Quantum control of single spins and single photons in diamond

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Quantum control of single spins and single

photons in diamond

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Quantum control of single spins and single

photons in diamond

Proefschrift

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

op gezag van de Rector Magniﬁcus prof. ir. K.C.A.M Luyben, voorzitter van het College voor Promoties,

in het openbaar te verdedigen op 27 maart 2012 om 12:30 uur

door

Toeno van der Sar

natuurkundig ingenieur geboren te Rotterdam.

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Copromotor: Dr. ir. R. Hanson

Samenstelling van de promotiecommissie: Rector Magniﬁcus voorzitter

Prof. dr. ir. L.P. Kouwenhoven Technische Universiteit Delft, promotor Dr. ir. R. Hanson Technische Universiteit Delft, copromotor Prof. dr. Y.V. Nazarov Technische Universiteit Delft

Prof. dr. ir. C.H. van der Wal Rijksuniversiteit Groningen Prof. dr. M.A.G.J. Orrit Universiteit Leiden

Prof. dr. J.T.M. Walraven Universiteit van Amsterdam Dr. J.J.L. Morton University of Oxford, Engeland

Published by: Toeno van der Sar

Cover design by: Wolfgang Pfaﬀ and Toeno van der Sar Printed by: Ipskamp Drukkers, Enschede

An electronic version of this thesis is available at www.library.tudelft.nl/dissertations This work is part of the research programme of the Foundation for Fundamental Research on Matter (FOM), which is part of the Netherlands Organisation for Scientiﬁc Research (NWO).

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Chapter 1 Introduction

1.1 The quantum computer

The development of the quantum computer is one of the greatest fundamental and technological challenges for physicists in the 21st century. A quantum computer is a computer that harnesses the power of the quantum properties of its elementary building blocks, which are called quantum bits or qubits. In contrast to bits in a classical computer, which can be either 0 or 1, qubits can be in an arbitrary superposition of 0 and 1 [1]. This essentially allows quantum computations to be performed on both 0 and 1 states simultaneously, and thus potentially at a higher speed than in a normal computer.

During the 1990’s, several quantum algorithms were developed that today still serve as prime motivators for the development of a quantum computer [1]. Undoubt-edly the main drive has come from Shor’s algorithm, an algorithm which would allow exponentially faster integer factorization than possible with the most eﬃcient known classical algorithm. It therefore has the potential to revolutionize the currently om-nipresent data encryption techniques which rely on integer factorization. Another famous quantum algorithm which strongly outperforms its classical counterpart is Grover’s algorithm. This algorithm, of which a small-scale version will be presented in chapter 7 of this thesis, can be used to search an unsorted database. It has been proven to provide quadratic speed-up over any possible classical counterpart.

The theory of quantum information has made much progress in the last decades, illustrated by the development of quantum software such as quantum algorithms. However, experimentally we are far from realising a quantum computer in which at least a few thousands of qubits coherently interact [2]. Although there does exist a wide variety of prototypical qubit systems, ranging from micron-sized superconduct-ing rsuperconduct-ings to angstrom-sized atoms in vacuum traps, these are currently all limited to just a handful of qubits [2]. The current world-record number of 14 qubits that

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can be controllably entangled is held by ions in vacuum traps [3]. This is a result of the near perfect isolation of such a system from the outside world, which ensures the stability of the quantum information stored in these ions. However, it is diﬃcult to scale up ion-based systems, as the required technologies bear little resemblance to large-scale classical computing technologies used for industrial mass-production. In this sense, a solid-state platform for quantum computation is more attractive, as it can beneﬁt from the vast knowledge obtained from years of developing scalable integrated circuit technologies in the chip industry.

1.2 Hybrid systems

A major challenge in using a solid-state system as a platform for a quantum computer is decoherence, which is the irreversible loss of quantum information that is encoded in a qubit. Decoherence of a solid-state qubit is caused by uncontrolled interaction of the qubit with its environment. To minimize such interaction, it is attractive to encode the qubit in the magnetic moment (spin) of a single impurity atom which is trapped in the lattice of a host material. This type of qubit combines the good isolation and associated long coherence time of a trapped atom with the versatility and potential scalability of a solid-state based system.

Nuclear spins are generally more robust to decoherence than electron spins be-cause of their smaller magnetic moment and accompanying smaller interaction with the environment. Nuclei are therefore more attractive for storing quantum informa-tion. However, the weak interaction between diﬀerent nuclei, even if they are only a few lattice sites apart, makes coupling them very diﬃcult. Electrons, on the other hand, interact much stronger but correspondingly have shorter coherence times.

An appealing idea is to develop a hybrid system, consisting of qubits of diﬀerent species each optimized for its speciﬁc task. Nuclei store quantum information, pho-tons mediate the transfer of quantum information between distant spin registers, and electrons form an interface between spin and photon states, mediate coupling between nuclear qubits, or act as fast processing qubits.

1.3 The nitrogen-vacancy center in diamond

In recent years, the nitrogen-vacancy (NV) center in diamond (Fig. 1.1) has attracted much attention for application in quantum technologies such as quantum informa-tion technology and ultra-sensitive metrology [4]. Even at room temperature, the electron spin of the NV center is stable enough to be coherently manipulated with high ﬁdelity. In addition, the electron spin can be initialized and read out by simple optical means. The NV center is an intrinsically hybrid quantum system: its elec-tron spin is always coupled to the nuclear spin of its own nitrogen atom [5]. This

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Introduction

photon

electron spin

N nuclear spin

13

_{C nuclear spins}

C C C C V C C N C C

Figure 1.1: The nitrogen-vacancy (NV) center in diamond. The NV center consists of a substitutional nitrogen atom adjacent to a vacancy in the diamond carbon lattice. It is an intrinsic hybrid system: Its electron spin is always coupled to the nuclear spin of its own nitrogen atom, and can furthermore be coupled to nuclear spins of nearby13_{C atoms.} The NV center has spin-dependent optical transitions, allowing its quantum state to be transmitted over long distances through the use of photons.

spin register can be extended by using nuclear13_{C spins in the vicinity of the NV}

center [6]. Furthermore, the NV center has spin-dependent optical transitions allow-ing the quantum state of the register to be transmitted over long distances through the use of photons [7]. Entanglement between distant registers can in principle be created by letting photons from two diﬀerent registers interfere on a beamsplitter and detecting the number of photons at the output ports [8]. Establishing a large-scale quantum network by coupling distant NV centers via photonic channels, and creating a hybrid quantum architecture in which the strengths of diﬀerent types of quantum information carriers are optimally exploited are the fundamental goals motivating the experiments described in this thesis.

1.4 Thesis overview

Chapter 2 provides an introduction to the NV center in diamond. We start by describing its electronic structure and how single NV centers can be detected and studied using ﬂuorescence microscopy. We then describe the spin properties of the NV center, including quantum control of the NV center electron spin and nuclear spin, and characterization of spin decoherence. Finally, we describe how cavity quantum electrodynamics can be used to control the optical properties of the NV center.

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To entangle distant NV centers via photonic channels it is crucial to optimize the optical properties of the NV center. This can be achieved by coupling the NV center to an optical cavity. In chapter 3 we describe the development of a nanopositioning technique that allows positioning of diamond nanocrystals that contain a single NV center with nanometer precision. In chapter 4 we use this technique to couple single NV centers contained in diamond nanocrystals to photonic crystal cavities. In chapter 5, we study the eﬀect of a nanocrystal on the optical properties of photonic crystal cavities.

The main source of decoherence for solid-state spin systems is the uncontrolled spin bath environment. In chapter 6, we describe how changes in the coherence properties of a single NV center can be used to probe the dynamics of the mesoscopic spin bath surrounding the NV center. Furthermore, using the NV center as a sensor, we demonstrate quantum control of the spin bath itself.

To fully exploit the hybrid nature of the NV center, a universal set of quantum gates for a hybrid spin system is required. In chapter 7, we present the develop-ment and the impledevelop-mentation of decoherence-protected quantum gates for a hybrid electron - nuclear spin register. We demonstrate the power of these gates by imple-menting Grover’s quantum search algorithm.

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BIBLIOGRAPHY

Bibliography

[1] M. A. Nielsen and I. Chuang, Quantum Computation and Quantum Information, Cambridge University Press, 2000.

[2] T. D. Ladd, F. Jelezko, R. Laﬂamme, Y. Nakamura, C. Monroe, and J. L. O’Brien, Quantum computers, Nature 464(7285), 45 (Mar. 2010).

[3] T. Monz, P. Schindler, J. Barreiro, M. Chwalla, D. Nigg, W. Coish, M. Harlander, W. Hänsel, M. Hennrich, and R. Blatt, 14-Qubit Entanglement: Creation and Coherence, Physical Review Letters 106(13), 130506 (Mar. 2011).

[4] D. D. Awschalom, R. Epstein, and R. Hanson, Diamond age of spintronics, Scientiﬁc American (October) (2007).

[5] B. Smeltzer, J. McIntyre, and L. Childress, Robust control of individual nuclear spins in diamond, Physical Review A 80(5), 050302 (Nov. 2009).

[6] P. Neumann, N. Mizuochi, F. Rempp, P. Hemmer, H. Watanabe, S. Yamasaki, V. Jacques, T. Gaebel, F. Jelezko, and J. Wrachtrup, Multipartite entanglement among single spins in diamond, Science 320(5881), 1326 (June 2008).

[7] E. Togan, Y. Chu, A. S. Trifonov, L. Jiang, J. Maze, L. Childress, M. V. G. Dutt, A. S. Sørensen, P. R. Hemmer, A. S. Zibrov, and M. D. Lukin, Quantum entanglement between an optical photon and a solid-state spin qubit, Nature

466(7307), 730 (Aug. 2010).

[8] S. Barrett and P. Kok, Eﬃcient high-ﬁdelity quantum computation using matter qubits and linear optics, Physical Review A 71(6), 060310 (June 2005).

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Chapter 2 The nitrogen-vacancy center in

diamond: background and

measurement techniques

The intrinsic hybrid nature of the NV center makes it a prime candidate for appli-cation in hybrid quantum technologies. The experiments described in chapters 3-7 of this thesis focus on controlling the quantum properties of single NV centers with the ultimate goal of creating a large-scale hybrid quantum network. In this chapter, we describe the properties of the NV center which are crucial to understanding these experiments. We start by explaining how single NV centers are detected and stud-ied using photoluminescence microscopy, and how the NV center electron spin and the nuclear spin of its own nitrogen atom can be controlled by oscillating magnetic ﬁelds. We describe the vast diﬀerence between the coherence and control timescales of the electron and nuclear spin, which forms the primary challenge for using the NV center as a fully functional electron-nuclear quantum processor. We end the chapter with a description on how the optical properties of NV centers can be ma-nipulated to increase the emission of coherent photons, with the goal of achieving measurement-based entanglement between distant NV registers.

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2.1 Electronic structure of the NV center

To understand how single NV centers are detected and how the NV center spin state can be read out using photoluminescence microscopy, we start with introducing the electronic level structure. The electronic structure of the NV center1 _{has been the}

subject of intensive theoretical and experimental studies in recent years [1–4]. Here we limit ourselves to the details of the level structure that are relevant for the experiments described in this thesis.

mS = 0 ±1 mS = mS = mS = 0 ±1 D GS = 2.87 GHz D ES = 1.4 GHz 637 nm Singlet states Electronic ground state Electronic excited state 0 ±1 0 ±1 a) b) PSB PSB

Figure 2.1: Electronic level structure of the NV center. a, The ground state of the NV center is a spin triplet with zero-field splitting DGS = 2.87 GHz which is connected to the excited state by an optical transition with a zero-phonon-line (ZPL) at 637 nm. At room temperature, the excited state can be described as a spin triplet with zero-field splitting DES = 1.4 GHz. To explain the properties of the NV center relevant for the experiments described in this thesis, it suffices to summarize the singlet states states into one level. b, Energy level diagram explaining the optical spin readout and initialization mechanism. The red lines indicate optical transitions, the dashed lines indicate the other primary (dark) transitions. Spin polarization into the ms = 0 state is a result of spin-dependent decay via the singlet states. Decay from the excited state into the singlet states happens primarily from the ms =±1 states, and decay from the singlet states into the ground state happens primarily into the ms= 0 state.

It is now established that there are six electrons associated with the NV center [1]: three from the dangling bonds of the vacancy, two from dangling bonds of the nitrogen atom, and one additional electron which is attracted from somewhere else in the diamond, presumably another nitrogen atom. The electronic ground state is a spin triplet, of which the ms = 0 and ms =±1 states, where ms denotes the quantum number of the spin projection along the symmetry axis (z-axis) of the

1_{The NV center can exist in two charge states: the neutral state NV}0_{and the negatively charged}

state NV−. In this thesis we deal excusively with the NV−center, and for brevity we will denote

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The nitrogen-vacancy center in diamond: background and measurement techniques

NV center, are split in energy due to spin-spin interaction by a zero-ﬁeld splitting D = 2.87 GHz (Fig. 2.1a).

The electronic excited state is also a spin triplet but an orbital doublet. It is connected to the electronic ground state by a strong optical transition with a zero-phonon line (ZPL) at 637 nm (1.945 eV). At room temperature, rapid inter-orbital transitions within the excited state lead to an eﬀective averaging of the spin properties of the orbitals [5]. As a result, the orbital doublet nature can be neglected and the electronic excited state can be described as a single spin triplet, of which the spin states ms= 0 and ms =±1 are split by a zero-ﬁeld splitting DES = 1.4 GHz (Fig. 2.1a).

It is now believed that there exist at least three singlet states that lie between the electronic ground and excited state [6]. Recent ensemble measurements have shown that there is an optical transition at 1046 nm between two of these states [7]. Since this transition is estimated to be∼ 104 times weaker than the 637 nm ZPL, passage through the singlet states is essentially a dark process. As we will describe in section 2.2, passage through the singlet states is strongly spin-dependent. Therefore the spin state of the NV center can be detected through the spin-dependent pho-toluminescence rate which is used as the standard method for room-temperature readout of the spin state of the NV center.

2.2 Spin properties of the optical transitions

One of the most remarkable properties of the NV center is that its electron spin can be initialized and read out by oﬀ-resonant excitation, even at room temperature. In this section we introduce the mechanism responsible for optical spin polarization and readout.

2.2.1 Optically induced spin polarization

Due to a spin-dependent relaxation mechanism between electronic ground and ex-cited state, the NV center electron spin polarizes into the ms= 0 state under optical excitation. During optical cycling, spin-ﬂips mainly occur through decay via the singlet states (Fig. 2.1b) [1, 6]. Decay from the excited state into the singlet states occurs primarily for the ms=±1 states, and decay from the singlet states into the electronic ground state occurs primarily into the ms= 0 state. After just a few op-tical cycles the system therefore mainly occupies the ms= 0 state. By following the optical excitation with a few microseconds of dark waiting time to allow deshelving of remaining population in the singlet states, the electron qubit is initialized in the ms= 0 state with a typical spin polarization between 80% and 95%. (Appendix B describes a detailed characterization of the electron spin polarization).

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2.2.2 Spin-dependent photoluminescence

The spin-dependent relaxation mechanism between electronic excited and ground state responsible for spin polarization also allows optical detection of the NV center spin state. Because the system is more likely to decay via the dark singlet states if it is in the ms=±1 states than if it is in the ms= 0 state, the photoluminescence (PL) is spin-dependent and can be used to detect the spin state of the NV center. This will be described in more detail in section 2.4.2.

2.3 Detecting single NV centers

The 637 nm ZPL of the NV center is accompanied by a broad phonon sideband (PSB), both in emission and absorption (Fig. 2.1a). The PSB is associated with transitions between electronic ground and excited state which, in addition to the emission/absorption of a photon, involve the excitation/absorption of a (local) vi-bration of the NV center [8]. These vibronic states, which have short≪ 1ps lifetimes corresponding to tens of nanometer spectral widths, give the PSB its characteristic spectral shape (Fig. 2.4a).

The PSB allows efficient excitation of the NV center by off-resonant laser light at e.g. 532 nm. The short optical lifetime (∼ 12 ns in bulk diamond), in com-bination with a high quantum efficiency for radiative relaxation (η ∼ 1), allows relatively straightforward detection of the photoluminescence of single NV centers. NV centers were first observed individually in 1997 by Gruber et al. using a confocal microscope [9].

2.3.1 Confocal microscopy

Nowadays, the confocal microscope has become the standard tool to optically isolate and study single NV centers. Fig. 2.2 schematically shows one of the confocal mi-croscopes built for the experiments described in this thesis. NV centers are excited by a 532 nm laser which is focused to a diﬀraction limited spot by a high numerical aperture (NA) objective (typically NA=0.95). Photoluminescence originating from the sample is separated from the excitation light by a dichroic mirror and optical ﬁlters. A photoluminescence map of a sample is made by scanning the position of the excitation laser across the sample and detecting the position-dependent photolu-minescence using avalanche photodiodes. A confocal microscope employs a pinhole to increase the signal-to-background ratio of the detected photoluminescence by blocking out-of-focus light.

The diﬀraction limit limits the spatial resolution of a confocal microscope to the order of a wavelength. NV centers can therefore only be optically isolated if the distance to neighbouring NV centers is large enough. For bulk diamond samples

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The nitrogen-vacancy center in diamond: background and measurement techniques Laser MO AOM APD APD S ample FC FC BS PBS fast steering mirror L L M M dichroic mirror L L

Figure 2.2: Schematic of a confocal microscope used to study single NV cen-ters. A 532 nm laser (Coherent Compass 315M, frequency doubled Nd:YAG) is used for excitation. An acousto-optical modulator (AOM, Crystal Technologies 3200-121), in double-pass configuration to increase the extinction ratio to about 60 dB, can be used to create laser pulses with a∼10 ns risetime. A fast steering mirror (FSM, Newport FSM-300-01) scans the laser over the sample. Two lenses image the FSM onto the back focal plane of a microscope objective (MO). The objective (NA=0.95, Olympus MplanApo50x) focuses the laser to a∼500 nm diffraction limited spot. The same microscope objective collects the photoluminescence (PL). The excitation light is separated from the PL by a dichroic mirror and further suppressed by an optical filter. The PL is focused onto the core of a single mode fiber by a fiber coupler (fc) and detected using an avalanche photodiode (APD, Perkin & Elmer SPCM-AQR-14-FC). The fiber core acts as the confocal detection pinhole. For samples with low background fluorescence, a multimode fiber is used to increase the collection efficiency. The beamsplitter (BS) is used for antibunching measurements only and is usually taken out of the beam path.

1 µm

a) Gold marker b)

5µm

NV centers Gold marker

Figure 2.3: NV centers in diamond nanocrystals. a, Scanning electron microscope (SEM) picture of diamond nanocrystals, ∼ 10 − 200 nm in size, dispersed on a glass coverslide. About 99% of the nanocrystals do not contain NV centers, and about 0.1-1% contain a single NV center. The slide is covered with∼5 nm of chromium to allow SEM imaging. The gold marker is used for position reference. b, Photoluminesence originating from single NV centers in diamond nanocrystals detected by a confocal micrscope such as the one shown in Fig. 2.2.

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650 700 750 0 500 1000 In te n si ty (a .u .) Wavelength (nm)

Single NV center photoluminescence spectrum Photoluminescence autocorrelation function

a) b) zero-phonon line N u mb e r o f co in ci d e n ce s g 2(τ ) Time delay, τ (ns) 0 200 400 0 50 100 150 200 0.0 0.5 1.0 1.5

Figure 2.4: Optical properties of a single NV center a, Room temperature photo-luminescence spectrum of a single NV center in a diamond nanocrystal. The spectrum is characterized by the zero-phonon line around 637 nm, which is accompanied by a broad phonon sideband. b, Autocorrelation function of single NV center luminescence, measured by splitting the photoluminescence to two detectors in a Hahnbury-Brown-Twiss setup. Finite APD dead time prevents the use of a single APD to measure correlations for short (tens of nanoseconds) time diﬀerences.

this implies using a diamond with a low enough concentration of NV centers [9]. Alternatively, we can use single NV center-containing diamond nanocrystals which are dilutely dipersed on a substrate (Fig. 2.3a)2_{. Fig. 2.3b shows a confocal}

photo-luminescence map of a sample containing diamond nanocrystals. NV centers show up as diﬀraction limited luminescing spots and are easily recognized by the photo-luminescence spectrum showing the distinct zero-phonon line and the broad PSB (Fig. 2.4a).

To determine if the photoluminescence detected from a diﬀraction-limited spot originates from a single NV center, we can check if the statistics of the photolumines-cence corresponds to that of a single photon emitter. A single photon emitter emits only one photon when it decays from the excited state to the ground state. This single photon nature can be revealed by measuring the second order auto-correlation function g2(τ ) of the emitted photoluminescence, deﬁned as

g2(τ ) = ⟨I(0)I(τ)⟩

⟨I2_(t)⟩ (2.1)

where I(t) is the detected photoluminescence intensity at a time t.

2_{Diamond nanocrystals used for the experiments described in chapters 3, 4, and 5 were bought}

from the company Mikrodiamant in Switzerland. The (synthetically grown) bulk diamond samples used for the measurements described in chapters 6, and 7 were obtained from Element 6 in the UK

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For large time diﬀerences there is no correlation in the emitted photolumines-cence, so g2(τ → ∞) = 1. However, a single photon emitter will have an antibunch-ing dip in its autocorrelation function at zero time diﬀerence (g2_{(0) = 0), because it}

cannot emit two photons at the same time. Fig. 2.4b shows a measurement of the autocorrelation function of NV center photoluminescence. The anti-bunched nature of single NV center emission was ﬁrst observed by Kurtsiefer et al. [10].

2.4 NV centers as spin qubits

On top of the remarkable optical spin initialization and readout, the NV center has excellent room temperature spin coherence properties. It is the unique combination of these properties that has attracted much attention to NV centers for application as solid-state qubits in recent years. In this section we will describe how NV centers can be used as qubits. First we will describe how the magnetic fields used for spin manipulation are generated and delivered to the NV center. In the next subsection we focus on using the electron spin of the NV center as spin qubit. We start by describing the energy spectrum, how spin transitions can be optically detected, and how the spin state can be coherently controlled by pulses of ac magnetic field. We will further discuss electron spin coherence and dynamical decoupling techniques which prolong the coherence time. In the final subsection we focus on using the nuclear spin of the NV centers own nitrogen nucleus as a second qubit. We will describe the coupling mechanism between the nuclear spin and the electron spin, and how the nuclear spin can be optically initialized and coherently manipulated.

2.4.1 Spin manipulation - Experimental setup.

The quantum state of a spin can be manipulated by short bursts of ac magnetic ﬁeld applied at the spin resonance frequency [12]. For bulk diamond samples, the currents generating these ﬁelds are delivered to the NV center by means of a stripline or coplanar waveguide that is fabricated on the surface of the diamond by electron beam lithography (Fig. 2.5). NV centers are typically located within∼30 µm of the stripline. For samples containing diamond nanocrystals dispersed on a coverglass, a shorted coax cable can be brought in close proximity to the sample (within 100 µm).

Fig. 2.6 schematically shows the measurement electronics used for spin manipu-lation experiments. An Arbitrary Waveform Generator (AWG) is at the core of the electronic setup. For the manipulation of the electron spin, two distinct channels are used to generate the pulses and provide the I/Q modulation to the microwave frequency carrier signal. The MW signal is produced by a vector signal generator, of which the frequency can be set to match the electronic qubit transition frequency. The relative phase of the pulses is determined by the amplitude ratio between the

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12 µm Waveguide Diamond C C C C C C N V C C C C C C N V C C C C C C N V

Qubit control pulses IN

OUT PCB

Figure 2.5: Delivering qubit control pulses to an NV center. Pulses for qubit control are input on a printed circuit board (PCB) and delivered to an NV center through a coplanar waveguide fabricated on the surface of a diamond. The middle picture is a photoluminescence map showing the waveguide and several NV centers which show up as white spots. Picture adapted from [11].

AWG AMP 0.8 - 4.2GHz 25 W AOM APD P7889 MW source I Q G PCB AMP 1MHz - 1GHz 30 W Att. COMPUTER _LASER IN OUT Vector magnet

Figure 2.6: Spin manipulation setup. An arbitrary waveform generator (AWG, Tex-tronix AWG5014) is at the core of the electronic setup, synchronizing the input signals to the instruments through multiple channels. The AWG provides the I/Q modulation of a Rohde & Schwartz SMBV 100A vector signal generator (MW source). To increase the MW source on/off ratio, the pulse modulation input (labeled G) is used to turn off the source when there are no pulses. The MW bursts are amplified by a high power amplifier (Amplifier Research 25S1G4). Pulses of ac current in the radio frequency range (∼MHz) are directly generated by the AWG. After amplification (Amplifier Research 30W1000B), the RF and MW bursts are combined and sent to the sample via the printed circuit board. The power is finally dissipated in an attenuator. The AWG also provides the modulation to the AOM to generate laser pulses, and triggers the data-acquisition card (FastComTec

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two channels. By setting the phase diﬀerence to π/2, a spin is rotated around or-thogonal axes. After I/Q modulation, the microwave signal is sent through a high power ampliﬁer.

For the manipulation of the nuclear spin, with typical spin resonance frequencies of the order of MHz (which we will refer to as radio frequency or RF), the signal can be directly generated by the AWG. This signal is sent through a separate high power amplifier. After amplification, the RF and MW signals are combined and fed to the on-diamond stripline through a printed circuit board. A static magnetic field can be applied to the NV center by a vector magnet. This field can be accurately aligned along the NV center axis which is crucial to obtain high fidelity spin readout and initialization at large magnetic fields.

2.4.2 The NV center electron spin as a spin qubit

The optically induced spin polarization and readout, in combination with the zero ﬁeld-splitting, allows quantum control of the electron spin state of the NV center at room temperature and at zero magnetic ﬁeld. In this section we describe how the electron spin can be used as a room-temperature qubit. The spin state can be manipulated using magnetic resonance techniques, and this can be detected through the spin-dependent photoluminescence. We describe characterization of the electron spin coherence properties, and how the coherence of the electron spin can be used to detect and study the dynamics of the spin bath environment, which is the subject of the experiments described in chapter 6.

Energy spectrum of the electron spin in a magnetic ﬁeld

The energy spectrum of the electron spin of the NV center in the electronic ground state is described by the Hamiltonian

H0= DGSSz2+ γeB· S (2.2)

where S = [Sx, Sy, Sz], the Si are the Pauli spin operators, and γe = 2.8 MHz/G is the gyromagnetic ratio of an electron spin with total spin S = 1. The first term describes the zero-field splitting and the second term the Zeeman interaction with a magnetic field B. The energy spectrum of eq. 2.2 is shown in Fig. 2.7 as a function of a magnetic field applied along the NV symmetry axis. The zero-field degenaracy of the ms=±1 states is lifted by a magnetic field.

Optical detection of spin resonances under CW excitation

The spin-dependent photodynamics of the NV center allows optical detection of transitions between the ms = 0 and the ms =±1 states. Under continuous wave optical excitation, the NV center primarily occupies the ms= 0 state. Transitions

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m_s= 0 m_s= -1 m_s= +1

Ener

gy

Magnetic field along z 1028 G D_GS

Figure 2.7: Energy of the NV center electron spin in a magnetic field applied along the NV center symmetry (z) axis. A magnetic fields lifts the zero-field degen-eracy of the ms=±1 states. The energies of the ms=±1 states shift with 2.8 MHz/G.

to the ms=±1 states can be induced by applying a oscillating magnetic field per-pendicularly to the spin quantization axis and in resonance with the spin transition frequency [12]. These transitions can be detected by a decrease in the photolumi-nescence rate. Fig. 2.8a shows an optically detected magnetic resonance (ODMR) spectrum at a small magnetic field applied along the symmetry axis. The applica-tion of a small field allows us to selectively address a specific transiapplica-tion and to define a qubit in e.g. the spin levels ms= 0 and ms=−1.

A zoom-in on one of the electron spin transitions (Fig. 2.8b) shows a three-dip structure associated with hyperﬁne coupling to the14_{N nuclear spin of the NV center}

(which has total spin I = 1). The nuclear spin is in a mixed state so that all three nuclear spin-dependent electron spin transition frequencies are visible. In the next subsection we will dicuss the coupling to the nuclear spin in more detail.

Continuous wave optical and microwave excitation experiments such as those shown in Fig. 2.8 allow us to probe the energy spectrum of the NV center in a simple manner. Once the spin transition frequencies are determined, we are in a position to start using the electron spin as a qubit by using pulsed microwave excitation to rotate the spin state over a controlled angle [12]. Although pulsed spin experiments are possible under CW optical illumination [13], the optical excitation into the electronic excited state strongly limits the spin lifetime. It is therefore more eﬀective to manipulate the spin in the dark and only use short pulses of optical illumination to initialize and readout the spin.

Coherent manipulation of the electron spin

In a dark spin manipulation experiment we ﬁrst reset the NV center into the ms= 0 state by a∼ 2µs green laser illumination followed by ∼ 1 µs dark waiting time to allow deshelving of the singlet states. Then we apply the desired pulse sequences of

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The nitrogen-vacancy center in diamond: background and measurement techniques 2.75 2.80 2.85 2.90 2.95 210k 220k 230k 240k C o u n ts Frequency (GHz) Frequency (GHz) 2.785 2.790 2.795 2.800 230k 240k 250k C o u n ts a) b)

Figure 2.8: ODMR spectra of an NV center in a ﬁeld of Bz = 29 G. a, Detected

photoluminescence as a function of the frequency of an applied microwave ﬁeld under continuous optical excitation. ODMR spectrum showing the two dips associated with the electron spin transitions from ms = 0 to the ms = ±1 states. b, A zoom-in on the

ms= 0↔ ms=−1 ODMR dip of a shows three dips associated with the three diﬀerent spin states of the host14_{N nuclear spin.}

ac magnetic field to manipulate the spin. The final spin state is read out by counting the spin-dependent number of photons during the first 0.3− 2 µs of a subsequent laser pulse. Typical spin-dependent photoluminescence timetraces used for room temperature electron spin readout are shown in Fig. 2.9.

The state of the electron spin qubit, defined by e.g. the electron spin levels ms= 0 and ms=−1, is controlled by applying pulses of AC magnetic field at the transition frequency. Fig. 2.10a shows Rabi oscillations of the electron spin between the ms = 0 and the ms =−1 states. By applying pulses of the right length and amplitude, we can create any coherent superposition of spin states. A π/2 pulse on the|0⟩ state creates the 1/√2(|0⟩+|1⟩) state, and a π pulse flips a qubit between |0⟩ and|1⟩. When consecutive pulses are applied, the relative phase of the microwave signal sets the rotation axis.

Electron spin Rabi frequencies up to fR ∼ 50 MHz are easily achieved. The ability to reach Rabi frequencies such that f2

R ≫ A2, where A = 2.16 MHz is the hyperfine coupling to the 14N nitrogen nuclear spin, allows us to rotate the electron spin irrespective of the nuclear spin state. By applying a sufficient magnetic field, the ms = 0 ↔ ms = +1 transition is well separated in frequency from the ms= 0↔ ms=−1 transition so that the level ms= +1 is not affected and remains idle. This is the case in most experiments described in this thesis, and to simplify the notation we therefore denote the relevant electron spin qubit levels |ms = 0⟩ and|ms=−1⟩ as |0⟩ and |1⟩ respectively.

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C o u n ts 0 0.5 1 1.5 2 2.5 500 1000 1500 2000 2500 3000

Photon arrival time (µs)

ms = |-1

m_s = |0

Photoluminescence time traces

Figure 2.9: Electron spin readout. The electron spin state is read out by counting the spin-dependent number of photons during the ﬁrst∼1 µs of a 532 nm laser excitation pulse. Because the system is more likely to pass through the singlet states when the electron spin is in the ms=±1 states, it emits less photons than when it is in the ms= 0 state. After about 2µs, the system has equilibrated. By turning the laser oﬀ at this point and waiting about 1µs to allow deshelving of the singlet states, the system is re-initialized into the

ms= 0 state.

Free induction decay of the electron spin

The dominant decoherence channel of the electron spin of an NV center is the coupling to the bath of external spins. In ’Ib’ diamond samples, the spin bath is formed by electron spins associated with single substitutional nitrogen atoms (P1 centers). In pure, ’2A’ diamond samples, the bath is formed by nuclear spins of13_C

naturally present (with abundance 1.08%) in diamond.

The spin bath causes a randomly fluctuating magnetic field at the location of the NV center. This field can be described with a Gaussian probability distribution function of variance b2_{. If the spin is brought into a superposition, the fluctuating}

Zeeman splitting causes dephasing of the NV center in a time T₂∗=√2/b. This can be measured in a Ramsey measurement (Fig. 2.10b), which is well described by

P = exp(−1 2b 2_τ2₎1 3 1 ∑ k=₋₁ (Pkcos(2π(δf + kA)τ + ϕ) (2.3)

where δf is a detuning, and the Gaussian decay corresponds to the Gaussian prob-ability distribution of the fluctuating field [12]. The three observed frequencies cor-respond to the three different nuclear spin state dependent electron spin resonance frequencies.

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The nitrogen-vacancy center in diamond: background and measurement techniques 0 1 2 3 4 5 0.0 0.2 0.4 0.6 0.8 1.0 Interpulsedelay τ (µs) Var._τ P(m s =0) Pulse length t (ns) P(m s =0) el. laser π 2 π2 Var. t el. laser 0 100 200 300 400 500 600 0.0 0.2 0.4 0.6 0.8 1.0 a) b)

Figure 2.10: Room temperature control and coherence of the NV center elec-tron spin. a, Rabi oscillations of the NV center elecelec-tron spin obtained by applying a pulse of variable length at the center resonance frequency of the hyperfine-split electron spin ms = 0 ↔ ms =−1 transition. A magnetic field of ∼30 G was applied along the symmetry axis. The Rabi frequency fR = 30 MHz well exceeds the hyperfine frequency

A = 2.16 MHz, ensuring nuclear spin-independent excitation of the electron spin. b,

Ram-sey measurement of an NV center in a IIa sample. The inter-pulse delay of two π/2 pulses is swept. The three frequencies corresponding to the three diﬀerent spin states of the host 14

N nuclear spin shape the oscillatory pattern. To create an artiﬁcial detuning, the phase of the second π/2 pulse is changed as a function of the inter-pulse delay, at a rate of 3.7 MHz. The Gaussian envelope function decays at T2∗ = 2.8 µs, a typical value for a IIa sample. 10 100 1000 0.6 0.5 0.7 P(el .=|0>) 0.9 0.8 1.0 N=1 N=2 N=4 N=8 N=16

Free evolution time (µs) Free evolution time

a) b) N = 4 N = 1 N = 2 N = 8 N =16 π 2

= pulse on electron = pulse on electronπ

Figure 2.11: Dynamical decoupling of the NV center electron spin. a, Pulse sequences showing how the number of pulses applied during the total free evolution time increases in a dynamical decoupling measurement shown in b. b, Dynamical decoupling of the electron spin of the NV center for diﬀerent number of applied pulses N (this data is described in more detail in chapter 7).

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slow compared to the timescale T2∗. The decay is caused by the averaging over many

repetitions of the same sequence required to build up suﬃcient statistics.

Hahn echo and dynamical decoupling of the electron spin

Dephasing due to fluctuations which are slow on the timescale of a single repetition, but vary between different repetitions, can be canceled by applying refocusing pulses. In a Hahn echo measurement, a single π pulse is applied exactly half-way between two π/2 pulses (see Fig. 2.11a with N=1). After the first π/2 pulse, the spin may acquire a phase due to magnetic field fluctuations. The π pulse essentially reverses the direction of phase accumulation, and therefore has a refocusing effect. This leads to an echo of the spin state after the second period of free evolution, the well-known spin echo effect [12]. If the magnetic field is constant during the total time 2τ of a single repetition of the measurement, the refocusing is perfect, and averaging over many repetitions does not lead to loss of coherence (in contrast to a Ramsey measurement). A spin echo experiment therefore yields a much longer time scale T2, as can be seen in Fig. 2.11b with N = 1.

The decay in a Hahn echo measurement is caused by fluctuations of the magnetic field on the time-scale of the inter-pulse delay. We can decouple the spin from these fluctuations by applying more π pulses during the same total free evolution time. By flipping the spin more frequently, the field becomes more static on the timescale of the inter-pulse delay and magnetic field fluctuations are averaged out more effectively. This is the principle of dynamical decoupling, a very effective method to preserve spin coherence in a slowly fluctuating magnetic environment. Universal dynamical decoupling of a single spin was first demonstrated by de Lange et al. [11], where the NV center electron spin was effectively decoupled from a spin bath dominated by P1 centers. Fig. 2.11 shows dynamical decoupling of the NV center electron spin in the presence of artificially induced magnetic field noise (this data is described in more detail in chapter 7).

Because a spin echo experiment is sensitive to magnetic ﬁeld ﬂuctuations caused by dynamics of the spin bath, a single NV center can be used as a sensor to probe the state of the spin bath. In chapter 6 we describe experiments on the control of the quantum dynamics of the mesoscopic spin bath environment, using a single NV center as the sensor.

2.4.3 The nitrogen nuclear spin as a spin qubit

The electron spin of the NV center is always coupled to the nuclear spin of its own nitrogen atom (Fig. 2.12a). The NV center is therefore an intrinsic, hybrid 2-qubit system. In chapter 7 we will describe the development and implementation of decoherence-protected quantum gates for hybrid spin systems. Here, we introduce the necessary background for understanding how the nuclear spin of the NV center

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The nitrogen-vacancy center in diamond: background and measurement techniques Frequency (GHz) A mplitude (a.u .) b) c) 0.95 1.00 1.444 1.446 1.448 1.450 0.95 1.00 |0↑ |1↑ |0↓ |1↓ π var. freq. el. nucl. laser |1↑ |0 ↑ |1↓ |0↓ C C C C C V NV center C N 1.4 GHz 2.9 MHz 5.1 MHz a)

Figure 2.12: The NV center as a hybrid 2-qubit system. a, The NV center electron spin (orange) is coupled to the nuclear spin (purple) of its own nitrogen nucleus. b, Top Panel: Electronic resonance frequency of the|0 ↑⟩ ↔ |1 ↑⟩ transition, 1.4495 GHz at 507 G. The measurement is performed by sweeping the frequency of a 9µs MW pulse with a 55 kHz Rabi frequency. The absence of other resonances is due to the optically induced spin polarization into the|0 ↑⟩ state. Taking into account the noise in the data, we estimate a nuclear spin polarization of > 97%. Lower panel: Electronic resonance frequency of the

|0 ↓⟩ ↔ |1 ↓⟩ transition. The measurement is performed by ﬁrst applying a π pulse on the |0 ↑⟩ ↔ |0 ↓⟩ transition, followed by a 9µs MW pulse with a 55kHz Rabi frequency of which

the frequency is swept. c, Energy levels at Bz = 510 G of the two-qubit system formed by the NV center’s electron spin coupled to the nuclear spin of its own nitrogen atom. For brevity, the ms= 0 and the ms=−1 levels encoding the electron spin qubit are denoted by|0⟩ and |1⟩ respectively. Similarly, the mI = +1 and the mI = 0 states encoding the nuclear qubit are denoted by| ↑⟩ and | ↓⟩ respectively. At Bz= 510 G, transitions within the two-qubit subspace are far detuned from transitions to the ms= +1 and the mI =−1 states so the system is well approximated by a four-dimensional Hilbert space.

can be used as a qubit. We ﬁrst describe the Hamiltonian of the electron spin -nuclear spin 2-qubit system, followed by a description of initialization and control of the nuclear spin and characterization of the nuclear spin coherence properties. We then introduce the key challenge for obtaining 2-qubit quantum gates for hybrid spin registers. All measurements presented in this subsection are done on the same NV center as the one used for the measurements in chapter 7.

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Hamiltonian of the NV center electron spin coupled to its host nitrogen nuclear spin

The14_{N isotope, the most common species (99.63% natural abundance), carries spin}

I = 1. The three dips observed in the ESR spectrum of Fig. 2.8b are associated with hyperﬁne coupling to the 14_{N nuclear spin of the NV center’s own nitrogen atom.}

The energy spectrum can be understood by analyzing the the Hamiltonian of the coupled system

H = H0+ P Iz2+ γnB· I + A//SzIz+ A⊥(SxIx+ SyIy) (2.4) Here, P = 4.95 MHz is the nuclear quadrupolar splitting, γn = 0.30 MHz/G is the nuclear gyromagnetic ratio, and A// = 2.16 MHz and A⊥ = 2.1 MHz are the hyperﬁne coupling parameters.

The flip-flop terms (the terms containing SxIx and SyIy) in eq. 2.4 can be ne-glected as long as the applied field Bz is such that the electronic spin transitions are far off-resonant from the nuclear spin transitions. This is the case in the exper-iments described in this thesis, where the electron (nuclear) spin resonances in the electronic ground state are of order GHz (MHz). Terms in Eq. (2.4) containing only nuclear spin operators do not affect the electron spin transition frequencies. The three resonance frequencies observed in the ESR spectrum of Fig. 2.8 consequently differ by A//.

The observation of all three resonance frequencies in Fig. 2.8 is a result of the nuclear spin being in a mixed state. When the nuclear spin is in a mixed state it cannot be used as a qubit. However, the nuclear spin can be polarized by a process involving ﬂip-ﬂops with the electron spin in the electronic excited state in combination with the optically induced electron spin polarization.

Optically induced nuclear spin polarization.

The nuclear spin state is efficiently polarized by optical illumination at a magnetic field Bz≈ 510 G [14,15]. Nuclear spin polarization is due to electron spin - nuclear spin flip-flops in the electronic excited state. Although at Bz ≈ 510 G flip-flops in the ground state are suppressed due to a large energy mismatch between electron and nuclear spin, flip-flops are allowed in the electronic excited state due to the different zero-field splitting DES = 1.4 GHz and the resulting degeneracy of the ms= 0 and the ms=−1 spin states at this field. Since optical cycling polarizes the electron spin in the ms = 0 state, these flip-flops primarily flip the electron in the ms= 0→ ms=−1 direction. The nuclear spin consequentially flips primarily in the ’opposite’ direction, and gets polarized into the mI = +1 state. This polarization process is very efficient, a result of the large hyperfine coupling in the electronic

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The nitrogen-vacancy center in diamond: background and measurement techniques 2.9355 2.9360 2.9365 2.9370 2.9375 0.8 0.9 1.0 Amp lit u d e (a .u .) Amp lit u d e (a .u .) 5.0965 5.0970 5.0975 5.0980 0.8 0.9 1.0 Frequency (MHz) Frequency (MHz) b) a) |1↑ |1↓ |0↑ |0↓ π var. freq. nucl. laser el. var. freq. nucl. laser el.

Figure 2.13: Optically detected nuclear magnetic resonance at Bz =507 G. a,

Nuclear spin resonance frequency of the |0 ↑⟩ ↔ |0 ↓⟩ transition. The measurement is performed by sweeping the frequency of a 2 ms RF pulse with a 0.25 kHz Rabi frequency.

b, Nuclear resonance frequency of the |1 ↑⟩ ↔ |1 ↓⟩ transition. The measurement is

performed by ﬁrst applying a π pulse on the|0 ↑⟩ ↔ |1 ↑⟩ transition, followed by a 2 ms RF pulse with a 0.25 kHz Rabi frequency of which the frequency is swept. The number between parentheses indicates the uncertainty in the last digit. From the diﬀerence between the two NMR frequencies we determine A = 2π· 2.16089(9) MHz.

0 10 20 30 40 50 60 0.0 0.2 0.4 0.6 0.8 1.0 Pulse length, t (µs) P ( n u cl .= | ↑ ) P ( n u cl .= | ↑ ) a) b) var. t nucl. laser el. nucl. laser el. 0.5 0.0 1.0 1.5 2.0 2.5

Free evolution time, t (ms)

π/2 π/2 π π 0 0.5 1.0 var. t

Figure 2.14: Control and coherence of the nitrogen nuclear spin at Bz≈510 G.

a, Rabi oscillations of the nitrogen nuclear spin. After optical initialization into the|0 ↑⟩ state, an RF pulse applied at the|0 ↑⟩ ↔ |0 ↓⟩ resonance frequency rotates the nuclear spin over an angle determined by the pulse length. b, Ramsey fringes of the nuclear spin. After initialization, a selective π pulse on the|0 ↑⟩ ↔ |1 ↑⟩ transition prepares the system in the |1 ↑⟩ state. Two π/2 pulses with variable inter-pulse delay t are applied on the|1 ↑⟩ ↔ |1 ↓⟩-nuclear transition. The phase of the second π/2-pulses is swept as a function of t to create an artificial detuning. To increase the signal contrast we finally apply a selective π-pulse on the|0 ↑⟩ ↔ |1 ↑⟩-transition. The resulting photoluminescence oscillates between the brightest (|0 ↑⟩) and darkest (|1 ↑⟩) photoluminescence levels. The data is fitted to∼ exp[−T/T2,nucleus∗ ]. We find T2,nucleus∗ = 5.3 ms, limited by the electronic

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excited state AES_⊥ ∼ 50MHz [2, 16]3. Since AES_⊥ is larger than the optical decay rate Γ∼ 16 MHz, a ﬂip-ﬂop is likely to happen within just one optical cycle. The polarization of the nitrogen nuclear spin is directly observed in an ESR experiment performed at 507 G by the presence of just a single resonance frequency (Fig. 2.12b).

Control and coherence of the nuclear spin

Using optical illumination at Bz = 510 G to initialize the nuclear spin into the mI = +1 state, the nuclear spin can be used as a second qubit. The level scheme of the electron spin - nuclear spin two qubit system is shown in Fig. 2.12c. For brevity, we denote the ms= 0 and the ms=−1 levels encoding the electron spin qubit by |0⟩ and |1⟩ respectively. Similarly we denote the mI = +1 and the mI = 0 states encoding the nuclear qubit by| ↑⟩ and | ↓⟩ respectively.

The nuclear spin transition frequencies can be accurately detected by sweeping the frequency of a low power RF pulse across the nuclear spin resonance frequencies (Fig. 2.13). The difference between the |1 ↑⟩ ↔ |1 ↓⟩ and the |0 ↑⟩ ↔ |0 ↓⟩ transition frequencies is equal to the hyperfine coupling constant A//. This allows us to determine A//= 2.16089(9) MHz, consistent with previously reported values [15]. Once the nuclear spin resonance frequencies are known, nuclear Rabi oscillations can be driven by applying radio frequency (RF) ac magnetic fields at the spin res-onance frequencies (Fig. 2.14a). To determine the nuclear spin dephasing time we perform a Ramsey measurement as shown in Fig. 2.14b, which yields T_2,nucleus∗ = 5.3 ms.

Nuclear spin readout

Similar to the electron spin readout, nuclear spin readout is achieved by measuring the spin-dependent photoluminescence rate. The photoluminescence rate depends on the nuclear spin state as a result of the same excited state flip-flop mechanism responsible for nuclear spin polarization [17] described above. When a nuclear spin transition is induced, changing e.g. the system from its initialized state |0 ↑⟩ to the|0 ↓⟩ state, optical illumination will result in an excited state flip-flop with the electron spin. The flip-flop changes the electron spin state from ms= 0 to ms=−1, and hence the system passes through the singlet states. The photoluminescence rate of the|0 ↓⟩ is therefore lower than that of the |0 ↑⟩ state. The photoluminescence rate of the|1 ↓⟩ is even lower as the system has to pass through the singlet states twice. Fig. 2.15 shows photoluminescence timetraces after the system has been prepared in each of the four different levels of the two qubit system shown in Fig. 2.12c.

3_{The hyperﬁne coupling constant is larger in the electronic excited state than in the electronic}

ground state as a result of a larger overlap of the orbital of the electron with the nitrogen nucleus. [2]

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The nitrogen-vacancy center in diamond: background and measurement techniques C o u n ts 0 0.5 1 1.5 2 2.5 500 1000 1500 2000 2500 3000

Photon arrival time (µs)

|1_↑

|0

↑

|1_↓

|0_↓

Photoluminescence time traces

Figure 2.15: System readout. Photoluminescence time traces after preparing the

system in each of the four states depicted in Fig. 2.12c. Due to ﬂip-ﬂop process in the electronic excited state, the photoluminescence not only depends on the electron spin state, but also on the nuclear spin state (as described in more detail in the text). This allows optical detection of both electron and nuclear spin transitions.

Rabi oscillations nuclear spin

Pulse sequence Rabi oscillations nuclear spin Ramsey electron spin

Pulse sequence Ramsey electron spin

0 5 10 15 20 25 0.0 0.2 0.4 0.6 0.8 1.0 Time (µs) 0.0 0.2 0.4 0.6 0.8 1.0 P ( n u cl .= | ↑ ) P ( e l. = | 0 ) el. nucl. laser a) b) π π/2 π/2 var. time el. nucl. laser

Figure 2.16: Comparison between the timescales of the electronic and the

nuclear spins. a, The electron spin dephasing time, T2∗= 3.55(7) µs as determined in a

Ramsey experiment, is much smaller than the typical time it takes to rotate the nuclear spin (∼ 10 µs for a nuclear π pulse). b, Pulse sequences of the measurements shown in a. The electron spin Ramsey measurement is performed by sweeping the inter-pulse delay of two π/2 pulses applied at the |0 ↑⟩ ↔ |1 ↑⟩ transition frequency. The pulse power corresponds to a ∼ 50 MHz Rabi frequency. The oscillation is a result of an artiﬁcial detuning created by sweeping the phase of the last π/2 pulse as a function of the inter-pulse delay. The nuclear Rabi oscillations shown in a are measured by ﬁrst applying a π pulse on the|0 ↑⟩ ↔ |1 ↑⟩ transition and then sweeping the duration of an RF pulse applied at the|1 ↑⟩ ↔ |1 ↓⟩ transition.

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From Fig. 2.15 it can be seen that diﬀerent population distributions can give the same photoluminescence rate. Therefore, to unambiguously determine the dif-ferent level occupation probabilities of an unknown state, we need to redistribute the population in diﬀerent ways and compare the detected photoluminescence of the resulting states. This is described in detail in appendix B, as well as how to do full two-qubit state tomography on the coupled electron spin- nuclear spin system.

Decoherence and control timescales for the electron and the nuclear spins

The gyromagnetic ratio of an electron spin is a factor∼ 1000 times larger than that of a nuclear spin, resulting in largely different decoherence and control timescales of both qubit types. Due to the larger gyromagnetic ratio, the electron responds quickly (typically, on a timescale of 1 ns – 0.1 µs, Fig. 2.10a) to an applied magnetic field. However, the electron is strongly coupled to the environment for the same reason, and decoheres quickly. The nuclear spin, due to the much smaller gyromagnetic ratio, is coupled to the environment much weaker, has much longer coherence time, but responds to the external control fields very slowly (on a timescale of 10 µs or longer, Fig. 2.14a).

Due to the large difference in timescales, an entangling gate between the elec-tronic and the nuclear spins has low fidelity if we do not incorporate protection against decoherence into the gate: the electron spin would dephase long before the nuclear spin responds to the magnetic control field. This is clearly illustrated by com-paring a Ramsey measurement of the electron spin to a measurement of nuclear Rabi oscillations (Fig. 2.16). In chapter 7 we will demonstrate how decoherence-protected quantum gates on such a hybrid electron-nuclear spin system can be implemented by integrating dynamical decoupling techniques into the gate operation.

The intrinsic hybrid nature of the NV center makes it an excellent candidate for application in hybrid quantum networks in which each qubit species plays a role that suits it best. Because the quantum state of the NV center electron spin can be mapped onto the state of an emitted photon, as was recently been demonstrated by Togan et al. [18], distant NV centers can in principle be entangled through a process called measurement-based entanglement (MBE). This will be introduced in the next section, as it is the main motivator for the experiments described in chapters 3, 4, and 5.

2.5 Towards measurement-based entanglement of

distant NV centers

The ability to entangle the spin states of two distant NV centers would be a true breakthrough for their application in quantum information technologies. Although

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NV centers can be entangled by their dipolar interaction [19], this method is lim-ited to NV centers with a mutual separation on the nanometer length scale set by the dipolar interaction strength. For long distance entanglement, as required for e.g. large quantum communication networks, it is therefore necessary to employ a diﬀerent entanglement scheme.

Long-distance entanglement is possible by making use of photonic channels. Several schemes exist for probabilistic measurement-based entanglement of distant qubits. These schemes are generally some variation of the DLCZ scheme [20, 21]. In this approach, two qubits are stimulated to emit single photons which are entangled with the qubit they originate from. These photons are then made to interfere on a beamsplitter which serves to erase the ’which-path’ information. The outputs of the beamsplitter are monitored by two detectors. Depending on the scheme, a cer-tain combination of detector clicks projects the two qubits onto an entangled state (Fig. 2.17a). 640 660 680 700 720 740 In te n si ty (a .u .) Wavelength (nm) NV center spectrum at 7K Zero-phonon line ↓↑ − ↑↓ a) b)

Figure 2.17: Towards measurement-based entanglement between distant NV centers. a, Principle of measurement-based entanglement. Two NV centers are made to emit photons which are entangled with the spin state of the NV center they originate from. The beamsplitter erases the which path information after which a certain combination of detector clicks projects the two distant NV spins onto an entangled state. b, Low temperature spectrum of an NV center. Even at low temperature, emission into the zero phonon line is only about 4% of the total emission. The low coherent photon emission rate forms a key challenge for MBE of NV centers.

Measurement-based entanglement (MBE) between two atomic qubits was demonstrated in 2007 by Moehring et al. [22], but MBE with solid-state qubits has not been demonstrated yet. Long-distance entanglement of solid-state qubits would be a major breakthrough for quantum information technologies due to the

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larger potential for scalability. NV centers are receiving much attention in this re-spect as they meet all the requirements for MBE: its spin state can be controlled with high ﬁdelity, it has a long coherence time, and it has spin-dependent optical transitions which can be used to entangle the spin state with the state of an emitted photon [18].

The observation of spin-photon entanglement is a major step towards MBE. However, generating spin-spin entanglement is more challenging, as it requires the simultaneous generation of two spin-photon entangled pairs from two NV centers. The spin-spin entanglement success probability scales as∼ Psp,1Psp,2, where Psp,iis the success probability of generating an entangled spin-photon pair with NV center i. To illustrate the experimental challenge for MBE of distant NV centers, we consider numbers from the experiments of Togan et al. [18]. Here, Psp≈ 10−6, yielding 10 entangled spin-photon pairs per second if the measurement is repeated at a 10 MHz repetition rate. With such low Psp, the success rate of creating spin-spin entangled pairs from two such NV centers is prohibitively low, on the order of 10−5per second at a 10 MHz repetition rate.

For all NV center MBE protocols two key challenges for attaining a large Psp are the low photon collection efficiency and the relatively low emission into the zero-phonon line (Fig. 2.17b). Recent experiments [23, 24] have demonstrated that solid immersion lenses (SILs), directly milled into diamond, improve the collection efficiency by more than an order of magnitude. This has led to the first observation of NV center two photon interference in the experiments by Bernien et al. [25], which is another crucial step towards MBE.

The experiments described in chapters 3, 4, and 5 of this thesis are motivated by another approach to attain a large Psp: Coupling the NV center to an optical cavity. Coupling to an optical cavity provides a unique advantage over using SILs, as it can be used to increase the relative emission into the zero-phonon line.

2.5.1 Cavity QED

Spontaneous light emission can be controlled by enhancing or suppressing the vac-uum fluctuations of the electromagnetic field at the location of the light source [26]. The vacuum fluctuations of the electromagnetic field can be enhanced over the free space value by confining the field to a very small volume in e.g. a photonic cavity. This can be seen from Fermi’s golden rule, which describes the spontaneous emis-sion rate Γ0 of an emitter in the ’weak coupling’ limit. In the weak coupling limit

the emission process is irreversible, similar to spontaneous radiation into free space. Fermi’s Golden rule is

Γ0=

2π

~2|⃗p · ⃗Evac|

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where p is the dipole moment of the emitter, g(ω) is the optical mode density, and Evacis the amplitude of the vacuum electric ﬁeld.

Fermi’s golden rule provides us with a tool to engineer the spontaneous emission rate of an emitter and can therefore be used to reshape the spectrum of the NV center. By embedding the NV center in an optical cavity which has a mode resonant with the ZPL, emission into the ZPL is increased, while oﬀ-resonant emission into the phonon sideband is suppressed due to the absence of available states (a photonic bandgap).

The enhancement is strongest if the emitter is placed in a cavity with a high quality factor in which the electromagnetic field is squeezed to the smallest possible mode volume (which is on the order of a wavelength). A challenge is to position the emitter into the mode maximum, as this requires nanometer-precise positioning accuracy of either the emitter or the cavity. Chapter 3 describes the development of a nanopositioning technique which can be used to position nanometer-sized objects at an arbitrary location, and chapter 4 describes how we have used this technique to couple a single NV center in a diamond nanocrystal to a high quality, small mode volume photonic crystal cavity. Finally, chapter 5 presents simulations and measure-ments of the effect of a nanoparticle on the optical propterties of a photonic crystal cavity, and quantifies the maximum attainable Purcell enhancement of emission into the zero-phonon line when an NV center in a diamond nanocrystal is coupled to a PC cavity.

Quantum control of single spins and single photons in diamond

Quantum control of single spins and single

photons in diamond

Quantum control of single spins and single

photons in diamond

Proefschrift

Toeno van der Sar

Contents

Chapter 1

Introduction

1.1

The quantum computer

1.2

Hybrid systems

1.3

The nitrogen-vacancy center in diamond

photon

electron spin

N nuclear spin

C nuclear spins

1.4

Thesis overview

Bibliography

Chapter 2

The nitrogen-vacancy center in

diamond: background and

measurement techniques

2.1

Electronic structure of the NV center

2.2

Spin properties of the optical transitions

2.2.1

Optically induced spin polarization

2.2.2

Spin-dependent photoluminescence

2.3

Detecting single NV centers

2.3.1

Confocal microscopy

2.4

NV centers as spin qubits

2.4.1

Spin manipulation - Experimental setup.

2.4.2

The NV center electron spin as a spin qubit

2.4.3

The nitrogen nuclear spin as a spin qubit

2.5

Towards measurement-based entanglement of

distant NV centers

2.5.1

Cavity QED

_{C nuclear spins}