Quantum measurement and entanglement of spin quantum bits in diamond

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ement and Entanglement of Spin

Qu

antum Bits in Diamond

W

olfgang Pfa

Uitnodiging

voor het bijwonen van de openbare verdediging van mijn proefschrift:

Quantum Measurement

and Entanglement of

Spin Quantum Bits in

Diamond

Vrijdag 20 december 2013

12:30u in de Aula TU Delft

Mekelweg 5 te Delft

Om 12:00u zal ik het onderwerp van mijn promotie kort toelichten.

Wolfgang Pfaff Kloksteeg 23c 2611 BL Delft +31(0)639682651 wolfgangpfff@gmail.com Paranimfen: Hannes Bernien Reinier Heeres

Feest

‘t Boterhuis

Markt 15-17a te Delft

Vrijdag 20 december 2013

vanaf 21:00u

Q

uantum Measurement and Entanglement of Spin

uantum Bits in Diamond

Wolfgang Pfaff

Casimir PhD series 2013-34

ISBN 978-90-8593-174-4

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of spin quantum bits in diamond

Proefschrift

ter verkrijging van de graad van doctor aan de Technische Universtiteit Delft,

op gezag van de Rector Magni�cus Prof. ir. K.C.A.M. Luyben, voorzitter van het College voor Promoties,

in het openbaar te verdedigen op vrijdag 20 december 2013 om 12:30 uur door

Wolfgang PFAFF

Diplom-Physiker, Universität Regensburg, Duitsland geboren te Bad Kissingen, Duitsland

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Prof. dr. ir. R. Hanson

Prof. dr. ir. L.P. Kouwenhoven Samenstelling promotiecommissie: Rector Magni�cus,

Prof. dr. ir. R. Hanson,

Prof. dr. ir. L.P. Kouwenhoven, Prof. dr. ir. J.E. Mooij,

Prof. dr. Y.M. Blanter, Prof. dr. D. Bouwmeester, Prof. dr. A. Wallra�,

Prof. dr. ir. L.M.K. Vandersypen,

voorzitter

Technische Universiteit Delft, promotor Technische Universiteit Delft, promotor Technische Universiteit Delft

Technische Universiteit Delft

University of California, Santa Barbara Eidgenössische Technische Hochschule Zürich Technische Universiteit Delft, reservelid

ISBN: 978-90-8593-174-4

Casimir PhD Series Delft-Leiden 2013-34

Printed by Gildeprint Drukkerijen — www.gildeprint.nl

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I��

W. Pfa� & H. Bernien

The non-classical features of quantum measurement and entanglement are of great funda-mental interest as well as important ingredients for future technologies powered by quantum mechanics, such as quantum computers and quantum networks (chapter 1.1). A promising candidate platform for constructing and studying elementary building blocks for such tech-nologies are spins in diamond (chapter 1.2). This thesis presents a set of experiments in which quantum entanglement between non-interacting quantum bits — formed by single spins in diamond — is generated by quantum measurements, showing a way of prototyping quantum registers and networks (outlined in chapter 1.3).

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1.1 Quantum computers and quantum networks

Quantum mechanics is widely regarded as one of the most successful theories in physics, and to the best of our knowledge no experiment has been performed thus far that contradicts it. This accuracy is remarkable because today’s experimental possibilities go well beyond of what the founders of quantum theory thought was achievable — Schrödinger for instance famously wrote1_:

“we never experiment with just one electron or atom or (small) molecule. In thought-experiments we sometimes assume that we do; this invariably entails ridiculous consequences [...]

it is fair to state that we are not experimenting with single particles, any more than we can raise Ichthyosauria in the zoo.”

This statement clearly does not hold any more, as experimental techniques to study single particles have been developed with great success in the recent decades — acknowledged by the recent awarding of the 2012 Nobel Prize in Physics to Serge Haroche and David J. Wine-land “for ground-breaking experimental methods that enable measuring and manipulation of individual quantum systems.”

Nonetheless, quantum mechanical predictions — and in particular their “ridiculous

con-sequences”, such as quantum entanglement2,3— prevail, and we are lead to believe that

seemingly exotic features of quantum theory are not merely side-e�ects of mathematical modelling, but real phenomena that can be observed. This development has in turn triggered great interest in the possibility to use these phenomena in new technologies that can outper-form counterparts that operate solely on the basis of classical physics.

As experimental quantum physicists we are interested in studying and controlling the quintessential features of quantum mechanics on the smallest scale. These e�orts are made in the hope that we can resolve fundamental disputes and questions about the interpretation of quantum theory4,5_{, as well as demonstrate the feasibility of new quantum technologies.} In particular, quantum information processing can help to overcome limitations faced in classical computing. An illustrating example is quantum physics itself: assume we want to simulate a molecule of 500 degrees of freedom, where each of those dimensions has at least two possible states. The full state information consists of the complex amplitudes of all

2500_{terms in the general superposition state — more numbers than the estimated total}

of atoms in the observable universe and certainly intractable by any classical computer. However, if controlled and preserved well enough, a quantum system that can be mapped

to the very molecule to be described can be used for the task6 _{— an idea that has been}

generalized to universal quantum simulators7_.

Furthermore it has become clear that also more general computational problems can be treated more e�ciently when generalizing the classical computer to a quantum Turing

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machine8,9_{, where the classical bit that takes values of either 0 or 1 is replaced by the} quantum bit (qubit) that can be in superpositions of basis states |0i and |1i. This idea has since given raise to the new discipline of quantum information processing10,11_{. Potential} applications that generated of lot of interest include the factorisation of large numbers12 and the searching of unsorted databases13_.

The implementation of quantum information protocols demands very good control over

individual and composite quantum systems14_{. Notably, quantum entanglement between}

qubits can be employed to achieve a performance-enhancement over classical protocols15,16_. Furthermore, the ability to perform projective quantum measurements is desirable for a number of applications, for instance quantum error correction10_.

For the communication of quantum information and for distributed quantum computing

quantum networks have been devised17_{. If entanglement between (remote) nodes of such a}

network can be established, quantum states and operations can be transmitted via teleport-ation18,19_{. The entanglement required can be generated, for instance, by interference and} measurement of photons20_.

The realisation of an extended network of quantum registers or computers that are linked by entanglement would allow us to study quantum physics at a truly macroscopic scale and test its limits. At the same time this step could enable radically new applications in computation and communication. The focus of this work is the generation of entanglement between non-interacting solid-state qubits by quantum measurement, in order to show a pathway for creating prototypes of such quantum networks.

1.2 Spins in diamond as building block for quantum technologies

We implement quantum registers with single spins that are associated to the Nitrogen Vacancy (NV) colour defect in diamond. The NV centre has gained much interest in ex-perimental quantum physics since the �rst observation of a single such centre in 199721_: the defect behaves much like a single ion or small molecule, but comes “pre-packaged” in a robust solid-state environment that it does not strongly interact with. The NV centre displays quantum phenomena that can be accessed with relatively simple experimental arrangements, and under moderate environment conditions.

The NV shows photoluminescence under excitation with visible light and is a stable single-photon emitter22_{. It has an electronic spin that can be initialized and read out optically, and}

manipulated with standard magnetic resonance techniques23_{. Remarkably, these properties}

are available even at room temperature. As a result, the NV has been used for experiments in various aspects of quantum science, from sensitive metrology24–27_{, studies of single-spin} decoherence28–30_{, to fundamental tests of quantum mechanics}31–33_{, to name only a few.}

Recent critical advances show the potential for NV centres in quantum information processing: It has been shown that nuclear spins interacting with the NV electronic spin can be used as qubits34–36_{, opening up the possibility for building local multi-qubit quantum}

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single photon emission

laser excitation

entanglement by photon interference and measurement

spin manipulation

by magnetic resonance pulses

or ?

Figure 1.1 | Vision of a macroscopic quantum network based on spins in diamond. Nodes consist of single NV centres in diamonds that are separated by macroscopic distances (metres up to kilometres). The NV electronic spin (purple) can be measured optically, and serves as the interface for a quantum register of nuclear spins (orange, green). Entanglement between nodes is established by interference and measurement of photons that are correlated with the electron spins.

registers. Further, at cryogenic temperatures the electronic spin can be interfaced coherently with photons37_{, enabling linking of such registers to macroscopic networks. Finally, the} electron spin and nuclear spin states can be determined in a single shot by projective quantum measurements38,39_{. On these grounds we believe that the NV centre is a promising platform} for implementing quantum registers and networks. A cartoon of the architecture we envision is shown in Fig. 1.1.

1.3 Thesis overview

In chapter 2 we will outline the relevant physical properties of the NV centre: we �rst discuss how the electronic spin can serve as the central qubit of a nuclear spin register, and how it can be used as the optical interface to this register. Furthermore, we explain how entanglement between remote NV centres can be established.

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Chapter 3 explains how our experiments are performed. We discuss the principles of addressing single NV centres, and measurement and manipulation of electronic and nuclear spins. We further lay out how we control and use the optical interface of the NV centre. Finally, we discuss the experimental setup used and how we con�rm the creation of entangled states.

In chapter 4 we demonstrate how entanglement between nuclear spins coupled to the same NV centre can be generated by a quantum measurement on the NV electronic spin. This method enables the creation of entangled states of high �delity and allows us to violate Bell’s inequality for the �rst time with spins in a solid.

We demonstrate generation of entanglement between two NV centre electronic spins that are separated by a macroscopic distance of three metres in chapter 5. We show how quantum interference and measurement of indistinguishable single photons from the NV centres can project the spins into a Bell-state.

We demonstrate deterministic quantum teleportation between remote spin qubits in chapter 6. Combining remote entanglement with a local Bell-state measurement we are able to teleport the state of a nuclear spin qubit onto an electronic spin over a macroscopic distance.

We �nally summarise the work presented in chapter 7 and discuss the future of this line of experiments.

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1.4 Bibliography

[1] E. Schrödinger. Are there quantum jumps? Part II. Brit. J. Phil. Sci. 3, 233 (1952). [2] S. J. Freedman and J. F. Clauser. Experimental Test of Local Hidden-Variable Theories.

Phys. Rev. Lett.28, 938 (1972).

[3] A. Aspect, P. Grangier and G. Roger. Experimental Tests of Realistic Local Theories via Bell’s Theorem. Phys. Rev. Lett. 47, 460 (1981).

[4] M. Schlosshauer, J. Ko�er and A. Zeilinger. A Snapshot of Foundational Attitudes Toward Quantum Mechanics. Stud. Hist. Phil. Mod. Phys. 44, 222 (2013).

[5] G. A. D. Briggs, J. N. Butter�eld and A. Zeilinger. The Oxford Questions on the foundations of quantum physics. Proc. R. Soc. Lond. A 469, 20130299 (2013).

[6] R. P. Feynman. Simulating physics with computers. Int. J. Theor. Phys. 21, 467 (1982). [7] S. Lloyd. Universal Quantum Simulators. Science 273, 1073 (1996).

[8] P. Benio�. Quantum mechanical hamiltonian models of turing machines. J. Stat. Phys. 29, 515 (1982).

[9] D. Deutsch. Quantum theory, the Church-Turing principle and the universal quantum computer. Proc. R. Soc. Lond. A 400, 97 (1985).

[10] M. A. Nielsen and I. L. Chuang. Quantum Computation and Quantum Information. Cambridge University Press, Cambridge (2001).

[11] N. D. Mermin. Quantum computer science: an introduction. Cambridge University Press, Cambridge (2007).

[12] P. W. Shor. Polynomial-Time Algorithms for Prime Factorization and Discrete Logar-ithms on a Quantum Computer. SIAM J. Comput. 26, 1484 (1997).

[13] L. K. Grover. A fast quantum mechanical algorithm for database search. Proceedings of the twenty-eighth annual ACM symposium on Theory of computing 212–219 (1996). [14] D. P. DiVincenzo. The Physical Implementation of Quantum Computation. Fortschr.

Phys.48, 771 (2000).

[15] C. H. Bennett and D. P. DiVincenzo. Quantum information and computation. Nature 404, 247 (2000).

[16] R. Jozsa and N. Linden. On the role of entanglement in quantum-computational speed-up. Proc. R. Soc. Lond. A 459, 2011 (2003).

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[18] C. H. Bennett et al. Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels. Phys. Rev. Lett. 70, 1895 (1993).

[19] D. Gottesman and I. L. Chuang. Demonstrating the viability of universal quantum computation using teleportation and single-qubit operations. Nature 402, 390 (1999). [20] L.-M. Duan, M. D. Lukin, J. I. Cirac and P. Zoller. Long-distance quantum communication

with atomic ensembles and linear optics. Nature 414, 413 (2001).

[21] A. Gruber et al. Scanning Confocal Optical Microscopy and Magnetic Resonance on Single Defect Centers. Science 276, 2012 (1997).

[22] C. Kurtsiefer, S. Mayer, P. Zarda and H. Weinfurter. Stable Solid-State Source of Single Photons. Phys. Rev. Lett. 85, 290 (2000).

[23] F. Jelezko, T. Gaebel, I. Popa, A. Gruber and J. Wrachtrup. Observation of coherent oscillations in a single electron spin. Phys. Rev. Lett. 92, 076401 (2004).

[24] J. R. Maze et al. Nanoscale magnetic sensing with an individual electronic spin in diamond. Nature 455, 644 (2008).

[25] G. Balasubramanian et al. Nanoscale imaging magnetometry with diamond spins under ambient conditions. Nature 455, 648 (2008).

[26] F. Dolde et al. Electric-�eld sensing using single diamond spins. Nature Phys. 7, 459 (2011).

[27] G. Kucsko et al. Nanometre-scale thermometry in a living cell. Nature 500, 54 (2013). [28] G. de Lange, Z. H. Wang, D. Riste, V. V. Dobrovitski and R. Hanson. Universal Dynamical

Decoupling of a Single Solid-State Spin from a Spin Bath. Science 330, 60 (2010). [29] C. A. Ryan, J. S. Hodges and D. G. Cory. Robust Decoupling Techniques to Extend

Quantum Coherence in Diamond. Phys. Rev. Lett. 105, 200402 (2010).

[30] B. Naydenov et al. Dynamical decoupling of a single-electron spin at room temperature. Phys. Rev. B83, 81201 (2011).

[31] V. Jacques et al. Experimental Realization of Wheeler’s Delayed-Choice Gedanken Experiment. Science 315, 966 (2007).

[32] G. Waldherr, P. Neumann, S. F. Huelga, F. Jelezko and J. Wrachtrup. Violation of a temporal bell inequality for single spins in a diamond defect center. Phys. Rev. Lett. 107, 090401 (2011).

[33] R. E. George et al. Opening up three quantum boxes causes classically undetectable wavefunction collapse. Proc. Natl. Acad. Sci. U.S.A. 110, 3777 (2013).

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[34] F. Jelezko et al. Observation of Coherent Oscillation of a Single Nuclear Spin and Realization of a Two-Qubit Conditional Quantum Gate. Phys. Rev. Lett. 93, 130501 (2004).

[35] M. V. G. Dutt et al. Quantum register based on individual electronic and nuclear spin qubits in diamond. Science 316, 1312 (2007).

[36] P. Neumann et al. Multipartite Entanglement Among Single Spins in Diamond. Science 320, 1326 (2008).

[37] E. Togan et al. Quantum entanglement between an optical photon and a solid-state spin qubit. Nature 466, 730 (2010).

[38] P. Neumann et al. Single-Shot Readout of a Single Nuclear Spin. Science 329, 542 (2010). [39] L. Robledo et al. High-�delity projective read-out of a solid-state spin quantum register.

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T�� NV ��

��

In this chapter we will outline the physical principles for using the NV centre as the elementary building block for quantum registers and quantum networks. In particular, we address the nature of the electronic spin and its suitability for serving both as a central qubit and an optical interface (chapter 2.2), and how a nuclear spin register around the electronic spin is formed (chapter 2.3). We �nally lay out the principles for establishing entanglement between remote NV centres using a photonic channel (chapter 2.4).

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a b C N V conduction band valence band a1’ a1 ex ey

Figure 2.1 | Basic structure of the NV centre. a, The nitrogen-vacancy defect in the carbon-matrix of diamond is formed by a substitutional nitrogen atom (N) next to a la�ice vacancy (V). b, Molecular orbitals (labels denote their symmetry) and their filling in the orbital ground state of NV . These orbitals are linear combinations of the hybridised sp3 orbitals of the nitrogen and the 3 carbon atoms that transform according to the irreproducible representations of the C3 symmetry group1,2_{. The lowest-lying orbital a}0

1is located inside the valence band.The relevant physics of the NV centre is mainly governed by the occupation of the orbitals in the band gap. In the ground state of NV the orbital configuration is a2₁e2. Optical excitation can promote one electron to the first orbital excited state a1₁e3.

2.1 Introduction

The nitrogen-vacancy defect (NV) is a colour-centre in diamond consisting of a substitutional nitrogen atom and a neighbouring lattice vacancy (Fig. 2.1a). In its neutral charge state (NV0₎ the defect hosts 5 electrons — 3 from the dangling bonds of the vacancy’s next-neighbour carbon atoms, and 2 donor electrons from the nitrogen. In this work we are mainly concerned with the negatively charged state (NV ), where an additional electron is captured from the environment.

The 6 electrons of NV occupy the available molecular orbitals in the ground state as shown in Fig. 2.1b. The two orbitals that have the highest energy are degenerate and host one unpaired electron each to form a spin triplet. Upon optical excitation one electron from the orbital below can be transferred to one of the two highest energy orbitals. Importantly, the energies of the ground as well as the �rst orbital excited state are located inside the band gap of diamond, resulting in ion-like properties of the defect.

In the following we will give only a brief overview of the features of the NV centre relevant for the experimental work presented. For more details we point the reader to the numerous reviews published very recently, covering both the fundamental properties of the defect3,4

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as well as the current applications5–11_.

2.2 The electronic spin: central qubit and optical interface

In the simpli�ed model of non interacting electrons both the ground state con�guration a2 1e2 and the �rst excited con�guration a1

1e3have degenerate spin singlet and triplet states. This degeneracy is lifted by the Coulomb interaction which leads to spin triplet (S=1) ground states3_A

2which are separated by 1.945 eV from the spin triplet excited states3E(Fig. 2.2)1,2.

The exact positions of the singlet states are not yet determined2 _{and are for this work}

summarised to one intermediate level. Adding spin-orbit and spin-spin interactions to the model splits the ground and excited triplet states.

Central qubit

Within the orbital ground state the ms =±1 levels are separated by a zero �eld splitting

D_{⇡ 2.88 GHz from the ms} =0 state. The ground state is described by the Hamiltonian

HGS=DS2_z+ eB · S, (2.1)

where Si are the Pauli spin operators and e=2.802 MHz/G is the gyromagnetic ratio. A

magnetic �eld Bz parallel to the N-V axis splits the ms =±1 states by the Zeeman e�ect

(Fig. 2.2d). For moderate �eld strengths the degeneracy between ms= 1 and ms=+1 is

lifted and their transitions to ms=0 can be selectively driven by applying microwaves with the frequency of the corresponding transition12_{. We can de�ne an e�ective two level system}

which serves as a qubit with ms=0 := |0i and ms= 1 := |1i (equally well ms=+1 can be

de�ned as |1i).

These qubit states are very robust. Coherence times of a few milliseconds can be observed even at room temperature and at low temperatures coherence times of single NVs beyond 10 ms for single NVs13_{and up to 0.5 s for ensembles}14_{have been observed. There are two} main reasons for these long times. First, the electronic energy levels lie deep within the large band gap of diamond (5.5 eV) and are therefore well isolated from the bulk electronic states. Second, the diamond lattice consists to 98.9% of spinless12_{C, leading to only slow dephasing} from a �uctuating nuclear spin bath.

The limit on the coherence time is set by the remaining magnetic impurities. These form a spin bath which creates a �uctuating magnetic �eld at the NV location which leads to dephasing15_{. For type Ib diamonds the spin bath is dominated by nitrogen defect centres (P1)} that have an electronic S = 1/2 spin. In type IIa diamonds the bath is given by the remaining 13_{C nuclear spins (I = 1/2).}

Optical interface

The spin-orbit and spin-spin interaction split the excited state triplet into four levels, two of which are doubly degenerate (E1,2and Ex, ). The spin of the Ex, states is ms=0. All other

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states are superpositions of ms=+1 and ms= 1. The transitions between the ground state and the excited state are in the visible (637 nm) and follow the selection rules given in table 2.11_{. At low temperature they can be resonantly excited using a laser of the corresponding} frequency and this way di�erent spin states can be selectively addressed.

A1 A2 E1 E2 Ex E 3_A 2,ms= 1 + + 3_A 2,ms=0 y x 3_A 2,ms=+1 + +

Table 2.1 | Optical selection rules. Transitions between the ground and excited states occur under the emission or absorption of a linearly polarised (x,y) or circularly polarised

( +, ) photon.

The NV centre can also be excited o�-resonantly with higher energy. In this way a phonon level above the excited state will be populated that quickly decays non-radiatively into3_E (see �g. 2.2). During the emission the reverse process happens by which the NV either directly decays into the ground state (zero phonon line, ZPL) or via a phonon level above the ground state (phonon side band, PSB).

From the excited triplet state the NV centre can also decay to the metastable singlet states.

This coupling is stronger for states with ms=±1 components than for ms=0 states. This

di�erence results in a spin dependent �uorescence rate that allows to determine the spin

state by averaging the �uorescence intensity over many experimental runs3_{. The singlets}

decay preferentially into ms =0, leading to spin polarisation under o�-resonant excitation. The initialisation �delity using this technique has been reported to range from 42–96%, depending on the experimental setting4_{. Higher initialisation �delity and spin readout in} a single shot can be achieved at low temperatures using spin selective resonant excitation (chapter 3.2).

The excited levels can be shifted by applying strain as well as by applying electric �elds to the NV centre16,17_{. Electric �elds along the N-V axis do not e�ect the spacing between} the levels but o�set the whole spectrum. Electric �elds perpendicular to this axis break the C3 symmetry and change the splittings between the levels (Fig. 2.2c). This also a�ects the spin components of the levels which are increasingly mixed with higher strain. The ability to tune the frequency of the emitted photons is crucial in order to link distant NV centres via a photonic channel.

2.3 Nuclear spins as quantum register

Nuclear spins in solids are promising candidates for quantum register qubits because of their long coherence times18,19_{and the availability of well-established nuclear magnetic} resonance techniques to manipulate them20,21_{. In particular, nuclear spins in diamond that}

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m_s = 0 m_s = ±1 3_E 3_A 2 E_1,2(m_s= ±1) Ex,y (ms = 0) A1 (ms = ±1) A₂(m_s= ±1) 1.945 e V 2 ! 2.802 MHz/G ω0 /2π PSB PSB singlet states 0 2 4 6 8 10 Lateral strain (GHz) −10 −5 0 5 10 En er gy (GH z) 2.88 GHz Magnetic field ‖ z Energy 0 |0〉 |1〉 a b c d

Figure 2.2 | Electronic level structure and optical excitation of NV . a, The ground state triplet3A2can be optically excited either resonantly to the3Eexcited or o�-resonantly into higher lying levels in the phonon side band (PSB) that quickly decays to3E. Emission of photon can either occur directly or in to the PSB above the ground state. b, The ground state is split by the zero field spli�ing into one ms=0 and two degenerate ms=±1 states. Spin-spin and spin-orbit interactions split the excited state into four levels two of which are doubly degenerate. The labels indicate the symmetry of the state under C3 transformations. Arrows indicate the allowed optical transitions. c, The excited state spli�ings are e�ected by strain (or equivalently by electric fields) that is applied perpendicular to the N-V-axis. With increasing strain the spin states of the levels are increasingly mixed. d, Magnetic fields

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couple strongly to the electronic spin of an NV centre have shown great potential for use in quantum registers in a row of proof-of-principle experiments in the recent years19,22–28_.

We de�ne nuclear spin qubits in the orbital ground state of NV . The key ingredient that allows us to do so is the hyper�ne interaction between nuclear spins and the NV electronic spin. This coupling causes a splitting of the electronic spin states ms=_{±1, with each of the} sub-levels corresponding to a nuclear spin eigenstate that we can use as qubit basis state.

The exact nature of the basis states of a particular nuclear spin depends on the details of the hyper�ne interaction, which is in general given by20

Hhf=µeµn X µ, =x, ,z Sµ " 8_{3 |} (rn)|2+ * ₁ |r rn|3 +! µ 3 * _n µn |r rn|3 +# I , (2.2)

where both Fermi contact and dipolar coupling are included. S and I are the electronic and

nuclear spin operators, respectively, µe,nare the magnetic moments of the electron and

nucleus, respectively, (r) is the electron wavefunction, rnis the position of the nucleus, n is a unit vector along the r rnaxis, µ is the Kronecker delta, and h·i denotes the average. For the case of small magnetic �elds — corresponding to the experimental settings used throughout this thesis — this interaction can be simpli�ed: the largest energy involved is the zero-�eld splitting in z direction, D ⇡ 2 ⇥ 2.88 GHz, that de�nes the quantisation of the electron spin. One can make a secular approximation and neglect terms in the Hamiltonian that contain Sxor S , because transitions of the electronic spin due to hyper�ne interaction is strongly suppressed due to the large energy mismatch. If the dominating in�uence on the nuclear spin is the interaction with the electron, the quantisation axis in ms=±1 is entirely determined by the direction of the hyper�ne �eld of the electron at the point of the nucleus. The resulting simpli�ed interaction Hamiltonian is then

Hhf=Sz X

z I ⌘ ASzIz0, (2.3)

where z0_{is the quantisation direction of the nuclear spin for ms} ₌ _{±1, and} _µ _{are the}

components of the hyper�ne tensor (implicitly given in (2.2)). We can see that our qubit basis states are nuclear spin eigenstates that are de�ned by the splitting due to the e�ective magnetic �eld of the electronic spin.

We are particularly interested in nuclear spins for which the interaction strength exceeds the line width of the electronic spin transitions, which is typically on the order of ⇠ 100 kHz for our samples. In this case it is possible to perform electron spin rotations between ms=0 $ ±1 that are conditional on the nuclear spin state, allowing for the implementation of a controlled-NOT (CNOT) gate. Suitable nuclei are the one of the nitrogen host atom of the NV centre as well as those of close-by13_{C atoms.}

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The nitrogen host

Each NV centre has a nitrogen atom with nuclear spin. Most commonly, with a natural abundance of 99.3%, the nitrogen is of the isotope14_{N with a nuclear spin of IN}₌_{1. There is} also the possibility for a15_{N (I = 1/2), either by chance or engineering}29,30_{. We will only}

discuss the14_{N case here because all experiments presented have been performed using}

naturally occurring NV centres with the14_{N isotope.}

The system Hamiltonian for electronic and nitrogen nuclear spin in the orbital ground state can be written as

He,N=DS_z2+ eB · S QIN2z+ NB · IN AkSzINz A_?⇣SxINx+ S IN ⌘, (2.4)

where Siand I_Niare the i-components of spin-1 operators for electron and nucleus, S and IN. Note that the quantisation axes (z-axes) for both electron and nitrogen spin are parallel. e,N are the electron and nitrogen nuclear spin gyromagnetic ratios, and Q is the quadrupolar splitting of the nitrogen nuclear spin. The hyper�ne interaction can be divided into a parallel

component Akand a perpendicular component A?. In the secular approximation,

He,N=DS2_z+ eBzSz QIN2z+ NBzINz ANSzINz, (2.5)

where we neglect all o�-diagonal terms of the14_{N spin because its quantisation is fully}

governed by its quadrupolar splitting, independent of the electron spin projection. The resulting level structure is shown in Fig. 2.3.

We can, for instance, identify mI= 1 = |1iNand mI=0 = |0iN, leaving the third nuclear spin projection idle. The splitting of the two basis states is — up to an additional Zeeman splitting — determined by the sum of the quadrupolar splitting and the hyper�ne interaction

strength, N0 =Q+ AN =2 ⇥ 7.132 MHz, and qubit rotations can be performed using

magnetic resonance pulses in the radio-frequency (RF) domain.

13

_{C spins}

Besides the spin of the nitrogen host, the NV electronic spin also couples to13_{C nuclear spins} (I = 1/2) in the environment. In non-puri�ed diamond, the13_{C isotope occurs with a natural} abundance of 1.1% within the otherwise spin-free12_{C material, and the resulting spin bath is} to a large extent responsible for the dephasing of the electronic spin31_{. However, individual}

13_{C atoms that are located only a few lattice sites away from the NV centre experience a}

strong hyper�ne interaction32_{, and are usable as qubits.}

For a strongly coupled carbon nuclear spin coupled to the NV centre electronic spin we can write the Hamiltonian in the orbital ground state as

He,N,C=DS2_z+ eBzSz QIN2z+ NBzINz ANSzINz+ CB · IC+ ACSzICz0, (2.6)

where we have again made the secular approximation. For small magnetic �elds, the quant-isation axis z0_{of the carbon spin in the ms}₌_{±1 manifold is determined by the hyper�ne �eld}

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Figure 2.3 | Spin level structure in the orbital ground state. The electronic ms = 0

state is separated from the ms=±1 levels by the zero-field spli�ing, D = 2 ⇥ 2.878 GHz

(at low temperature). In presence of an external magnetic field the ms = ±1 levels are

split by 2 eBz, where Bz is the z-component of the field, and e =2 ⇥ 2.802 MHz/G is

the electron gyromagnetic ratio. The mI =±1 states of the of the nitrogen spin (I = 1

for14N, the case depicted here) are lowered with respect to the mI = 0 level due to a

nuclear quadrupole spli�ing, with Q = 2 ⇥ 4.946 MHz. Hyperfine interaction between the electronic and the14N spin splits the mI=±1 levels for ms=±1, with a coupling constant

AN =2 ⇥ 2.186 MHz. We ignore the magnetic field spli�ing of the nuclear spin at this

point. In the presence of a strongly coupled13C nuclear spin (I = 1/2) the ms =_{±1 levels are} further split by a coupling constant ACthat depends on the la�ice site that the13C atom occupies. A typical set of qubit basis states are indicated.

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of the electron at the lattice site of the13_{C atom. Contrary to the spin-1}14_{N the carbon spin} has no pre-determined quantisation for ms =0 — its quantisation axis is then determined by the alignment of the externally applied magnetic �eld. Therefore, unless the magnetic �eld is aligned with the e�ective hyper�ne �eld at the location of the nucleus, the quantisation

axes can be di�erent for ms =0 and ms =±1. An illustration of the level diagram for one

13_{C spin coupled to the NV is shown in Fig. 2.3.}

We identify qubit basis states as mI =+1/2 = |0iCand mI = 1/2 = |1iC. The splitting

depends on the hyper�ne interaction strength, and can range from tens of kHz for13_{C atoms} a few sites away, up to more than 100 MHz for neighbour sites of the vacancy24,32_.

2.4 Remote entanglement via a photonic channel

The optical interface of the NV centre does not only allow to access the local register but also provides a route to connect remote registers by entanglement. In a measurement-based scheme33–36_{a combined detection of photons emitted from both emitters projects the spins} into an entangled state13,37,38_{. This type of protocols are particularly well suited for the case} of the NV centre as the �delity of the entangled state is not directly e�ected by the success probability and is therefore robust against photon loss.

The general scheme of entanglement generation is shown in Fig. 2.4. First, both centres are caused to emit a photon that is entangled with the electronic spin39–42_{. The overall} spin-photon state is then

↵

= 1

2⇥|0i | i + |1i ↵⇤ , (2.7)

where |0i, |1i are two electronic spin states and | i, ↵ are two orthogonal photonic states, that could be, for instance, di�erent polarization-, frequency-, time-bin-, or number-states. Such entangled states can be created by using the spin-dependent optical transitions of the NV centre at low temperatures.

In order to create entanglement between the two spins, both photons are overlapped on a beamsplitter. The beamsplitter transforms a photon | i in the input mode according to43_:

| ia ! _{p2(| i}1 1+ | i2), | ib ! p2(| i1 1 | i2), (2.8)

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a b

a

1 2

b

c

Figure 2.4 | Measurement-based creation of remote entanglement. a, Each spin is prepared in a state that is entangled with an emi�ed photon. b, The two photons are overlapped on a beamspli�er. If the photons are indistinguishable the beamspli�er erases the which-path information. c, Detection of certain photonic states projects the spins into an entangled state.

The full state of the two photons and two spins is ↵

a⌦ ↵b = p2(|0i1 a| ia+ |1ia ↵a)⌦p2(|0i1 b| ˜ib+ |1ib ˜↵b)

= 1

4[|0ia(| i1+ | i2)+ |1ia( ↵1+ ↵2) ⌦ |0ib(| ˜i1 | ˜i2)+ |1ib( ˜↵₁ ˜↵₂)]

= 1

4[|00i (| ˜i1 | ˜i2+ | ˜i1| i2 | i1| ˜i2)

+ |11i ( ˜↵1 ˜↵₂+ ˜↵₁ ↵₂ ↵₁ ˜↵₂)

+ |01i ( ˜↵1 ˜↵2+ ˜↵1| i2 | i1 ˜↵2)

+ |10i ( ˜ ↵1 ˜ ↵₂+ | ˜i1 ↵2 ↵1| ˜i2)]. (2.9)

For indistinguishable photons (| i = | ˜i, ↵ = ˜↵) terms such as | ˜i1| i2 | i1| ˜i2

cancel. This is called two-photon quantum interference, or Hong-Ou-Mandel e�ect44_{, and is} essential for the success of the protocol. Then equation 2.9 becomes:

↵

a⌦ ↵b = 1₄[|00i (|2 i1 |2 i2)

+ |11i ( 2 ↵1 2 ↵2)

+ (|01i + |10i)( ↵1 ↵2)

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Detecting a certain photonic state projects the two emitters into the corresponding spin state. Assume that | i and ↵ are two orthogonal polarization of the photon. Then the detection of one photon in each output port of the beamsplitter (Fig. 2.4c) projects the two

spins into the entangled Bell state =1/p2(|01i |10i).

In chapter 5 we use such a measurement-based protocol to create entanglement between two NV centres that are separated by three meters. The photon states used in that experiment

are number states, | i = |1iphotonand ↵ = |0iphoton. With the detectors used states as

|2 i1 =_|2i1,photonand | i1 ↵₂ =_|1i1,photon_|0i2,photoncannot be distinguished. Therefore, the spins are projected into a mixed state. This issue can be overcome with an adaptation of the protocol as suggested by Barret and Kok36_{, consisting of two rounds of the protocol} with a spin �ip in between. Detection of exactly one photon in each round then projects the two spins into an entangled state (chapter 5).

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2.5 Bibliography

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[2] M. W. Doherty, N. Manson, P. Delaney and L. C. L. Hollenberg. The negatively charged nitrogen-vacancy centre in diamond: the electronic solution. New J. Phys. 13, 5019 (2011).

[3] F. Jelezko and J. Wrachtrup. Single defect centres in diamond: A review. phys. stat. sol. (a)203, 3207 (2006).

[4] M. W. Doherty et al. The nitrogen-vacancy colour centre in diamond. Physics Reports 528, 1 (2013).

[5] V. Acosta and P. Hemmer. Nitrogen-vacancy centers: Physics and applications. MRS Bull38, 127 (2013).

[6] L. Childress and R. Hanson. Diamond NV centers for quantum computing and quantum networks. MRS Bull 38, 134 (2013).

[7] L. T. Hall, D. A. Simpson and L. C. L. Hollenberg. Nanoscale sensing and imaging in biology using the nitrogen-vacancy center in diamond. MRS Bull 38, 162 (2013). [8] S. Hong et al. Nanoscale magnetometry with NV centers in diamond. MRS Bull 38, 155

(2013).

[9] M. Lončar and A. Faraon. Quantum photonic networks in diamond. MRS Bull 38, 144 (2013).

[10] D. M. Toyli, L. C. Bassett, B. B. Buckley, G. Calusine and D. D. Awschalom. Engineering and quantum control of single spins in semiconductors. MRS Bull 38, 139 (2013). [11] J. Wrachtrup, F. Jelezko, B. Grotz and L. McGuinness. Nitrogen-vacancy centers close

to surfaces. MRS Bull 38, 149 (2013).

[12] F. Jelezko, T. Gaebel, I. Popa, A. Gruber and J. Wrachtrup. Observation of coherent oscillations in a single electron spin. Phys. Rev. Lett. 92, 076401 (2004).

[13] H. Bernien et al. Heralded entanglement between solid-state qubits separated by three metres. Nature 497, 86 (2013).

[14] N. Bar-Gill, L. M. Pham, A. Jarmola, D. Budker and R. L. Walsworth. Solid-state electronic spin coherence time approaching one second. Nature Commun. 4, 1743 (2013).

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[15] V. V. Dobrovitski, A. E. Feiguin, R. Hanson and D. D. Awschalom. Decay of Rabi oscillations by dipolar-coupled dynamical spin environments. Phys. Rev. Lett. 102, 237601 (2009).

[16] P. Tamarat et al. Stark Shift Control of Single Optical Centers in Diamond. Phys. Rev. Lett.97, 83002 (2006).

[17] L. C. Bassett, F. J. Heremans, C. G. Yale, B. B. Buckley and D. D. Awschalom. Elec-trical Tuning of Single Nitrogen-Vacancy Center Optical Transitions Enhanced by Photoinduced Fields. Phys. Rev. Lett. 107, 266403 (2011).

[18] M. Steger et al. Quantum Information Storage for over 180 s Using Donor Spins in a 28Si "Semiconductor Vacuum". Science 336, 1280 (2012).

[19] P. C. Maurer et al. Room-Temperature Quantum Bit Memory Exceeding One Second. Science336, 1283 (2012).

[20] C. P. Slichter. Principles of magnetic resonance. Springer, New York, 3rd edition (1990). [21] L. Vandersypen and I. Chuang. NMR techniques for quantum control and computation.

Rev. Mod. Phys.76, 1037 (2005).

[22] F. Jelezko et al. Observation of Coherent Oscillation of a Single Nuclear Spin and Realization of a Two-Qubit Conditional Quantum Gate. Phys. Rev. Lett. 93, 130501 (2004).

[23] M. V. G. Dutt et al. Quantum register based on individual electronic and nuclear spin qubits in diamond. Science 316, 1312 (2007).

[24] P. Neumann et al. Multipartite Entanglement Among Single Spins in Diamond. Science 320, 1326 (2008).

[25] P. Neumann et al. Single-Shot Readout of a Single Nuclear Spin. Science 329, 542 (2010). [26] L. Jiang et al. Repetitive readout of a single electronic spin via quantum logic with

nuclear spin ancillae. Science 326, 267 (2009).

[27] G. D. Fuchs, G. Burkard, P. V. Klimov and D. D. Awschalom. A quantum memory intrinsic to single nitrogen-vacancy centres in diamond. Nature Phys. 7, 789 (2011). [28] T. van der Sar et al. Decoherence-protected quantum gates for a hybrid solid-state spin

register. Nature 484, 82 (2012).

[29] D. M. Toyli, C. D. Weis, G. D. Fuchs, T. Schenkel and D. D. Awschalom. Chip-scale nanofabrication of single spins and spin arrays in diamond. Nano Letters 10, 3168 (2010).

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[30] K. Ohno et al. Engineering shallow spins in diamond with nitrogen delta-doping. Appl. Phys. Lett.101, 2413 (2012).

[31] J. R. Maze, J. M. Taylor and M. D. Lukin. Electron spin decoherence of single nitrogen-vacancy defects in diamond. Physical Review B 78, 94303 (2008).

[32] B. Smeltzer, L. Childress and A. Gali. 13C hyper�ne interactions in the nitrogen-vacancy centre in diamond. New J. Phys. 13, 025021 (2011).

[33] C. Cabrillo, J. Cirac, P. Garcia-Fernandez and P. Zoller. Creation of entangled states of distant atoms by interference. Phys. Rev. A 59, 1025 (1999).

[34] L.-M. Duan, M. D. Lukin, J. I. Cirac and P. Zoller. Long-distance quantum communication with atomic ensembles and linear optics. Nature 414, 413 (2001).

[35] C. Simon and W. T. M. Irvine. Robust Long-Distance Entanglement and a Loophole-Free Bell Test with Ions and Photons. Phys. Rev. Lett. 91, 110405 (2003).

[36] S. D. Barrett and P. Kok. E�cient high-�delity quantum computation using matter qubits and linear optics. Physical Review A 71, 60310 (2005).

[37] D. L. Moehring et al. Entanglement of single-atom quantum bits at a distance. Nature 449, 68 (2007).

[38] J. Hofmann et al. Heralded Entanglement Between Widely Separated Atoms. Science 337, 72 (2012).

[39] B. B. Blinov, D. L. Moehring, L.-M. Duan and C. Monroe. Observation of entanglement between a single trapped atom and a single photon. Nature 428, 153 (2004).

[40] E. Togan et al. Quantum entanglement between an optical photon and a solid-state spin qubit. Nature 466, 730 (2010).

[41] W. B. Gao, P. Fallahi, E. Togan, J. Miguel-Sanchez and A. Imamoglu. Observation of entanglement between a quantum dot spin and a single photon. Nature 491, 426 (2013). [42] K. De Greve et al. Quantum-dot spin-photon entanglement via frequency

downconver-sion to telecom wavelength. Nature 491, 421 (2013).

[43] T. Legero, T. Wilk, A. Kuhn and G. Rempe. Time-resolved two-photon quantum interference. Applied Physics B Lasers and Optics 77, 797 (2003).

[44] C. K. Hong, Z. Y. Ou and L. Mandel. Measurement of subpicosecond time intervals between two photons by interference. Phys. Rev. Lett. 59, 2044 (1987).

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E��

In this chapter we introduce the experimental techniques that allow us to employ the NV centre as a building block for quantum registers and networks. We �rst show how we can optically address single NVs (chapter 3.1), and how we can use a combination of magnetic resonance pulses and spin-resolved optical excitation at low-temperatures to initialize, manipulate, and measure both the NV electronic spin (chapter 3.2) as well as nuclear spins in the environment (chapter 3.3). We discuss how we can pre-select during the experiment on the correct charge-state of the NV centre and on the optical transitions being on resonance with the laser frequencies (chapter 3.4). We show the suitablility of the NV centre for generating remote entanglement by demonstrating coherent control (chapter 3.5) and deterministic tuning of optical transitions (chapter 3.6). We conclude with a technical description of the experimental setup used (chapter 3.7) and an overview over the methods used to prove entanglement generation (chapter 3.8).

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3.1 Addressing single NV centres

For the experiments presented in the following chapters we employ single NV centres in bulk diamond, cooled to liquid helium temperatures. We �rst locate and pre-select centres of interest in a simpler setup at room-temperature, and subsequently fabricate structures for enhancing the collection e�ciency, tuning optical resonances, and manipulating spins.

Samples

We use naturally occurring NV centres in high purity type IIa chemical-vapor deposition grown diamond with a h111i crystal orientation obtained by cleaving a h100i substrate. The diamonds are supplied by Element 6.

Localising NV centres

Using confocal microscopy, Gruber et al. were able to detect single NV centres for the �rst time in 19971_{. To date this technique remains the standard tool for addressing single} NV centres optically. We use a home-built confocal microscope at room temperature for basic characterisation. To �nd NVs we focus the beam of a green laser (532 nm) onto the sample using a microscope objective with high numerical aperture (typically, NA = 0.95). The same objective captures the �uorescence from the sample. A dichroic mirror separates the red-shifted phonon side-band emission from the excitation beam into a detection path (Fig. 3.1a).

By scanning the sample in all three dimensions using piezo-electric positioners and monitoring the position-dependent �uorescence via an avalanche photo diode (APD) we are able to localise emitters within the sample (Fig. 3.1b). We test whether such a candidate is indeed an NV centre: The spectrum of the NV emission exhibits a characteristic zero-phonon line (ZPL) at wavelength ⇡ 637 nm and a dominant red-shifted phonon side-band (PSB) which can be recognised easily (Fig. 3.1c). We can see whether the emission detected comes from a single NV centre (or more general, from a single-photon emitter) by measuring the

second-order autocorrelation function 2_{( )}_{using a Hanbury Brown-Twiss con�guration}4_.

By directing the emission onto a 50:50 beamsplitter with an APD located at each output port we can record two-photon coincidences on the APDs as a function of the delay between the two events. This corresponds, up to normalisation, to 2_{( )}_{(Fig. 3.1d). A single emitter} shows 2_{(0) < 1/2.}

Solid immersion lenses

One of the main limitations that are encountered when performing optical measurements on single NV centres in bulk diamond is the low collection e�ciency that we estimate to be typically on the order of (0.1–0.5)%. A large fraction of the ‘lost’ photons arises from total internal re�ection at the diamond-air interface due to the large mismatch of the refractive

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a b c d BS FM MO DM APD APD spectro-meter 625 675 725 wavelength (nm) PL (a.u.) ZPL PSB 0 100 τ (ns) 0 1 g 2(τ )

1 µm

x y z _{sample and} scanner

Figure 3.1 | Identifying single NV centres. a, Setup for sample characterisation. A mi-croscope objective (MO; Olympus MPlanApo50x) focuses the green excitation laser (Coherent Compass 315M, frequency doubled Nd:YAG) onto the sample, mounted on a piezo scanning stage (Physik Instrumente). A dichroic mirror (DM; Semrock) separates the fluorescence spectrally into the detection path. Via a mechanically switchable mirror (FM) the emission is guided either to a spectrometer (Princeton Instruments Acton) or to a beam spli�er (BS) followed by two APDs (Perkin Elmer SPCM-AQR-14-FC) in HBT configuration. The signals of the APDs are recorded by a time-tagging module (FastComTec P7889). b, Confocal map of a typical sample. The colour map indicates fluorescence level, where blue is high intensity. The high-intensity spots correspond to emission coming from single NV centres. (Figure adapted from2_{.) c, Spectrum of the emission of a typical NV centre. The zero phonon line is} clearly visible at = 637 nm. d, Second-order autocorrelation function. Taking into account background luminescence (do�ed line), the anti-bunching dip at = 0 reaches a value close to 0. The solid line is a fit to a three-level model taking into account relaxation via the singlet-levels3_.

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150 _kcts/se cond 10 1 0.1 10 µm a b RF Gate

Figure 3.2 | Solid immersion lenses. a, Scanning electron microscope micrograph of a typical device a�er fabrication. Next to the lens are a gold strip line for applying magnetic resonance pulses (‘RF’) and gold gates for applying DC voltages in order to tune optical resonances by the DC Stark e�ect. b, Confocal scan with green laser excitation. The bright spot in the centre of the SIL is the emission from the NV centre. (Figure b is adapted from7_.) devices with SILs we estimate an enhanced typical collection e�ciency on the order of (2–5)%.

In order to deterministically etch a solid immersion lens (SIL) around a speci�c NV centre we �rst de�ne a 200 nm thick gold coordinate system onto the surface via electron beam lithography. This coordinate system is visible in our confocal setup, in which we determine the relative positions of the NV centres, as well as in the focused ion beam system (FEI Strata DB 235), which is used to etch the SILs at these positions. The NV centres in question are typically located 5–15 µm underneath the diamond surface. We use a 30 kV gallium ion beam to mill the lenses. The desired lens pro�le is approximated by etching concentric rings of varying diameters. After writing the lenses we clean the sample for 30 min in a boiling mixture of equal parts of perchloric, sulfuric and nitric acid. We subsequently remove residual surface contamination by reactive-ion etching in an oxygen-plasma.

Additionally we de�ne a 200 nm thick gold microwave strip line for applying magnetic resonance pulses and electronic gates around the lenses via electron beam lithography (Fig. 3.2).

3.2 Electron spin measurement and control

In order to use a single NV centre and nuclear spins in its environment as a quantum register we need to be able to perform basic operations on all involved spins. In the following we will �rst show how we can use spin-resolved optical excitation at low temperatures to initialise and read out the NV electron spin in a single shot. Together with control of the spin in the optical ground state by magnetic resonance pulses in the microwave domain, these techniques give us all required operations needed to use the electronic spin as a single qubit as well as an ancilla qubit for addressing nuclear spins. The optical toolset that provides these techniques has been established to a large part by the work of Robledo et al.7_.

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a m_s= 0 m_s= ±1 MW E_1/2 A₁ E_y E_x A₂ λexc _inte ns it y (a .u .) 0 5 10 15 frequency (GHz) b E_1/2 E_y E_x A₁ A2 + 470.4515 × 103 Figure 3.3 | Resonant excitation at low temperature. a, Spectroscopy scheme. We sweep the wavelength of the excitation laser while simultaneously driving the spin in the ground state. We monitor the PSB fluorescence during this sequence. When an allowed transition is excited a peak in fluorescence is detected. b, Typical excitation spectrum of a low-strain NV when exciting well below saturation power (a few nW). The data shown was recorded at a sample temperature of T = 4.2 K.

Optical spectroscopy at low temperatures

At liquid-helium temperature the optical transitions of the NV centre are narrowed close

to their lifetime-limited values of a few MHz8,9_{. The optically allowed transitions and}

excited level structure can be determined using resonant optical excitation in the zero-phonon line combined with driving the spin in the ground state with microwaves. In a low-temperature confocal microscope (Fig. 3.15) we sweep the wavelength of the excitation laser while monitoring the �uorescence of the phonon side-band emission, separated spectrally from resonant emission and laser re�ection by a dichroic �lter. Additionally we continuously drive the ms =0 $ ms=±1 spin transition in the ground state to repopulate all spin states and therefore enable detection of transitions associated with any spin state (Fig. 3.3). The resulting trace shows the allowed transitions as depicted in Fig. 3.3b, and reveals the strain of the NV centre (see Fig. 2.2).

Electron spin initialisation by optical spin pumping

As described in chapter 2 the NV electronic spin is slightly mixed in the optically excited states. Therefore, under resonant excitation of a single transition, the �uorescence decays with time because of shelving into the other spin state (Fig. 3.4). Such shelving can be used for high-�delity preparation of the electronic spin, as earlier demonstrated with quantum dots10_.

We estimate the optical pumping e�ciency and spin preparation �delity from the aver-aged time dependence of the �uorescence during optical pumping (Fig. 3.4b). In particular,

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A_1/2 E_x/y m_s= 0 m_s= ±1 a b A₁ E_x 0 10 20 30 40 50 200 400 600 800 1000 Time (µs) Time (µs) kcts/second 00 1 2 3 4 5 20 40 60 80 100 kcts/second 0

Figure 3.4 | Optical spin pumping of the electron spin. a, Pumping scheme. We excite

a transition associated either with ms = 0, Ex or E (bright arrows) or with ms = ±1,

typically A1or A2(dark arrows). Additional to spin-conserving optical cycling (solid arrows) spin-non-conserving relaxation (dashed arrows) occurs with a finite probability. In case of a spin-flip, the excitation laser is then o� resonance and the spin remains ‘trapped’ in the other spin state. b, Typical averaged fluorescence time trace for excitation on Ex, and on A1(inset), with excitation powers roughly equal to the saturation power, Psat⇡ 6 nW. The signal magnitude at a given time is proportional to the probability that the spin has not yet flipped a�er that time. Black solid lines are fits to single-exponential decays, yielding spin flip times of (8.1 ± 0.1) µs for Exand (0.39 ± 0.01) µs for A1. From the initial and saturation fluorescence levels we infer lower bounds of the initialisation fidelities of (99.2 ± 0.1)% for

pumping into ms=_{±1 and (99.7 ± 0.1)% for pumping into m}s=0. (Figures adapted from

Robledo et al.7₎

under excitation of the ms=0 associated transition, E_xor E (ms=±1 associated transition, typically A1or A2) to a single exponential with an o�set, Ae t/t1+B. The �t values contain

in-formation about the preparation �delity into ms=±1 (ms =0): Owing to possibly imperfect preparation, the amplitude A of the exponential is less than or equal to the �uorescence rate

that would result from a perfectly-prepared ms=±1 (ms=0) state. Because of background

counts from ambient light and detector dark counts, the o�set B is greater than or equal to

the �uorescence rate from remaining undesired population in ms=0 (ms=±1) following

optical pumping. The ratio B/(A + B) is thus greater than or equal to the fraction of optically

active population remaining in the ms =0 (ms =_{±1) state, and 1 B/(A + B) is a lower}

bound for the preparation �delity.

The precise value of the preparation error depends on the NV strain and on the laser intensity: both factors heavily a�ect o�-resonant excitation of transitions associated with the spin state that is to be prepared — and thus the degree to which we pump back into the ‘wrong’ spin state. Typical values for the upper bound of the preparation error range between 0.1% and 1%.

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a 0 5 10 15 20 25 30 35 40 counts 0.0 0.2 0.4 0.6 0.8 1.0 occ urr en ce ms= 0: mean = 8.18 counts P(0) = 0.10 ms= ±1: mean = 0.08 counts P(0) = 0.98 ms= ±1 ms= 0 0 10 20 30 40 50 time (µs) 0.0 0.2 0.4 0.6 0.8 1.0 fi de lit y F_{at t = 18 µs}ro = (94.7±0.2)% ms= 0 ms= ±1 Fro b

Figure 3.5 | Electron spin single-shot readout. a, Histograms of the number of PSB

photons obtained during a 50 µs long readout a�er preparation into ms=0 and ms =±1.

b, Time-dependency of the readout fidelity. Error bars (one statistical standard deviation) are smaller than the symbols. The data shown contains 5000 repetitions of the experiment

and was recorded using a low-strain NV centre (Ex E spli�ing ⇡ 2 GHz) with excitation

on E at saturation power (⇡ 5 nW).

Single-shot readout of the electron spin

Spin-resolved optical excitation allows for reading out the electronic spin in a single shot: The presence or absence of �uorescence under excitation of a cycling transition (we choose either Exor E ) reveals the spin state. In low-strain NV centres spin mixing9,11and phonon-induced transitions12_{are suppressed at low temperatures (T < 10 K). This allows, provided} su�ciently high collection e�ciency of the optical setup and e�cient rejection of re�ected laser light, for detection of a few emitted photons before the spin �ips.

Characterisation of the single-shot readout is shown in Fig 3.5. Following preparation

of the spin in either ms =0 or ms=_{±1 by optical spin-pumping, we apply a readout laser}

pulse. For preparation in ms=_{±1 almost no photons are detected during the readout pulse,}

whereas for ms = 0 we typically detect a few (Fig. 3.5a). For an unknown spin state we

therefore assign ms =±1 to an event where no photons have been collected, and ms =0

in case of least one photon. Assuming perfect initialisation, the �delity of ms=0 readout is given by the probability to detect at least one photon following preparation in ms =0, F0=1 P (0|ms =0), where P (0|ms=0) is the probability to obtain no photons provided

initialisation of ms = 0. Conversely, F_±1 = P (0|ms = ±1). The readout �delity for an

unknown spin state is the mean of the two,

Fro=1

2(F0+ F±1) . (3.1)

Imperfect initialisation of the spin state can only lead to a decrease (increase) of the probability to detect a photon after nominal preparation of ms=0 (ms=±1). The �delity obtained from the calibration as described is therefore a lower bound.

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The optimal readout time is determined by the excitation power and by the time-dependent

probability to detect a photon in the case the spin is in ms = ±1. Due to detector dark

counts and o�-resonant driving of transitions associated with ms =±1 during readout this

probability increases with time. Therefore, in the time regime where a spin originally in

ms=0 has �ipped (and thus 1 P (0|ms=0) cannot increase anymore) the readout �delity

drops. Typical optimal readout times for excitation with saturation power are on the order of a few tens of microseconds (Fig 3.5).

The best readout �delity achievable depends, besides the collection e�ciency of the setup used, on the strain7_{. The value of the lateral strain determines the degree to which the spin} in the excited state is mixed, and therefore how many photons we can expect to detect for

ms =0 before a spin-�ip occurs. Additionally the strain determines how far the closest

transition associated with ms=±1 is detuned from the readout transition, and therefore the role of o�-resonant excitation that leads to lower values of F_±1. Typical readout �delities for pre-selected NVs with the experimental setups used are in the range of (90–97)% for readout on either Exor E .

Using the SSRO calibration as obtained from data as shown in Fig. 3.5 we can infer the actual electron spin state. We assume (unknown) populations c0and c1=1 c0of the ms =0 and ms =±1 states, respectively, prior to readout. The probabilities p0and p1with which we measure ms=0 and ms=±1, respectively, are related to the readout �delities de�ned above via p0 p1 ! =E c0 c1 ! = _{1 F}F0 1 F±1 ±1 F±1 ! c0 c1 ! . (3.2)

The readout correction to obtain c0and c1is then given by the inverse of the matrix describing the readout error, E 1_.

The electron readout is in principle a projective quantum measurement7_{. However, during} the integration time required for optimal discrimination of the two spin states (typically on the order of 10 µs at saturation power) the spin can also be pumped optically from ms=0 ! ms =±1. This results in a destructive readout for projection into ms=0, meaning that deterministic, high-�delity projective measurements can currently not be achieved without higher detection e�ciencies or improvement of the readout protocol.

Manipulation of the electron spin by magnetic resonance

In order to manipulate the electron spin, we drive it by applying microwaves on resonance with an electron spin transition in the ground state. We typically restrict our view to the two-level system de�ned by |0i := ms=0 and |1i := ms= 1. By applying a small magnetic �eld

along the NV axis (a few Gauss) we split the ms =_{±1 levels. Keeping the Rabi frequencies}

with which we drive the spin between ms=0 $ 1 well below the splitting ensures that

the driving between ms=0 $ +1 is negligible and we can ignore the ms=+1 level.

The electron spin transitions are split further due to hyper�ne interaction with close-by nuclear spins, such that there is a distinct resonance frequency for each nuclear spin

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a b 0 1 2 3 4 5 6 7 8 MW frequency (MHz)+2823 0.6 0.7 0.8 0.9 1.0 P( cl ic k) _m s = -1 m_I = m_s = 0 MW frq. ms = +1 2γ_eB_z -1 0 +1 MW SSRO MW SSRO 0 100 200 300 400 500 MW pulse length (ns) 0.0 0.2 0.4 0.6 0.8 1.0 F( |0 ⟩)

Figure 3.6 | Electron spin manipulation. a, Electron spin resonance. Following

initial-isation of the electron into ms=0 we sweep the frequency of a MW pulse applied across

the ms = 0 $ 1 transition. The probability to measure ms =0, P (click), drops when

the pulse is on resonance. The ms =+1 manifold is split from the ms =0 manifold by a

static magnetic field with z-projection Bzby 2 eBz ⇡ 100 MHz, where e=2.8 MHz/G is

the electron gyromagnetic ratio. Indicated are the14N spin projections associated with

each transition observed. b, Electron spin Rabi oscillation. We sweep the duration of a MW

pulse on resonance with the (ms,mI) = (_{0, 1) $ ( 1, 1) transition. The readout has been}

corrected for imperfect SSRO such that we obtain the fidelity of the electron spin with |0i as measurement result. The solid line is a sinusoidal fit. For the data shown here the14N

spin was initialised into mI = 1. The error bars (one statistical standard deviation) are

smaller than the symbols.

con�guration. In case of only the14_{N (I = 1) and no}13_{C spins coupled strongly to the}

NV electron there are thus three resonances between ms = 0 $ 1, corresponding to

populations of the14_{N spin with spin projections mI} ₌_{{ 1,0,+1}.}

We can determine the values of the resonance frequencies in the following way: after

initialising the electron spin into ms = 0 we apply a weak microwave (MW) pulse of

constant length, followed by read out of the electron spin. Sweeping the MW frequency we observe a drop in the probability to detect a photon during readout at each resonance frequency (Fig. 3.6a). Using the information obtained like this we �x the MW frequency to the desired transition and are then able to rotate the electron spin with variable Rabi frequency (adjustable by MW power applied), rotation angle (adjustable by pulse duration) and rotation axis (adjustable by the phase of the MW �eld). As an example, a Rabi oscillation measurement is depicted in Fig. 3.6b.

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3.3 Nuclear spin measurement and control

In the following we explain how we can use a combination of conditional electron spin rotations and projective quantum measurements on the electron for initialisation and single-shot readout of nuclear spin states. Together with radio-frequency magnetic resonance pulses that allow manipulation of the nuclear spin states these methods complete the toolset required for basic operations on a nuclear spin quantum register.

Projective measurement

We use the hyper�ne (HF) interaction to access nuclear spins coupled to the electronic spin. Due to the HF splitting in the ms= 1 state it is possible to apply microwave pulses to the electron that are resonant with only one nuclear spin con�guration as can be seen, for the

case of only the14_{N spin, from the ESR spectrum in Fig. 3.6a. A -pulse on the electron}

between its states |0ieand |1iethat acts only for a particular nuclear spin state is therefore an implementation of a CNOT gate (or To�oli-gate in the case of more than one nuclear spin).

Such operations on the electron spin allow the measurement of the nuclear spin state (Fig. 3.7a). After initialising the electronic state into |1iewe can map a particular nuclear

eigenstate onto |0ie by performing a -rotation controlled by that eigenstate. Looking

only at the14_{N spin for brevity, an initial state |1ie}₍ _|0iN_{+ |1iN}₎_{becomes the entangled}

state |1ie|0iN+ |0ie|1iN. Measurement of the electronic state |0ieconstitutes then a

measurement of the nuclear spin state |1iN. In this simple case, a measurement outcome |1ie also gives a readout result for the nuclear spin, |0iN. In the case of more than one nuclear spin, however, in general only the measurement outcome |0iegives a complete readout result because |1ieis associated with more than one eigenstate of the nuclear spin register.

CNOT implementations

Ideally, one would want to implement the controlled rotation on the electron spin with a MW -pulse whose Rabi frequency is much smaller than the detuning of the transition closest to the transition driven. In practice the transitions are broadened by dephasing, meaning that a very weak -pulse implemented using a square MW pulse does not invert the full population (Fig. 3.7b).

For a combination of better inversion and acceptable selectivity we employ a stronger square pulse that performs a rotation on resonance and a 2 rotation on the neighbouring transition, which is detuned by the hyper�ne splitting ANof the14_{N spin (Fig. 3.7b). The} on-resonance Rabi frequency required to achieve a Rabi frequency 2 on the neighbouring

transition follows from the condition 2 = ( 2_{+ A}2

Quantum measurement and entanglement of spin quantum bits in diamond

Uitnodiging

Quantum Measurement

and Entanglement of

Spin Quantum Bits in

Diamond

Vrijdag 20 december 2013

12:30u in de Aula TU Delft

Mekelweg 5 te Delft

Feest

‘t Boterhuis

Markt 15-17a te Delft

Vrijdag 20 december 2013

vanaf 21:00u

Q

uantum Measurement and Entanglement of Spin

uantum Bits in Diamond

Wolfgang Pfaff

Casimir PhD series 2013-34

ISBN 978-90-8593-174-4

of spin quantum bits in diamond

Wolfgang PFAFF

Contents

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1.1 Quantum computers and quantum networks

1.2 Spins in diamond as building block for quantum technologies

1.3 Thesis overview

1.4 Bibliography

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2.1 Introduction

2.2 The electronic spin: central qubit and optical interface

Central qubit

Optical interface

2.3 Nuclear spins as quantum register

The nitrogen host

C spins

2.4 Remote entanglement via a photonic channel

2.5 Bibliography

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3.1 Addressing single NV centres

Samples

Localising NV centres

Solid immersion lenses

1 µm

3.2 Electron spin measurement and control

Optical spectroscopy at low temperatures

Electron spin initialisation by optical spin pumping

Single-shot readout of the electron spin

Manipulation of the electron spin by magnetic resonance

3.3 Nuclear spin measurement and control

Projective measurement

CNOT implementations

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_{C spins}

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