV level and thus probing the nanoscale spectroscopy of small numbers of electronic and nuclear spins. In addition, the move to single NV centres entirely removes the requirement for microwave pulsing since eliminating the contribution from off-axis NVs in the ensemble is not required. Finally, we interrogated the nuclear transitions of the P1 centre in order to investigate the potential to apply T 1 spectroscopy to lower frequency noise, in this case at 50 MHz rather than the 1.5 GHz of the electron spin transitions thus demonstrating its broadband applicability. 3.1 The P1 Centre in Diamond The P1 centre consists of a substitutional nitrogen impurity within the carbon lattice. It contains an electron spin- 1 and exists at a significant density 2 ( 1ppm) in type 1B diamond. This leads to a nearest-neightbour dipoledipole interaction with the NV centre spin on the MHz scale. This is stronger than the ensemble average interaction with the 13 C bath, the other most common spin defect in diamond, and thus the P1 centre is the dominant cause of decoherence within the environmental bath in these types of diamonds. In addition to the electron spin, nitrogen also contains a nuclear spin. For the following investigation the nitrogen spins were at natural abundance meaning the dominant isotope (99.6% abundance) was 14 N which has a nuclear spin-1. The hyperfine coupling (between the electron and nuclear spin) leads to a rich ESR spectrum and thus it is an ideal test target for NV relaxometry-based 93

94 magnetic resonance spectroscopy. The unpaired electron spin of the P1 centre sits on 1 of the 4 carbonnitrogen (C-N) bonds surrounding the substitutional nitrogen. Due to the diamond-cubic lattice, when we apply a well-aligned magnetic field along the NV quantisation axis, there are 2 possible orientations for the P1 centre nuclear-electron separation with respect to the background field. There is a 1 in 4 possibility that the unpaired electron spin is to be found on the single C-N bond aligned with the z-oriented magnetic field. This state will be referred to as an on-axis P1 centre. Additionally, there is a 3 in 4 chance that the unpaired electron sits on 1 of the 3 bonds equally misaligned with the background magnetic field which will be referred to as off-axis. This leads to separate hyperfine states P1 Hamiltonian The spin Hamiltonian for an on-axis P1 centre is H P1 /h = γ e BS (t) z ( +A S (t) x I x (t) where γ e = GHz/T and γ N γ N BI (t) z + S (t) y + A S (t) z I (t) y I z (t) ) ( ) + Q I (t) 2 (3.1) = khz/t are the gyromagnetic ratios of the electron and of the 14 N nucleus, respectively, A = MHz and A = MHz are the axial and transverse hyperfine coupling parameters, and Q = 3.97 MHz is the nuclear quadrupole coupling parameter [86]. Finally, S ( ) (t) = S x (t), S y (t), S z (t) and I ( ) (t) = I x (t), I y (t), I z (t) are the electron and nuclear spin operators of the P1 centre. Again we will use the superscript (t) to refer to a target spin being interrogated, in this instance the P1 centre, 94 z

95 while the superscript (p) will refer to the probe NV centre spin. The electron spin operators will always be denoted by S and the nuclear spin operators by I. B is the strength of the magnetic field aligned along the z axis. For an off-axis P1 centre, the magnetic field is now forming an angle with the intrinsic quantisation axis of the P1 centre. To express the Hamiltonian in the same z-basis where z is the direction of the magnetic field, one must apply a rotation of the spin operators. This leads to a Hamiltonian of the same form [1, 66] H P1 /h = γ e BS (t) z + A (S x (t) I x (t) γ N BI (t) z + S (t) y I y (t) + A S z (t) I z (t) ) + Q ( I (t) z ) 2 (3.2) where the apparent hyperfine and quadrupole coupling parameters are modified according to A = 1 9 (A + 8A ), A = 1 9 (4A + 5A ) and Q = Q P1 Centre Energy Levels We are able to find the energy levels of the P1 centre by solving for the eigenvalues of the Hamiltonian (Eqn. 3.1). To first order in A ωe these energies 95

96 are E h E h E h E h E h E h ( + 1 2, +1 ) = ω e 2 + A 2 + Q + ω N ( + 1, 0 ) = ω e A2 2ω e ( + 1 2, 1 ) = ω e ( 1 2, +1 ) = ω e ( 1 2, 0 ) = ω e 2 A 2 + A2 + Q ω N 2ω e 2 A 2 A2 + Q + ω N 2ω e 2 A2 2ω e ( 1, 1 ) = ω e A 2 + Q ω N (3.3) where the ket m (t) S, m(t) I indicates the state of the unperturbed P1 spin system. The states whose energy contains a term A 2 /2ω e are in fact perturbations on the electron and nuclear spin states stated since the transverse hyperfine coupling (A ) causes state mixing. Throughout this work we will always be looking at the P1 centre under the influence of a significant magnetic field (B 400G) leading to the electron + 1 and 1 being split by more 2 2 than 1GHz. In this instance the spin mixing is negligible and will be ignored. In Eq. (3.3) we introduced the Zeeman shift for the electron ω e = γ e B and for the nucleus ω N = γ N B. For all measurements in this chapter we used a type-ib diamond from Element 6 with a 14 N concentration specified to be < 200 ppm. The upper limit of 200 ppm corresponds to a median distance between an NV centre and the nearest P1 centre of 1.7nm. 96

97 3.1.3 Determining the Frequency Spectrum of the P1 Centre Evolution In order to calculate the spectrum of the P1 centre in diamond, and hence the resonance points for T 1 spectroscopy, we need to understand the frequencies at which the P1 centre evolves. To do this we consider the autocorrelation function of the dynamics of the P1 centre. The evolution operator of the P1 spin target is simply U P1 = exp( ih P1 ). (3.4) where H P1 is the Hamiltonian defined in Eqn The autocorrelation function of the x-basis of the P1 centre is } S x (t)s x (0) = e Γ(t) 2 t Tr {U t (t)s x U t (t)s x. Here the exponential decay simply simulates decoherence of the x-projection of the electron spin over a timescale Γ 2. The axial component, z, has a longer decay parameter, Γ 1, due to the quantisation of the P1 centre and hence this is the decay chosen for the autocorrelation function of the P1 centre s z- component. Taking the fourier transform of the autocorrelation function of the P1 centre s x-projection, we find the evolution frequencies of the P1 centre s lateral component. Similarly, the frequencies from the z-component of the P1 autocorrelation function were also found. All of the evolution frequencies of the P1 centre are shown in Table We then compared these frequencies to the energy levels of the P1 centre (from Eqn. 3.3), to deduce which transitions these frequencies correspond to. The states between which these transitions correspond to are also shown 97

98 Evolution Frequency P1 States Transition Component ω e + A + A2 2ω e + 1 2, , +1 Γ sing EPR ω e + A2 ωe + 1 2, 0 1 2, 0 Γ sing EPR ω e A + A2 2ω e + 1 2, 1 1 2, 1 Γ sing EPR ω e 2ω N + A2 ωe + 1 2, 1 1 2, +1 NA x,y Q ω N + A 2 + A2 2ω e 1 2, 1 1 2, 0 Γ NMR x,y Q + ω N + A 2 A ω e 2, , +1 Γ NMR x,y Q + ω N A 2 Q ω N A 2 x,y x,y x,y 1 2, 0 1 2, +1 Γ NMR x,y + 1 2, , 0 Γ NMR x,y ω e Q ω N + A 2 + A2 ωe + 1 2, 0 1 2, +1 Γ double EPR ω e + Q ω N A 2 + A2 + 1 ωe 2, 1 1 2, 0 Γ double EPR z z Table 3.1: The spectral components of the P1 centre s evolution, found via taking the Fourier transform of the autocorrelation function. The second column describes the spin states that each frequency is a transition between. The third column is the type of transition that corresponds to. The final column is which component of the autocorrelation function produces that evolution frequency. in Table 3.1. The first three rows are the transitions corresponding to a single electron spin flip. This type of transition we will label Γ sing EPR. However there are additional evolution frequencies also. The fourth row corresponds to a single electron spin flip along with a double transition of the nuclear spin. Requiring 3 separate spin transitions, the strength of this transition is highly suppressed. Due to this, and also because it closely overlaps with the far stronger nuclear spin zero transition of the form Γ sing EPR, this transition can be neglected. From the lateral (x,y) components of the autocorrelation function we 98

99 also find 4 transitions involving a single flip of the P1 nuclear spin. These transitions we label Γ NMR. These will be investigated later in this chapter. Finally, we find a pair of transitions from the longitudinal (z) component of the auto-correlation function. These transitions contain an electron spin transition along with a flip of the nuclear spin. We will refer to these as double EPR transitions, labelled Γ double EPR. 3.2 Electronic Paramagnetic Resonance on the P1 Centre in Diamond Initially we investigated conducting EPR on the P1 centres surrounding a single NV spin probe. From the previous calculation of the P1 evolution frequencies, we can see that there are 5 frequencies of the P1 centre corresponding to electronic spin transitions which should be detectable with the NV centre via T 1 spectroscopy. As discussed, the P1 centre can exist in two distinct hyperfine states and hence these 5 transitions can occur in either the aligned or misaligned states leading to a total of 10 transitions. By matching the energy transitions of the NV (Eqn. 2.40) to the various transitions of the P1, we solved for the resonant magnetic fields. These are shown in Table 3.2. Several individual NV centres were studied within the type IB diamond and gave consistent results. All measurements shown in the following have been obtained with the same NV centre, at room temperature. To acheive the EPR spectrum, the magnetic field was scanned in the range B G, corresponding to transition frequencies ω NV MHz. A probe 99

100 m (t) I Symmetry ω NV (B r ) (MHz) (on/off axis) Theory Exp. on (1) (1) off (1) (1) on (1) 1408(1) off (1) 1418(1) on (1) off (1) (1) off (1) 1460(1) on (1) 1468(1) off (1) (1) on (1) (1) Table 3.2: Summary of the theoretical and experimental EPR transition frequencies of P1 centres in diamond on resonance with a probe NV spin. The first column indicates the 14 N nuclear spin projection, m (t) I, for the singlequantum transitions, and the initial and final projections for the doublequantum transitions. The second column indicates the symmetry axis of the P1 centre. The theoretical uncertainties are estimated based on the uncertainty on the value of D NV, which is the dominant source of error here. The experimental values are extracted from fitting the spectra in Figs. 3.1a) and 3.1b) with a sum of Lorentzian functions, with the uncertainty indicated being the standard error given by the fit. evolution time of τ 1 = 40 µs was used to monitor the NV spin relaxation rate and detect cross-relaxation events as B is varied. We chose this evolution time after initial measurements showed this approximately optimised the sensitivity given the strength of the observed transitions. Fig. 3.1a) shows the normalised signal I s (τ 1 )/I r (τ 1 ) plotted against ω NV 100

101 (a) Magnetic field strength, B (G) I s (τ 1 )/I r (τ 1 ) On-axis P1 Off-axis P1 NV transition frequency, ω NV (MHz) (b) I s (τ 1 )/I r (τ 1 ) (c) m S (t) = Single quantum m I (t) Double quantum (d) I s (τ)/i r (τ) τ 1 ω NV = 1300 MHz ω NV = 1380 MHz ω NV (MHz) m S (t) = Wait time, τ (ms) Figure 3.1: (a) T 1 -EPR spectrum of P1 centres in diamond with τ 1 = 40 µs. (b) High resolution spectra of the double transitions (dotted regions in a.). Vertical lines indicate the theoretical frequencies for each allowed transition. Dotted (dashed) lines corresponding to the on-axis (off-axis) P1 centres with colors as defined in (c). (c) Energy levels of a single P1 centre showing the relevant transitions. (d) Full T 1 relaxation curve measured off resonance (black markers) and on a P1 resonance (blue markers). Inset: zoom-in of the on-resonance data for short evolution times. Dashed line indicates τ used in (a,b). (bottom axis) and B (top axis). The vertical lines in Fig 3.1a) correspond to the analytical transition values previously calculated where the blue lines are the single quantum transitions, Γ sing EPR, while the orange lines correspond to the transitions also involving a flip of the nuclear spin, Γ double EPR. The dotted 101

102 lines correspond to the transitions for an on-axis P1 centre while the dashed lines correspond to the transitions for an off-axis P1. The pulse sequence was repeated times and the vertical error bars represent one standard deviation in the shot noise. Horizontal errors depend on the fit of the ODMR taken at each point. Due to using pulsed ESR with a long Rabi period of 2 µs the horizontal errors are smaller than the data points in Fig. 3.1a) and b). In order to confirm the presence of both double transitions, zoomed data with lower shot noise was taken around these points. This is shown in Fig. 3.1b). These measurements confirm the presence of five transitions corresponding to a single electron spin flip seen in previous P1 studies [66, 67, 85] as well as the 4 transitions corresponding to the flip of the electron and nuclear spin. The weaker of these double transitions (those for an on-axis P1) had only been experimentally found by Hall et al. with their ensemble NV measurement [1]. This is the first detection of these transitions at the nanoscale of a small number of P1 centre surrounding a sinlge NV. Fig. 3.1c) is a schematic of the P1 energy levels in order to show which transitions are allowed. It should be noted that each P1 centre switches between all four symmetry axes on a time scale of a few ms [87], which is much shorter than our total measurement time of approximately 1 minute per data point. Therefore, even a single P1 centre will produce all of these resonance lines in the T 1 -EPR spectrum, with the on-axis lines being three times weaker than the off-axis lines, since all four possible axes have an equal rate of occurrence. To confirm that the increased NV relaxation rates were from on-resonance noise, we measured full T 1 relaxation curves for two different NV transition 102

103 frequencies, ω NV observed, and ω NV = 1300 MHz where no resonance with the P1 centres is = 1380 MHz which corresponds to the single-quantum transition with m (t) I = 1 of the on-axis P1 centres. This data is plotted in Fig. 3.1d). While the off-resonance data shows a single exponential behaviour with a characteristic decay rate Γ 1,ph = 220 ± 20 s 1 associated with phonon-dominated relaxation [53], the on-resonance data is closer to the bi-exponential behaviour predicted by a classical on-resonance noise source. This is the signature of a resonance process, where only one of the NV transitions (here m (p) S = 0 1) is driven by the environment while the other transition (m (p) S = 0 +1) remains unaffected. Thus we get two separate timecales of equilibration, firstly between the on-resonance 0 and 1 states governed by the dipole-dipole interaction and then the far from resonance +1 state is thermalised on the phonon timescale. From the dipole-dipole Hamiltoinian, we find that the spin relaxation rate caused by a single target electron spin at resonance, located at a distance r from the NV spin and forming an angle θ with the external magnetic field (Fig. 3.2a), is given by Γ EPR 1,r = 1 ( ) 2 ( ) µ0 γ NV γ e h 3 sin 2 2 Γ 2 2 θ (3.5) 2 r 3 where Γ 2 is the total dephasing rate of the NV-target spin system (see derivation in Sec ). In the experiment, a given P1 centre is on resonance with the NV only a small fraction of the time for a given resonant magnetic field, due to the four equiprobable symmetry axes, two electron spin states and three nuclear spin states. This results in an effective relaxation rate weaker than that stated by Eq. (3.5), e.g. by a factor 24 for the transition probed in Fig. 3.1d). Taking this into account, and assuming a typical dephasing 103

104 rate Γ 2 = 10 6 s 1 and angle θ = π/4, we infer a distance r 10 nm to the nearest neighbouring P1 centre, which contribute most to the signal [1]. It should be noted that this estimate predicts a significantly lower value for the P1 density (< 1 ppm) than that stated by the diamond manufacturer ( 200 ppm). However this estimate produces a low-bound for the density of P1 centres for three reasons. Firstly, the derivation of the decay curve assumes a magnetic noise that is constant in amplitude. In reality, when an environmental spin transfers its spin excitation to the NV centre, it is pumped into a state that does not produce on-resonance environmental fluctuations. This reduces the overall background noise strength from the nearest neighbor P1 at longer evolution times. Thus our decay is better fitted by a stretched exponential rather than a bi-exponential. Our bi-exponential fit underestimates that decay at very short evolution times (see inset of Fig. 3.1d), when the nearest neighbor should be closest to a thermally mixed state and hence overestimates the separation to the nearest P1. Secondly, during the laser pulse initialisation, the on-resonance T 1 decay will act to partially polarise the nearest P1 centres in its electron spin state that does not cause T 1 decay. While we did not quantify this in the electron spin case, we will discuss this ability for environmental spin polarisation in Chapter 5. Finally, even for a very short evolution time of 1µs, there is significant decay from that found off-resonance (see inset in Fig. 3.1d). This suggests that the NV centre is unable to be fully polarised due to the strong cross-relaxation process. This inefficient polarisation could similarly be seen as an initial measurement delay so that we are not able to interrogate the system at a true τ = 0 after electron initialisation. The impact here would be a significantly reduced 104

105 on-resonance decay rate compared to that from a thermal mixture. 3.3 EPR Spectrum of the P1 Centre In this section we conduct a full analysis of the autocorrelation function to compute the full EPR spectrum from the P1 centre, extending the method previously used to obtain the resonance frequencies in order to probe the strength of the transitions. In the previous section, we used a fully quantum mechanical approach to treat the case where the NV spin interacts with a single spin- 1 2 target. To treat the more complex case of the P1 center, which comprises a spin- 1 2 electron and a spin-1 nucleus, we employ a semiclassical approach where the NV quantum dynamics is calculated under the classical magnetic field generated by the target spin system. We already used this formalism to calculate the evolution frequencies of the P1 whereas here we use this to calculate the full spectrum. In Ref. [1], it has been shown that the two approaches give identical results for the case of a single spin- 1 2 target. According to the semiclassical approach, the NV relaxation rate at a given background magnetic field B can be expressed as [1]: Γ 1 (B) = 4b 2 (ω) [ Γ (p) 2 Γ (p) 2 ] 2 + [ωnv (B) ω] 2 P B (ω) dω (3.6) where Γ (p) 2 is the dephasing rate of the NV spin, ω NV (B) is the transition frequency of the NV spin, P B (ω) is the normalized distribution of transition frequencies of the target spin system (ie: the magnetic spectrum of the environment) at field B, and b is the mutual coupling strength between probe 105

106 and target. The task of determining the relaxation rate of the NV spin thus reduces to computing the associated coupling strengths, b, and frequency spectra, P B (ω), of the environment. This is achieved via examination of the Hamiltonian components associated with the NV-target interaction and self-interactions within the target system, respectively. If the target is a single spin- 1 2 with gyromagnetic ratio γ t, the coupling strength is obtained from the term of the dipole dipole interaction that corresponds to the resonance condition calculated in Section 2.2.2: ( ) ( ) µ0 γ NV γ t h 3 sin 2 b ± = 2 θ 1 ± 1, (3.7) 2 r 3 where the sign ± refers to the two possible resonances. For the P1 center, Eq. 3.7 can be used to model the single quantum EPR transitions (only b + in that case, with γ t = γ e ), the hyperfine-enhanced NMR transitions (with γ t = γ e ) which will be discussed later, and the direct NMR transitions (with γ t = γ N ). For the double quantum EPR transitions, the relevant term in the dipole-dipole interaction leads to an interaction strength [1] ( ) ( ) µ0 γ NV γ t h 3 sin 2θ b ± = 8. (3.8) 2 r 3 To determine the dynamic behavior of the P1 environment, we compute the autocorrelation functions associated with the field components of the target spin system. This is the same as how we determined the evolution frequencies Single Quantum EPR Transitions In the case of the single quantum transitions of the P1 centre, the relaxation of the NV spin is caused by its coupling to the lateral components of the 106

107 P1 spin. Thus, we compute the autocorrelation function associated with the lateral dynamics of the P1 spin, } S x (t)s x (0) single = e Γ(t) 2 t Tr {U t (t)s x U t (t)s x = e Γ(t) 2 t cos (ω t,i t) (3.9) {ω t,i } where the sum runs over the three Larmor precession frequencies {ω t,i } corresponding to the single-quantum EPR transitions, as given in Table. (3.1), assuming an on-axis P1 centre: (m (t) I one for each possible nuclear spin state = 0, ±1). The corresponding spectrum may then be found by computing the Fourier transform of the autocorrelation function, which gives P single (ω) = {ω t,i } Γ (t) 2 (Γ (t) 2 ) 2 + (ω ω t,i ) 2. (3.10) Inserting Eqs. (3.7) and (3.10) into Eq. (3.6) gives the relaxation rate as a function of the static magnetic field strength, B, about the single-quantum EPR transitions, Γ EPR 1 (B) = ( {ω t,i } ) 2 ( ) 2 µ 0 γ NV γ eh 2 3 sin 2 θ 2 r 3 [Γ (p) 2 +Γ(t) Γ (p) 2 +Γ(t) 2 2 ]2 +[ω NV (B) ω t,i (B)] 2. (3.11) This spectrum comprises three Lorentzian peaks corresponding to the three possible P1 nuclear spin states (m (t) I = 0, ±1). The amplitude of each peak, that is, the NV relaxation rate on resonance with a single-quantum EPR transition of the P1, matches that obtained using the fully quantum mechanical approach for a single spin- 1 2 (see Eq. (2.23)). Moreover, Eq. (3.11) shows that the line width of each resonance is governed by the total dephasing rate Γ 2 = Γ (p) 2 + Γ (t)

108 3.3.2 Double Quantum Transitions For the double-quantum EPR transitions of the P1 centre, the relaxation of the NV spin is caused by its coupling to the axial components of the P1 spin. Thus we compute, { } S z (t)s z (0) double = e Γ(t) 1 t Tr U T (t)s z U T (t)s z e Γ(t) 1 t 1 {ω t,j } 4 ( A ω e (3.12) ) 2 ( ) sin 2 ωt,j t (3.13) 2 where {ω t,j } are the two frequencies corresponding to the double-quantum EPR transitions, as given in Table. (3.1), and we retained terms up to ( ) A 2 order O in the prefactor. Here the damping factor corresponds to ωe 2 longitudinal relaxation since it applies to the z spin component, with a decay rate denoted as Γ (t) 1. Like before, computing the Fourier transform gives the associated spectrum, P double (ω). Inserting P double (ω) and Eq. (3.8) into Eq. (3.6) gives the field-dependent relaxation rate about the double-quantum EPR transitions, Γ EPR,double 1 (B) = ( ) 2 µ 0 γ NV γ eh ( 2 3 sin 2θ 2 {ω t,j } [Γ (p) 2 +Γ(t) r 3 ) 2 ( Γ (p) 2 +Γ(t) 1 ) 2 A ω e(b) 1 ]2 +[ω NV (B) ω t,j (B)] 2. (3.14) Compared with the single-quantum transitions, the relaxation rate is further damped by a factor of order A 2 /ω2 e. This is representative of the fact that the double-quantum transitions occur via a two-step process. For example, using the m (p) S, m(t) S, m(t) I basis, the transition 0, + 1, 0 1, 1, is a two-step transition via an off-resonant intermediate state 0, + 1 2, 0 0, 1, +1 1, 1, +1. As written here, the first step is enabled by

109 transverse hyperfine coupling within the P1 centre (strength A ), while the second step is enabled by the dipolar interaction between the NV and P1 electron spins (strength ω d ). However, although the initial and final state have the same energy at the resonant field B r, the intermediate state is detuned by an energy dominated by the electron Zeeman shift ω e = γ e B r and hence we see this significant damping term. At the field where these resonances occur (B 500 G), the prefactor is (A /ω e ) , which is why the double-quantum transitions appear significantly weaker than the single-quantum transitions in the T 1 -EPR spectrum. Apart from this suppression factor, the angular dependence also differs from that of the single-quantum transitions, because they rely on a different term of the dipolar interaction. This is illustrated in Fig 3.2b). For the situation where the relaxation is dominated by a single target electron spin, this provides a way to extract the position (r, θ) of the target relative to the probe. This opens the possibility of extracting spatial information on a target surface electron-nuclear spin system with a hyperfine splitting such as nitroxide spin labelled proteins [88]. In the present experiment, it was not feasible to implement such a spatial resolution scheme as the high density of P1 centres mean that several different impurities contribute to the signal Comparison with Double Electron Electron Resonance In order to contrast T 1 -EPR with the most common method of T 2 -EPR, a DEER spectrum was obtained from the same NV probe. This is shown in Fig. 3.2c) which exhibits only the five transitions associated with the 109

110 single-quantum P1 transitions as seen in the T 1 -EPR spectrum. The doublequantum transitions within the P1 centre are not addressable via a simple oscillating magnetic field and are therefore not seen in the DEER spectrum. (a) target (c) NV target π/2 τ/2 π τ/2 π/2 π B θ r NV PL (a. u.) Driving frequency (MHz) (b) T 1 -EPR (111) 0 (d) DEER (111) B 180 Single-quantum Double-quantum 180 Figure 3.2: (a) Definition of r and θ. (b) Plot of the interaction strength of the T 1 -EPR technique as a function of θ for single-quantum (red) and double-quantum (grey) transitions. The angles to a (100) and (111) surface are shown. (c) DEER spectrum (pulsing scheme above the graph) recorded at B 580 G with τ = 1.5 µs. (d) Plot of the signal intensity of the DEER technique as a function of the angle θ. In addition to probing transitions that DEER is unable to, T 1 -EPR could also act in a complementary fashion to DEER due to their different angular dependencies as discussed. The different angular dependencies are shown in 110

111 Fig. 3.2b) and d). The most common surface orientation for single crystal diamond samples is (100), where DEER has no sensitivity to spins on the surface directly above the NV, whereas T 1 -EPR has non-zero sensitivity for such spins. On the other hand, for (111) surfaces, T 1 -EPR has no sensitivity to spins located above the NV while DEER is at a maximum. Each technique is sensitive to spins at different angles and hence could be used in a complementary fashion to map the entire region around a single NV probe. 3.4 Hyperfine-Assisted Nuclear Magnetic Resonance Having shown the ability of T 1 spectroscopy to interrogate electron spins, we now look at whether such a technique can investigate spin species at a significantly lower frequency and whether it can interrogate nuclear spin transitions. To do this, we now turn to the nuclear spin transitions of the P1 centres, in which only the nuclear spin projection ( m (t) I = 1) changes, whilst the electron spin projection is conserved ( m (t) S = 0). These occur at transition frequencies from 40 to 70 MHz on either side of the GSLAC of the NV centre, determined by the hyperfine, quadrupole, and Zeeman couplings of the P1 nuclear spins. The exact forms of these are shown in Table. 3.1 and are labeled Γ NMR. Assuming direct dipole-dipole interaction between the NV spin and the P1 nuclear spin, the NV relaxation rate at resonance can be predicted using Eq. (3.5) by simply replacing the electron gyromagnetic ratio, γ e, by the 14 N gyromagnetic ratio, γ N. This leads to a decay rate weaker by a factor 111

112 of ( γ N / γ e ) than for the EPR transitions. This would lead to an on-resonance interaction significantly weaker than the phonon background meaning there would not be the required signal to noise ratio to measure these transitions. However, in the present case there exists another interaction mechanism that leads to a greatly enhanced relaxation rate, which we will refer to as hyperfine-enhanced NMR. Similar to double-quantum EPR, hyperfineenhanced NMR occurs via a two-step process, here involving a double flip of the P1 electron spin, e.g. 0, + 1, 0 0, 1, +1 1, + 1, +1, using the notation m (p) S, m(t) S, m(t) I. In this example, the first step is enabled by transverse hyperfine coupling within the P1 centre (strength A ), while the second step is enabled by dipolar coupling between the NV and P1 electron spins. The intermediate state is detuned from the initial and final states by an energy dominated by the electron Zeeman shift ω e = γ e B r. Thus it is the low-frequency noise of the EPR spectrum, caused by transitions of the nuclear spin, that is actually being interrogated rather than the NMR spectrum directly. In semi-classical terms, hyperfine-enhanced NMR can be interpreted as a modulation of the P1 electron spin precession caused by the hyperfine-coupled P1 nuclear spin precession, which acts as a beat frequency in the NV-P1 electron-electron interaction Theory on the competing interaction pathways Here we compare the strength of the two possible mechanisms for the P1 NMR spectrum: these low-frequency components of the P1 electron spectrum, as measured via the NV-P1 electron coupling; and the direct NMR 112

113 spectrum of the P1 nuclear spin. In what follows, we discuss the origin of these signals, and show that it is only the former that produces an appreciable signal. These results demonstrate that electron-mediated enhancement of NMR signals is a viable mechanism for vastly improved sensing of nuclear magnetic resonance. Hyperfine-enhanced NMR transitions. In determining the low frequency components of the P1 EPR spectrum near 1024 G, we proceed as ( ) A 2 in the EPR spectrum previously but retain terms of order O, where ωe 2 ω e 2.88 GHz at 1024 G is the P1 electron Larmor frequency. Furthermore, we ignore terms of frequency near ω e (those that cause the EPR transitions probed earlier), since these are too high to resonate with the NV frequency at this field. The relevant components of the autocorrelation function are given by S x (t)s x (0) hyp = e Γ (t) 2 t {ω t,k } ( A ω e ) 2 ( ) ωt,k t cos 2 (3.15) where {ω t,k } are the four frequencies corresponding to the NMR transitions, as given in Table 3.1, and we discarded the higher order terms in A ω e and A ωe in the prefactor. Note that although Eq. (3.15) refers to the autocorrelation function of the P1 electron spin, the dephasing rate here, Γ (t) 2, is that of the P1 nuclear spin. This is because the NMR frequencies {ω t,k } do not depend (at first order) on the P1 electron Larmor frequency, ω e, and therefore are not affected by the associated fluctuations. Inserting this into Eq. (3.6) gives the field-dependent relaxation rate about these hyperfine-enhanced NMR 113

114 transitions, Γ NMR,hyp 1 (B) = ( ) 2 ( µ 0 γ NV γ eh 2 3 sin 2 θ 1±1 2 {ω t,k } [Γ (p) 2 +Γ (t) r 3 ) 2 ( Γ (p) 2 +Γ (t) 2 ) 2 A ω e(b) 2 ] 2 +[ω NV (B) ω t,k (B)] 2. (3.16) In comparison with the single-quantum EPR transitions, the hyperfine-enhanced NMR transitions result in NV relaxation rates that are suppressed to an order of (A /ω e ) 2. The ± sign comes from a pair of distinct types of dipole-dipole interactions which will be discussed in detail in section Direct NMR transitions. We may apply the same approach as above to calculate the NMR spectrum associated with the direct coupling between the NV spin and the nuclear spin of the P1. The autocorrelation function is given by I x (t)i x (0) direct = e Γ (t) 2 t {ω t,k } cos (ω t,k t). (3.17) It is readily apparent that these dynamics are not suppressed like those in the hyperfine-enhanced case. Despite this, the resulting effect on the NV relaxation of the direct NMR transitions is much lower than those of the hyperfine-enhanced NMR, due to the differences in coupling to the NV spin. Inserting this into Eq. (3.6) gives the field-dependent relaxation rate about these direct NMR transitions, Γ NMR,direct 1 (B) = ( {ω t,k } ) 2 ( ) 2 µ 0 γ NV γ N h 2 3 sin 2 θ 1±1 2 r 3 [Γ (p) 2 +Γ (t) Γ (p) 2 +Γ (t) 2 2 ] 2 +[ω NV (B) ω t,k (B)] 2. (3.18) At the field where the transitions occur (B 1000 G), the suppression factor from the low-frequency EPR transition is (A /ω e ) As a result, the resonances in hyperfine-enhanced NMR are expected to be

115 weaker than in single-quantum EPR, but only 4 times weaker than in double-quantum EPR (due to the field being twice as large). However this is still vastly stronger than the direct NMR which suffers from the significant reduction in the dipole-dipole interaction strength due to the electron gyromagnetic ratio of the P1 centre being replaced by the far weaker nuclear gyromagnetic ratio of 14 N. Crucially, the hyperfine assisted NMR transitions are a factor of 10 5 stronger than if probed via direct dipolar interaction between the NV electron spin and the P1 nuclear spin Measurement of the P1 Nuclear Spin Spectrum To probe the NMR transitions experimentally, we chose a longer probe evolution time, τ 2 = 600 µs and normalised the signal using a reference probe evolution time τ 1 = 1 µs. The T 1 -NMR spectra recorded in the range MHz before the GSLAC (ω NV > 0) and after the GSLAC (ω NV < 0) are shown in Figs. 3.3a) and 3.3b), respectively, each revealing four wellresolved transitionss. A full T 1 relaxation curve recorded at one of the resonances (ω NV = +42 MHz, Fig. 3.3c) confirms that the interaction is significantly weaker than in the T 1 -EPR spectrum, with an induced decay rate Γ 1,r = 1.4(2) 10 3 s 1, but is far greater than if probed via direct dipole-dipole coupling alone. Each family of P1 centres (on and off axis) gives rise to four nuclear transitions on either side of the GSLAC (Fig. 3.3d), resulting in a maximum of eight transitions in both spectra. The theoretical values of the transitions are given in Table 3.3 and indicated as vertical lines in Figs. 3.3a) and 3.3b), showing excellent agreement with the values obtained from the experimental 115

116 (a) I s (τ 2 )/I s (τ 1 ) On-axis P1 Off-axis P1 Magnetic field strength, B (G) ω NV > (c) I s (τ)/i r (τ) τ 1 τ 2 ω NV = 70 MHz ω NV = 42 MHz NV transition frequency, ω NV (MHz) Wait time, τ (ms) A 2 /ω e +2ω N 2ω N 2Q (b) I s (τ 2 )/I s (τ 1 ) On-axis P1 Off-axis P1 Magnetic field strength, B (G) NV transition frequency, ω NV (MHz) ω NV < (d) m S (t) = m S (t) = 1 2 (t) m I ω NV > 0 ω NV < 0 Figure 3.3: (a,b) T 1 -NMR spectra of P1 centres in diamond after a wait time τ 2 = 600 µs, normalised to a reference wait time, τ 1 = 1 µs. In (a) the magnetic field is B < 1024 G, while in (b) B > 1024 G (see inset). (c) Full T 1 relaxation curve measured off resonance (black markers) and on resonance (blue markers). The vertical dashed line indicates the probe times τ 1 and τ 2 used in (a,b). (d) Energy levels of a single P1 centre showing the NMR transitions. Red crosses indicate transitions with the weakest relaxation rate, Γ NMR,hyp 1,r, that are not resolved. spectra. Those resonance with NV transition frequencies > 50 MHz are the on-axis P1 centres while those with NV transition frequencies < 50 MHz are the off-axis P1 centres. Not all nuclear transitions are resolved experimentally though, which is clearer for the on-axis P1 centres. There are 4 theoretical transitions in each case but only two are visible. Those transitions that do not appear in the experimental spectra are indicated by red crosses in the P1 energy schematic shown in Fig. 3.3d). 116

117 P1 Axis m (t) S m (t) I Before GSLAC ω NV (B r ) (MHz) m (t) I After GSLAC ω NV (B r ) (MHz) (on/off) Theory Exp. Theory Exp. on off (1) 51.6(1) (1) NR (1) (1) 60.8(1) (1) (1) 61.2(1) (1) NR (1) 54.9(1) (1) (1) (1) 42.0(1) (1) 42.5(1) (1) (1) (1) 45.6(1) (1) 45.8(1) Table 3.3: Summary of the analytic and experimental T 1 -NMR resonances of P1 centres in diamond measured by a probe NV centre. Uncertainties are based on the uncertainties in the hyperfine parameters [86]. The experimental values are extracted from the spectra in Fig The first column indicates the symmetry axis of the P1 centre. The second column indicates the electron spin projection (m (t) S ) and the third column indicates the transition for the 14 N nuclear spin projection (m (t) I ). NR, not resolved On the suppressed hyperfine-enhanced NMR transitions The reason for only detecting half of the transitions is due to the different dipolar coupling terms. Here we present an analysis of a pair of the hyperfineenhanced NMR transitions in order to explain the different decay strengths seen and why only half of the transitions are detected in these experiments. The P1 hyperfine interaction leads to transitions which flip both the elec- 117

118 tron and nuclear state of the P1. Similarly, the dipole-dipole interaction is responsible for flipping the electron spin of both the P1 centre and the NV. The hyperfine-enhanced NMR transitions are achieved via a doubletransition within the NV-P1 system involving one dipole-dipole interaction and one hyperfine interaction leaving the P1 electron in its initial state while both the NV and P1 nuclear spin are flipped. Consider the following NMR transitions within the NV-P1 system (once again with the entries in the kets being the NV electron spin, P1 electron spin and P1 nuclear spin respectively): 0, + 1, , + 1, 0 2 0, + 1, 0 2 1, + 1, If we write out the full double transition via an intermediate state along with each type of transition we have 0, + 1, +1 Dipole 1, 1, +1 Hyperfine 1, + 1, , + 1, 0 Hyperfine 2 0, 1, +1 Dipole 2 1, + 1, The first of these transitions has a dipole-dipole transition of the form 0, + 1, 2 m(t) I ( ) 2. 1, 1, 2 m(t) I, which has a spatial dependence of The second has 3 sin 2 θ r 3 a dipole-dipole transition of the form 0, 1, 2 m(t) I 1, + 1, 2 m(t) I ( ) which 3 sin has a spatial dependence of 2 θ 2 2. r 3 The total decay rate is the linear sum of all the contributing decays of atoms in the bath. Hence integrating these functions across all space gives the comparative strength of the transitions. Doing this in spherical coordinates 118

119 gives: 2π π ( 3 sin 2 θ ) 2 [ 16π r 2 sin θdrdθdφ = 6 ] 0 0 r min 2π π 0 0 r min r 3 15r 3 min ( ) 3 sin 2 2 θ 2 r 2 sin θdrdθdφ = 16π r 3 15r 3 min ( ) 2 3 sin So those transitions depending on 2 θ r are expected to be on average a 3 factor of on 32π / 16π 5rmin 3 15rmin 3 ( ) 3 sin 2 θ 2 2. r 3 ( The transitions depending on = 6 times stronger than those transition that depend ) 2 3 sin 2 θ 2 r were not resolved in our NMR 3 measurements because of this expected weaker transition strength. This analysis assumes an ensemble average of bath spin positions and for our single NV case it will depend on the exact position of bath spins. A longer probe time could be employed to detect those weaker transitions, however the sensitivity decreases for τ beyond the optimal measurement time of half the off-resonance T 1. At first instance it may seem we have neglected the possibility of the other possible intermediate state: 0, + 1, +1 Hyperfine 2 0, 1, 0 Dipole 2 1, + 1, 0 2 0, + 1, 0 Dipole 2 1, 1, 0 Hyperfine 2 1, + 1, However these transitions are not possible as the non-axial hyperfine does not cause transitions involving the s, + 1, +1 or 2 s, 1, 1 states. 2 m (p) m (p) 119

120 3.4.4 Evaluation of the P1 Hamiltonian Parameters The measurements of the nuclear spin transitions energies allows us to determine various Hamiltonian parameters of the P1 centres. For the on-axis P1 centres, the four resonance frequencies within the P1 system are ω t ( + 1 2, , +1 ) = A 2 A2 2ω e + Q ω N ω t ( + 1 2, , 1 ) = A 2 Q ω N ω t ( 1 2, 0 1 2, +1 ) = A 2 Q + ω N ω t ( 1 2, 0 1 2, 1 ) = A 2 + A2 2ω e + Q + ω N (3.19) where ω e = γ e B r and ω N = γ N B r are the electron and nuclear Zeeman shifts at the corresponding resonant field, and the kets denote the P1 states m (t) S, m(t) I. The double arrow accounts for the transitions both before and after the GSLAC, leading to eight resonances in total. From the experimental spectra, all frequency values of the four strongest transitions (with decay rate Γ NMR,hyp 1,r+ ) can be determined. Therefore, one can use Eqs. (3.19) to directly deduce the values of A, A, Q and γ N, as illustrated in Figs. 3.3a) and 3.3b). Here we find A = 114.2(1) MHz, A = 91(3) MHz, Q = 3.9(1) MHz and γ N = 1.9(5) MHz/T. These values are all in good agreement with the literature values obtained from ensemble measurements on P1 centres [86]. The estimate of the nuclear gyromagnetic ratio though is poor with an error of over 25%. This is due to the gyromagnetic ratio causing only a very small shift of the resonances compared to the other parameters. This illustrates that our technique has the capacity to measure the hyperfine and quadrupole coupling parameters as well as the gyromagnetic ratio of the target system 120

121 when interrogating only a small number of target spins surrounding a single NV centre. 3.5 Summary In this chapter we carried out a first demonstration of T 1 spectroscopy using a single NV centre. This extends the previous work to the nanoscale and allows for microwave free magnetic resonance spectroscopy. In addition to the ability to sense electron spins, we have shown the broadband nature of the technique and demonstrated that a nuclear spin hyperfine-coupled to a proximal electron spin can have its T 1 signal significantly enhanced. This introduces a novel method for interrogating magnetic noise across a wide frequency range. 121

122 Chapter 4 Nuclear Magnetic Resonance via T 1 Relaxometry Much of the work in this chapter has been peer-reviewed and published in Refs. [2, 84, 89]. 4.1 Introduction Nuclear magnetic resonance (NMR), and its associated imaging technologies were among the few greatest scientific discoveries of the 20th century. Such measurements gave us a new understanding across a whole range of scientific disciplines. Perhaps best known, it allowed for non-invasive biological sensing and imaging of hydrogen atoms, including detection of structure via analysis of the chemical shifts. This gave scientists a never before seen look into the processes of the human body, allowing for major advances in healthcare such as early detection of tumours. 122

123 Measurements of the P1 centre in diamond demonstrated the ability of T 1 spectroscopy to detect electron spins at the nanoscale as well as nuclear spins with a significant hyperfine interaction. While these measurements cover almost 2 orders of magnitude in frequency, bare nuclear spins (ie: those without a significant hyperfine coupling to a nearby electron spin) have a low gyromagnetic ratio and therefore come into resonance at an NV quantisation of < 5 MHz. In addition to the importance of showing detection at nuclear spin frequencies, the previous detection of nuclear spin transitions within a P1 centre utilised the electron to report the nuclear spin resonances, vastly increasing the interaction strength. Without the presence of the electron there is no ability to utilise the hyperfine-enhanced sensing that we have done previously and hence we will need to show that the NV can detect a far weaker signal. This chapter covers the detection of nuclear magnetic resonance via T 1 spectroscopy. Sec. 4.2 discusses the theoretical underpinnings of NMR via T 1 spectroscopy. We firstly consider the potential sensitivity of such a technique, investigating the interaction strength with a single nuclear spin. Next, we investigate the impact of the NV centre s hyperfine interaction with its nitrogen nuclear spin near the ground state level anti-crossing (GSLAC). Finally we consider a specific sensing geometry, that of a semi-infinite layer of nuclear spins external to a flat surface, to show the sensitivity we expect for a simple experimental demonstration which we aim to implement. In Sec. 4.3, we conduct an experimental study into the feasibility of conducting T 1 nano-nmr. This includes investigating the energy levels of the NV centre and the existence of factors that would cause a change in T 1 123

124 due to factors other than nuclear spin signals. We also investigate the NV readout mechanism and the NV s sensitivity to fluctuating fields near the GSLAC. Finally, in Sec. 4.4, we experimentally demonstrate T 1 -NMR on an organic sample external to the diamond and compare this new technique to the current state of the art technique. 4.2 Theory of NMR via T 1 Spectroscopy Interaction with a Single Nuclear Spin Before considering the NV-target spin system in all its complexity for T 1 - NMR, we initially considered a simplified system without hyperfine shifts of the NV states. We did this to make an initial estimate of the interaction strength between a target nuclear spin and the NV centre in order to determine whether nano-nmr via this method would be realistically achievable. To test this we considered the time taken to measure a single nuclear spin. This is similar to in Section however we do not limit ourselves to the small interaction strength regime and now numerically calculate our optimal measurement time for each resonance and at various angles and positions. For these calculations, the hyperfine interaction of the NV electron spin with its 14 N (or 15 N) nucleus was neglected. We first consider a single nuclear spin- 1 2 with gyromagnetic ratio γ t located a distance r from the NV spin and forming an angle θ with the direction of the external magnetic field (see Fig. 4.1a). The eigenstates of the non-interacting system are labeled as m (p) S, m(t) I where m(p) S and m (t) I are the projection of the NV electron spin 124

125 probe and target nuclear spin, respectively. Since γ t < γ NV, there are two resonances at magnetic field strengths B ± r, before and after the GSLAC. In this basis, the first resonance (B r ) corresponds to the transition 0, 1 2 1, while the second one (B+ r ) corresponds to 0, , 1 2, assuming γ t > 0. In the case where γ t < 0, the signs of the nuclear spin states would be reversed. As done in chapter 2, by solving the evolution of the system starting in either the state 0, 1 or 0, + 1, we can express the 2 2 relaxation rate Γ NMR 1,r± is fulfilled. We obtain Γ NMR 1,r± = 1 Γ 2 induced on the NV spin when each resonance condition ( ) 2 ( ) µ0 γ NV γ t h 3 sin 2 2 θ 1 ± 1 (4.1) 2 2 where Γ 2 is the total dephasing rate of the system which is the sum of the dephasing rates of the NV and nuclear spin (ie: Γ 2 = Γ 2 (NV) + Γ 2 (Nuc)). As a prototypical system, we consider a proton ( 1 H) spin, which has a gyromagnetic ratio γ t = MHz/T. The dephasing rate is assumed to be dominated by that of the NV spin since nuclear spins interact much more weakly with their environment than electron spins. We will take Γ 2 = 10 6 Hz, which corresponds to a typical dephasing rate for a near-surface NV centre [90]. Figs. 4.1b) and 4.1c) show the relaxation rates normalised to the phonon background, Γ NMR 1,r± /Γ 1,ph, computed as a function of r and θ using the above parameters along with a background phonon relaxation rate for the NV of Γ 1,ph = 200 s 1. The solid black line indicates the contour Γ NMR 1,r± /Γ 1,ph = 1, showing that the induced relaxation rate Γ NMR 1,r± r 3 reaches the phonon background relaxation rate Γ 1,ph for distances as large as 4.21 nm at θ = π/2 for Γ NMR 1,r+ and 3.68 nm at θ = 0, π for Γ NMR 1,r. To estimate the acquisition time that would be required experimentally 125

126 Figure 4.1: (a) The target spin, a bare nuclear spin, near a single NV centre. θ refers to the polar angle and r to the distance from the NV probe to the relevant nuclear spin. (b,c) Relaxation rate Γ NMR 1,r± induced by resonant interaction with a single 1 H nuclear spin, calculated as a function of r and θ at the resonant field B r + in (b) and Br in (c). The values are normalised by the background relaxation rate set to be Γ 1,ph = 200 s 1 and the ratio Γ NMR 1,r± /Γ 1,ph is plotted on a log scale. Black lines are contours corresponding to Γ NMR 1,r± /Γ 1,ph = 0.2, 1 and 7 from top to bottom. This translates into a total acquisition time of 5 min, 20 s and 1 s, respectively, in order to detect the interaction with a signal-to-noise ratio of 1. to detect a single proton spin, we need to compare the change of PL signal at resonance with the measurement noise. Assuming the noise is dominated by photon shot noise, the signal-to-noise ratio when measuring the PL signal after a wait time τ (i.e., I s (τ)) can be expressed as Rtro T tot 3C SNR(τ) ( τ 4 e Γ 1,phτ 1 e ) Γ 1,rτ (4.2) where R is the photon count rate under continuous laser excitation, t ro is the read-out time, T tot is the total acquisition time and C is the T 1 contrast as defined in Section Taking the derivative of this with respect to τ we are able to solve for τ opt, the optimal evolution time. In the limit 126

127 Γ 1,r Γ 1,ph we find that τ opt = (2Γ 1,ph ) 1, however in general the optimum wait time is shorter and depends on Γ 1,r. In this case we optimise τ for the given interaction strength Γ 1,r. We define the minimum acquisition time, T tot,min, as the time needed to obtain an optimised ratio SNR(τ opt ) equal to unity. Using typical experimental conditions, namely R = s 1, t ro = 300 ns and C = 0.25, we find that 20 s are required to detect an interaction such that Γ 1,r = Γ 1,ph. In Figs. 4.1b) and 4.1c) two other contours are shown corresponding to an acquisition time of 1 s and 5 minutes. The latter case enables the detection of a proton spin located up to 5 nm away from the NV probe while still allowing acquisition of a full spectrum in a few hours. To further investigate this, we look at detecting a single proton external to a particular surface. Fig. 4.2a) shows the interaction strength between an NV centre 2 nm below a (100) surface and a single proton on the surface. The strength is plotted for positions (x,y) from -5 nm to 5 nm. The NV is ( ) 1 positioned below (x,y) = (0,0) and is rotated arccos o away from the surface normal towards the +x direction as shown by the red arrow. The interaction at B r +, in the top panel, is significantly stronger than at Br, in the bottom panel. The B r + interaction plot contains a single lobe which reaches a maximum at (x,y) = ( 0.52, 0) nm with a value of 5.46 khz. In contrast, the Br plot contains a pair of lobes on either side of the (0,0) point since this corresponds to the magic angle and no interaction for this resonance point. The maximum interaction is 890 Hz at (1.1, 0) nm. The calculation is re-plotted in (b) however this time for a (111) surface with the NV aligned with the surface normal. These plots display the sym- 127

128 (a) (100) Surface (b) (111) Surface NV NV Hz NV NV Figure 4.2: (a) The on resonance signal, Γ ± r in khz between a 2nm deep NV centre and a single proton at that position on a (100) surface with Γ 2 = 1 MHz. The top (bottom) panel shows the interaction at B r + (Br ). The NV centre is rotated 54.7 o from the surface normal in the positive x-direction. (b) The same data for (b) but for a (111) surface with the NV directed along the surface normal. metry of the dipole-dipole interaction, displaying rotational symmetry about the surface normal. Here the B r + resonance has no interaction with a proton at (0,0) and reaches a maximum at (x 2 + y 2 ) = 1.63 nm with a value of 300 Hz. In contrast, the Br resonance is at a maximum at (0,0) with a strength of 3.86 khz and drops to zero at (x 2 + y 2 ) = nm, once again corresponding to the magic angle. In each case, the interaction strength at the B r + resonance retains significant interaction strength over a 128

129 far greater area of the surface compared to the Br interaction which shows much stronger localisation. Importantly, maximum interaction strengths in each instance are at least comparable to standard room-temperature phonon T 1 times (Γ 1,ph 200 Hz). This suggests single protons would be able to be detected via T 1 relaxometry with a 2 nm deep NV centre. In addition to this, we see the stark difference in angular sensitivities for the pair of resonance points. By comparing the strength at B r + to that at Br, spatial imaging of nuclear spins would be feasible. While shallow NV centres can suffer from instability, in recent years studies have been done with NV centres less than 2 nm deep [18]. From these calculations we conclude that single proton spin detection would be a possibility in a reasonable experimental timeframe. This motivates further investigation into T 1 -NMR NV state-mixing at the GSLAC Due to the low gyromagnetic ratio of nuclear spins, the points of resonance between the NV and the target nuclear spin will occur at a comparatively low quantisation of the NV of less than 5 MHz. For example, 1 H nuclear spins, with a high gyromagnetic ratio for nuclear spins of MHz/T, will come into resonance at 4.4 MHz while 13 C, with a lower gyromagnetic ration of MHz/T. will come into resonance at 1.1 MHz. Due to the non-axial hyperfine coupling, A, between the NV electron spin and the nitrogen nuclear spin (2.7 MHz for 14 NV and 3.65 MHz for 15 NV) being of a similar strength to the NV quantisation at these points, these off-diagonal 129

130 components of the Hamiltonian will cause state mixing between the NV spin levels. This occurs where the quantisation of the NV centre ground state approaches zero at the ground state level anticrossing (GSLAC). To investigate the possibility of nuclear spin detection via T 1 spectroscopy, we looked at the energy levels of the NV centre near the GSLAC which occurs at a magnetic field of 1024 G. It was important to investigate the energy levels of the NV centre and which transitions were able to be addressed by an environmental nuclear spin leading to an increased T 1 decay. We found the NV energy levels by solving for the eigenvalues of the NV centre Hamiltonian from Eqn For this we excluded the +1 level since it is at a large quantisation of 5.7 GHz, far from resonance. For a 14 NV, the energy levels are E h ( 0, +1 ) =Q ω N (4.3) E h ( 0, 0 ) =S 1 1 (4.4) E h ( 1, +1 ) =S (4.5) E h ( 0, 1 ) =S 2 2 (4.6) E h ( 1, 0 ) =S (4.7) E h ( 1, 1 ) =A + D + Q + ω N + ω NV, (4.8) 130

131 where S 1 = 1 ( ) D + Q ωn + ω NV A (4.9) 2 1 = 1 4A ( ) 2 A D Q + ω N ω NV (4.10) S 2 = 1 2 (D + Q + ω N + ω NV ) (4.11) 2 = 1 2 (S 2 ) 2 4 (DQ A Dω N + Qω NV + ω N ω NV ). (4.12) For a 15 NV, the energy levels are where S 3 = 1 4 E ( ) 0, + 1 ω N 2 = (4.13) h 2 E ( 0, 1 h 2 ) =S3 3 (4.14) E ( 1, + 1 h 2 ) =S3 + 3 (4.15) E ( ) 1, 1 1 ( ) 2 = A + 2D + ω N + ω NV, (4.16) h 2 ( 2D A + ω NV ), (4.17) 3 = 1 4 (A 2D 2ω NV ) 2 4 ( 2DωN 2A 2 A ω N ω 2 N + ω Nω NV ). (4.18) The non-axial hyperfine, A, induces state mixing around the GSLAC and thus the m s, m I spin states are not eigenstates for all magnetic fields. Those states which mix are labelled with an asterisk and the labels m s, m I are the correct spin state for that energy at magnetic fields much lower than the GSLAC. Those states with no asterisk are eigenstates of the system across all magnetic fields. For the 14 NV, near the GSLAC there are two eigenstates which are superpositions of 0, 0 and 1, +1 and two eigenstates which are 131

132 Relaxation rate, Γ 1 (s -1 ) superpositions of 0, 1 and 1, 0. For the 15 NV, near the GSLAC there are two eigenstates which are superpositions of 0, 1 2 and 1, (a) (b) Magnetic Field (G) Magnetic Field (G) (c) 1 H NV B θ r 13 C (d) C 1 H θ = 0 θ = π/4 θ = π/2 θ = 0 θ = π/4 θ = π/ Relaxation rate, Γ 1 (s -1 ) Magnetic field strength, B (G) Figure 4.3: (a,b) The transition frequencies from the initial state for a 14 NV (a) and a 15 NV (b). The levels in red are those that can be coupled to via a dipole-dipole interaction with a background spin. (c) The target spins considered are 1 H, generally found external to the diamond, and 13 C, found within the diamond lattice. (d) Simulated T 1 -NMR spectrum of a single 1 H spin (black lines) or a single 13 C spin (blue lines) as detected by a 14 NV. The target spin is assumed to be at a distance r = 3 nm from the NV spin. Apart from the state mixing, another implication of the non-axial hyperfine in this region is that the NV nuclear spin is polarised. The laser pumping causes transitions of the electron spin into the 0 state while conserving the nuclear spin state. Meanwhile the hyperfine causes mixing between the 0, m I and the 1, m I + 1 states. Overall this leads to the NVs being 132

133 polarised into the 0, +1 and 0, +12 states for 14 NV and 15 NV respectively. Fig. 4.3 shows the energy transitions from the respective initial state across the GSLAC for a 14 NV (a) and a 15 NV (b). A dipole-dipole interaction with a background spin is able to cause a single flip of the NV electron spin. Hence only those states containing a proportion of 1, +1 and 1, can be coupled to for 14 NV and 15 NV respectively. Those transitions which can be coupled to via a single NV electron flip are shown in red while the other transitions are shown with grey dotted lines. The Hamiltonian of the spin- 1 2 the Zeeman term target nuclear spin is simply composed of H/h = γ t B. I (t) = γ t B z I (t) z where I (t) are the spin- 1 2 operators, B is the magnetic field and the second line assumes a field aligned in the z-axis, along the NV quantisation axis. From this we can solve for the magnetic fields corresponding to the resonance points B ± r γ t = ω NV where ω NV corresponds to those addressable transitions shown in red in Fig. 4.3(a) and (b). On these plots the resonance points with a 1 H nuclear spin and shown in dark blue and the resonance points with a 13 C nuclear spin are shown in light blue. In calculating the interaction strength in Sec , the hyperfine interaction of the NV electron spin with its 14 N (or 15 N) nucleus was neglected. To account for this effect, we numerically simulated T 1 -NMR spectra of a 133

134 single 1 H spin while considering the full hyperfine structure of the 14 NV centre. In the simulation, the NV spin is initialised in the state 0, +1 while the target spin is initialised in a thermally mixed state. The system s evolution is computed under the same assumptions as in Sec , with a total dephasing rate Γ 2 = 10 6 Hz. The population remaining in the NV 0 state as the system evolves is then used to infer the relaxation rate Γ 1,r. The simulation was done via evolution under the Lindblad equation from Eq. (2.16). The super-operator formalism was used to allow timesteps longer than the dephasing time of both the NV and the target nuclear spin. Fig. 4.3d) shows Γ 1,r as a function of the magnetic field strength B across the GSLAC for a single 1 H spin located at a distance r = 3 nm with various angles θ (see Fig. 4.3c). Also shown for comparison is the case where the target is a single 13 C spin. In both cases, three peaks are observed in the spectrum. The two side peaks correspond to the NV-target resonances and occur at fields B r + = G and Br = G for the 1 H case, and B r + = G and Br = G for the 13 C case. This matches with the analytical values calculated via the energy transitions shown in (a). The splitting between the two resonances is directly related to the gyromagnetic ratio with a correction due to the level avoided crossing causing an asymmetry about the central feature. Importantly, the peaks in T 1 decay are easily distinguishable showing spectroscopic capabilities. The comparison of the different angles θ illustrates the different angular dependences for the two resonances as expressed in Eq. (4.19). Here we see directly that T 1 -based NMR spectroscopy would enable not only identification of unknown spin species but also quantification of their densities, via the 134

135 strength of decay, as well as calculation of angular positions, by comparing the decay rate for each resonant transition. We note that the width of the resonances that is, the spectral resolution scales with the dephasing rate Γ 2 [1], in our case 1 MHz. It can therefore be improved by several orders of magnitude by engineering NV centres in high-purity diamond crystals [91]. The central feature common to the spectra of both species at B = G is specific to the GSLAC of the NV rather than the target spins themselves. It occurs at the magnetic field where the initial NV state 0, +1 crosses with one of the eigenstates containing a superposition of 1, +1 and 0, 0. These states can be linked via an oscillating magnetic field and thus, at resonance, these can be coupled via a static transverse magnetic field. The impact of this crossing and other crossings within the 14 NV and 15 NV systems are discussed in Sec In this instance, the non-axial component of the effective field produced by the target spin is what is causing these decays. In reality this crossing will be sensitive to any non-axial field caused by environmental noise rather than the target nuclear spins alone. Therefore this transition is not relevant for determining the nuclear spin species. These calculations show that nuclear spin detection could be theoretically carried out by T 1 spectroscopy at a distance of 3 nm in a reasonable experimental timeframe. This shows that T 1 -NMR would be able to be done at a sensitivity approaching the single spin level Detecting an ensemble of nuclear spins While single nuclear spin sensing would be the ultimate goal, as a simpler proof of principle we initially planned to detect an ensemble of nuclear spins. 135

136 We considered protons external to the diamond. Here we look at the interaction strengths expected from an ensemble of nuclear spins. Once again, in order to allow an analytic calculation, we have neglected the hyperfine interaction of the NV centre with its nitrogen spin to calculate the interaction strength. We are able to formulate the exact value by modifying this by a constant that takes into account the state mixing at each resonance position. In the experimental demonstration of T 1 -NMR, we will look at detecting the protons within a semi-infinite sample external to the diamond and hence we now calculate the expected interaction strength for such a measurement. Our aim is to consider what depth we can expect an NV to be sensitive to protons within objective lens oil external to the diamond beyond the background noise. To do this, we take an NV interacting with a single environmental proton in the correct initial state to cause decay. The induced decay rate after and before the GSLAC, Γ i 1,sig,±, when at the resonance point, is [84] Γ i 1,sig,± = 1 2Γ 2 ( ) 2 ( ) µ0 γ NV γ t h 3 sin 2 2 θ i 1 ± 1 (4.19) 4π where θ i is the polar angle of separation of the i th relative to the NV quantisation axis, r i is the separation distance, and Γ 2 = Γ 2,NV + Γ 2,H Γ 2,NV. The total decay rate for an ensemble of hydrogen atoms is then just a linear sum over all the decay rates contributed by the individual protons, r 3 i Γ 1,sig,± = i Γ i 1,sig,± (4.20) = ρ oil 4Γ 2 ( ) 2 µ0 γ NV γ t h 4π V ( ) 3 sin 2 2 θ 1 ± 1 dv (4.21) where ρ oil is the density of protons within the oil and V is the volume occupied 136 r 3

137 by the protons. There is a factor of 1 included since the average population 2 of each proton in the correct level to cause a transition is 0.5 due to it being a thermally mixed 2-level system. Analytically we will consider two special cases. Firstly we consider detection using an NV beneath a (100) surface. This is the surface used for our later experiments as well as being the predominant surface preparation of bulk diamond samples. In order to complete the integral, we need to transform into the (x, y, z) basis of the (100) surface, where ˆ z is the outward surface normal. This means that the integral will be over a semi-infinite layer z > d where d is the NV depth within the diamond. These transformations are: ( ) 3 sin 2 2 ( θ 3x 2 + y 2 2 2yz + 2z 2) 2 = r 3 (x 2 + y 2 + z 2 ) 5 (4.22) ( ) 3 sin 2 2 ( θ 2 x 2 y 2 2 2yz ) 2 = r 3 (x 2 + y 2 + z 2 ) 5. (4.23) Solving the integrals with these substitution gives: ( ) 3 sin 2 2 ( θ 3x 2 + y 2 2 2yz + 2z 2) 2 dv = z>d r 3 z>d (x 2 + y 2 + z 2 ) 5 dv ( 3x 2 + y 2 2 2yz + 2z 2) 2 = = d d (x 2 + y 2 + z 2 ) 5 dxdydz (92y y 3 z + 564y 2 z yz z 4 ) ( ) 9 π 1 2 dydz 128 y 2 + z 2 19π = d 8z dz 4 = 19π 24d 3 137

138 and ( ) 3 sin 2 2 ( θ 2 x 2 y 2 2 2yz ) 2 dv = z>d (x 2 + y 2 + z 2 ) 5 dv ( x 2 y 2 2 2yz ) 2 z>d r 3 = = d d (x 2 + y 2 + z 2 ) 5 dxdydz (28y y 3 z + 276y 2 z yz 3 + 3z 4 ) ( ) 9 π 1 2 dydz 128 y 2 + z 2 3π = d 8z dz 4 = π 8d. 3 The total decay rate for the NV on resonance after (+) and before (-) the GSLAC for an NV beneath a (100) surface is therefore: Γ 1,sig,+ = ρ ( ) 2 oil µ0 γ NV γ t h 19π (4.24) 4Γ 2 4π 24d 3 Γ 1,sig, = ρ ( ) 2 oil µ0 γ NV γ t h π 4Γ 2 4π 8d. (4.25) 3 This shows that, for a (100) surface detecting a sample with a constant density of nuclear spins external to the surface, the higher-field resonance should be a factor of 19 3 stronger than the lower-field resonance. By doing the same for the (111) surface with the NV normal to the surface we find Γ 1,sig,+ = Γ 1,sig, = ρ oil 4Γ 2 ( µ0 γ NV γ t h 4π ) 2 π 4d 3. (4.26) Interestingly, when integrating over the semi-infinite volume for the (111) surface, we find the same interaction strength for each resonance. This is in spite of there being no interaction with a nuclear spin directly above the 138

139 Decay Rate (khz) Ratio of Signal (a) (100) (b) (100) π Surface Angle 0 0 π Surface Angle Figure 4.4: (a) The interaction strength for an NV 10 nm beneath a surface, interacting with a (100 nm) 3 cube of protons external to that surface. This is plotted against the surface angle corresponding to the angle between the NV quantisation axis and the surface normal. The dephasing rate was taken to be Γ 2 = 1 MHz and the proton density of the oil was taken to be 50 nm 3. The B r + (Br ) resonance is plotted as the solid (dotted) blue line. (b) The ratio of the interaction strengths Γ NMR Tot,+ /ΓNMR Tot, as a function of surface angle. surface for the high-field resonance, B r +. In contrast, Br is at an interaction maximum for this angle. Hence, despite leading to the same total interaction strength, these two interactions are contributed by nuclear spins within different volumes. The Br resonance interrogates those nuclear spins within a small volume directly above the NV while the B r + interrogates nuclear spins shifted some distance from that point as shown previously in Fig This would allow interrogation of different volumes around the NV centre. Beyond this analytical calculation, a numerical calculation was done simulating a (100nm) 3 cube of proton spins above an NV centre 10 nm beneath the surface. The dephasing was once again taken to be 1 MHz and the density of the oil was taken to be 50 nm 3. The interaction strength is plotted 139

140 in Fig. 4.4(a) as a function of the surface angle, the angle between the NV quantisation axis and the surface normal. Importantly, the expected values closely match the analytical calculations for those surfaces discussed and they clearly exceed the phonon relaxation rate of 200 Hz. Additionally, the ratio of strength of the two transitions is plotted in Fig. 4.4(b). This shows a range of ratios with the pair of resonances giving the same total interaction at a surface angle of 0, corresponding to the (111) surface through to Γ NMR Tot,+ being almost a factor of 12 larger than Γ NMR Tot, at a surface angle of π Sensing Volume The final analysis before moving onto the experimental implementation was to consider what sensing volume would be interrogated. We want to understand at what distance our sensing region is confined to in order to put a limit on the ensemble size we are looking at. Although interrogating individual nuclear spins would be ideal, nanometre scale interrogation of ensemble samples could still significantly advance our understanding of certain important structures. We calculated the sensing volume numerically. To do this we considered a hemisphere of radius r centred directly above the NV centre. We did this for a (100) surface as this is what we will look at experimentally in Sec The signal was integrated in spherical coordinates and the signal from the protons within a sensing volume of the given radius is shown in Fig. 4.5(a). As the sensing region grows larger the interaction strength asymptotes towards those values plotted in Fig. 4.4(a) for a (100) surface. Fig. 4.5(b) shows the fraction of the total signal within that sensing region. For the case of 140

141 (a) (b) Radius of Sensing Region (nm) Radius of Sensing Region (nm) Figure 4.5: (a) The signal produced by the protons within the hemispherical sensing region with the given radius for each resonance for an NV 10 nm below the surface. (b) The fraction of the total signal produced by the protons within the sensing region for each resonance. the resonance at B r +, half the signal is contributed by the protons within a hemispherical region of radius 10.3 nm and 95% of the signal is contributed by the protons within a radius of 35.4 nm. Similarly, at the Br transition, half the signal is contributed by the protons within a hemispherical region of radius 12.1 nm and 95% of the signal is contributed by the protons within a radius of 37.2 nm. This corresponds to 50% of the signal being contributed by the nearest protons while 95% of the signal is contributed by the nearest protons. 141

142 4.3 Feasibility Study towards T 1 -NMR Before conducting T 1 -NMR, it is imperative to understand the behaviour of the NV centre near the GSLAC. We investigated a number of phenomena near the GSLAC in order to better understand how T 1 -NMR would be acheived. The major questions we needed to answer were: 1. How does the readout process work for the NV centre near the GSLAC? 2. How well are the NV centre electron spin and nuclear spin polarised across the GSLAC? 3. Are the transitions of the NV centre addressable via an interaction with a background spin at < 5 MHz quantisation? 4. How does the background relaxation rate change across the GSLAC and does this impact the measurement process? 5. Is a weak oscillating AC field, mimicking a nuclear spin signal, able to be measured via T 1 spectroscopy at the frequencies of nuclear spin resonances? To answer these questions we investigated the GSLAC of both isotopes of NV centre in various samples. In addition to these questions about the nature of the NV centre around the GSLAC, we needed to ensure that our experimental setup was able to access this magnetic field regime while still allowing alignment and good magnetic field stability. 142

143 4.3.1 Optically Detected Magnetic Resonance at the GSLAC We started by performing an ODMR study across the GSLAC. Initially we looked at individual NV centres deep (d 200 nm) within an isotopically purified diamond ([ 12 C] > 99.99%). We did this in order to eliminate as much environmental noise as possible and investigate the transitions within the NV centre under optimal conditions. If the NV centre was not well behaved across the GSLAC under these conditions then introducing a nearby surface for external spin sensing would be futile. As discussed previously, an oscillating magnetic field acts to drive the NV centre in the same way as a dipole-dipole interaction with an environmental spin, and hence the ODMR would show the transitions that could be addressed for the sake of T 1 -NMR. Fig. 4.6 shows the ODMR spectra across the GSLAC of both a 14 NV (left panels) and a 15 NV (right panels) in this sample. Overlayed on these plots are the theoretical transitions possible assuming no polarisation of the nuclear spin. The data in the top plots is the high-energy transition of the electron spin state, The single dominant transition shows that the NV nuclear spin is efficiently polarised for both NV isotopes, into +1 for the 14 NV and for the 15 NV. The only exception to this is as the 15 NV reaches a magnetic field of B 1026 G where the presence of a second transition line suggests imperfect polarisation of the NV nuclear spin. This polarisation of the nuclear spin has been well documented at the excited state level anti-crossing [83, 92], but has not been previously quantified experimentally across the GSLAC. Using the relative strengths of the electron spin 0 +1 transitions, we find that the nuclear spin is 143

144 Figure 4.6: ODMR spectrum of a 14 NV (left) and 15 NV (right), measured as a function of the axial magnetic field strength across the GSLAC. The top panels show the high energy transitions of the electron, 0 +1, while the bottom panels show the low energy transitions Overlaid are theoretical plots of all potential transitions. Nuclear spin polarisation makes some of the transitions dominant (shown as solid lines), while the strength of the rest are negligible (dashed lines). polarised to 90% for the 14 NV across the whole range of fields scanned here. For the 15 NV, the nuclear spin is polarised to 90% up to fields of B 1026 G but then becomes completely unpolarised above B 1028 G. Of particular interest to T 1 -NMR is the low frequency electronic transitions on the bottom panels in Fig The 14 NV shows an avoided crossing, seen by asymptotic behaviour of the energy levels, at B 1022 G and at a frequency of 5 MHz, matching the transitions plotted previously in Fig This frequency shift is due to the nuclear quadrupole moment 144

145 (Q = 4.95 MHz) shifting this avoided crossing away from ω = 0 MHz. The consequence of this is that signals of frequencies of 4 6 MHz would show only very weak contrast in T 1 spectroscopy for the transition at Br, the transition at a magnetic field lower than the GSLAC. Thus protons, whose gyromagnetic ratio causes a signal of 4.4 MHz, would be very difficult to interrogate at the Br transition. Conversely, the presence of the nuclear quadrupole moment leads to transitions with little state mixing down to sub-mhz frequencies. This would be essential to interrogate nuclear species with significantly lower gyromagnetic ratios than protons, such as 13 C, which would produce a signal of 1.1 MHz, or 2 H, which would produce a signal of 0.67 MHz. Additionally, beyond the crossing at B 1024 G there is a single clean transition with limited state mixing, ideal for T 1 -NMR. It is this transition that would provide the simplest initial detection for T 1 -NMR. In the bottom-right panel of Fig. 4.6, the low-frequency transitions of the 15 NV also show asymptotic behaviour of the two transitions however this is at close to zero frequency. Once again the plot closely mirrors the theoretical transitions overlayed on the graph. The impact of the lack of quadrupole moment is that any signals below 2 MHz will be very difficult to interrogate due to state mixing strongly limiting the transition strength. However for signals 3 MHz, there is a pair of clean transitions where the state mixing does not significantly limit the detection ability. This allows for both resonances to be interrogated and suggests the possibility for angular information about the environmental spins to be investigated as was detailed in section In addition to this information, the ODMR trace of the 15 NV allowed us 145

146 to measure the non-axial hyperfine coupling, A, for a single NV. Taking our analytic forms of the NV energy levels in eq. 4.13, and solving for when the pair of transitions from the initial state to each of the mixed states have the same magnitude we find analytic expressions for both the magnetic field and transition frequency at which these states cross: B = A 2D 2 (γ N + γ N V ) ( ) ( A 2 ω = 2 + γn 2 D A ). γ N + γ NV 2 Since we know the parameters of the system and can take the axial hyperfine, A, from the ODMR splitting, we are able to calculate the perpendicular hyperfine parameter for this NV from the transition frequency at the crossing point. We find a value of A = 3.69 ± 0.03 MHz in good agreement with the ensemble-averaged value of A = 3.65 ± 0.03 MHz [93]. The most important result from this ODMR data is that the NV still behaves well as a quantum system with initialisation and readout of the spin state possible at transitions low enough for NV cross-relaxation with environmental nuclear spins to be detected. In addition, the readout process was confirmed to be still simply a measure of the population within the electron 0 state. The fact that we see a second transition for the 15 NV at fields higher than 1026 G shows that the contrast occurs for both NV nuclear spin states, whether or not they are in the nuclear spin state that corresponds to an eigenstate. The second transition seen at these fields is a transition between the 0, 1 2 and the 1, 1 2 states suggesting that both the 0, and 0, 2 1 states are bright states while the 1, and 1, 2 1 are both dark states. This is despite the 0, 1 2 and 1, states undergoing state 146

147 mixing across the GSLAC and thus neither being eigenstates of the system T 1 Decay Features at the GSLAC caused by NV level crossings In order to detect environmental spin species via T 1 spectroscopy, we have to be certain that any T 1 features at the correct frequency for a particular spin species do not come from other interactions between the environment and the NV. In particular, non-axial magnetic fields were found to be able to drive transitions when certain levels crossed with the initial state. That was seen by the central feature present in Fig. 4.3d), where the non-axial magnetic field provided by the environmental nuclear spins caused a T 1 decay. Here we provide a theoretical analysis of the crossing features present near the GSLAC as well as experimental measurements to determine their strength. The 14 NV energy levels as a function of the aligned field are shown in Fig 4.7. The initial state, 0, +1 has 3 separate crossings at magnetic fields of G, G, G. There is no transverse field component in this case. The lower plots are zoomed in on each crossing with the black lines being the energy levels under no transverse field while the red lines have a transverse field of 0.1 G. At G the initial state 0, +1 crosses with an eigenstate that is α 0, 1 + β 1, 0 where α α 0.76 and β β At G the initial state 0, +1 crosses with the 1, 1 state. These two transitions show a slight avoided crossing under the application of the non-axial field ( 10 khz). This indicates that the states are only very weakly coupled and there was no experimental evidence of these crossings causing an increased 147

148 (a) Magnetic Field (G) Magnetic Field (G) Figure 4.7: The energy levels of a 14 NV across the GSLAC. Below, the energy levels near each crossing with an applied transverse field of B x = 0 G (black) and B x = 0.1 G (red). decay rate of either the electron or nuclear spin of the NV centre. When we consider the states that are being coupled this is unsurprising as all of these transitions require multiple spin flips. The major crossing that needs to be considered for a 14 NV is the one at G. Here the initial state 0, +1 crosses with an eigenstate that is α 1, +1 +β 0, 0 where α α 0.21 and β β Due to the transition being a single transition of the electron or nuclear spin, this crossing has a significantly stronger coupling for even a slight non-axial magnetic field as shown by the avoided crossing being far larger ( 300 khz) under a 0.1 G 148

149 Figure 4.8: a) Single-point T 1 measurements taken across the GSLAC for a shallow implanted 14 NV clearly showing the strong T 1 decrease at 1024G. b) T 1 curves out to 100µs at the 1024 G crossing and away from the 1024G crossing. The NV is unable to be polarised at the crossing. c) PL scans across the GSLAC recorded with 14 NV centers in various diamond samples. (i) Deep NV center in isotopically purified CVD diamond (ii) Deep NV center in natural-isotopic-content CVD diamond. (iii) Shallow NV center in CVD diamond as in (a) and (b). (iv) Deep NV center in type-ib diamond. The curves are vertically offset from each other for clarity. transverse field. To investigate this, 14 NV centres in a number of different NV samples were investigated by single point T 1 measurements taken across the GSLAC. 149

150 Fig. 4.8(a) shows data from a shallow ( 10 nm) implanted 14 NV in a CVD diamond. The single point T 1 is taken across this crossing feature for evolution times of 2 µs, 10 µs and 1 ms. There is a decay centred at the crossing feature. The lack of signal at 2 µs shows that the NV centre experiences a strong enough decay due to this crossing that it is unable to be fully polarised. To confirm this, a full T 1 curve out to 100 µs taken near the crossing point (at 1024 G) and away from the crossing point (at 1020 G). These are shown in Fig. 4.8(b) with the measurement away from the crossing showing a T 1 decay with a characteristic time of 300 µs. In contrast, the data taken at the crossing shows such a strong T 1 decay that there is no ability to even polarise the NV centre. Importantly this shallow implanted NV centre is critical for the ability to detect nuclear spins external to the diamond. The crossing feature has a full width half maximum of 1.3 G, covering transitions 1.5 MHz. Since protons come into resonance at a transition strength of 4.4 MHz this suggests the ability to image protons with this shallow NV centre. These measurements across the GSLAC were conducted on 4 different samples and the results are plotted in Fig. 4.8(c). In each case the timestep was optimised to give the greatest contrast for the central feature. The four plots are: (i) a deep NV in isotopically purified CVD diamond; (ii) a deep NV in CVD diamond with a natural isotopic abundance ( 1.1% 13 C); (iii) a shallow NV in CVD diamond with natural isotopic abundance; (iv) a deep NV in type 1B diamond with N density of 200 ppm. The deep NV centres in CVD diamond showed a central feature of 0.4 G width regardless of the presence of 13 C. In contrast, the near surface NV showed this broadened to 150

151 1 G. This suggests that the presence of 13 C within the crystal is a minor consideration in comparison to the surface when looking at the contribution of low-frequency noise. This is unsurprising as the gyromagnetic ratio of 13 C is very weak ( γ = MHz/T) meaning the major noise contribution is simply its Lamor frequency since the interactions between carbon-13 spins produce only sub-mhz noise. The width of this feature is important in determining whether a particular spin species can be detected. This strong background T 1 decay due to the crossing feature would lead to an inability to detect nuclear spins within this feature. The blue line in Fig. 4.8(c) shows the 14 NV-proton resonance point at B + r. Importantly, for the deep NVs within CVD diamond and the near-surface NV, this resonance point is clearly separated from the central T 1 feature. The NV centre within the type 1B diamond shows a much broader central transition. While the NV shown in Fig. 4.8(c)iv) shows the NVproton resonance may be outside the central feature, a number of other NV centres showed an even broader feature. For this reason we used implanted NV centres within CVD diamond to perform our subsequent NMR experiments. We similarly investigated the presence of crossings within the 15 NV. Fig 4.9 shows the energy levels of the system with 2 crossings of the initial 0, state. At G, the NV centre crosses with the eigenstate α 0, β 1, where α α 0.97 and β β The zoomed lower plot shows a strong ( 50 khz) avoided crossing under B = 0.1 G transverse field. For the purposes of T 1 -NMR, this crossing feature is unimportant since it occurs at an NV transition of 13 MHz while the gyromagnetic ratio of nuclear 151

152 (a) Magnetic Field (G) Magnetic Field (G) Figure 4.9: The energy levels of a 15 NV across the GSLAC. Below, the energy levels near each crossing with an applied transverse field of B x = 0 G (black) and B x = 0.1 G (red). spins causes their resonance points to be 5 MHz. We postulate that this crossing is what causes the loss of nuclear spin polarisation at these fields for the 15 NV previously seen in Fig. 4.6(b) due to the transition causing a nuclear spin flip predominantly due to α α β β. In contrast, the other crossing for a 15 NV, occurring at G is vitally important for T 1 -NMR. At this magnetic field the 0, crosses 152

153 Population in m s = Magnetic Field (G) = 0 G = 0.05 G Figure 4.10: Simulated single point T 1 spectra for a 15 NV centre when interacting with a single proton at a distance of 1nm and a polar angle of θ = π with an evolution time of 20 µs. The red shows the spectrum with no 2 misaligned field with the signal showing the hydrogen resonance. The blue is simulated with a misaligned field of 0.05G where the crossing feature is dominant and overlaps with the hydrogen signal. with the 1, 1 2 state. Although this transition requires both an electron and nuclear spin flip, the zoomed data shows an avoided crossing of 50 khz under a 0.1 G non-axial field suggesting a strong feature. Most importantly, this crossing coincides with the high-field NV-proton resonance. Analytically these two points are separated by only 0.03 G, far smaller than the magnetic field measurement error. In addition, the broadening of these two features means that they will be indistinguishable. Fig shows a theoretical calculation of the NV electron spin bright state population after an evolution time of 20 µs across the high-field NV-proton resonance. In each case the NV centre is interacting with a single proton a distance of 1 nm away with a polar angle of separation θ = π. The red curve shows the evolution under 2 153

154 zero transverse magnetic field. In this situation there is no impact of the crossing feature and the NV-proton resonance can be seen by the reduced population near G. Note the low contrast due to the short evolution time. The blue curve is the same simulation but now with a 0.05 G transverse field. Here the NV-proton signal is far outweighed by a significant decay from the crossing feature. This transverse field corresponds to just a o misalignment of the magnetic field, significantly better than the alignment achieved by our magnet optimisation procedure. Thus it is expected that the NV-proton high-field resonance will not be measurable for a 15 NV. Even if perfect alignment were possible, the random orientation of bath spins would provide a transverse field far too strong for the NMR signal to be measurable above this intrinsic crossing. Avoiding this feature will be vital for proton NVR with 15 NV AC Magnetometry via T 1 Relaxation Having studied these important features at the GSLAC, we decided to test the ability of the NV centre to detect fluctuating AC fields at frequencies comparable to those of nuclear spin species. In order to test the possibility of detecting fluctuating magnetic fields near the GSLAC, we generate a local magnetic field by running an oscillating current through a wire placed in proximity to the diamond. To mimic nuclear spin detection, we apply signals at various frequencies: 8 MHz, 4 MHz (approximate for 1 H or 19 F nuclear spin species), 1 MHz (approximate for 13 C), and 500 khz (approximate for 2 H or 17 O). The alternating current is modulated in amplitude and phase to ensure that the NV center is not coherently driven but experiences noise 154

155 Figure 4.11: T 1 spectra under the application of a randomly fluctuating noise source with 8MHz, 4MHz, 1MHz and 500KHz frequency. The plots are separated for ease of view. from a randomly fluctuating current around a given frequency, similar to the signal from an ensemble of target nuclear spins. The amplitude of the current is adjusted to obtain an RMS strength of 1 µt, which corresponds approximately to the field generated by a dense organic sample of nuclear spins located at a 5 nm standoff distance [51]. The probe time is set to 10 µs to maximize the PL contrast. The resulting spectra, measured on a deep 14 NV center in an isotopically purified CVD diamond, are shown in Fig While all frequencies are clearly detected after the GSLAC, the 4- and 8-MHz peaks are significantly weaker before the GSLAC. This effect is due to the NV transitions being very weak in this region because of the avoided 155

156 crossing, as discussed previously (see Sec ). Past the GSLAC, however, there is no issue measuring any frequency, and thus NMR spectroscopy would be possible in this region for most commonly found nuclear-spin species. We note that the width of the resonances is governed here by the amplitude of the applied field (1 µt) through power broadening. This not only demonstrates that T 1 spectroscopy can be used to sense fluctuating AC fields down to 500 khz, but demonstrates the feasibility of T 1 -NMR on nuclear spin species down to a gyromagnetic ratio of γ 4.9 MHz/T. 4.4 Demonstration of NMR via T 1 Relaxometry Having shown that NMR via T 1 relaxometry should be feasible, we then aimed to undertake NMR on nuclear spins external to the diamond. Since we were already using an oil objective, the high density of protons within the objective lens oil provided a convenient target on which to conduct a first demonstration. This has been done for previous NMR demonstrations using T 2 based detection [18, 51, 72]. The diamond sample comprised a high purity CVD homoepitaxial layer, grown in a Seki AX6500 diamond reactor. The sample was implanted firstly with 3.5 kev 15 N + ions and then with 3.5 kev 14 N + ions. Both implants were at a dose of cm 2 and this was followed by annealing in vacuum at 800 C for 4 hours and acid cleaning in a boiling mixture of sulphuric acid and sodium nitrate. The growth was on a (100) surface, so that the NV centres 156

157 have their symmetry axis forming a 54.7 angle to the surface normal. The NV centres were identified as either 15 NV or 14 NV via their characteristic hyperfine splittings under low-power ODMR [42]. (a) (b) Figure 4.12: (a) Experimental setup for proton NMR detection. The oil external to the surface containing a proton density of 50 nm 3 is the measurement target. (b) The angular dependency of the two resonance transitions showing the high-field transition at B res + having a strong resonance interaction at the angle of a (100) surface while the low-field transition at Bres having zero interaction for a proton directly above a (100) surface. The experimental setup is shown in Fig. 4.12(a) with the target protons contained within the objective lens oil external to the (100) surface of the diamond. Fig. 4.12(b) shows the interaction strength as a function of polar angle of separation θ for each transition. The low-field transition, shown by the dashed blue line, goes to an interaction strength of zero at the angle of a (100) surface. In contrast, the high-field transition, shown by the solid blue line, has a significant interaction at that angle. One demonstration we planned to undertake was to demonstrate the potential for spatial imaging via the different interaction strengths at the pair of transition. 157

158 I s (τ) Driving Frequency (MHz) (a) (b) Approx. Magnetic Field (G) τ = 1µs τ = 2µs Magnet Position Figure 4.13: (a) ODMR spectra for a 15 NV centre across the GSLAC. The expected NV-proton resonances are shown by the pair of blue lines. (b) Single-point T 1 measurement taken across the expected high-field NV-proton transition showing a significant feature even at an evolution time of 1µ s Proton NMR using a 15 NV We first aim to demonstrate T 1 -NMR detection of a proton bath using a 15 NV centre. To do this we identified a 15 NV by taking low-power ODMR and finding an NV with the characteristic hyperfine splitting of 3.03 MHz. 158

159 We then conducted a sweep across the GSLAC and took ODMR at each point. This ODMR for each magnet position is plotted in Fig. 4.13(a) from a magnetic field of 1021 G to G. This matches the expected transitions from Fig. 4.3(b). The NV-proton resonance points are shown in blue. We then took single point T 1 data near the expected high-field transition of this NV centre. The data for 1 µs and 2 µs is shown in Fig. 4.13(b). This shows a strong decay of 2 MHz which corresponds to an NV depth of 3 nm. This is an uncharacteristically shallow for the implant energy used and is at a depth that would likely lead to optical instability of the NV (ie blinking). This was not observed. As previously discussed, the high-field NV-proton transition coincides with the crossing of the 0, and 1, 1 2 states of the NV centre. Residual non-axial fields can cause these states to become mixed, leading to a strong T 1 decay of the NV centre. Thus we conclude that the strong decay we observe is highly likely due to the NV crossing feature rather than an NMR signal from the proton bath external to the diamond. There exists, however, a way to shift the NV crossing feature away from the high-field NV-proton resonance. This can be done by selecting an NV with a strong hyperfine coupling to an environmental background spin. In natural diamond, 1.1% of the carbon lattice consists of 13 C which has a spin- 1 2 nuclear sin with a gyromagnetic ratio of MHzT 1. The NV centre electron spin can experience a hyperfine coupling with a background 13 C up to tens of MHz in strength [94]. Fig. 4.14(a) shows a low-power ODMR conducted on a 15 NV which is strongly coupled to an environmental 13 C. In this instance, A C shows the axial hyperfine interaction with the 13 C of 159

160 Transition Frequency (MHz) ODMR Signal (a) A C A C 0.94 A NV A NV (b) Driving Frequency (MHz) Magnetic Field (G) Figure 4.14: (a) ODMR spectrum of a 15 NV coupled to a background 13 C. The hyperfine couplings to the NV nuclear spin, A NV, and the 13 C, A C, are shown. (b) The theoretical transition frequencies of a 15 NV centre with a 4.1 MHz axial and 3 MHz non-axial hyperfine coupling to an environmental 13 C. The NV-proton resonances are shown in blue while the red cross shows the position of the NV crossing feature. 4.1 MHz while A NV shown the typical NV hyperfine interaction of 3.03 MHz for a 15 NV. This corresponds to a 13 C from family E, F from Ref. [94]. 160

161 If we consider the NV hamiltonian from Eqn. 1.2 we can analytically solve the NV energy levels with additional terms for the hyperfine and zeeman shift caused by the 13 C H 13 C NV /h = H NV /h + A 13 C S z I z + A 13 C (S x I x + S y I y ) γ13 CB z I z (4.27) where H NV is the bare NV Hamiltonian, A 13 C and A 13 C are the axial and nonaxial hyperfine interactions respectively, S and I are the NV electron and 13 C spin operators respectively and γ13 C is the 13 C gyromagnetic ratio. Near the GSLAC, the 13 C will be efficiently polarised under optical pumping so that the initial state of the system will be m NV s, m NV I, m 13 C I = 0, + 1, We then solved for the eigenvalues of the system in order to find the energy levels. This then gave us the energy transitions within the system. These are plotted in Fig. 4.14(b) with the states able to be coupled via a dipolar interaction with an environmental spin shown by the solid black lines. We have discarded those transitions that require a flip of the 13 C. Also plotted, in red, is the proton transition leading to the two NV-proton resonance points marked with the dashed and solid blue lines. The crossing of the 0, + 1, with the 1, 1 2, is marked by the red cross. This is the equivalent transition that was overlapped with the NV-proton transition for the 15 NV not coupled to a carbon-13. In this instance we can see that the NV-proton transition and this crossing point are separated by 0.7 G. This separation would make these two decay points easily distinguishable, allowing the NVproton transition to be probed separately to the NV crossing feature. Thus we conducted NMR on the NV centre whose ODMR is shown in Fig. 4.14(a). The ODMR spectrum map as a function of B around 1024 G for this NV is shown in Fig. 4.15(a). The theoretical data from Fig.4.14(b) is overlayed 161

162 (a) 8 Magnetic field, B (G) Driving freq. (MHz) ω H ω X (b) PL (a. u.) 1 PL (a. u.) A B C τ = 1 µs τ = 20 µs Magnetic field, B (G) (c) (d) τ = 100µs τ = 20µs PL (a. u.) PL (a. u.) ω NV ω H (MHz) ω NV ω H (MHz) Figure 4.15: (a) ODMR spectra taken across the GSLAC for the 15 NV centre coupled to a 13 C from Fig with the theoretical transitions overlaid on the graph. (b) T 1 -NMR spectrum across the GSLAC with τ = 1 µs (black) and τ = 20 µs (blue). (c,d) T 1 -NMR spectra around the 1 H resonances with τ = 100 µs (c, B r ) and τ = 20 µs (d, B + r ). In each case the x-axis is the detuning from the expected resonant frequency. Solid lines are Lorentzian fits. 162

163 across the experimental data. The magnetic field values quoted are correct for the theoretical transitions. In contrast, the experimental data is plotted by magnetic field point. In the lower field regime (B G) there is a disagreement between the theory and experiment. This is likely due to the steps of the magnet not being a linear step in field strength and thus the field they are plotted against is not correct. While the axial hyperfine parameter, A 13 C, can be found via a low-power ODMR such as that shown in Fig. 4.14(a), the non-axial hyperfine parameter A 13 C can not be found in this way. This non-axial parameter affects the frequency at which the two transitions cross at G. We found the best fit to the data was setting this to A 13 C the crossing frequency of 3.5 MHz. A lower value of A 13 C frequency at which the transitions cross. = 3.0 MHz in order to match leads to a lower For example a value of A 13 C = 2 MHz gives a transition frequency of 3.0 MHz while a value of A 13 C gives a transition frequency of 3.9 MHz. = 4 MHz Thus we estimate the non-axial hyperfine coupling between the NV and the environmental 13 C to be 3.0 ± 0.4 MHz. Across the range of magnetic fields plotted in Fig.4.15a), the Larmor frequency of 1 H is shown by the green line. The two resonance points, ω NV = ω H, are shown as green dots. To probe those resonances, the PL intensity is measured as a function of B, using wait times τ = 1 µs and τ = 20 µs (Fig b). The resulting T 1 -NMR spectrum reveals three main features. The first two dips at B G and B G are seen only for τ = 20 µs and correspond to the cross-relaxation resonances with 1 H. The third, stronger dip at B G, visible even at short time τ = 1 µs, is the 163

164 crossing feature related to the GSLAC structure of the 15 NV centre induced by residual transverse magnetic fields. Here we see that the coupling to the 13 C shifts this feature away from the HV-proton resonance at B + r allowing it to be interrogated separately. Zoomed-in spectra of each 1 H resonance are shown in Figs. 4.15(c) and 4.15(d). They are plotted against the frequency difference, ω NV ω H, where ω NV is obtained by fitting the ODMR spectrum and ω H is the expected 1 H Larmor frequency, i.e. ω H 4.36 MHz in Fig. 4.15(c) and ω H 4.37 MHz in Fig. 4.15(d). Lorentzian fits to the data (solid lines) show that the observed resonances match expectations within experimental uncertainty. We note that the low-field resonance is significantly weaker than the highfield one, as can be seen in Fig. 4.15(b), which is why a longer probe time of τ = 100 µs was employed in Fig. 4.15(c). This difference is due to the different angular dependencies of the interactions after and before the GSLAC. From Eqn. 4.24, we are able to estimate an independent depth measure for each transition in the following way. Through taking a 3-point T 1 spectrum, including both a short-time (τ = 1 µs) and a long time (τ = 3 ms) probe time, we were able to fit our data to Eq. 2.4 and extract both Γ + 1,sig and Γ 1,sig. Those can be used to obtain two independent estimates of the NV depth, which yields d = 9.3 ± 1.3 nm from the low-field resonance and d = 11.0 ± 0.7nm from the high-field resonance. These values are within uncertainty from each other, and are consistent with typical depth ranges observed with similar implantation energies [72]. Alternatively, the two on-resonance decay rates can be used to probe the angular distribution of target spins. A numerical simulation of the signal expected 164

165 from a bath of protons external to a flat surface gave an angle of 44 ± 13 between the NV quantisation axis and the surface, which is consistent with the expected angle of 54.7 for our (100) surface. From the inferred depth, we deduce that 50% of the signal is generated by the closest protons, corresponding to a detection volume of (10 nm) 3 [18, 51] Proton NMR using a 14 NV Sensing experiments generally use implanted 15 NV centres as they can be distinguished from native centres which are 99.6% 14 NV. This allows the shallow NV centres to easily be differentiated from those deep within the crystal. However, the lack of a nuclear quadrupole moment limits the ability for a 15 NV to interrogate signals 1 MHz as required for many nuclear species. Additionally, the lack of hyperfine coupling to an environmental 13 C for most 15 NVs, especially in an isotopically purified sample, makes detection of protons difficult due to the overlap of the high-field NV-proton transition with the NV crossing feature. This is especially true for a (100) surface where the lower-field transition experiences no interaction with a proton external to the surface directly above the NV. On the other hand, we have previously seen that the nuclear quadrupole moment of 14 NV allows transitions to be investigated down to the sub-mhz regime without the presence of strong state mixing. In addition to proton detection, this would be vital for T 1 detection of nuclear spin species with lower gyromagnetic ratios than 1 H. Here we demonstrate proton NMR using a 14 NV spin probe. Due to the strong state mixing of the 14 NV energy levels near the NV- 165

166 (a) 12 Driving freq. (MHz) ω H 0.7 PL (a. u.) Magnetic field, B (G) (c) (b) PL (a. u.) ω NV 2.7 MHz 4.4 MHz PL (a. u.) τ 1 µs 200 µs Evolution time, τ (μs) ω NV ω H (MHz) Figure 4.16: (a) ODMR spectra for a 14 NV centre across the GSLAC with theoretical transitions overlaid. (b) T 1 -NMR spectrum around the 1 H resonance, obtained with τ = 200 µs and plotted as a function of the detuning from resonance, ω NV ω H. Solid line is a Lorentzian fit. (c) Full T 1 curves taken on resonance (ω NV = 4.4 MHz, blue data) and away from the resonance (ω NV = 2.7 MHz, grey data). Solid lines are exponential fits. proton resonance before the GSLAC, only the transition after the GSLAC was probed. T 1 -NMR detection of protons using a 14 NV centre is demonstrated in Fig The ODMR spectrum map across the NV GSLAC is shown in Fig. 4.16a), while the NMR spectrum around the higher-field NV- 1 H resonance is shown in Fig. 4.16b), using a probe time τ = 200 µs. 166

167 The measured transition frequency of this resonance is within uncertainty of the expected 1 H transition of ω H 4.37 MHz. Full T 1 relaxation curves out to an evolution time of 200 µs were recorded on and off this resonance feature (Fig. 4.16(c), confirming that the signal is caused by a simple decrease in T 1 lifetime. The off-resonance data was taken with the NV centre at a lower quantisation energy of 2.7 MHz. This is at a level where surface noise, such as electron-electron interactions, could significantly reduce the T 1. However, the significant reduction in T 1 on resonance with the protons shows the presence of a very strong signal at 4.4 MHz which does not exist at lower frequencies. This is characteristic of an NMR resonance feature due to the quantised energy levels of the nuclear spin targets. In addition to the full T 1 curves, we conducted a comparative oil and air measurement with a 14 NV. A control measurement was taken with the NV surface exposed to air with an evolution time of 200 µs. The data is shown in grey in Fig A best-fit Lorentzian with a central transition limited to within experimental error ( 2%) of the theoretical transition frequency shows no decay. In comparison, the oil measurement on the same NV (shown in blue) shows a contrast of 1.7% at 4.42 ± 0.06MHz, within errors of the theoretical transition. Due to these checks, as well as the results with various NV centres containing both isotopes of nitrogen, we conclude that the additional 4.4 MHz signal measured in all of these cases is indeed due to hydrogen nuclear spins within the oil external to the diamond. 167

168 PL (norm.) Air Oil ω NV (MHz) Figure 4.17: Comparative oil (blue) and air (grey) measurement around the expected NV-proton resonance for a 14 NV with an evolution time of 200 µs Sensitivity of T 1 -NMR and Comparison to T 2 - based Methods Here we compare the sensitivity of T 1 spectroscopy for nuclear spin detection to the current state of the art of T 2 -based spectroscopy. We do this via a comparison of the signal-to-noise ratio (SNR) derived under identical conditions, in the case of a shallow NV centre detecting an ensemble of nuclear spins as in the geometry of our NMR experiments. In Chapter 2, we derived the decay from a classical noise source for an NV where the background T 1 is governed by the decay of phonons. When we approach the GSLAC with a shallow NV centre, the background decay is no longer governed by phonons but by surface magnetic noise. As this does not cause transitions between the m s = +1 and -1 states, the form of the decay curve will be different. Here we redo that calculation where the dominant off-resonance decay is magnetic noise rather than phonon noise. This will 168

169 allow us to compare T 2 and T 1 methods. To analyse the spin dynamics of near-surface NV centres, we therefore consider the model of Fig. 4.18, where the three spin states of the NV electronic ground state, 0, ±1, are linked via transitions rates k + and k associated with the transitions and 0 1, respectively. The corresponding populations are denoted n 0, n +1 and n 1, respectively. Those two transition rates depend on the magnetic field, which forms the basis for our method of T 1 spectroscopy. +1 k + 1 k 0 Figure 4.18: Model used to describe the population dynamics within the NV electronic ground state, i.e., between the spin states 0, + 1 and 1. Solving the rate equations yields the populations as a function of time, 169

170 which gives for n 0, n 0 (τ) = (3n 0(0) 1) ξ (3n +1 (0) 1) k + (3n 1 (0) 1) k e (κ+ξ)τ 6ξ + (3n 0(0) 1) ξ + (3n +1 (0) 1) k + + (3n 1 (0) 1) k e (κ ξ)τ 6ξ (4.28) where the initial populations satisfy the closed-system condition n 0 (0) + n +1 (0) + n 1 (0) = 1, and we defined κ = k + + k ξ = k+ 2 + k 2 k + k. The PL intensity at the start of the readout laser pulse following a wait time τ, I s (τ), can be expressed as I s (τ) = I A n 0 (τ) + I B [n +1 (τ) + n 1 (τ)] = I B + (I A I B )n 0 (τ) (4.29) where I A and I B < I A are the PL rates associated with spin states 0 and ± 1. By inserting Eq. (4.28) into Eq. (4.29), we obtain an equation for the PL intensity as a function of τ, the rates k ±, the initial populations following the initialisation pulse, and the PL rates. We note that near the GSLAC (B 1024 G), k + is governed by magnetic noise at high frequency ( 5.74 GHz) while k is sensitive to low-frequency magnetic noise ( 10 MHz). As a consequence, one generally has k k +, which results in a simple expression for I s (τ), [ I s (τ) I ] Ce Γ 1,tot τ, (4.30) where I 0 and C are constants, and we defined Γ 1,tot = 2k. 170

171 For T 1 spectroscopy, the measurement sequence consists of a 3 µs laser pulse followed by a wait time τ assumed to be much longer than 3 µs. The useful signal I s (τ) is obtained by counting the photons within a read-out time t ro = 300 ns. As a result, the PL signal is acquired only for a fraction t ro /τ of the total experiment time, T tot. Therefore total number of photons detected can be expressed as N (Γ 1,sig, τ) = RT tot t ro τ [ ] 1 C + Ce (Γ 1,int+Γ 1,sig)τ (4.31) where R is the photon count rate under continuous laser excitation, Γ 1,int is the intrinsic longitudinal relaxation rate, Γ 1,sig is the relaxation rate induced by the on-resonance target spins (i.e., the signal ), and C is the contrast of the T 1 relaxation curve. The change in the number of photons caused by the presence of Γ 1,sig is N signal (τ) = N (0, τ) N (Γ 1,sig, τ) = RT tott ro C e ( Γ 1,intτ 1 e ) Γ 1,sigτ. (4.32) τ The photon shot noise associated with the measurement is N noise (τ) = N (Γ 1,sig, τ) RTtot t ro τ (4.33) where we used the approximation C 1. Taking a series solution for a small signal regime (Γ 1,sig Γ 1,int ) and only keeping the first order terms in Γ 1,sig, the SNR is then SNR T1 (τ) = N signal(τ) N noise (τ) RT tot t ro Ce Γ 1,intτ Γ 1,sig τ. (4.34) 171

172 To find the maximum probe time we solve where the derivative equals zero. That is dsnr T1 (τ) dτ = RTtot t ro C τ 2 Γ 1,sige Γ1,intτ (2Γ 1,int τ 1) (4.35) = 0 (4.36) giving an optimal probe time of τ = 1 2Γ 1,int. Thus the signal to noise ratio becomes SNR T1,max RT tot t ro Ce Γ 1,sig. (4.37) 2Γ 1,int Likewise, we can write the number of photons detected in a T 2 -based sensing scheme as N (Γ 2,sig, τ) = RT tot t ro τ [ 1 C + Ce (Γ 2,intτ) 2 e (Γ 2,sigτ) 2] (4.38) where Γ 2,int is the intrinsic transverse relaxation rate in the absence of signal (under the considered dynamical decoupling sequence, e.g. XY8-N), Γ 2,sig is the additional transverse relaxation rate induced by the target spins, and all other constants are defined as before (in particular C is still the contrast of the T 1 relaxation curve). Note that the time dependence is now Gaussian rather than exponential [72]. In the small signal regime (Γ 2,sig Γ 2,int ), the SNR is maximised for a probe time τ = 3 2Γ 2,int and the signal-to-noise ratio is SNR T2,max ( ) 3/2 3 RT tot t ro Ce 3/4 (Γ 2,sig ) 2. (4.39) 2Γ 2,int We now consider a particular sensing situation which allows us to express the signal as measured in T 1 or T 2 spectroscopy. Namely, we look at the case of a single NV centre located near the diamond (100) surface, detecting a 172

173 semi-infinite volume of nuclear spins as shown in Fig.4.12(a). The induced longitudinal relaxation rate for the resonance past the GSLAC is given by Eq. (4.24), that is, Γ 1,sig = ρ 4Γ 2 ( µ0 γ NV γ t h 4π ) 2 19π 24d 3 (4.40) where d is the NV depth, ρ is the density of target nuclear spins, γ t is the gyromagnetic ratio of the target spins. In the T 2 -based locking technique, the induced decoherence rate is [72] ( ) 2 (Γ 2,sig ) 2 µ0 γ NV γ t h 5 = ρ 4π 48πd. (4.41) 3 We can thus write the SNR in both sensing schemes as 1 SNR T1,max A 19πe (4.42) Γ deph Γ1,int ( ) 3/2 SNR T2,max A 5(3/e)3/4 1 48π2. (4.43) 3/2 Γ 2,int The constant common to both, A, is given by A = ( ) 2 3C ρ oil µ0 γ NV γ t h RT tot t ro. (4.44) 4 d 3 4π Evaluating the numeric factors and using the conventional notations T 2 = 1/Γ 2, T 1 = 1/Γ 1,int and T 2 = 1/Γ 2,int, we obtain the ratio between the SNRs, SNR T1,max SNR T2,max T T 2. (4.45) (T 2 ) 3/2 Note that here T 2 is the extended decoherence time under dynamical decoupling. For a near-surface NV centre in a bulk diamond (d 10 nm), typical values are T2 = 2 µs, T 1 = 2 ms and T 2 = 200 µs under dynamical 173

174 decoupling. This yields a ratio SNR T 1,max SNR T2,max 1. In other words, our all-optical T 1 -based spectroscopy technique is as sensitive as the existing T 2 -based approach which requires complex microwave pulse sequences. We note that the sensitivity of the T 1 approach could be improved in a number of ways, for instance by using quantum logic to improve readout fidelity [73] or dynamical decoupling to extend the dephasing time T 2. (a) T 1 NMR (b) XY8 NMR τ τ τ τ PL (a. u.) d = 10.7 ±0.1 nm Coherence (a. u.) d = 10.5 ±0.1 nm ω NV ω H (MHz) ω τ ω H (MHz) Figure 4.19: Comparison of microwave-free T 1 -NMR to the XY8 protocol. (a,b) Spectra from PMMA obtained with a single 14 NV centre by using: (a) microwave-free T 1 relaxometry after the GSLAC, with a probe time τ = 100 µs; (b) an XY8-N dynamical decoupling sequence with N = 256 microwave pulses at a field B = 300 G. The corresponding pulse sequence is depicted above each graph, with laser pulses in green and microwave pulses in red or blue corresponding to 0 or 90 relative phase. In (a), the spectrum is constructed by varying the NV frequency, ω NV, via varying the magnetic field strength. In (b), it is constructed by scanning the probe frequency ω τ = π/τ, where τ is the inter-pulse delay, inclusive of the finite π pulse duration. Error bars represent the photon shot noise (one standard deviation). In order to verify this and further compare the two approaches, we conducted a comparative measurement using a single shallow 14 NV centre, where 174

175 the diamond was coated with a layer of PMMA. Fig shows the hydrogen spectrum measured via T 1 relaxometry (a) and via an XY8-256 sequence (b). Both spectra were acquired in a total time of about 2 hours, and show a clear feature at the 1 H frequency with a similar signal-to-noise ratio. In addition, one can compare the NV depth, d, inferred through fitting the appropriately normalised data from each method, assuming the signal comes from a semi-infinite bath of protons on a (100) surface. Using a proton density of ρ = 56 nm 3, we find d = 10.7 ± 0.1 nm from the T 1 data, against d = 10.5 ± 0.1 nm from the XY8 data, indicating a high level of consistency between the two approaches. Although the sensitivity of the two techniques is comparable, and can easily be found for a general NV simply by adjusting the above calculation for the specific experimental geometry considered, the spectral resolution of the two techniques difers significantly. The spectral resolution of T 1 spectroscopy is currently limited by the dephasing rate Γ 2 = 1/T2, while it is limited by T 2 with the XY8 method. Importantly, for XY8 spectroscopy, the spectral resolution can be improved further using correlation spectroscopy [95, 96, 97] with recent results showing this can be limited simply by the total experiment time [79, 80]. We note that both the sensitivity and spectral resolution of the T 1 approach could be dramatically improved by optimising T2, which motivates further work towards understanding and mitigating the decoherence of near-surface NV centres [19, 98, 99]. 175

176 Chapter 5 Nuclear Spin Hyperpolarisation via Direct Cross-Relaxation This chapter contains preliminary measurements that contributed to and were published in Ref. [3]. As discussed in Chapter 1, the major limitation to improving the spatial resolution of conventional NMR and EPR sensing techniques is the poor signal to noise. This is particularly true for the case of induction-based NMR. This lack of signal occurs because the induction measurement is sensitive to the net magnetic field produced by the nuclear spins and hence spins with opposite polarisation will cancel. Even considering protons, with their high gyromagnetic ratio, at room temperature (300 K) and in the presence of a strong magnetic field of 10 T, the thermal polarisation is only around 40 spins per million protons. Thus the signal to noise could be increased by orders of magnitude if the nuclear spins being sensed could be strongly polarised. The act of polarising nuclear spins beyond their thermal polarisation is 176

177 called hyperpolarisation and there has been significant work in this area in recent years [78, 100, 101, 102, 103]. The main method is to transfer the polarisation from polarised electronic spins to surrounding nuclear spins. However current methods require low temperatures, high magnetic fields or complex chemistry limiting their potential application. In contrast, the NV centre gives an easy way to achieve a polarised electron spin at room temperature and low magnetic field. If we could efficiently transfer this polarisation to a nuclear spin bath, this would overcome many of the complexities of current hyperpolaristion technologies. To this end much work has been done with NV centres to achieve this [78, 103, 104, 105, 106, 107, 108, 109, 110]. However the different energy scales of the NV centre electron spin splitting compared to the nuclear spin splitting has meant this work has focused on techniques requiring microwave driving such as Hartmann-Hahn in order to transfer the population from the NV to the nuclear spin bath. This relies on cross-relaxation occurring in the rotating frame of the qubits, requiring precise driving strengths and leaving the polarisation transfer rate highly susceptible to inhomogeneities within the sample. What we have demonstrated via our NMR measurements is that it is in fact possible to tune the magnetic field to the point where direct crossrelaxation between the NV electron spin and the nuclear spins in the environment occurs. That is, it is possible to conduct cross-relaxation in the laboratory frame rather than the rotating frame. While we have demonstrated spectroscopic detection of nuclear spins by seeing the electron spin lose polarisation, this process also acts to polarise the nuclear spin bath. 177

178 5.1 Detection and Hyperpolarisation of 13 C Spins In Chapter 4 we demonstrated the detection of protons external to the diamond substrate however we did not show the spectroscopic detection of 13 C nuclear spins. We were using a sample with natural carbon abundances and therefore, given the 1.1% abundance of 13 C and the comparatively small distances to these nearby nuclear spins, the signal from the 13 C spins should have been significantly stronger than the signal from the external protons. An obvious first step would have been to demonstrate T 1 -NMR on the 13 C within the diamond. While measuring external protons has significantly more applicable interest than 13 C within the diamond, we did attempt to measure the signal from the 13 C. The initial results were inconclusive, far removed from the strong signal expected in comparison to that measured from the protons. While there were difficulties in measuring the comparatively weak gyromagnetic ratio of the 13 C, leaving the resonance far closer to the central crossing feature of the 14 NV spectrum (see Fig. 4.11), this seemed unable to explain the lack of signal observed. We postulated that we were in fact polarising the surrounding 13 C spin bath leading to a far weaker signal. The gyromagnetic ratio of 13 C is MHzT 1. This means that the resonance points occur when the NV centre is at a splitting of 1.1 MHz. As with the case of the protons, at the resonance after the GSLAC, the 0 1 transition of the NV will be driven by a flip of the 13 C nuclear spin As we continually re-initialise our NV centre, unless 178

179 (a) (b) Laser τ τ RF Driving π Figure 5.1: (a) Single point T 1 measurements on the same single NV. The blue data is taken with the standard sequence used for the measurements throughout this thesis shown in Fig. 2.1d). The orange data is taken with a π-pulse at every second evolution showing the clear 13 C resonances. (a) The pulse sequence to ensure the nuclear spin bath is not polarised. By applying a π-pulse at the beginning of every second evolution time, the initial NV state is switched between 0 and 1 each time. This switches the state which the nuclear spins are polarised towards in each evolution and thus prevents the buildup of polarisation. there is another T 1 process to return the nuclear spin bath to its thermal population, the nuclear spin bath will become polarised in the 1 2 spin state. Importantly, the nuclear spin bath gets polarised in the state which cannot cross-relax with the NV centre and thus the increased NV relaxation mechanism is removed. Therefore, if a high level of nuclear spin polarisation 179

180 is achieved, the T 1 time returns to the off-resonance background. In order to test this hypothesis we ran two separate T 1 -spectroscopic scans on a 14 NV deep within a CVD diamond across the GSLAC. The diamond has low density of nitrogen leading to a spin-bath dominated by 13 C. Firstly we ran the standard T 1 spectroscopic scan used previously within this thesis which could produce nuclear spin polarisation with an evolution time of τ = 4 µs. This scan is shown in blue in Fig. 5.1a), showing no clear NMR resonances with just the single dip at the central 14 NV crossing point visible. In addition to this, a second scan was undertaken with the measurement sequence altered to include a π-pulse on the NV centre at the beginning of every second evolution time. This sequence is shown in Fig. 5.1b). The π-pulse acts to intialise the NV centre electron spin into the 1 spin state as opposed to 0 for no π-pulse. This reverses the direction of polarisation build-up in the nuclear spin bath for every subsequent run and thus prevents polarisation of the spin bath. We took only the signal from those evolutions with no π-pulse on the NV centre and plotted them in 5.1a). In comparison to the blue curve, there are now a pair of clear resonance dips on either side of the central feature with the splitting corresponding to the 13 C gyromagnetic ratio. This is clear evidence of poalrisation build-up within the bath and it is only once this polarisation is removed by interleaving the π-pulses that the T 1 spectroscopic measurement is possible. Importantly the detection of protons within oil, presented in Chapter 4, would not be strongly effected by this process due to the intrinsic T 1 of the protons within the oil being too short to sustain significant polarisation build-up. Similarly, the P1 centres measured in Chapter 3 would have a 180

181 T 1 far too short for this polarisation to be significant. When conducting T 1 spectroscopy it is important to consider whether the targets could be polarised and whether the application of a depolarisation scheme like that employed here is necessary. Significant further experimental work on this method of nuclear spin polarisation has been undertaken by Mr. David Broadway along with theoretical analysis by Dr. Liam Hall. They were able to achieve polarisation of 13 C within a CVD diamond of greater than 99% up to a distance of 21 nm from the NV centre. This corresponds to a fold increase on thermal polarisation for a total of spins. In addition, they were able to achieve polarisation of proton spins within PMMA external to the diamond surface. The level of polarisation achieved was 50% average polarisation over a (26 nm) 3 volume. Upon further analysis it has been shown that this direct lab-frame cross relaxation gives an order of magnitude coupling improvement over the rotating frame coupling of, for example, Hartmann Hahn type schemes. Indeed, with high quality implanted diamond, nanoscale fabrication to increase the diamond surface area and a well controlled magnetic field across a broad area, the scale up of such technology to produce significant volumes of contrast agents is feasible. This work is detailed in Ref. [3]. While the use of T 1 -NMR may be limited to situations where the application of T 2 -NMR is too difficult, as the likelihood of achieving the same spectral resolution appears unlikely, nuclear spin hyperpolarisation is an area where this direct cross-relaxation could provide a significant improvement over current technologies. 181

182 Chapter 6 Conclusion and Outlook The work of this thesis has investigated T 1 -based magnetic resonance spectroscopy. In particular we have considered how the NV can be tuned into resonance with environmental spin species in order to sense the mutual crossrelaxation that both the probe NV and the target spin experience. Through this investigation we have been able to show that nanoscale T 1 magnetic resonance spectroscopy with the NV centre is possible on both electron and nuclear spin targets and considered the sensitivity, spatial resolution and possible imaging capabilities of such a technique. In Chapter 2, we covered the theoretical background of T 1 spectroscopy. Through modeling the target spins as a classical noise source we were able to form an analytical solution for the relaxation curve of the NV centre. The double exponential form of this solution came from the pair of resonant levels of the 3-level NV electron spin system equilibrating over a shorter timeframe than the off-resonance level due to the cross-relaxation with the target spin. Beyond this we calculated the sensitivity and linewidth expected 182

183 for interrogating a single target spin by completing a fully quantum anaylsis of the problem. These calculations showed that sensing single electron and nuclear spins beyond the diamond substrate was theoretically achievable. Having introduced and analysed T 1 spectroscopy theoretically, in Chapter 3 we were able to demonstrate this technique on the nanoscale by interrogating the EPR spectrum of the P1 centre in diamond using a single NV spin probe. Furthermore, by extending the measurement to the nuclear spin transitions of the P1 centre, we were able to show the broadband applicability of the technique. Importantly, this measurement at frequencies of MHz was of environmental noise that is difficult to interrogate via other sensing modalities with the NV centre. The existence of hyperfine-enhanced NMR shows that nuclear spins coupled strongly to an electron spin can have their signal significantly increased by having their transitions effectively reported via this proximal electron spin. This suggests the possibility of significantly extending the effective sensing volume of the NV centre for nuclear spin detection by utilising electronic spin labels to differentiate and target particular nuclear spins within samples of interest. Chapter 4 extended the previous measurements to the regime of nuclear spins not strongly coupled to a proximal electron spin. Importantly this showed that direct laboratory frame cross-relaxation between the NV centre and nuclear spins is possible and opens up a new method for nanoscale NMR. This was shown to be similarly sensitive as T 2 -based techniques and, although it produces significantly worse linewidths currently, the easier experimental implementation and removal of the requirement for microwave pulsing suggests this technique could find a niche in samples where strong 183

184 microwave pulses may be difficult to implement such as in-vivo or in-vitro NMR. To enable the NMR spectrocopy in Chapter 4 we undertook a study of the NV centre near the GSLAC. This improved knowledge of the behaviour of the NV in this regime allows the NV centre to be utilised at significantly lower energy quantisations than previously. This regime is particularly interesting due to the ability to directly tune the NV to environmental magnetic fluctuations. While we used this for NMR there are also other applications such as controlling the NV centre indirectly, via driving bath electronic spins as we also investigated during my time in the Hollenberg group (see Appendix A). Finally, Chapter 5 summarised the important implication that laboratory frame cross-relaxation has for hyperpolarisation of nuclear spins. The possibility of increasing the polarisation rate compared to Hartmann-Hahn type techniques makes this an area where directly tuning of the NV into resonance with nuclear spins could have significant implications. The outlook for quantum sensing is bright, with many possible areas of importance. In particular, the NV centre in diamond represents a unique method for achieving nanoscale high-precision magnetic sensing at ambient temperatures. The work of this thesis provides another avenue for nanoscale spectroscopy with the NV. While it is always a difficult task predicting where scientific fields will head in the future however there are three separate areas where I feel that this work could prove important. The first of these areas is in nanoscale EPR studies, working as a complementary technique to DEER. In this instance I do not foresee T 1 spectroscopy 184

185 as outperforming DEER although it could provide similar spectral resolution and sensitivity. It is the access to a different term of the dipole-dipole interaction that is important, meaning the techniques are sensitive to different angular separations. While sensitivity to different angular separations from the NV centre could be achieved via rotating the background field while applying DEER, the zero-field splitting and loss of fluorescence at certain misaligned fields make this difficult. T 1 spectroscopy would provide a different way of gaining angular information and mapping the full volume around a particular NV. The second area where T 1 spectroscopy could be important is in NMR sensing in environments where microwave pulsing is difficult or dangerous to achieve. There has been recent work in attaching a nanodiamond directly to the end of an optical fibre for initialisation and readout. The purpose of this is to better employ NV sensing in difficult environments. For example this would remove the need for optical windows in cryostats for low-temperature studies and would allow easy optical access to NV centres within organisms. However coupling the NV to a microwave driving field, while retaining the small size of the device, is a significant challenge. Thus although T 1 spectroscopy is unlikely to compete with T 2 -based schemes in terms of spectral resolution, in instances where microwave-free operation is required, nano- NMR could be achieved in this way. However, I believe the main area where this work will be important is in hyperpolarisation of nuclear spins. The transfer process via pulsing methods requires precise tuning of the pulses and, while it was thought the narrower linewidth from T 2 -based processes would significantly improve the polarisa- 185

186 tion transfer rate, recent work by Liam Hall suggests this is not the case and in fact direct tuning, as in T 1 spectroscopy, has a stronger polarisation rate despite the comparative simplicity of the implementation. Further research is required in this area however this looks to be a very promising potential application of this work. In conclusion, T 1 spectroscopy represents a promising new avenue for interrogating environmental noise with the NV centre in diamond. The microwave free, broadband nature of the technique suggests it could be utilised for important applications in the future. 186

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206 Appendix A Environmentally Mediated Spin Manipulation I will use this appendix to detail one piece of work not directly related to spin spectroscopy that I was involved in during this thesis. The work of this chapter is detailed in Ref. [4]. In our attempts to understand the crossing feature present in the 15 NV NMR spectrum, discussed in Sec , we discovered a novel control mechanism for the NV centre. Before understanding that the feature was simply a crossing due to the NV system, the large T 1 drop at the field where the proton resonance should occur opened up some interesting possibilities. One possible explanation for this feature was that it was indeed the proton resonance but this was significantly strengthened by the presence of reporter electron spins on the surface. The existence of surface electron spins has been utilised previously for imaging strongly interacting protons [111]. This would be the same hyperfine-enhancednmr mechanism we investigated when mea- 206

207 suring the nuclear transitions of the P1 centre in Chapter 3. In order to determine if this was the case, we attempted to apply a strong, continuous microwave drive at the resonance of a free electron spin to decouple the NV from the environmental electrons and also from the hyperfine-enhanced proton signal. While this was not the reason for the T 1 feature, we were able to demonstrate that the NV itself could be coherently controlled via driving of the electronic spin bath. It should be noted that while I, along with Jean- Phillippe Tetienne, realised that this driving was possible and developed the simple theoretical model, the experiments presented here were undertaken by Scott Lillie. When the NV centre is near the GSLAC, the energy splitting of the levels is decreased from the GHz scale to the scale of a few MHz or below. If an environmental spin (see Fig. A.1a) is driven with a rabi oscillation frequency, Ω env, matching the NV centre s splitting, then the fluctuating field produced by the environmental spin will drive Rabi oscillations on the NV centre in a Hartmann-Hahn type double resonance. This driving is achieved despite the applied microwaves being on the GHz scale as opposed to the NV splitting being on the MHz scale. Fig. A.1b) shows the energy levels of the two systems and how the dressed state splitting of the environmental electron matches the NV centre splitting when this environmentally mediated resonance is achieved. Fig. A.1c) shows the comparable energy levels of the environmental spin, being driven at its resonance of 2.87 GHz which causes rabi oscillations on the NV centre at a splitting of a few MHz near to the GSLAC. The strength of the field caused by the environmental electrons which then 207

208 Figure A.1: Schematic of environmentally mediated resonance. (a) The environmental electron spins (red) are driven by an on-resonance microwave field of frequency ω MW which causes a fluctuating field of frequency Ω env at the NV. (b) The energy levels of the environmental spin and the NV. The frequency Ω env comes from the dressed state splitting of the environmental spin which, when it matches the NV splitting, causes coherent driving of the NV spin. (c) Transition energy of the NV and the environmental electrons showing that when near the GSLAC of the NV centre, the resonance driving field for the electrons is 2.87 GHz. drives the NV centre depends on the spatial distribution of environmental spins since the dipole-dipole interaction mediates the driving of the NV. However to model this phenomenon, we considered the electronic spin bath to be composed of a single macrospin. We drive the macrospin with a microwave field: B MW = B 1 cos (ω MW t) ê (A.1) where B 1 is the amplitude and ω MW is the frequency of the driving field, and ê denotes the vector component of the field perpendicular to the macrospin 208

209 quantisation axis. The probability of finding the macrospin transitioned from its initial state,, after time τ is simply given by the Rabi formula: P (τ) = ( Ωenv Ω env ) 2 1 cos (Ω env τ) 2 (A.2) where Ω env = Ω2 env + (ω MW ω env ) 2 is the environmental spin rabi frequency, ω env is the resonance frequency of the environmental spin and Ω env = γ eb 1 2 is the driving strength of the microwave driving field. From this we can treat the effective field of the macrospin as a classical field acting on the NV centre. This field is therefore: B env = B env cos (Ω env t) ê (A.3) where B env = γ e α ({ r i }) ( Ωenv Ω env ) 2 is the amplitude of the field arising from the macrospin. The factor α ({ r i }) accounts for the dependence of the environmental macrospin on the relative position of the environmental spins and the NV, r i. Applying the Rabi formula for a second time, this time to the NV spin initialised in 0, gives the probability of finding the NV in the 1 state after time τ as P 1 (τ) = ( ΩNV Ω NV ) 2 1 cos (Ω NV τ) 2 (A.4) where Ω NV = Ω2 NV + (Ω env ω NV ) 2 is the Rabi frequency of the NV with an effective driving strength of Ω NV = γ e α ({ r i }) ( Ωenv Ω env ) 2 / 2. This gives a general formula for the NV centre s population as a function of the microwave driving time, driving frequency and driving strength. We have only considered transitions between the 0 and 1 states of the NV due to the +1 electronic spin state being far from resonance. 209

210 (a) (b) Figure A.2: a) Directly driven ODMR on the NV centre at ω showing the single transition energy. b) The pulse sequence for EMR driving. In order to experimentally interrogate this we investigated a single near surface ( 20 nm deep) 15 NV. To simplify the complex hyperfine structure of the NV centre near the GSLAC, we applied a static magnetic field of B G, so that the two NV transitions were degenerate (the point referred to in Chapter 4 as ω ), leading to a single NV transition frequency of 2.65 MHz. Fig. A.2a) shows a standard ODMR of this NV centre showing the single transition. Eqn. A.4 suggests an EMR matching condition when the environmental spin Rabi frequency matches the NV transition frequency. That is ω NV = Ω env which gives ω NV = Ω 2 env + (ω MW ω env ) 2. (A.5) This condition gives the driving strength as a function of detuning in order to hit the NV centre resonance. There also exists an optimal point where the environmental spins are driven resonantly (ω MW = ω env ) and this EMR matching condition is met where the NV contrast is maximised. To map the EMR driving as a function of environmental driving strength and frequency, we start at this optimal point and run the pulse sequence 210

211 shown in Fig. A.2b). We then set a time τ which maximises the contrast for the NV corresponding to a π-pulse. Holding this driving time τ fixed and varying both Ω env (y-axis) and ω MW (x-axis) gives the photoluminescence map shown in Fig. A.3a). The dotted line shows the EMR matching condition. Fig. A.3b) is the theoretical map expected from eqn. A.4, where the probability has been scaled to the maximum experimental PL contrast. The resonant branches emanating from the optimal driving point (ω MW = ω env = 2868 MHz and Ω env = ω N V = 2.65 MHz) arise from the ability to recover the matching condition when the environmental spins are driven off-resonance by reducing the driving strength, Ω env. The experimental data is found to be in good overall agreement with the theoretical model, as indicated by the line cuts presented in Figs. A.3c) and A.3d). Having understood how the NV driving depends on the power and frequency at which the environmental electrons are driven, we now moved to demonstrate the driving dynamics of this method. Utilising the optimal driving parameters, an EMR driven Rabi curve on the NV was measured by varying the driving pulse duration (Fig. A.4a). An oscillation with a period of 3 µs is observed demonstrating the coherent driving of the NV centre. This Rabi period depends specifically on the distribution of the environmental electron spins. EMR Rabi driving can be achieved for any pair of (ω MW, Ω env ) where the matching condition is satisfied as shown by Fig. A.4b). Having demonstrated the ability for coherent control we then applied this driving technique to quantum control schemes fundamental to quantum information and sensing. We demonstrated that Ramsey (Fig. A.4c) and 211

212 (a) (b) (c) (d) Figure A.3: a) Experimental EMR photo-luminescence (PL) map as a function of the driving frequency, ω MW, and driving strength, Ω env. The dashed line corresponds to the EMR matching condition. b) Theoretical EMR PL map. c) Driving frequency (1 MHz to 3.5 MHz from top to bottom) and d) driving strength (2864 MHz to 2868 MHz from top to bottom) line cuts of the above EMR PL maps with experiment (black) and theory (red). spin-echo sequences (Fig. A.4d) could be achieved via EMR driving also. The EMR-driven Ramsey sequence shows an oscillation at the NV energy splitting of 2.65 MHz. This comes from the phase shift of the EMR driven pulse. In directly driven Ramsey, the driving fields are phase locked and hence the initial and final π -pulses rotate about the same axis in the NV s 2 212

213 Figure A.4: a) Optimally driven EMR Rabi oscillation on the NV centre. b) A map of Rabi oscillations as the driving strength on the environmental electrons is varied. The line corresponding to the Rabi curve in (a) is shown. A driving frequency ω MW = 2868 MHz is used for driving strengths Ω env > 2.65 MHz and ω MW is reduced for Ω env < 2.65 MHz such that the EMR matching condition is satisfied. rotating frame. For EMR, the phase of the driving field is static during the evolution time and hence it pulses the NV around the same axis in the lab frame. Hence we see an oscillation at the NV centre s evolution frequency as it accumulates phase relative to the pulsing axis. Importantly we do not see this in the spin-echo sequence (Fig. A.4d) because this is a static accumulation that is rephased by the spin-echo sequence. The EMR spinecho curve shows revivals at 1.1MHz, corresponding to the Larmor precession of the surrounding 13 C spin bath [112]. Importantly, comparing this data to 213