![12 Quantum Information Processing with Diamond
between two nodes of a quantum network. In this proposal, illustrated in Fig. 1.3,
two ground states of a three-level atom are assumed to function as a stationary
storage qubit. Each of the ground states, labeled |↑› and |↓›, is connected to a
shared excited state through optical transitions, creating a so-called Λ system.
One of these transitions is coupled to a classical ‘control field’, while the other is
coupled to a mode of an optical cavity. As discussed in more detail below, the
cavity is an important part of this proposal, as it serves the dual purpose of
enhancing the vacuum coupling rate between the excited state and the second
ground state (|↓›) and of allowing the light emitted by the transition to be efficiently
collected. When the classical field is used to drive the transition between the first
ground state (|↑›) and the excited state, the cavity-coupled optical transition
quickly causes the excitation to coherently emit a ‘cavity photon’and transition to
the second ground state (|↓›). This is effectively a Raman, or Λ, coupling between
|↑› and |↓›, and we refer to this type of system as an ‘optically coupled spin’below.
For a suitably designed system, when the classical drive is applied, the spin
coherence between the two ground states is transferred onto the state of the photon
emitted into the cavity mode. Assuming the initial spin state is a superposition
α|↑›+β|↓› of the ground states and there are initially no cavity photons, the state
of the system before and after the classical drive can be expressed as
[1.1]
where |0› and |1› refer to cavity photon populations equal to 0 and 1, respectively.
This cavity photon state can be outcoupled, transmitted through a low-loss optical
channel and reversibly reabsorbed by an identical atomic spin system in a cavity
at a spatially distant location. In this way, quantum states can be transferred
between distant qubits, and a quantum network can be created in which nodes are
coupled via an optical quantum bus.55
In a related proposal in 1999, Cabrillo et al.56
introduced a scheme for entangling
two optically coupled spins by building an experiment in which the detection of a
photon created after simultaneous excitation of two spins cannot be uniquely
correlated with one of the two spins. This proposal has evolved into a canonical
1.3 Illustration of transfer of a quantum state between two distant
qubits via a photonic channel.](https://image.slidesharecdn.com/2368622-250601235359-ad6bf92e/85/Quantum-Information-Processing-With-Diamond-Principles-And-Applications-1st-Edition-Steven-Prawer-28-320.jpg)
![Principles of QIP using diamond 13
implementation in which the emission from each system is interfered in a beam
splitter whose outputs are monitored by photon counters. This concept is referred
to as measurement-based entanglement, and was extended in 2001 by Duan,
Lukin, Cirac and Zoller,57
who described how measurement-based entanglement
can be used to implement a set of gates sufficient to build a quantum computer –
this is often referred to as the DLCZ scheme for quantum computing, and is
illustrated in Fig. 1.4. The measurement-based entanglement central to these
proposals relies on simultaneously exciting two optically coupled spins using
weak classical control fields, such that each spin has a low probability of making
the Λ transition and generating a cavity photon. If the output from each cavity is
interfered in a 50:50 beam splitter or waveguide coupler, and if the cavity photons
generated by each system are indistinguishable, then upon detection of a photon
at one of the output ports of the interference device, it is impossible to determine
from which spin it originated. Under these conditions, it can be shown that the
two spin systems, labeled A and B, are entangled. In the simplest ideal
implementation considered here, the system has initial and final states
[1.2]
where φ is a phase shift associated with differences in the optical path lengths of
the arms of the interference device, and the choice of+or−in ‘±’is determined by
which detector measures a photon. If a sufficiently large number of spins can be
entangled in this way, a rich class of applications may become possible. Of
particular interest are quantum computing and simulation schemes involving
cluster states, which can introduce the fault tolerance necessary for building
realistic QIP hardware.58, 59
At the time of the initial proposals described above, trapped ions or neutral
atoms were assumed to be the medium of choice for realizing the cavity–spin
system. Indeed, in recent years, formidable progress has been made in trapping
1.4 Illustration of scheme for measurement-based entanglement
between two spin qubits interacting indirectly through photon
interference.](https://image.slidesharecdn.com/2368622-250601235359-ad6bf92e/85/Quantum-Information-Processing-With-Diamond-Principles-And-Applications-1st-Edition-Steven-Prawer-29-320.jpg)
Quantum Information Processing With Diamond Principles And Applications 1st Edition Steven Prawer
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- 5. Quantum Information Processing with Diamond
- 6. Related titles: Quantum optics with semiconductor nanostructures (ISBN 978-0-85709-232-8) Optical switches: Materials and design (ISBN 978-1-84569-579-8) Semiconductor lasers: Fundamentals and applications (ISBN 978-0-85709-121-5)
- 7. Woodhead Publishing Series in Electronic and Optical Materials: Number 63 Quantum Information Processing with Diamond Principles and Applications Edited by Steven Prawer and Igor Aharonovich amsterdam • boston • cambridge • heidelberg • london new york • oxford • paris • san diego san francisco • singapore • sydney • tokyo Woodhead Publishing is an imprint of Elsevier
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- 9. xi (* = main contact) Contributor contact details Editors S. Prawer Melbourne Materials Institute School of Physics University of Melbourne Room 159 Parkville, VIC 3010, Australia E-mail: s.prawer@unimelb.edu.au I. Aharonovich School of Physics and Advanced Materials University of Technology Sydney Ultimo, NSW 2007, Australia E-mail: igor.aharonovich@uts. edu.au Chapter 1 P. E. Barclay Institute for Quantum Information Science University of Calgary Calgary, AB T2N 1N3, Canada E-mail: pbarclay@ucalgary.ca Chapter 2 H. Weinfurter Faculty of Physics LMU Munich Schellingstr. 4 D-80799 Munich, Germany E-mail: h.w@lmu.de Chapter 3 R. Kalish Physics Department and Solid State Institute Technion, Israel Institute of Technology Technion City Haifa 32000, Israel E-mail: kalish@technion.ac.il Chapter 4 J. M. Smith Department of Materials University of Oxford Parks Road Oxford OX1 3PH, UK E-mail: jason.smith@materials.ox.ac.uk
- 10. xii Contributor contact details Chapter 5 J. T. Choy, B. J. M. Hausmann, M. J. Burek, T. M. Babinec and M. Lončar* Harvard University Maxwell Dworkin 127 33 Oxford Street Cambridge, MA 02138, USA E-mail: loncar@seas.harvard.edu Chapter 6 E. Neu Universität des Saarlandes Fachrichtung 7.2 Experimentalphysik Campus E2.6 66123 Saarbrücken, Germany and Department of Physics University of Basel Klingelbergstrasse 82 CH-4056 Basel, Switzerland E-mail: elke.neu@unibas.ch C. Becher* Universität des Saarlandes Fachrichtung 7.2 Experimentalphysik Campus E2.6 66123 Saarbrücken, Germany E-mail: christoph.becher@physik. uni-saarland.de Chapter 7 A. W. Schell, J. Wolters, T. Schröder and O. Benson* AG Nanooptik Institut für Physik Humboldt-Universität zu Berlin Newtonstr. 15 12489 Berlin, Germany E-mail: oliver.benson@physik.hu-berlin.de Chapter 8 G. Dutt* and M. U. Momeen Department of Physics and Astronomy University of Pittsburgh 100 Allen Hall 3941 O’Hara Street Pittsburgh, PA 15260, USA E-mail: gdutt@pitt.edu Chapter 9 L. P. McGuinness Institute for Quantum Optics Albert-Einstein-Allee 11 University of Ulm 89081 Ulm, Germany E-mail: liam.mcguinness@uni-ulm.de Chapter 10 P. Maletinsky* Department of Physics University of Basel Klingelbergstrasse 82 CH-4056 Basel, Switzerland E-mail: patrick.maletinsky@unibas.ch V. Jacques Laboratoire Aim′e Cotton, CNRS Universit′e Paris Sud and ENS Cachan 91405 Orsay, France E-mail: vincent.jacques@ens-cachan.fr
- 11. Contributor contact details xiii Chapter 11 R. Amsüss Vienna Center for Quantum Science and Technology Atominstitut Technische Universität Wien Stadionallee 2 1020 Vienna, Austria S. Saito and W. J. Munro* NTT Basic Research Laboratories NTT Corporation 3-1 Morinosato-Wakamiya Atsugi Kanagawa 243-0198, Japan E-mail: bilmun@qis1.ex.nii.ac.jp Chapter 12 K. Fox and S. Prawer* Melbourne Materials Institute School of Physics University of Melbourne Room 159 Parkville, VIC 3010, Australia E-mail: kfox@unimelb.edu.au; s.prawer@unimelb.edu.au Chapter 13 I. Aharonovich School of Physics and Advanced Materials University of Technology Sydney Ultimo, NSW 2007, Australia E-mail: igor.aharonovich@uts.edu.au S. Prawer Melbourne Materials Institute School of Physics University of Melbourne Room 159 Parkville, VIC 3010, Australia E-mail: s.prawer@unimelb.edu.au
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- 17. xxi Foreword Diamond exhibits several unique physical properties: a wide band gap, high thermal conductivity, hardness and inertness. These unique characteristics mean that diamond plays a unique role in many technologies. During the last decade, a new field of application of diamond related to quantum engineering has emerged. This book reports the fascinating progress that has been achieved in this rapidly developing interdisciplinary field of research. A number of leading scientists have reported how the unique properties of diamond and methods for controlling those properties have created the potential for using ‘quantum diamond’in a wide range of new technologies. There are three major applications of diamond in the context of the control of single defects. The first is related to the generation of scalable quantum registers for quantum information-processing protocols. The second field relates to the generation of single-photon emitters using color centers as a light source. This application is crucial for linear-optics quantum computing and the rapidly growing field of quantum communications. The third area is the application of diamond defects for novel imaging and sensing technologies. The book is organized into a number of parts. The first part covers the background to novel diamond technologies and includes an overview of single- photon sources for quantum cryptography, quantum information processing with defects and the basic principles of nanoengineering of diamond. The ability to engineer color centers on demand and with high spatial accuracy is crucial for many quantum-based technologies. This is why the field of ion implantation is reviewed in depth in the book. The second part of the book covers different application areas associated with single color centers in diamond. Some of the applications are already at a very advanced stage. As an example, diamond-based single-photon sources are now commercially available. Part II also contains an overview of diamond sensor technology, such as novel diamond-based microscopy techniques and neuroimaging, that will potentially come to the market in the near future. Applications related to quantum information processing include scalable quantum networks, integrated diamond photonics and quantum memories based on ensembles of diamond spins. Finally, the interdisciplinary field of quantum
- 18. xxii Foreword applications in biological systems is also highlighted, paving the way to one of the most fascinating applications of diamond – direct sensing in living cells. The field of quantum applications of diamond is at quite an early stage of development. This book summarizes the successes in this field during the last decade and shows potential avenues for the future. Written in an accessible style by world-leading experts in the field, the book will be of interest to a broad community of scientists working in the fields of solid-state quantum physics, magnetic resonance, quantum optics and biophysics, as well as to graduate students and those starting out on research in this important field. Professor Fedor Jelezko Ulm University, Germany
- 19. 3 © 2014 Elsevier Ltd 1 Principles of quantum information processing (QIP) using diamond P. E. BARCLAY, University of Calgary, Canada DOI: 10.1533/9780857096685.1.3 Abstract: Impurities in diamond are promising candidate qubits for applications in quantum information processing. Electron and nuclear spins associated with these impurities combine long coherence times with optical addressability, and have been used to demonstrate several important elements of proposed quantum information processing systems. Here we review recent experimental progress in the optical manipulation of impurities in diamond, and efforts to build more complex quantum systems involving coupled qubits. Key words: diamond, quantum, nuclear spins, nitrogen-vacancy, impurities, qubits, colour centre, electronic, photonic networks. 1.1 Introduction Since the first experimental demonstration in 1997 by Gruber et al. of optically detected magnetic resonance of single nitrogen–vacancy (NV) color centers in diamond,1 researchers have made rapid progress towards utilizing impurities in diamond for applications in quantum information science. Today, impurities in diamond, and NV centers in particular, are leading candidates for solid-state implementations of quantum information processing (QIP) devices.2–4 This progress has been driven by improvements in the properties of the available material, and by increasingly sophisticated experimental techniques for probing and manipulating the electronic and nuclear spin degrees of freedom of atom-like impurities in diamond. Inthesubsequentsections,wereviewexperimentalprogresstowardsperforming photonic-network-based quantum information processing with diamond. We begin by discussing the properties of diamond impurities which make them suitable for qubits, and introduce approaches to creating diamond impurities suitable for QIP. We then discuss how these impurities can be used for photonic- network-based QIP, in which NV centers separated by microns or meters are entangled using quantum optical interference and measurement. 1.2 The role of diamond impurities in quantum information processing (QIP) The study of impurities in crystalline materials has long been a topic of interest to solid-state and optical physicists. Impurities play a crucial role in determining the
- 20. 4 Quantum Information Processing with Diamond electronic properties of semiconductors,5 and are responsible for the color of nominally transparent crystals such as diamond. These latter types of optically active impurities are referred to as color centers. Color centers are candidate qubits when they possess appropriate electronic properties and symmetry within the crystal lattice for optical transitions to be used to manipulate and read out their electronic spin states. In practice, it is also desirable for these spin states to have sufficiently long lifetimes and coherence properties to enable implementation of the operations needed for the proposed quantum information processing.6 Beginning in 1974, Davies and colleagues performed many of the first studies of the optical and electronic properties of the spectral lines associated with diamond color center emission,7–10 laying the groundwork for much of the research describedbelow.Byobservingtheeffectofexternalfieldsonthephotoluminescence properties of diamond color centers, in concert with group-theoretic and other first-principles arguments, researchers have been able to identify spectral features with specific diamond impurities. Since then, over 100 color centers have been identified in diamond.11 Most of these diamond color centers have been studied in bulk, owing to relatively weak optical transitions, which make it experimentally challenging to observe emission from a single impurity. Although ensembles of color centers may be used as elements in quantum information processing – see for example the progress towards creating quantum memories in diamond,12, 13 silicon14 and rare-earth-doped crystals and glasses15 – it is the unprecedented ability of researchers to manipulate single color centers which makes diamond an attractive platform for quantum information processing. In recent years, experimentalists have demonstrated manipulation of single color centers with exquisite sensitivity. The quantum state of single electron spins associated with diamond color centers can be manipulated using combinations of optical, microwave and RF fields.16, 17 These spins have sufficiently long coherence times to enable observation of their coherent coupling to nearby nuclear spins, enabling the first ever demonstration of the measurement of a single nuclear spin.18, 19 Using related techniques, quantum information can be reversibly mapped from electronic to nuclear spins,20, 21 where it can be stored for record lengths of times with high fidelity.22 Below, we discuss these properties in more detail, and highlight material science issues which must be considered for implementation of practical quantum information processing devices. 1.3 Types of diamond color center Color centers in diamond consist of a combination of defects in the diamond crystal lattice, which include common impurities such as nitrogen and configurations of missing carbon atoms, referred to as vacancies. A given color center may be found in several possible charge states, depending on the possibility
- 21. Principles of QIP using diamond 5 of accepting or donating charge from or to other carbon atoms or defects in the diamond lattice. If the localized electronic states of a defect support optical transitions between excited and ground states such that the defect is optically active, it may be referred to as a color center. These optical transitions may be accompanied by the generation of phonons through coupling between the change in the electronic wave function during an optical transition and the surrounding elastic crystal lattice. For the purposes of quantum information processing, the desirable properties of color centers include the potential for their quantum state to be initialized, manipulated and read out. These operations are critical elements of the DiVincenzo criteria for a physical system to be useful for quantum computing.23 The QIP scheme considered in Section 1.6 requires execution of these operations at the single-color-center level using optical fields, and at a sufficiently high rate to overcome decoherence of the spin states. Linear-optics QIP schemes24 relying on the generation, interference and detection of single photons do not require the ability to manipulate spin degrees of freedom, but benefit from generation of single photons at a high rate with a high level of indistinguishably. This requires optical transitions with short radiative lifetimes, high radiative efficiency and, ideally, transform-limited linewidths. With the benefit of experimental evidence, we can understand how the intrinsic physical properties of diamond allow color centers in diamond to be excellent candidate qubits. The theoretical basis for this was studied by Weber et al.,25 in the context of understanding the desirable coherence properties that have been observed in studies of NV centers. Among crystalline materials, diamond has one of the largest electronic band gaps, of 5.5eV. This allows localized electronic excited and ground states of impurities to be isolated from the valence and conduction bands of the surrounding diamond lattice, even for energies at optical wavelengths. If the diamond lattice consists of the isotopically abundant 12 C, which possesses zero nuclear spin, the ground state of the color center spins will not be subjected to magnetic-field fluctuations from the nuclear spins of the surrounding atoms. Finally, the large Young’s modulus of diamond results in a high Debye temperature, and a relatively low room temperature phonon population compared with other crystals, decreasing the effect of thermal excitation of crystal lattice phonons on the spin coherence properties. Of the many color centers identified in diamond, only a small number have been observed at the single-impurity level. The first single color center to be measured was the NV color center,1 which consists of a nitrogen impurity adjacent to a vacancy in an otherwise perfect diamond carbon lattice. Since then, optical emission from single nickel-related defects,26 silicon–vacancy defects27, 28 and chromium29 impurities has been observed. These impurities all possess optical transitions with relatively large dipole moments characterized by short spontaneous-emission lifetimes. In practice, these transitions allow photoluminescence measurements in which 104 –106 photons per second can be
- 22. 6 Quantum Information Processing with Diamond collected from a single color center using a standard confocal microscope equipped with a high-numerical-aperture (NA=0.4–0.9) objective.30 This emission can be efficiently measured using single-photon counters. (Owing to the relatively high detector dark count rates of non-Si-based single-photon counters, this is more challenging for emitters in the near-IR band. Progress in the development of superconducting photon counters31 promises to alleviate this difficulty.) Depending on the interaction between the emitter and phonons in the crystal lattice, the emission may be spread over a wide bandwidth owing to phonon-assisted emission. (Isolating color center emission from background emission or dark counts becomes more challenging as the transition linewidth broadens.) 1.4 Key properties of nitrogen–vacancy (NV) centers Of the observed color centers in diamond, negatively charged NV centers have proven to date to be the most promising for QIP applications. The negatively charged NV center, NV− , has so far been the focus of research attention, and will be considered exclusively in the remainder of this chapter. The electronic structure of a negatively charged NV center, shown in Fig. 1.1, consists of a 3 A2 spin triplet ground state connected to a manifold of 3 E excited states by optical transitions with a zero-phonon-line (ZPL) wavelength of 637nm. In many experiments, NV centers are excited incoherently using a green source at 532nm. This source excites the NV centers to high-energy states not shown in Fig. 1.1, which decay nonradiatively to the excited states discussed here, before decaying to the ground state and emitting a photon. In addition to this zero-phonon process, phonon sidebands at longer and shorter wavelengths can be generated during emission and absorption via these transitions. The ground states of the NV− electronic spin triplet are labeled by ms =0,±1. The ms =±1 and ms =0 states are split in energy by 2.87GHz by the crystal field. The Zeeman effect can be used to tune this splitting with an external magnetic field. In some experiments, the ms =−1 and ms =0 states have been tuned close to 1.1 Electronic structure of NV center.
- 23. Principles of QIP using diamond 7 resonance using a 1000G magnetic field, where they undergo spin mixing.32 The 3 E excited state is also a spin triplet; however, its fine structure is extremely sensitive to temperature and stress.33 A remarkable property of incoherent excitation of NV− is that this method can be used to initialize and read out the spin ground state. Under 532nm excitation, the NV− normally decays via spontaneous emission back to its initial ground state. This is the result of the spin-preserving nature of the incoherent optical excitation and subsequent phononic decay to the 3 E states. However, with a small probability, an intersystem crossing may occur, in which population is transferred from the ms =±1 to the ms =0 ground states. This is mediated by a combination of nonradiative decay and an optical transition at 1042nm between a pair of singlet states of the NV− , and has been the topic of significant study in recent years.34 As a result, after a sufficiently long period of continuous excitation (approximately 1–10μs), the NV population will be shelved in the ms =0 ground state. Because of the relatively long lifetime (approximately 100ns) of the 1042nm transition, the brightness of the emission from an NV center initialized in the ms =0 state will be higher than for one in the ms =±1 state. This difference in brightness is the mechanism by which the NV spin state is read out optically. This behavior is exhibited at both room and cryogenic temperature, and is critical to all experiments involving optical measurement and control of NV centers. Using the incoherent optical initialization and readout techniques described above, in combination with microwave fields, the coherence times of the electronic spin ground states of single NV centers can be measured. By increasing the purity of the diamond carbon lattice – reducing nitrogen impurities to below the ppb level, and ensuring a 12 C isotope content of 99.7% – coherence times exceeding a millisecond can be observed at room temperature. Achieving these purities is now possible in diamond samples grown synthetically using chemical vapor deposition (CVD).35 When the background magnetic-field fluctuations associated with impurities are sufficiently small to enable long electronic spin coherence times, the electronic spin can be used as a sensitive probe of the state of the small number of nuclear spins that have not been removed. Recent experiments have demonstrated coherent coupling between single NV electronic spins and nuclear spins associated with 13 C impurities near the NV center. Coherence times exceeding 1 s have been demonstrated for the nuclear spins, making them attractive candidates for storage of quantum information.22 At low temperature, the ZPL linewidth narrows, as the effect of thermally excited crystal phonons is reduced. At temperatures below 10K, the ZPL can have a linewidth as narrow as 13MHz,36 limited by the spontaneous-emission lifetime of the excited state. Emission into the zero-phonon line accounts for approximately ηzpl ∼3.5% of the NV emission. Phonon sideband emission accounts for the remainder, and is the result of Franck–Condon coupling between changes in the electronic waveguide during the optical transition which couple to the surrounding crystal lattice. From an the point of view of experimental measurement, phonon
- 24. 8 Quantum Information Processing with Diamond sideband emission both poses challenges and provides opportunities. It decreases the spectral brightness of the NV emission, making it more challenging to distinguish photons emitted by an NV center from background light or detector noise. Coupling NV centers to resonant optical structures such as cavities also becomes challenging, since cavity linewidths are typically much narrower than the sideband bandwidth. As a result, optical cavities only enhance a fraction of the NV emission. In the case of a cavity resonant with the zero-phonon line, the cavity–dipole coupling rate is essentially reduced by a factor of relative to the case where all of the emission is into the zero-phonon line, as is the case with atomic quantum emitters and certain quantum dot emitters. Conversely, phonon sideband emission can aid in measurements involving resonant excitation to measure the properties of NV centers. Using spectral filters to separate scattered excitation photons from emitted phonon sideband photons, the population dynamics of the NV centers can be studied as a function of wavelength and other parameters of the resonant excitation field. Optical transitions between ground states and specific excited states are governed by selection rules dependent on the initial and final spin states and on the polarization of the excitation field. The allowed transitions also depend on the local strain applied to the diamond crystal lattice. In the absence of strain, the excited states consist of a manifold of six excited states split in energy owing to spin–orbit and spin–spin interactions.37 Stress mixes these states, and creates two branches of excited state with well-defined linear momentum. In many diamond samples, this splitting exceeds the spontaneous-emission linewidth, and can easily be identified by low-resolution (>1GHz) spectroscopic measurements at cryogenic temperatures. At intermediate stress levels, the states become mixed.38 Careful control of this mixing is essential for implementing the Λ optical transitions necessary for the measurement-based entanglement schemes presented in the following section. By controlling the polarization of the excitation field, as well its wavelength, it is possible to resonantly excite specific optical transitions of the NV center.16, 39 This capability is essential for many recent impressive quantum optics experiments involving NV centers, including EIT, spin–photon entanglement and single-shot readout. 1.5 Techniques for creating NV centers Many of the experimental studies described above were done using ‘naturally occurring’ NV centers. These centers are incorporated into high-purity diamond samples during the CVD growth process, or are found in natural diamond without any special processing. Many proposed QIP applications of NV centers require or would benefit from the ability to locate them deterministically in the crystal lattice. In particular, the QIP schemes discussed in Section 1.3 require an ability to position NV centers relative to optical cavities or other NV centers, with which they can interact directly. An ability to control the density of NV centers is also desirable in
- 25. Principles of QIP using diamond 9 applications in which ensembles of NV centers are needed, for example electromagnetically induced transparency, magnetometry and quantum memory.12, 13 There are several methods for postprocessing diamond samples to create NV centers. Generally, the goal of these techniques is to create NV centers whose properties, for example optical linewidth and spin coherence time, are as close to ideal as possible, and to control the location and density of the NV centers created. In practice, it is necessary to balance these sometimes competing requirements. It can be challenging to introduce the elements needed to create an NV center – a vacancy and a nitrogen impurity – without introducing more imperfections than necessary and without degrading the spin coherence and optical transition properties. Similarly, neighboring NV centers will begin to interact as their density is increased, also modifying and/or degrading their optical and spin properties. Three commonly used techniques for creating NV centers are ion implantation, electron or positron irradiation, and selective nitrogen doping during the CVD growth of diamond. These techniques are typically combined with high- temperature annealing (at T>650°C), which causes vacancies to migrate throughout the diamond crystal. It is energetically favorable for vacancies to combine with nitrogen impurities to form NV centers, and if there is a sufficient numberofelectrondonorsinthecrystallattice,theNVcenterswillbepreferentially in the desired negatively charged state. 1.5.1 Ion implantation Ion implantation allows impurities such as N, Ga and He to be implanted into the diamond lattice. The implantation process also damages the diamond lattice, creating vacancies. Ideally, when the sample is annealed, the vacancies will either be removed or combine with nitrogen impurities to create NVcenters. Implantation depth profiles vary depending on the implantation element, but can be controlled by adjusting the energy of the ion beam (usually in the keV range). Typically, implantation depths between a few nanometers and 100nm are used. The profiles can be modeled using widely available software.40 A powerful aspect of ion implantation is the spatial control of the implanted ions that it provides. Ions can be implanted with a spatial resolution of less than 50nm either by using a focused source, for example a He ion microscope41 or a scanning tip,42 or by patterning a protective mask on the diamond surface with ∼30nm diameter holes.43 Thanks to the high spatial resolution of ion implantation, it is an attractive technique for QIP applications requiring the coupling of NV centers to nanophotonic devices such as waveguides and cavities in an on-chip photonic network, as discussed in Section 1.6. The challenges associated with this approach include minimizing the number of implanted ions needed to create a single NV center, and the relatively high uncertainty in the number of NV centers created per implantation site when one is attempting to create single NV centers. Implantation statistics are Poissonian,
- 26. 10 Quantum Information Processing with Diamond and the likelihood of creating zero or two NV centers when the target is a single NV center per site is high. It may also be difficult to remove excess vacancies that do not create NV centers, as they may combine with other impurities to form other non-NV complexes in the diamond lattice. Excess damage or residual impurities created during ion implantation degrade the properties of the resulting NV centers. Optically, this is manifested in spectral shifts and broadening of the NV ZPL, which is most easily observed at low temperatures. It can also create charge traps which prevent the NV centers from being preferentially observed in the NV− state.44 The spin coherence time may also be degraded. Gaining a better understanding of these effects is complicated by the proximity of the implanted NV centers to the diamond surface, which also has an effect on their properties.45 1.5.2 Irradiation Irradiation with an MeV-energy electron or positron flux is a technique commonly used for creating vacancies in diamond samples. As with ion-implanted samples, annealing is used to encourage the vacancies to migrate towards N impurities and form NV centers. The characteristic length scale of the damage profile produced by this technique is much longer than that for ion implantation, and the irradiation usually creates vacancies with a uniform density throughout the diamond sample (with a typical thickness of 250μm or larger). This technique has been used to create diamond samples with relatively high NV densities whose inhomogeneous linewidths are relatively low compared with ensembles created using ion implantation.46 It can be applied to both bulk and nanocrystalline diamond samples. 1.5.3 Diamond heterostructures Incorporation of thin layers of NV centers into otherwise high-purity diamond during the CVD growth process is a recently demonstrated alternative to the implantation techniques described above. This technique allows sheets of NV centers to be embedded near the surface of the diamond sample without introducing unnecessary damage or impurities into the crystal lattice. As has been demonstrated,47–49 NV centers within 100nm of the diamond surface with narrow optical linewidths (140MHz–1.2GHz) and long coherence times (T2 >600μs) have been created in this way. It has been demonstrated50 that the orientation of these centers within the crystal lattice can also be controlled. These NV centers can be coupled to microwave circuits51 or nanophotonic structures patterned on the diamond surface.52, 53 1.6 QIP with NV centers: diamond photonic networks As discussed above, the high precision with which the quantum state of NV centers can be controlled makes them attractive candidates for performing QIP.
- 27. Principles of QIP using diamond 11 Their relatively efficient coupling to photons further strengthens their suitability for these applications. Diamond-based QIP has become a major experimental and theoretical research activity, with groups around the world attempting to harness the desirable properties of NV centers to implement the resources needed for QIP. These efforts to utilize NV centers for QIP can be roughly divided into three categories: • QIP based on NV center single-photon sources; • QIP based on entanglement and coherent interactions between neighboring NV centers; • QIP based on long-range entanglement between NV centers within a quantum optical network. Below, we review progress towards realizing the latter approach. Optical-network-based QIP involves creating entanglement between multiple quantum systems, or nodes, without a direct interaction between the individual nodes. This scheme, illustrated in cartoon form in Fig. 1.2, relies on being able to coherently map the quantum state of a stationary ‘storage qubit’ onto a physical system that can be transmitted, called a ‘flying qubit’, and reversibly mapped back onto a new storage qubit. Photons are excellent flying qubits, thanks to their noninteracting nature and to the existence of nanoscale, microscale, fiber and free-space optical technology optimized for routing and manipulating light. Atomic and solid-state quantum emitters are natural systems for implementing stationary qubits, owing to their optical transitions, which, through judicious experimental design, can be used to manipulate and transmit information describing their internal electronic or nuclear spin quantum state. Early work on developing the theory underlying this QIP approach was jump- started by a proposal in 1997 from Cirac et al.54 for distributing entanglement 1.2 Cartoon of a photonic-network QIP device in which stationary storage qubits are coherently connected through on-chip photonic channels.
- 28. 12 Quantum Information Processing with Diamond between two nodes of a quantum network. In this proposal, illustrated in Fig. 1.3, two ground states of a three-level atom are assumed to function as a stationary storage qubit. Each of the ground states, labeled |↑› and |↓›, is connected to a shared excited state through optical transitions, creating a so-called Λ system. One of these transitions is coupled to a classical ‘control field’, while the other is coupled to a mode of an optical cavity. As discussed in more detail below, the cavity is an important part of this proposal, as it serves the dual purpose of enhancing the vacuum coupling rate between the excited state and the second ground state (|↓›) and of allowing the light emitted by the transition to be efficiently collected. When the classical field is used to drive the transition between the first ground state (|↑›) and the excited state, the cavity-coupled optical transition quickly causes the excitation to coherently emit a ‘cavity photon’and transition to the second ground state (|↓›). This is effectively a Raman, or Λ, coupling between |↑› and |↓›, and we refer to this type of system as an ‘optically coupled spin’below. For a suitably designed system, when the classical drive is applied, the spin coherence between the two ground states is transferred onto the state of the photon emitted into the cavity mode. Assuming the initial spin state is a superposition α|↑›+β|↓› of the ground states and there are initially no cavity photons, the state of the system before and after the classical drive can be expressed as [1.1] where |0› and |1› refer to cavity photon populations equal to 0 and 1, respectively. This cavity photon state can be outcoupled, transmitted through a low-loss optical channel and reversibly reabsorbed by an identical atomic spin system in a cavity at a spatially distant location. In this way, quantum states can be transferred between distant qubits, and a quantum network can be created in which nodes are coupled via an optical quantum bus.55 In a related proposal in 1999, Cabrillo et al.56 introduced a scheme for entangling two optically coupled spins by building an experiment in which the detection of a photon created after simultaneous excitation of two spins cannot be uniquely correlated with one of the two spins. This proposal has evolved into a canonical 1.3 Illustration of transfer of a quantum state between two distant qubits via a photonic channel.
- 29. Principles of QIP using diamond 13 implementation in which the emission from each system is interfered in a beam splitter whose outputs are monitored by photon counters. This concept is referred to as measurement-based entanglement, and was extended in 2001 by Duan, Lukin, Cirac and Zoller,57 who described how measurement-based entanglement can be used to implement a set of gates sufficient to build a quantum computer – this is often referred to as the DLCZ scheme for quantum computing, and is illustrated in Fig. 1.4. The measurement-based entanglement central to these proposals relies on simultaneously exciting two optically coupled spins using weak classical control fields, such that each spin has a low probability of making the Λ transition and generating a cavity photon. If the output from each cavity is interfered in a 50:50 beam splitter or waveguide coupler, and if the cavity photons generated by each system are indistinguishable, then upon detection of a photon at one of the output ports of the interference device, it is impossible to determine from which spin it originated. Under these conditions, it can be shown that the two spin systems, labeled A and B, are entangled. In the simplest ideal implementation considered here, the system has initial and final states [1.2] where φ is a phase shift associated with differences in the optical path lengths of the arms of the interference device, and the choice of+or−in ‘±’is determined by which detector measures a photon. If a sufficiently large number of spins can be entangled in this way, a rich class of applications may become possible. Of particular interest are quantum computing and simulation schemes involving cluster states, which can introduce the fault tolerance necessary for building realistic QIP hardware.58, 59 At the time of the initial proposals described above, trapped ions or neutral atoms were assumed to be the medium of choice for realizing the cavity–spin system. Indeed, in recent years, formidable progress has been made in trapping 1.4 Illustration of scheme for measurement-based entanglement between two spin qubits interacting indirectly through photon interference.
- 30. 14 Quantum Information Processing with Diamond atoms within cavities, and in creating measurement-based entanglement between atomic and optical systems.60–70 However, a challenge for performing practical QIP with a large number of atomic systems is scaling the technical overhead related to trapping atoms within cavities. For this reason, researchers have sought out solid-state implementations of atom–cavity systems, which use ‘artificial atoms’ such as quantum dots and NV centers to eliminate the complexities of optical trapping, at the cost of introducing new challenges related to the influence of a solid-state environment on the quantum state of an artificial atom. These systems also provide the dual challenge and opportunity associated with creating nanoscale solid-state cavities and photonic hardware, within which artificial atoms can be embedded and connected in a manner analogous to the sketch in Fig. 1.2. In order to successfully realize this vision, several technical requirements related to the NV centers and the cavity properties must be met, and in some cases new photonic technology must be developed. 1.6.1 Requirements The key element of the measurement-based entanglement protocols introduced above is a stationary qubit whose state can be coherently transferred to a photon. Here, we assume that the storage qubit is an electron spin associated with the ground state of an NV center.As discussed above, heralded entanglement between two spins can be generated by weakly exciting each spin such there is a small probability of emitting a photon on which the spin state is encoded, and then interfering the emitted fields from each spin on the input ports of a beam splitter and measuring the output ports of the beam splitter. Several conditions must be satisfied for entanglement to be generated upon detection of a photon at the beam splitter output. First, it must be possible to coherently transfer the spin state onto the degrees of freedom of an outgoing photon. In the simplest scheme, illustrated in Figs. 1.3 and 1.5, this is achieved using a Λ transition between two spin states. In the example illustrated in Fig. 1.3, if the spin is in the up or the down state, a single photon or a vacuum state, respectively, is emitted into the cavity. Polarization or other optical degrees of freedom may also be used in place of the photon for an appropriately designed system. Realizing Λ systems in atomic media is relatively straightforward owing to their well-defined optical transitions and selection rules. Solid-state systems such as NV centers are more challenging to work with in this regard, and not all spin systems can be optically coupled via a Λ transition. A second requirement is that the photons emitted from each artificial atom be indistinguishable.71 This ensures that the beam splitter erases ‘which path’ information, so that upon detection of a photon it is not possible to determine from which spin it was emitted. In practice, this requires that the optical transitions for each spin emit photons of the same wavelength, polarization and temporal wave function. As discussed below, finding two identical artificial atoms which emit
- 31. Principles of QIP using diamond 15 photons at the same wavelength can be challenging. Generating photons with indistinguishable temporal wave functions can be achieved by carefully designing the classical driving field used to excite the Λ system. The key parameters for this process are illustrated in Fig. 1.5. If the classical drive is turned on and off slowly compared with the optical transition rates g and Ω, and if the vacuum coupling rate between the cavity and the optical transition of the spin is sufficiently high compared with the spontaneous emission rate γ of the atom and the decay rate κ of the cavity photon, the temporal wave function of the outgoing flying qubit will follow the classical drive wave function. This process is often referred to as stimulated Raman adiabatic passage (STIRAP), and is sensitive to the properties of the cavity, such as the mode volume V and the quality factor Q, which determine the coupling and decay rates of the cavity photon. For the proposed schemes to work efficiently, it is necessary for g2 /κγ>1, corresponding to the coupled cavity– atom system operating in a Purcell enhanced regime, where emission from the atom into the cavity dominates over spontaneous emission processes.37 Finally, it is desirable to collect, transmit and detect the emitted photons with sufficientlyhighefficiencyfortheschemetobecapableofgeneratingentanglement at a rate faster than the decoherence rate of the spins. Among solid-state quantum emitters, diamond NV centers are a promising system for realizing this entanglement scheme. Their electron spin ground states have long coherence times, and can be coherently coupled through well-defined optical transitions, which form a Λ system under suitable strain conditions.39, 72 In principle, all NV centers have identical electronic and optical properties, although in practice these properties are affected by the local environment of the NV center. 1.5 Cavity-enhanced photon generation from a spin qubit with a Λ optical transition. g is the single-photon coupling rate between the cavity mode and the optical transition. κ and γ are the decay rate of the cavity photon and the spontaneous emission rate of the NV excited state, respectively. Q and V are the quality factor of the cavity and the mode volume, respectively. c(t) is the temporal wave function of the photon amplitude, and Ω(t) is the temporal wave function of the classical drive.
- 32. 16 Quantum Information Processing with Diamond For example, local variations in strain and in the charge environment can significantly shift the energies of the excited states of the centers and of the photons emitted during the spin–photon state transfer process. This can destroy the indistinguishability of the photons, preventing the generation of measurement- based entanglement. Another challenge posed by NV centers is that the majority of their emission is into phonon sidebands whose wavelengths span a range greater than 100nm; only 3.5% of NV emission at low temperature is into the spectrally narrow ZPL. Optical micro- and nano-cavities can address this limitation by resonantly enhancing the ZPL via the Purcell effect. In principle, cavities with a sufficiently high optical Q and small optical mode volume V may enhance the rate of spontaneous emission into the ZPL by several orders of magnitude, such that it becomes the dominant emission wavelength of the NV center.72 This has the additional benefit of coupling the NV emission into a well-defined cavity mode, from which it can be efficiently collected using an on-chip or fiber-based waveguide.74 From a practical point of view, enhanced collection efficiency of the NV emission is necessary for so-called ‘single-shot’ readout of the NV spin state, in which the NV photoluminescence is sufficiently bright that the spin state can be measured in a time short compared with its lifetime. Creating a quantum network connecting multiple NV centers on-chip is also challenging. A major difficulty is in creating multiple identical cavities, each of which is coupled to an NV center and part of a large on-chip photonic network. This difficulty is in part related to the relative infancy of diamond-based photonics research, and in part a more general challenge associated with creating complex photonic devices and circuits. As discussed below, progress in diamond photonics has been promising in recent years, and researchers’ abilities to create functional photonic circuits are constantly advancing with efforts to develop on-chip optical interconnects for classical computing applications. 1.6.2 Experimental progress and challenges In recent years, major experimental milestones in the realization of optical- network- and measurement-based QIP with NV centers have been demonstrated. In addition to the experiments described in Section 1.2 demonstrating the spin coherence and manipulation properties of NV centers, researchers have made significant progress with all-optical coherent manipulation and readout of NV center spins. Santori et al.39, 72 used optical fields to create coherent superpositions of NV spin states, an initial step towards coherently transferring the spin state to an outgoing photon. Faraday rotation of an optical field by a single NV center was demonstrated by Buckley et al.75 In 2010,Togan et al.76 demonstrated entanglement between a single NV center spin and an emitted photon. This experiment required a detailed understanding of the properties of the selection rules describing the
- 33. Principles of QIP using diamond 17 optical transitions of NV centers, careful management of the stress environment of the NV center in order to implement a Λ transition between two NV spin states, and precise timing to separate excitation photons from emitted photons. In 2011, a further breakthrough was made in Delft by Robledo et al.,77 who demonstrated single-shot readout of an NV spin. This experiment utilized a solid immersion lens, which was fabricated on a diamond chip using focused ion beam milling, around an NV center of interest. This immersion lens was essential for boosting the collection efficiency of photons from the NV center, allowing the electron spin state to be read out more quickly than in previous experiments. Unlike optical cavities, a solid immersion lens enhances the collection efficiency for the ZPL as well as for the phonon sidebands. This work was followed by a demonstration of photon indistinguishability using emission from two unique NV centers on the same chip,78 employing a DC electric field provided by on-chip electrodes to Stark-shift the NV ZPL emissions to the same wavelength. Most recently, measurement-based entanglement between two NV centers separated by a macroscopic distance (3m)79 has been demonstrated. This is the first experiment demonstrating measurement-based entanglement of two solid-state systems, and major step towards implementing QIP using NV centers. In order to scale these experiments sufficiently to enable QIP, it is necessary that hardware enabling efficient routing of photons between NV centers, beam splitters and detectors be developed. Several groups have recently succeeded in integrating NV centers into nanophotonic devices. Researchers at HP Laboratories have used an optical cavity to enhance the NV ZPL emission,52, 74, 80 while researchers at Harvard have developed on-chip circuitry for collecting and manipulating light coupled into the cavity field.81 A limiting factor in the use of these devices for QIP is the relatively poor quality of the optical transitions of the NV centers embedded in the nanostructures used for these tasks. Compared with the NV centers found in bulk, unpatterned material, these centers suffer from larger nonradiative dephasing and spectral diffusion. Overcoming these limitations is critical for these devices to play a key role in future measurement-based QIP on chip using NV centers. 1.7 Conclusion During the last decade, NV centers in diamond have enabled researchers to conduct quantum optics and spin manipulation experiments with a precision previously restricted to atomic systems. At the single-NV-center level, some important initial solid-state demonstrations of a host of quantum information storage and manipulation experiments suggest that NV centers have a promising future for implementing QIP protocols. The development of the integrated photonic technology necessary to efficiently use NV centers as a QIP resource is under way, and future work will likely tackle the challenge of coherently connecting a growing number of NV centers, with the ultimate goal of creating on-chip quantum networks.
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- 37. 21 © 2014 Elsevier Ltd 2 Principles of quantum cryptography/ quantum key distribution (QKD) using attenuated light pulses H. WEINFURTER, LMU Munich, Germany DOI: 10.1533/9780857096685.1.21 Abstract: Based on the very foundations of quantum physics, quantum information methods can enhance conventional communication methods and enable completely new types of information processing. Quantum cryptography or, more precisely, quantum key distribution is the first application that enables communications to be made secure against eavesdropping attacks. This chapter gives an overview of the principle of the method and of how to implement this method based on attenuated laser pulses. Key words: quantum information, quantum communication, quantum cryptography, secure communication, free space. 2.1 Introduction Communication is ubiquitous in our daily life, and the need for protection of data is paramount. Cryptography provides numerous methods to ensure that communications are secure against eavesdropping attacks; however, all of them rely on certain assumptions about the capabilities of the eavesdropper. For example, the security of public-key cryptosystems and of the frequently employed advanced data encryption standards AES for point-to-point communication is based on the assumption that the mere computational complexity of deciphering the encrypted message prevents an attack. But how can we make sure that the eavesdropper does not have algorithms or machines which are way beyond what is the publicly available state of the art? After all, we know that a quantum computer can easily break into current public-key systems – we just do not have a working machine, yet. The one-time pad is the unique exception to all these issues. There, the sender encodes every bit of the message with a new bit from a perfectly random key. If the message and key are of equal length, the randomness of the key results in perfect randomness of the cipher, the bit sequence which is sent to the receiver. If the cipher is fully random, it could in fact mean any message with equal probability. An eavesdropper knowing this cipher still cannot do any better than guessing the initial message – in fact, for this, the eavesdropper would not even have needed to tamper with the communication. This sounds great, but the sender and receiver
- 38. 22 Quantum Information Processing with Diamond now have to first share the secret key. As mentioned above, it is of equal length to the message, and thus the one-time pad is often referred to as ‘the perfect method to communicate securely, provided one can communicate securely’. The conventional method used to distribute keys securely for most of the last century has been the ‘trusted courier’, carrying a long random key from one location to the other. Here again assumptions have to be made, for example that we really can trust the courier. However, there is a little twist to the story: the key can be any random bit sequence, not a specific message anymore. This is where quantum cryptography or, better, quantum key distribution (QKD) enters the stage (Bennett and Brassard, 1984; for a review, see Gisin et al., 2002). Based on the laws of physics, it guarantees the security of key distribution. Even better, it gives an upper bound on the information an eavesdropper can have about the key. For the first time in the field of cryptography we can quantify and specify the security of the communication, a feat completely impossible in conventional cryptography. The technology required for employing quantum cryptography is largely identical to that for standard optical communication methods, and thus the first commercial providers are already offering turnkey- ready systems. Here, we give an overview of the underlying principle and introduce the state of the art of QKD based on attenuated light pulses. 2.2 Principles of quantum key distribution (QKD): the BB84 protocol As mentioned above, the one-time pad requires a perfectly random bit sequence as a key, which is known only to the legitimate users. QKD achieves this by employing basic principles of quantum physics, especially the Heisenberg uncertainty relationship. The quantum properties of light are used to encode the key for transmission, as any unauthorized measurement results in irreversible changes in the quantum system, thereby revealing the attack. QKD was introduced in 1984 by Bennett and Brassard based on some initial ideas of Wiesner (1983). In Bennett and Brassard’s scheme, single quantum systems are prepared in well-defined, randomly chosen states by the sender (Alice) and sent over a quantum channel to the receiver (Bob), who observes a particular property of the quantum system. Given some additional negotiations over a public communication channel, they can distill a key whose secrecy can be determined. Several different properties of light can be utilized for encoding; let us use the general notation of a qubit, i.e., a two-state quantum system. In analogy to its classical counterpart, which can have the value ‘0’ or ‘1’, a qubit can be in the two uniquely distinguishable states |0〉 or |1〉. In a way that is unimaginable for classical systems, Alice and Bob can now also prepare and observe so-called superposition states, in particular the (distinguishable) states and In an experiment, these states might correspond, for example, to horizontal and vertical linear polarizations of photons, single quanta
- 39. Principles of quantum cryptography/QKD 23 of light, or ±45° linear polarizations respectively (Fig. 2.1). Alice and Bob agree to associate the states |0〉 and |0 ̄ 〉 with the bit value ‘0’and the states |1〉 and |1 ̄ 〉 with the bit value ‘1’. For key distribution, Alice prepares a stream of photons, each randomly prepared in one of the four states, and sends them over to Bob, who now performs measurements randomly for every photon either for the set (basis) B0 ={|0〉, |1〉} or for the set B1 ={|0〉, |1〉}. Note that the operators associated in quantum mechanics with these two bases do not commute, and Heisenberg’s uncertainty relation comes into effect. Observing the state |0〉, say, in the basis B0 will give the unique result ‘0’. However, when the state |0〉 obtains in the basis B1 , the result is maximally unsharp and random, and Bob observes ‘0’ and ‘1’ with equal probability. Moreover, quantum mechanics tells us that for two such operators which cannot be measured simultaneously in theory, there simply does not exist an apparatus in the real world which could do the job. For the polarization encoding mentioned above, this means that Bob can orient his analyzer either to measure along the horizontal and vertical directions or to measure along the ±45° directions. He cannot do both at the same time. After the measurements, communication over the public channel starts. For each instance, Bob informs Alice (a) whether he observed the photon, i.e., that the photon was not lost during transmission or measurement, and (b) which of the two bases he used. Alice replies to say whether she used the same basis. She and Bob 2.1 Principle of BB84 quantum key distribution. (a) A single photon prepared by Alice randomly in one of four polarization directions is sent over the quantum channel to Bob. Bob analyzes the photon either along the vertical and horizontal polarizations or along the ±45° linear polarizations. Provided he detects the photon, he sends information about the measurement basis to Alice via the public channel. (b) If an eavesdropper (Eve) intercepts the line, there are instances where Eve will choose the other basis relative to Alice, and thus Bob can detect the photon in a polarization orthogonal to the one sent by Alice. This results in errors in the key bits, revealing the attack.
- 40. 24 Quantum Information Processing with Diamond erase all instances where they used different bases, as in those cases Bob’s result is not at all correlated with Alice’s preparation. The remaining results make up the so-called ‘sifted key’. Ideally, thanks to the perfect correlations that occur when the same basis is used, these strings of preparations by Alice and of measurement results obtained by Bob are identical and thus can be used as a key for communication. But is this key secure? What happens if an eavesdropper (Eve) tries to interfere? For example, she could cut the quantum channel and perform the same measurements as Bob does. Eve then uses these results to encode a new set of photons and now sends them to Bob. If Alice and Bob were to use only a single basis, i.e., only two perfectly distinguishable states, Eve could gather all information about the key. Alice and Bob would not see any difference whether Eve attacked or not. But in that case, they did not actually use quantum physics at all. They only encoded classical states in a quantum system. However, as Alice and Bob use the four states from mutually conjugate bases randomly, they force the eavesdropper to behave similarly. There is then a probability of 1:2 that if Alice and Bob use the same basis, Eve will measure (and reprepare) in the other basis. In this case Eve’s state will be uncorrelated with Alice’s state and, finally, Bob’s result will be uncorrelated with both Eve’s and Alice’s preparation. Thus, with a probability of 25%, Bob’s bit in the sifted key will differ from Alice’s bit – the eavesdropper has introduced a significant amount of noise. There are better strategies, but even the best one, acting on all sent photons simultaneously, will cause about 11% of errors (Gisin et al., 2002). This noise is not just a technical issue, like the ‘click’that eavesdroppers caused some tens of years ago in (ancient) telephone links; it is of fundamental character and directly linked to the uncertainty principle. According to the laws of quantum physics, eavesdropping will unavoidably cause noise in the sifted key. Alice and Bob can thus evaluate the noise in the sifted key by determining the quantum bit error ratio (QBER) given by the fraction of wrong bits in the sifted key. From this, they can deduce the amount of information the eavesdropper has. Thisinformationcannowbeshrunktoanegligiblelevelby‘privacyamplification’, essentially by shrinking the key according to the QBER. Thereby, finally, Alice and Bob obtain a perfectly random bit sequence whose security is quantifiable thanks to the laws of quantum physics. 2.3 Protocol extensions and alterations This original protocol was later supplemented by various extensions and alternative strategies. For example, Bennett proposed the B92 protocol, which used only two nonorthogonal states for encoding, together with a check that forced the eavesdropper to really resend some photons even in cases where the results were not unique (Bennett, 1992). Six-state protocols for qubits and protocols for higher-dimensional quantum systems have been designed to increase
- 41. Principles of quantum cryptography/QKD 25 the complexity of the protocol and thereby increase the noise that eavesdropping attacks cause (Bruss, 1998). Higher-dimensional systems also enable one to simultaneously confirm the security of the transmission and obtain a bit from every sent photon (Beige et al., 2002). Similarly, by unbalancing the frequency of use of B0 and B1 , the sifted key can be made almost as long as the number of detected photons (Lo et al., 2005a). The initial protocol, as in the description above, relies on the indivisibility of single quanta. This is evidently fulfilled when single photons are used for the encoding. Attenuated pulses of light may contain, with a certain probability, more than only a single photon. This gives an eavesdropper a chance to split the pulse and use the additional photon for an attack. Alice can then use only very feeble pulses, whose attenuation also depends on the losses along the quantum channel (Lütkenhaus, 2000). This would shorten the maximum link length dramatically. However, a simple twist in the original protocol, namely encoding ‘0’ with states from B0 and ‘1’ with states from B1 , renders the eavesdropper unable to obtain information from two-photon pulses; only very rare pulses with an even higher number of photons can then be used by her (Scarani et al., 2004). The so-called decoy protocols completely remove the photon number problem (Hwang, 2003; Lo et al., 2005b). Essentially, Alice and Bob use the basic principle of QKD, i.e., detection of eavesdropping is enabled by using nonorthogonal states, to also shield the photon number degree of freedom against attacks. Again, any information which might have leaked to an adversary can be quantified and removed by privacy amplification. Based on this protocol, the secret-key rate for attenuated-pulse QKD exhibits the same scaling as the ideal case of sending single photons. The probability of obtaining a key bit from a sent pulse (for BB84) is lower by a factor of about two or three and suffers mainly in extreme cases where there is very low transmission or there is detector saturation. Different quantum properties of light are used in two further groups of protocols. On the one hand, entanglement between a pair of photons distributed to Alice and Bob results in correlated measurement results between the two distant observers. Thus, without preparation, but with only measurements in a symmetric configuration, a key can be generated (Ekert, 1991). Its security can be tested by evaluating the entanglement, again with measurements in nonorthogonal directions. On the other hand, the quantum nature of the electric-field states can be utilized to encode and protect the key (Grosshans and Grangier, 2002). Electric- field variables again cannot be measured simultaneously with arbitrary precision, and thus offer similar opportunities for QKD to qubits. In conclusion, for key distribution QKD utilizes the perfect correlation between the preparation and the measurement result when the same basis is used, and it utilizes the randomness of the results when the measurement is done in a conjugate basis to secure the distribution against eavesdroppers.
- 42. 26 Quantum Information Processing with Diamond 2.4 Implementing QKD After learning about the first quantum communication protocol, the BB84 protocol, whichenablessecurekeyexchange,onemightthinkthatitwouldbestraightforward to set up an experiment. Yet it took quite a few years, and in particular also the initiative of the inventors Bennett and Brassard, who, together with Besette, Savail and Smolin, started experiments on QKD. The first secure quantum key between Alice and Bob was established back in 1991 in the laboratories of the IBM Research Center in Yorktown Heights (Bennett et al., 1992). In this setup, called ‘Aunt Martha’, attenuated light pulses were transmitted over 32cm between the sender and the receiver unit (Fig. 2.2). Based on the BB84 protocol, Bennett et al. demonstrated how Alice and Bob could indeed verify whether an eavesdropper had disturbed the transmission or whether it was possible to extract a secure key. The first experiment used a light-emitting diode as the light source and fast Pockels cells to choose the polarization direction. A key rate of a few hundred bits per second was achieved, and a number of eavesdropping attacks were simulated. It was even demonstrated how to correct residual bit errors and how to guarantee full security in the presence of (experimental) noise. This shining example became the model for numerous quantum cryptography systems developed worldwide. In the following sections, an overview of the current status of developments is given, which have led to the first commercial systems. The most important criteria for QKD systems are a high key rate and a long distance. Usually one cannot optimize both at the same time, and some compromises have to be made. No compromise, however, is acceptable when it comes to reliability and user-friendliness. To make QKD a real application, it is thus necessary to develop new optics, quite different from the standard quantum optics setups. These latter setups allow high flexibility, but they are quite expensive 2.2 Setup for the first quantum cryptography demonstration (Bennett et al., 1992b) (© C. Bennett).
- 43. Principles of quantum cryptography/QKD 27 and, owing to the many alignment options, are usually not stable enough for continuous operation. The distance between Alice and Bob is limited mainly by losses in the quantum channel and by the efficiency and noise of single-photon detectors. Losses or low efficiency reduce the number of detected photons and thus the number of bits in the raw key. Noise (dark counts) in Bob’s detectors results in a noise floor of bit errors, which are indistinguishable from those caused by eavesdropping attacks. It can be corrected for, but only at the cost of raw key material. In the case of low efficiency or high loss, this noise floor can easily reach the 11% level, at which no secure key can be distilled anymore. Any attempts to amplify the single-photon signal have to fail as well, since, according to the no-cloning theorem, any amplifier or repeater as used in a conventional optical communication scheme introduces the same noise as an eavesdropper would do. This would therefore ruin the remarkable advantages of quantum key distribution. Only a quantum repeater with intermediate quantum error correction and memory stages along the quantum channel could enable truly long-distance communication. Its basic components are being developed now. As it will take some time before we are able to use it, we have to rely on conventional means to transmit light. Two options for quantum channels are available, which determine the wavelength and consequently distinguish the complete system. Photons can be distributed either using glass fiber connecting Alice and Bob or using telescopes aligned mutually for optimal coupling. In the following, the two systems are compared, in the form in which they have been implemented in prototypes or commercial systems. Most of the systems under development rely on attenuated light pulses, as this is less expensive and enables high-rate systems. Before going into detail, we shall say a word about hacking QKD systems. In several demonstrations, it has been shown that key generation can be eavesdropped without being detected by Alice and Bob (Zhao et al., 2008; Gerhardt et al., 2011; Weier et al., 2011). This was possible because the systems concerned had one or other hardware feature which enabled the attack. Typically, particular features of the detector, for example timing issues or dead time, allowed manipulation of the quantum channel that was not revealed by the standard QBER analysis. It is evident that a QKD system requires very careful design to avoid any possible side channels and to stop any intrusion of an eavesdropper into the sender and receiver modules along the quantum channel. Moreover, it is crucial to continuously control the functioning of the system components, above all the detector. However, all quantum hacking can only utilize hardware features of the system; it is never the principle of QKD which can be threatened. 2.5 Fiber-based QKD Glass fiber systems make the best use of standard telecommunication fibers. These fibers are already available between the main communication centers or
- 44. 28 Quantum Information Processing with Diamond could be installed with reasonable effort. The standard telecommunication wavelengths are 1300 and 1550nm, where dispersion and loss, respectively, reach a minimum. State preparation, manipulation and analysis can be achieved with standard telecommunication components. So far, the disadvantage of this wavelength regime is the high noise and relatively low efficiency of current single-photon detectors (germanium and InGaAs avalanche diodes). Optimization of these detectors has enabled us to steadily increase the distance over the last few years up to a limit of about 200km. The superconducting detectors under development at the moment are effectively noise-free owing to their very high time resolution and are achieving increasingly higher efficiencies. Distances of 500km seem feasible with these new developments. Glass fiber is slightly birefringent. Over long distances, this effect adds up. Care has to be taken because this birefringence may vary, depending on the stress in or the temperature of the fiber. As a result, a well-defined initial polarization fluctuates strongly at the receiver and one has to compensate for the birefringence based on reference measurements. However, it is more advisable to define a new encoding for the qubit. The two-state system in this case is defined by two possible times at which the photon can be detected (‘time-bin coding’). A (variable) beam splitter determines the relative size of the amplitudes, and a phase shifter in one of the arms behind the beam splitter enables one to set any desired state. The two arms are recombined at a second beam splitter. If the lengths of the two arms differ by more than the coherence time of the light, no interference occurs at the second beam splitter, and the light exits in two time slots from this unbalanced interferometer. Only one output is chosen, but this does not matter at all for attenuated light pulses as this reduction still happens within Alice’s sender module. At Bob, an equivalent unbalanced interferometer is used to split and recombine the incoming amplitudes again, and, after application of Bob’s phase, it allows us to observe interference depending on Alice and Bob’s phases (with 50% efficiency).Accepting this reduction, one is thus able to observe interference over very large distances, almost independent of possible fluctuations along the quantum channel. A very reliable and stable system was developed at the University of Geneva. The group led by Nicolas Gisin and Hugo Zbinden found a clever extension of the basic principle which significantly increased the stability and quality of the system (Ribordy et al., 2000). In addition to using time-bin coding to reduce the influence of the fiber, they made the receiver, Bob, the source of the light pulses. He first generates bright, coherent pulses at two different times with a polarizing, unbalanced interferometer and sends them to Alice. She can now use the bright pulses to easily synchronize her actions, consisting of the application of one of four possible phase shifts, back reflection at a Faraday mirror and attenuation to the single-photon level. On the way back to Bob, all rotations of the light are undone, and only then does Bob apply his phase shift. Under the assumption that
- 45. Principles of quantum cryptography/QKD 29 all fluctuations occur on a much slower time scale than the time it takes the light to travel from Bob to Alice and back again, all disturbances cancel. Only the phase difference between Alice and Bob’s modulations remains and determines the result of the measurement. By using a polarizing interferometer together with a Faraday mirror (which rotates the polarization of the reflected light by 90°), this system does not suffer from the usual 50% reduction of time- bin-coding systems. From the measurement results, Alice and Bob can infer the mutual phase settings and obtain the key bits, which are now more or less immune to any disturbance. With such a so-called ‘plug & play’ system, QKD was demonstrated between the cities of Geneva and Lausanne over a distance of 67km at a rate of about 150bit/s. Even more remarkably, the glass fiber connecting Alice and Bob was a standard fiber used by Swisscom. The sender and receiver modules were integrated into 19 inch racks and placed in buildings belonging to Swisscom, which were far from being the air-conditioned laboratories of standard quantum optics experiments. This very reliable system was the basis for the development of the first commercial QKD system by the spin-off company ID Quantique. Today, Vectis offers secure point-to-point connection, easily integrable into standard communication networks. Similarly, SeQureNet is offering a newly developed system based on continuous-variable QKD. The development of dedicated electronics for operating single-photon detectors at very high rates enabled the team of Andrew Shields at Toshiba, UK, to develop the currently most capable system (Dixon et al., 2010). By implementing the concept of the decoy-based BB84 protocol, key rates of more than 1Mbit/s were achieved over a distance of 50km. It will be exciting to see what improvements beyond the current state of the art can be achieved, possibly with superconducting detectors or novel encoding schemes. 2.6 Free-space QKD If a direct line of sight is available, coupling the sender and receiver with telescopes becomes possible. High transmission through air can be achieved for wavelengths in the range from 780 to 850nm. Highly efficient, low-noise silicon avalanche photodiodes are available for this range. Free-space links are very flexible and can be used in a number of different possible scenarios, for example over short-range links of a few meters (e.g., to ATMs for authentication and key upload) or over medium-distance links of several kilometers directly connecting buildings in urban areas. Over such distances, near-IR wavelengths guarantee higher key rates thanks to the higher detector efficiency. But free-space links also offer new possibilities at very long distances. Provided there are trusted satellites, global key exchange becomes possible (Nordholt et al., 2002)). From a height of about 500–1000km, a sender on a satellite tracks a ground station and sends polarized light pulses, which in turn are collected by a large telescope on earth to exchange
- 46. 30 Quantum Information Processing with Diamond a secret key. If the satellite then flies over another ground station, a second secret key can be exchanged. Combining the two keys in the satellite gives a secure key between the ground stations and enables worldwide communication within a trusted network. For free-space QKD, all of the required components, in particular the laser diodes, are low-price, standard products. Instead of costly polarization modulators, it is thus more economic to use four differently oriented laser diodes. By activating only one of the four laser diodes at any time, the required polarized attenuated light pulses can be generated. The main problem that free-space links suffer from is air turbulence, which reduces the effective aperture of the telescopes significantly. Thus, to collect the maximum number of attenuated pulses, large receiver telescopes are required. Figure 2.3 shows a schematic illustration of a pair of sender and receiver modules developed for free-space QKD. In the sender, eight laser diodes are mounted on a ring around a gold-plated cone such that the light reflected from the cone into the quantum channel is already oriented along four different directions at two different brightness levels for decoy state encoding. The design guarantees high stability under harsh conditions. The power of such modules and their suitability for future applications has been demonstrated in various experiments, starting with a distance of 23.4km in the Alps (Kurtsiefer et al., 2002) in quite tough atmospheric conditions, such as temperatures down to −20°C and strong winds. The ultimate distance for links between stations on earth is offered by a test range between the Canary Islands of Tenerife and La Palma (Fig. 2.3). There, over a distance of 144km, QKD could be evaluated under conditions perfectly representative of links to satellites (Schmitt- Manderbach et al., 2007). In this experiment, the sender was implemented as a simple telescope with an output lens 15cm in diameter. The receiver was at the optical ground station of the European SpaceAgency, where a 1m mirror telescope collected the strongly scattered light, followed by standard polarization analysis. In a static scenario, the polarization is a very reliable and stable degree of freedom for encoding quantum states. No reduction in the degree of polarization due to air scattering was observed. Owing to the long link, the effective aperture was reduced to about 3cm, reducing the overall link efficiency by more than 30dB. Yet, despite this, the results can be compared favorably with links to satellites in low earth orbits about 1000km in height but with significantly less disturbing atmosphere along the path. While the above long-distance QKD scheme could operate only during the night, the groups of Richard Hughes at Los Alamos (Hughes et al., 2002) and Christian Kurtsiefer at the University of Singapore (Peloso et al., 2009) have also demonstrated the feasibility of daylight key exchange. Narrow filtering in the frequency and spatial domains and precise selection of the detection time are necessary for this.
- 47. 2.3 Schematic illustration of free-space quantum key distribution over a distance of 144km between the Canary Islands of La Palma and Tenerife. The insets schematically show details of the sender and receiver modules. Based on GPS timing signals, the Alice control unit fires one of eight laser diodes with one out of four polarizations and three different light intensities to enable security analysis based on the decoy protocol. In the receiver, the light is detected behind one of four polarization analyzer outputs and time-tagged for further communication. In spite of more than 32dB loss, a key rate of 12.8bit/s was achieved.
- 48. 32 Quantum Information Processing with Diamond 2.7 Future trends So, where can we use quantum cryptography today? What is possible, and where should future research be directed? Evidently, only point-to-point key distribution is possible with the scheme presented here. An entanglement-based scheme, which would be somewhat more involved, might enable the construction of networks based on multipartite entanglement; however, this is surely too far- fetched. Nevertheless, it will be necessary to develop network structures in order to best support current types of communication. Designed as a point-to-point communication link, a bridge (network layer 2) connecting two communication partners would surely best utilize the power of quantum key distribution. Standard encryptors may, for example, use the Advanced Encryption Standard (AES) for such connections to provide communication rates of up to 10Gbps, which is of course way beyond the capability of QKD. However, such conventional, widespread systems also have to provide some cipher key management, which is often done once per session only, using RSA-type or Diffie–Hellman codes (which, in turn, could be attacked by a future quantum computer). Key exchange using QKD now enables one, for the first time, to securely provide new keys for AES at a high rate of more than 1000 times per second. This enhances the security of the system dramatically. At a lower rate, encryptors could be configured to use one-time-pad encryption over the bridge and thus provide ultimate security against eavesdroppers. The first publicly known regular use of QKD-enhanced communication has been during elections in Geneva, Switzerland, since 2007. Encryptor systems as described above, made by ID Quantique, secured the connection between the ballot data entry center and the government repository where the votes were stored. The additional benefit of such a system is not so much protection from outside attempts to eavesdrop as the ability to verify that the data have not been corrupted in transit between entry and storage. Since then, a number of links have been installed (see http://www.idquantique.com/news-and-events/press- releases.html), for example to secure low-latency links from bank headquarters to branches or disaster recovery centers, and between data centers of network providers. Besides these point-to-point applications, several network structures have been demonstrated. The basic elements here are ‘trusted nodes’, which are interconnected by QKD links and provide high-level key management and rerouting for all nodes. After the first three-node Darpa Quantum Network in Boston (Elliot, 2006), the SECOQC demonstration network in the Vienna area combined six nodes with eight links (Peev et al., 2009). The nodes consisted of a combination of node modules responsible for all classical communication and QKD modules which could be integrated and exchanged in a highly flexible manner, depending on the particular environment and requirements, such as long distance or high rate. The next development step was taken with the Tokyo
- 49. Principles of quantum cryptography/QKD 33 Network in 2010, where even one-time-pad-encoded video conferences have proven to be feasible over 50km links (Sasaki et al., 2011). As an additional result, a standardization effort was initiated to define the structures of future QKD links and network modules. QKD links to satellites have been proposed for secure global communication. Developments towards this challenging goal have brought impressive results, with free-space link distances increasing successively from 20 to 144km, and with the development of daylight key distribution, enabled by precise filtering in all degrees of freedom to enable the registration of signals at the single-photon level in bright sunshine. It has been shown that today’s optical communication links can be enhanced by quantum communication methods. Additional improvements are necessary, as the link efficiencies have to be better by a few orders of magnitude relative to conventional optical links. Finally, a very recent experiment has shown that it is feasible to link an aircraft and an optical ground station (Nauerth et al., 2013). The goal of metropolitan networks consisting of fiber or short free-space-based QKD links, connected via satellite links to provide worldwide secure key exchange, is coming within reach. 2.8 Conclusion Quantum cryptography has become the first commercial application of the principles of quantum information. A novel level of security has been made possible by using the very basic principles of quantum physics.Any eavesdropper, allowed to perform any possible attack, will be revealed. Even better, QKD allows one to quantify the maximum amount of information which might have leaked to the eavesdropper. The systems which are operative at the moment use either a fiber channel or a connection through free space to transmit feeble light signals. Attenuated laser pulses can be used, supported by additional protocol features, and allow the design of robust, cost-effective systems. High-speed point-to-point connections, very long connections over more than 100m and networks in metropolitan areas have been developed and will be connected by satellite links, enabling global secure communication in the future. 2.9 References Beige, A., Englert, B.-G., Kurtsiefer, C. and Weinfurter, H. (2002) Secure communication with a publicly known key, Acta Phys. Pol. A 101, 357–366. Bennett, C. H. (1992) Quantum cryptography using any two nonorthogonal states, Phys. Rev. Lett. 68, 3121–3124. Bennett, C. H. and Brassard, G. (1984) Quantum cryptography: public-key distribution and coin tossing, Proceedings of IEEE International Conference on Computers, Systems and Signal Processing, 175–179. Bennett, C. H., Bessette F., Brassard, G., Salvail, L. and Smolin, J. (1992) Experimental quantum cryptography, J. Cryptol. 5, 3–28.
- 50. 34 Quantum Information Processing with Diamond Bruss, D. (1998) Optimal eavesdropping in quantum cryptography with six states, Phys. Rev. Lett. 81, 3018–3021. Dixon, A.R., Yuan, Z.L., Dynes, J.F., Sharpe, A.W. and Shields, A.J. (2010) Continuous operation of high bit rate quantum key distribution, Appl. Phys. Lett. 96, 161102. Ekert, A. K. (1991) Quantum cryptography based on Bell’s theorem, Phys. Rev. Lett. 67, 661–663. Elliot, C. (2006) The Darpa quantum network, in Quantum Communications and Cryptography, ed. A. V. Sergienko, CRC Press, pp. 83–102. Gerhardt, I., Liu, Q., Lamas-Linares, A., Skaar, J., Kurtsiefer, C. and Makarov, V. (2011) Full-field implementation of a perfect eavesdropper on a quantum cryptography system, Nature Commun. 2, 349. Gisin, N., Ribordy, G., Tittel, W. and Zbinden, H. (2002) Quantum cryptography, Rev. Mod. Phys. 74, 145–195. Grosshans, F. and Grangier, P. (2002) Continuous variable quantum cryptography using coherent states, Phys. Rev. Lett. 88, 057902. Hughes, R. J., Nordholt, J. E., Derkacs, D. and Peterson, C. G. (2002) Practical free-space quantum key distribution over 10km in daylight and at night, New J. Phys. 4, 43. Hwang, W.-Y. (2003) Quantum key distribution with high loss: toward global secure communication, Phys. Rev. Lett. 91, 057901. Kurtsiefer, C., Zarda, P., Halder, M., Weinfurter, H., Gorman, P. M., et al. (2002) A step towards global key distribution, Nature 419, 450. Lo, H.-K., Chau, H. F. andArdehali, M. (2005a) Efficient quantum key distribution scheme and proof of its security, J. Cryptol. 18, 133–165. Lo, H.-K., Ma, X. and Chen, K. (2005b) Decoy state quantum key distribution, Phys. Rev. Lett. 94, 230504. Lütkenhaus, N. (2000) Security against individual attacks for realistic quantum key distribution, Phys. Rev. A 61, 052304. Nauerth, S., Moll, F., Rau, M., Fuchs, C., Horwath, J., et al. (2013) Air to ground quantum communication, Nature Photon. 7, 382–386. Nordholt, J., Hughes, R., Morgan, G., Peterson, C. and Wipf, C. (2002) Present and future free-spacequantumkeydistribution,Proc.SPIE4635,Free-SpaceLaserCommunication Technologies XIV, 116–126. Peev, M., Pacher, C., Alléaume, R., Barreiro, C., Bouda, J., et al. (2009) The SECOQC quantum key distribution network in Vienna, New J. Phys. 11, 075001. Peloso, M. P., Gerhardt, I., Ho, C., Lamas-Linares, A. and Kurtsiefer, C. (2009) Daylight operation of a free space, entanglement-based quantum key distribution system, New J. Phys. 11, 045007. Ribordy, G., Gautier, J.-D., Gisin, N., Guinnard, O. and Zbinden, H. (2000), Fast and user- friendly quantum key distribution, J. Mod. Opt. 47, 517–531. Sasaki, M., Fujiwara, M., Ishizuka, H., Klaus, W., Wakui, K., et al. (2011) Field test of quantum key distribution in the Tokyo QKD network, Opt. Express 19, 10387–10409. Scarani, V., Acín, A., Ribordy, G. and Gisin, N. (2004) Quantum cryptography protocols robust against photon number splitting attacks for weak laser pulses implementations, Phys. Rev. Lett. 92, 057901. Schmitt-Manderbach, T., Weier, H., Fürst, M., Ursin, R., Tiefenbacher, F., et al. (2007) Experimental demonstration of free-space decoy-state quantum key distribution over 144km, Phys. Rev. Lett. 98, 010504.
- 51. Principles of quantum cryptography/QKD 35 Weier, H., Krauss, H., Rau, M., Fürst, M., Nauerth, S. and Weinfurter, H. (2011) Quantum eavesdropping without interception: an attack exploiting the dead time of single-photon detectors, New J. Phys. 13, 073024. Wiesner, S. (1983) Conjugate coding, SIGACT News 15, 78–88. Zhao, Y., Fung, C.-H. F., Qi, B., Chen, C. and Lo, H.-K. (2008) Quantum hacking: Experimental demonstration of time-shift attack against practical quantum key distribution systems, Phys. Rev. A 78, 042333.
- 52. 36 © 2014 Elsevier Ltd 3 Ion implantation in diamond for quantum information processing (QIP): doping and damaging R. KALISH, Technion, Israel Institute of Technology, Israel DOI: 10.1533/9780857096685.1.36 Abstract: The creation of specific luminescent centers in semiconductors, in particular the controlled creation of nitrogen–vacancy (NV) centers in diamond, relies heavily on ion implantation. Furthermore, the formation of vacancies and the creation of various photonic devices for the manipulation and transportation of the photons emitted by these centers rely on the bond breakage that accompanies ion implantation. In this chapter, we review the physics related to the slowing down of ions in diamond, stressing its implications for implantation doping, the controlled introduction of vacancies and the creation of sacrificial graphitic regions in diamond. Key words: ion implantation in diamond, NV center in diamond, ion-induced graphitization of diamond, photonic crystals in diamond. 3.1 Introduction For many applications, solid materials need to be modified by the introduction of foreign atoms. This can be done during growth of the material, by postgrowth in- diffusion of the desired impurities or by ion implantation. Ion implantation, being a violent process in which atoms are shot into the target at energies which exceed the binding energies of the atoms in the target, is always accompanied by displacement of host atoms, i.e., damage to the implanted target material. This damage is undesirable in most doping applications, and it has to be avoided. This is usually achieved by the choice of specific implantation schemes or by postimplantation annealing. Ion implantation into semiconductors has many advantages in cases in which control of the concentration and depth profile of impurities is required. The dopant atoms are shot into the semiconductor under specific implantation conditions (i.e., with a specific implantation energy and implantation fluence), thus enabling accurate design of the dopant profile. Ion implantation, when done at high fluences, is also used to create composite materials by loading the target with the required foreign atoms at high concentrations. However, it also has drawbacks, mainly due to the damage inflicted on the implanted material by the ions during their slowing down in the solid. This damage can, however, be beneficial for some applications, as will be described below.
- 53. Ion implantation in diamond for QIP 37 The general topic of ion implantation for material modification, which is mainly done for the doping of semiconductors, is well studied and well documented. Ion implantation into diamond has also been extensively studied, in the search for ways to achieve electronic doping of diamond, into which foreign atoms cannot be readily introduced by diffusion. Furthermore, the unique dual bonding configuration of carbon, which is sp3 bonded in diamond and sp2 bonded in graphite, has triggered much research into the physics involved in the conversion of sp3 to sp2 bonding of carbon.1, 2 Renewed interest has recently awakened in the ion implantation of diamond, mainly driven by the discovery that specific luminescent centers in diamond, based on impurity–vacancy complexes, can serve as qubits. This new application of ion implantation to create luminescent centers imposes very strict requirements on the implantation process. For many applications it is desirable to have single, well-isolated emission centers, located at well-defined positions in the diamond. Hence an entire new field of ion implantation has emerged. It includes single-ion implantations at predetermined locations, and often at low ion energies. The recent developments in this technology will be discussed below, but with reference to only a limited number of key publications in the field. Furthermore, the photons emitted by the luminescent centers in the diamond need to be efficiently propagated and manipulated in photonic structures, preferably constructed in the same diamond in which the photoluminescent centers reside. Hence a new discipline of creating photonic crystals in diamond has emerged. Much of it also hinges on ion implantation of diamond.3 The damage inflicted on the implanted material can be beneficial in cases in which modification of the crystallinity or the bonding configuration of the target atoms is required. This applies to the case of diamond, amongst other materials, because it is highly insulating, extremely hard and chemically inert owing to the short, strong sp3 bonds between the carbon atoms of which it is composed. Diamond can be drastically modified by breaking sp3 bonds, turning it into electrically conducting, chemically etchable, sp2 -bonded graphite. This finds application, amongst other things, in the creation of a sacrificial graphitic layer in diamond that can be chemically removed, enabling the production of thin diamond membranes and other submicron-sized structures. Damage-related graphitization can also be used to create fine, well-defined, electrically conductive graphitic regions in diamond. Also, the fact that ions slowing down in matter displace atoms can be used to introduce point defects (vacancies) in a controlled manner. This is of importance, for example, when vacancy-containing complexes (i.e., NV centers) need to be created in diamond for quantum applications. All these specific propertied of ion-damaged diamond will be discussed in detail below. It should be noted that the requirements for efficient doping of diamond by ion implantation, i.e., the removal of implantation-related defects from the vicinity of the implanted impurities, and the requirements for the formation of
- 54. 38 Quantum Information Processing with Diamond impurity–vacancy complexes are orthogonal. In the first case, the vacancies formed by the implantation must be removed from the implanted ions, whereas in the second case, the vacancies must be manipulated so as to be located at specific positions in the diamond; for the formation of NV centers, their presence near the implant is essential. In the following, we briefly describe the general concepts of importance for an understanding of the results of ion implantation as far as the damage and the final implant location are concerned. We then focus on implantation-related damage in the particular case of ion implantation of diamond for doping purposes. We summarize the current situation regarding doping of diamond as achieved by ion implantation for p- and n-type doping. The controlled introduction of damage that accompanies ion implantation in diamond finds application, amongst other things, in the creation of particular impurity–defect-related luminescent centers, as well as in the production of nanosized structures in diamond. These and various implantation/annealing schemes designed for the efficient exploitation of the various ion-induced modifications of diamond will be reviewed below. 3.2 Doping diamond 3.2.1 Doping during high-pressure high-temperature (HPHT) growth The phase diagram of carbon shows that under conditions of normal temperature and pressure (NTP), the thermodynamically stable form of bonding in carbon is sp2 , i.e., graphitic. This is in contrast to diamond, which is composed of sp3 - bonded carbon atoms. The latter can be formed only at high pressures and high temperatures. Nevertheless, stable diamond exists in nature under NTP conditions owing to the high potential barrier between the sp2 and sp3 configurations, which in practice prevents a spontaneous transition from sp3 -bonded carbon atoms (diamond) to sp2 -bonded atoms (graphite). Diamond can thus be formed in nature only under high-pressure high-temperature (HPHT) conditions, such as exist deep in the earth, in very large HPHT laboratory devices and during explosions. During the above thermodynamic processes, impurities present in the growth environment can be introduced into the diamond. The most common impurities (‘dopant atoms’) that are introduced into diamond during these ‘natural’processes are nitrogen, boron and some transition metals. These impurities are, apart from boron, which is a p-type dopant with a reasonably shallow level, useless as electronic dopants at practical temperatures owing to the deep levels they form in the diamond band gap. The status of the doping of diamond by application of ‘thermodynamic’conditions has been thoroughly studied for several decades, and many reviews on this have been published. Hence this topic will not be dealt with further here.
- 55. Another Random Document on Scribd Without Any Related Topics
- 56. sinful earthly love which he hath spoken. Such love would burn your soul to all eternity with fire that never could be quenched. If you can tear away all roots and traces of this from your heart, if by fasting and prayer and penance you can become worthy to be a bride of your divine Lord, then your prayers will gain power, and you may prevail to secure his eternal salvation. But listen to me, daughter,— listen and tremble! If ever you should yield to his love and turn back from this heavenly marriage to follow him, you will accomplish his damnation and your own; to all eternity he will curse you, while the fire rages and consumes him,—he will curse the hour that he first saw you." These words were spoken with an intense vehemence which seemed almost supernatural. Agnes shivered and trembled; a vague feeling of guilt overwhelmed and disheartened her; she seemed to herself the most lost and abandoned of human beings. "My father, I shall think no penance too severe that may restore my soul from this sin. I have already made a vow to the blessed Mother that I will walk on foot to the Holy City, praying in every shrine and holy place; and I humbly ask your approval." This announcement brought to the mind of the monk a sense of relief and deliverance. He felt already, in the terrible storm of agitation which this confession had aroused within him, that nature was not dead, and that he was infinitely farther from the victory of passionless calm than he had supposed. He was still a man,—torn with human passions, with a love which he must never express, and a jealousy which burned and writhed at every word which he had wrung from its unconscious object. Conscience had begun to whisper in his ear that there would be no safety to him in continuing this spiritual dictatorship to one whose every word unmanned him,— that it was laying himself open to a ceaseless temptation, which in some blinded, dreary hour of evil might hurry him into acts of horrible sacrilege; and he was once more feeling that wild, stormy revolt of his inner nature that so distressed him before he left the convent.
- 57. This proposition of Agnes's struck him as a compromise. It would take her from him only for a season, she would go under his care and direction, and he would gradually recover his calmness and self- possession in her absence. Her pilgrimage to the holy places would be a most proper and fit preparation for the solemn marriage-rite which should forever sunder her from all human ties, and make her inaccessible to all solicitations of human love. Therefore, after an interval of silence, he answered,— "Daughter, your plan is approved. Such pilgrimages have ever been held meritorious works in the Church, and there is a special blessing upon them." "My father," said Agnes, "it has always been in my heart from my childhood to be the bride of the Lord; but my grandmother, who brought me up, and to whom I owe the obedience of a daughter, utterly forbids me; she will not hear a word of it. No longer ago than last Monday she told me I might as well put a knife into her heart as speak of this." "And you, daughter, do you put the feelings of any earthly friend before the love of your Lord and Creator who laid down His life for you? Hear what He saith: 'He that loveth father or mother more than me is not worthy of me.'" "But my poor old grandmother has no one but me in the world, and she has never slept a night without me; she is getting old, and she has worked for me all her good days;—it would be very hard for her to lose me." "Ah, false, deceitful heart! Has, then, thy Lord not labored for thee? Has He not borne thee through all the years of thy life? And wilt thou put the love of any mortal before His?" "Yes," replied Agnes, with a sort of hardy sweetness, "but my Lord does not need me as grandmother does; He is in glory, and will never be old or feeble; I cannot work for Him and tend Him as I shall her. I cannot see my way clear at present; but when she is
- 58. gone, or if the saints move her to consent, I shall then belong to God alone." "Daughter, there is some truth in your words; and if your Lord accepts you, He will dispose her heart. Will she go with you on this pilgrimage?" "I have prayed that she might, father,—that her soul may be quickened; for I fear me, dear old grandmamma has found her love for me a snare,—she has thought too much of my interests and too little of her own soul, poor grandmamma!" "Well, child, I shall enjoin this pilgrimage on her as a penance." "I have grievously offended her lately," said Agnes, "in rejecting an offer of marriage with a man on whom she had set her heart, and therefore she does not listen to me as she is wont to do." "You have done right in refusing, my daughter. I will speak to her of this, and show her how great is the sin of opposing a holy vocation in a soul whom the Lord calls to Himself, and enjoin her to make reparation by uniting with you in this holy work." Agnes departed from the confessional without even looking upon the face of her director, who sat within listening to the rustle of her dress as she rose,—listening to the soft fall of her departing footsteps, and praying that grace might be given him not to look after her: and he did not, though he felt as if his life were going with her. Agnes tripped round the aisle to a little side-chapel where a light was always kept burning by her before a picture of Saint Agnes, and, kneeling there, waited till her grandmother should be through with her confession. "Ah, sweet Saint Agnes," she said, "pity me! I am a poor ignorant young girl, and have been led into grievous sin; but I did not mean to do wrong,—I have been trying to do right; pray for me, that I may overcome as you did. Pray our dear Lord to send you with us on this pilgrimage, and save us from all wicked and brutal men who
- 59. would do us harm. As the Lord delivered you in sorest straits, keeping soul and body pure as a lily, ah, pray Him to keep me! I love you dearly,—watch over me and guide me." In those days of the Church, such addresses to the glorified saints had become common among all Christians. They were not regarded as worship, any more than a similar outpouring of confidence to a beloved and revered friend yet in the body. Among the hymns of Savonarola is one addressed to Saint Mary Magdalen, whom he regarded with an especial veneration. The great truth, that God is not the God of the dead, but of the living, that all live to Him, was in those ages with the truly religious a part of spiritual consciousness. The saints of the Church Triumphant, having become one with Christ as He is one with the Father, were regarded as invested with a portion of his divinity, and as the ministering agency through which his mediatorial government on earth was conducted; and it was thought to be in the power of the sympathetic heart to attract them by the outflow of its affections, so that their presence often overshadowed the walks of daily life with a cloud of healing and protecting sweetness. If the enthusiasm of devotion in regard to these invisible friends became extravagant and took the language due to God alone, it was no more than the fervid Italian nature was always doing with regard to visible objects of affection. Love with an Italian always tends to become worship, and some of the language of the poets addressed to earthly loves rises into intensities of expression due only to the One, Sovereign, Eternal Beauty. One sees even in the writings of Cicero that this passionate adoring kind of love is not confined to modern times. When he loses the daughter in whom his heart is garnered up, he finds no comfort except in building a temple to her memory,—a blind outreaching towards the saint-worship of modern times. Agnes rose from her devotions, and went with downcast eyes, her lips still repeating prayers, to the font of holy water, which was in a dim shadowy corner, where a painted window cast a gold and
- 60. violet twilight. Suddenly there was a rustle of garments in the dimness, and a jeweled hand essayed to pass holy water to her on the tip of its finger. This mark of Christian fraternity, common in those times, Agnes almost mechanically accepted, touching her slender finger to the one extended, and making the sign of the cross, while she raised her eyes to see who stood there. Gradually the haze cleared from her mind, and she awoke to the consciousness that it was the cavalier! He moved to come towards her, with a bright smile on his face; but suddenly she became pale as one who has seen a spectre, and, pushing from her with both hands, she said faintly, "Go, go!" and turned and sped up the aisle silently as a sunbeam, joining her grandmother, who was coming from the confessional with a gloomy and sullen brow. Old Elsie had been enjoined to unite with her grandchild in this scheme of a pilgrimage, and received the direction with as much internal contumacy as would a thriving church-member of Wall Street a proposition to attend a protracted meeting in the height of the business season. Not but that pilgrimages were holy and gracious works,—she was too good a Christian not to admit that,— but why must holy and gracious works be thrust on her in particular? There were saints enough who liked such things; and people could get to heaven without,—if not with a very abundant entrance, still in a modest way,—and Elsie's ambition for position and treasure in the spiritual world was of a very moderate cast. "Well, now, I hope you are satisfied," she said to Agnes, as she pulled her along with no very gentle hand; "you've got me sent off on a pilgrimage,—and my old bones must be rattling up and down all the hills between here and Rome,—and who's to see to the oranges?—they'll all be stolen, every one." "Grandmother"—began Agnes in a pleading voice. "Oh, you hush up! I know what you're going to say. 'The good Lord will take care of them.' I wish He may. He has his hands full, with all the people that go cawing and psalm-singing like so many crows, and leave all their affairs to Him!"
- 61. Agnes walked along disconsolate, with her eyes full of tears, which coursed one another down her pale cheeks. "There's Antonio," pursued Elsie, "would perhaps look after things a little. He is a good fellow, and only yesterday was asking if he couldn't do something for us. It's you he does it for,—but little you care who loves you, or what they do for you!" At this moment they met old Jocunda, whom we have before introduced to the reader as portress of the Convent. She had on her arm a large square basket, which she was storing for its practical uses. "Well, well, Saint Agnes be praised, I have found you at last," she said. "I was wanting to speak about some of your blood-oranges for conserving. An order has come down from our dear gracious lady, the Queen, to prepare a lot for her own blessed eating, and you may be sure I would get none of anybody but you. But what's this, my little heart, my little lamb?—crying?—tears in those sweet eyes? What's the matter now?" "Matter enough for me!" said Elsie. "It's a weary world we live in. A body can't turn any way and not meet with trouble. If a body brings up a girl one way, why, every fellow is after her, and one has no peace; and if a body brings her up another way, she gets her head in the clouds, and there's no good of her in this world. Now look at that girl,—doesn't everybody say it's time she were married? —but no marrying for her! Nothing will do but we must off to Rome on a pilgrimage,—and what's the good of that, I want to know? If it's praying that's to be done, the dear saints know she's at it from morning till night,—and lately she's up and down three or four times a night with some prayer or other." "Well, well," said Jocunda, "who started this idea?" "Oh, Father Francesco and she got it up between them, and nothing will do but I must go, too." "Well, now, after all, my dear," said Jocunda, "do you know, I made a pilgrimage once, and it isn't so bad. One gets a good deal by
- 62. it, first and last. Everybody drops something into your hand as you go, and one gets treated as if one were somebody a little above the common; and then in Rome one has a princess or a duchess or some noble lady who washes one's feet, and gives one a good supper, and perhaps a new suit of clothes, and all that,—and ten to one there comes a pretty little sum of money to boot, if one plays one's cards well. A pilgrimage isn't bad, after all; one sees a world of fine things, and something new every day." "But who is to look after our garden and dress our trees?" "Ah, now, there's Antonio, and old Meta his mother," said Jocunda, with a knowing wink at Agnes. "I fancy there are friends there that would lend a hand to keep things together against the little one comes home. If one is going to be married, a pilgrimage brings good luck in the family. All the saints take it kindly that one comes so far to see them, and are more ready to do a good turn for one when one needs it. The blessed saints are like other folks, they like to be treated with proper attention." This view of pilgrimages from the material standpoint had more effect on the mind of Elsie than the most elaborate appeals of Father Francesco. She began to acquiesce, though with a reluctant air. Jocunda, seeing her words had made some impression, pursued her advantage on the spiritual ground. "To be sure," she added, "I don't know how it is with you; but I know that I have, one way and another, rolled up quite an account of sins in my life. When I was tramping up and down with my old man through the country,—now in this castle and then in that camp, and now and then in at the sacking of a city or village, or something of the kind,—the saints forgive us!—it does seem as if one got into things that were not of the best sort, in such times. It's true, it's been wiped out over and over by the priest; but then a pilgrimage is a good thing to make all sure, in case one's good works should fall short of one's sins at last. I can tell you, a pilgrimage is a good
- 63. round weight to throw into the scale; and when it comes to heaven and hell, you know, my dear, why, one cannot be too careful." "Well, that may be true enough," said Elsie, "though as to my sins, I have tried to keep them regularly squared up and balanced as I went along. I have always been regular at confession, and never failed a jot or tittle in what the holy father told me. But there may be something in what you say; one can't be too sure; and so I'll e'en school my old bones into taking this tramp." That evening, as Agnes was sitting in the garden at sunset, her grandmother bustling in and out, talking, groaning, and hurrying in her preparations for the anticipated undertaking, suddenly there was a rustling in the branches overhead, and a bouquet of rosebuds fell at her feet. Agnes picked it up, and saw a scrip of paper coiled among the flowers. In a moment, remembering the apparition of the cavalier in the church in the morning, she doubted not from whom it came. So dreadful had been the effect of the scene at the confessional, that the thought of the near presence of her lover brought only terror. She turned pale; her hands shook. She shut her eyes, and prayed that she might not be left to read the paper; and then, summoning all her resolution, she threw the bouquet with force over the wall. It dropped down, down, down the gloomy, shadowy abyss, and was lost in the damp caverns below. The cavalier stood without the wall, waiting for some responsive signal in reply to his missive. It had never occurred to him that Agnes would not even read it, and he stood confounded when he saw it thrown back with such apparent rudeness. He remembered her pale, terrified look on seeing him in the morning. It was not indifference or dislike, but mortal fear, that had been shown in that pale face. "These wretches are practicing on her," he said, in wrath, "filling her head with frightful images, and torturing her sensitive conscience till she sees sin in the most natural and innocent feelings."
- 64. He had learned from Father Antonio the intention of Agnes to go on a pilgrimage, and he longed to see and talk with her, that he might offer her his protection against dangers which he understood far better than she. It had never even occurred to him that the door for all possible communication would be thus suddenly barred in his face. "Very well," he said to himself, with a darkening brow, "let them have it their own way here. She must pass through my dominions before she can reach Rome, and I will find a place where I can be heard, without priest or grandmother to let or hinder. She is mine, and I will care for her." But poor Agnes had the woman's share of the misery to bear, in the fear and self-reproach and distress which every movement of this kind cost her. The involuntary thrill at seeing her lover, at hearing from him, the conscious struggle which it cost her to throw back his gift, were all noted by her accusing conscience as so many sins. The next day she sought again her confessor, and began an entrance on those darker and more chilly paths of penance, by which, according to the opinion of her times, the peculiarly elect of the Lord were supposed to be best trained. Hitherto her religion had been the cheerful and natural expression of her tender and devout nature, according to the more beautiful and engaging devotional forms of her Church. During the year when her confessor had been, unconsciously to himself, led by her instead of leading, her spiritual food had been its beautiful old hymns and prayers, which she found no weariness in often repeating. But now an unnatural conflict was begun in her mind, directed by a spiritual guide in whom every natural and normal movement of the soul had given way before a succession of morbid and unhealthful experiences. From that day Agnes wore upon her heart one of those sharp instruments of torture which in those items were supposed to be a means of inward grace,—a cross with seven steel points for the seven sorrows of Mary. She fasted with a severity which alarmed her grandmother, who in her inmost heart cursed the day that ever she had placed her in the way of saintship.
- 65. "All this will just end in spoiling her beauty,—making her as thin as a shadow," said Elsie; "and she was good enough before." But it did not spoil her beauty, it only changed its character. The roundness and bloom melted away, but there came in their stead that solemn, transparent clearness of countenance, that spiritual light and radiance, which the old Florentine painters gave to their Madonnas. It is singular how all religious exercises and appliances take the character of the nature that uses them. The pain and penance, which so many in her day bore as a cowardly expedient for averting divine wrath, seemed, as she viewed them, a humble way of becoming associated in the sufferings of her Redeemer. "Jesu dulcis memoria," was the thought that carried a redeeming sweetness with every pain. Could she thus, by suffering with her Lord, gain power like Him to save,—a power which should save that soul so dear and so endangered! "Ah," she thought, "I would give my life-blood, drop by drop, if only it might avail for his salvation!"
- 66. CHAPTER XX FLORENCE AND HER PROPHET It was drawing towards evening, as two travelers, approaching Florence from the south, checked their course on the summit of one of the circle of hills which command a view of the city, and seemed to look down upon it with admiration. One of these was our old friend Father Antonio, and the other the cavalier. The former was mounted on an ambling mule, whose easy pace suited well with his meditative habits; while the other reined in a high-mettled steed, who, though now somewhat jaded under the fatigue of a long journey, showed by a series of little lively motions of his ears and tail, and by pawing the ground impatiently, that he had the inexhaustible stock of spirits which goes with good blood. "There she lies, my Florence," said the monk, stretching his hands out with enthusiasm. "Is she not indeed a sheltered lily growing fair among the hollows of the mountains? Little she may be, sir, compared to old Rome; but every inch of her is a gem,—every inch!" And, in truth, the scene was worthy of the artist's enthusiasm. All the overhanging hills that encircle the city with their silvery olive- gardens and their pearl-white villas were now lighted up with evening glory. The old gray walls of the convents of San Miniato and the Monte Oliveto were touched with yellow; and even the black obelisks of the cypresses in their cemeteries had here and there streaks and dots of gold, fluttering like bright birds among their gloomy branches. The distant snow-peaks of the Apennines, which
- 67. even in spring long wear their icy mantles, were shimmering and changing like an opal ring with tints of violet, green, blue, and rose, blended in inexpressible softness by that dreamy haze which forms the peculiar feature of Italian skies. In this loving embrace of mountains lay the city, divided by the Arno as by a line of rosy crystal barred by the graceful arches of its bridges. Amid the crowd of palaces and spires and towers rose central and conspicuous the great Duomo, just crowned with that magnificent dome which was then considered a novelty and a marvel in architecture, and which Michel Angelo looked longingly back upon when he was going to Rome to build that more wondrous orb of Saint Peter's. White and stately by its side shot up the airy shaft of the Campanile; and the violet vapor swathing the whole city in a tender indistinctness, these two striking objects, rising by their magnitude far above it, seemed to stand alone in a sort of airy grandeur. And now the bells of the churches were sounding the Ave Maria, filling the air with sweet and solemn vibrations, as if angels were passing to and fro overhead, harping as they went; and ever and anon the great bell of the Campanile came pulsing in with a throb of sound of a quality so different that one hushed one's breath to hear. It might be fancied to be the voice of one of those kingly archangels that one sees drawn by the old Florentine religious artists,—a voice grave and unearthly, and with a plaintive undertone of divine mystery. The monk and the cavalier bent low in their saddles, and seemed to join devoutly in the worship of the hour. One need not wonder at the enthusiasm of the returning pilgrim of those days for the city of his love, who feels the charm that lingers around that beautiful place even in modern times. Never was there a spot to which the heart could insensibly grow with a more home-like affection,—never one more thoroughly consecrated in every stone by the sacred touch of genius.
- 68. A republic, in the midst of contending elements, the history of Florence, in the Middle Ages, was a history of what shoots and blossoms the Italian nature might send forth, when rooted in the rich soil of liberty. It was a city of poets and artists. Its statesmen, its merchants, its common artisans, and the very monks in its convents, were all pervaded by one spirit. The men of Florence in its best days were men of a large, grave, earnest mould. What the Puritans of New England wrought out with severest earnestness in their reasonings and their lives, these early Puritans of Italy embodied in poetry, sculpture, and painting. They built their Cathedral and their Campanile, as the Jews of old built their Temple, with awe and religious fear, that they might thus express by costly and imperishable monuments their sense of God's majesty and beauty. The modern traveler who visits the churches and convents of Florence, or the museums where are preserved the fading remains of its early religious Art, if he be a person of any sensibility, cannot fail to be affected with the intense gravity and earnestness which pervade them. They seem less to be paintings for the embellishment of life than eloquent picture-writing by which burning religious souls sought to preach the truths of the invisible world to the eye of the multitude. Through all the deficiencies of perspective, coloring, and outline incident to the childhood and early youth of Art, one feels the passionate purpose of some lofty soul to express ideas of patience, self-sacrifice, adoration, and aspiration far transcending the limits of mortal capability. The angels and celestial beings of these grave old painters are as different from the fat little pink Cupids or lovely laughing children of Titian and Correggio as are the sermons of President Edwards from the love-songs of Tom Moore. These old seers of the pencil give you grave, radiant beings, strong as man, fine as woman, sweeping downward in lines of floating undulation, and seeming by the ease with which they remain poised in the air to feel none of that earthly attraction which draws material bodies earthward. Whether they wear the morning star on their forehead or bear the lily or the sword in their hand, there is still that suggestion of mystery and power
- 69. about them, that air of dignity and repose, that speak the children of a nobler race than ours. One could well believe such a being might pass in his serene poised majesty of motion through the walls of a gross material dwelling without deranging one graceful fold of his swaying robe or unclasping the hands folded quietly on his bosom. Well has a modern master of art and style said of these old artists, "Many pictures are ostentatious exhibitions of the artist's power of speech, the clear and vigorous elocution of useless and senseless words; while the earlier efforts of Giotto and Cimabue are the burning messages of prophecy delivered by the stammering lips of infants." But at the time of which we write, Florence had passed through her ages of primitive religious and republican simplicity, and was fast hastening to her downfall. The genius, energy, and prophetic enthusiasm of Savonarola had made, it is true, a desperate rally on the verge of the precipice; but no one man has ever power to turn back the downward slide of a whole generation. When Father Antonio left Sorrento in company with the cavalier, it was the intention of the latter to go with him only so far as their respective routes should lie together. The band under the command of Agostino was posted in a ruined fortress in one of those airily perched old mountain-towns which form so picturesque and characteristic a feature of the Italian landscape. But before they reached this spot, the simple, poetic, guileless monk, with his fresh artistic nature, had so won upon his traveling companion that a most enthusiastic friendship had sprung up between them, and Agostino could not find it in his heart at once to separate from him. Tempest- tossed and homeless, burning with a sense of wrong, alienated from the faith of his fathers through his intellect and moral sense, yet clinging to it with his memory and imagination, he found in the tender devotional fervor of the artist monk a reconciling and healing power. He shared, too, in no small degree, the feelings which now possessed the breast of his companion for the great reformer whose purpose seemed to meditate nothing less than the restoration of the Church of Italy to the primitive apostolic simplicity. He longed to see
- 70. him,—to listen to the eloquence of which he had heard so much. Then, too, he had thoughts that but vaguely shaped themselves in his mind. This noble man, so brave and courageous, menaced by the forces of a cruel tyranny, might he not need the protection of a good sword? He recollected, too, that he had an uncle high in the favor of the King of France, to whom he had written a full account of his own situation. Might he not be of use in urging this uncle to induce the French King to throw before Savonarola the shield of his protection? At all events, he entered Florence this evening with the burning zeal of a young neophyte who hopes to effect something himself for a glorious and sacred cause embodied in a leader who commands his deepest veneration. "My son," said Father Antonio, as they raised their heads after the evening prayer, "I am at this time like a man who, having long been away from his home, fears, on returning, that he shall hear some evil tidings of those he hath left. I long, yet dread, to go to my dear Father Girolamo and the beloved brothers in our house. There is a presage that lies heavy on my heart, so that I cannot shake it off. Look at our glorious old Duomo;—doth she not sit there among the houses and palaces as a queen-mother among nations,—worthy, in her greatness and beauty, to represent the Church of the New Jerusalem, the Bride of the Lord? Ah, I have seen it thronged and pressed with the multitude who came to crave the bread of life from our master!" "Courage, my friend!" said Agostino; "it cannot be that Florence will suffer her pride and glory to be trodden down. Let us hasten on, for the shades of evening are coming fast, and there is a keen wind sweeping down from your snowy mountains." And the two soon found themselves plunging into the shadows of the streets, threading their devious way to the convent. At length they drew up before a dark wall, where the Father Antonio rung a bell. A door was immediately opened, a cowled head appeared, and a cautious voice asked,—
- 71. "Who is there?" "Ah, is that you, good Brother Angelo?" said Father Antonio, cheerily. "And is it you, dear Brother Antonio? Come in! come in!" was the cordial response, as the two passed into the court; "truly, it will make all our hearts leap to see you." "And, Brother Angelo, how is our dear father? I have been so anxious about him!" "Oh, fear not!—he sustains himself in God, and is full of sweetness to us all." "But do the people stand by him, Angelo, and the Signoria?" "He has strong friends as yet, but his enemies are like ravening wolves. The Pope hath set on the Franciscans, and they hunt him as dogs do a good stag. But whom have you here with you?" added the monk, raising his torch and regarding the knight. "Fear him not; he is a brave knight and good Christian, who comes to offer his sword to our father and seek his counsels." "He shall be welcome," said the porter, cheerfully. "We will have you into the refectory forthwith, for you must be hungry." The young cavalier, following the flickering torch of his conductor, had only a dim notion of long cloistered corridors, out of which now and then, as the light flared by, came a golden gleam from some quaint old painting, where the pure angel forms of Angelico stood in the gravity of an immortal youth, or the Madonna, like a bending lily, awaited the message of Heaven; but when they entered the refectory, a cheerful voice addressed them, and Father Antonio was clasped in the embrace of the father so much beloved. "Welcome, welcome, my dear son!" said that rich voice which had thrilled so many thousand Italian hearts with its music. "So you are come back to the fold again. How goes the good work of the Lord?"
- 72. "Well, everywhere," said Father Antonio, and then, recollecting his young friend, he suddenly turned and said,— "Let me present to you one son who comes to seek your instructions,—the young Signor Agostino, of the noble house of Sarelli." The Superior turned to Agostino with a movement full of a generous frankness, and warmly extended his hand, at the same time fixing upon him the mesmeric glance of a pair of large, deep blue eyes, which might, on slight observation, have been mistaken for black, so great was their depth and brilliancy. Agostino surveyed his new acquaintance with that mingling of ingenuous respect and curiosity with which an ardent young man would regard the most distinguished leader of his age, and felt drawn to him by a certain atmosphere of vital cordiality such as one can feel better than describe. "You have ridden far to-day, my son,—you must be weary," said the Superior, affably; "but here you must feel yourself at home; command us in anything we can do for you. The brothers will attend to those refreshments which are needed after so long a journey; and when you have rested and supped, we shall hope to see you a little more quietly." So saying, he signed to one or two brothers who stood by, and, commending the travelers to their care, left the apartment. In a few moments a table was spread with a plain and wholesome repast, to which the two travelers sat down with appetites sharpened by their long journey. During the supper, the brothers of the convent, among whom Father Antonio had always been a favorite, crowded around him in a state of eager excitement. "You should have been here the last week," said one; "such a turmoil as we have been in!"
- 73. "Yes," said another, "the Pope hath set on the Franciscans, who, you know, are always ready enough to take up with anything against our order, and they have been pursuing our father like so many hounds." "There hath been a whirlwind of preaching here and there," said a third, "in the Duomo, and Santa Croce, and San Lorenzo; and they have battled to and fro, and all the city is full of it." "Tell him about yesterday, about the ordeal," shouted an eager voice. Two or three voices took up the story at once, and began to tell it, all the others correcting, contradicting, or adding incidents. From the confused fragments here and there Agostino gathered that there had been on the day before a popular spectacle in the grand piazza, in which, according to an old superstition of the Middle Ages, Fra Girolamo Savonarola and his opponents were expected to prove the truth of their words by passing unhurt through the fire; that two immense piles of combustibles had been constructed with a narrow passage between, and the whole magistracy of the city convened, with a throng of the populace, eager for the excitement of the spectacle; that the day had been spent in discussions, and scruples, and preliminaries; and that, finally, in the afternoon, a violent storm of rain arising had dispersed the multitude and put a stop to the whole exhibition. "But the people are not satisfied," said Father Angelo; "and there are enough mischief-makers among them to throw all the blame on our father." "Yes," said one, "they say he wanted to burn the Holy Sacrament, because he was going to take it with him into the fire." "As if it could burn!" said another voice. "It would to all human appearance, I suppose," said a third. "Any way," said a fourth, "there is some mischief brewing; for here is our friend Prospero Rondinelli just come in, who says, when
- 74. he came past the Duomo, he saw people gathering, and heard them threatening us: there were as many as two hundred, he thought." "We ought to tell Father Girolamo," exclaimed several voices. "Oh, he will not be disturbed!" said Father Angelo. Since these affairs, he hath been in prayer in the chapter-room before the blessed Angelico's picture of the Cross. When we would talk with him of these things, he waves us away, and says only, 'I am weary; go and tell Jesus.'" "He bade me come to him after supper," said Father Antonio. "I will talk with him." "Do so,—that is right," said two or three eager voices as the monk and Agostino, having finished their repast, arose to be conducted to the presence of the father.
- 75. CHAPTER XXI THE ATTACK ON SAN MARCO They found him in a large and dimly lighted apartment, sitting absorbed in pensive contemplation before a picture of the Crucifixion by Fra Angelico, which, whatever might be its naïve faults of drawing and perspective, had an intense earnestness of feeling, and, though faded and dimmed by the lapse of centuries, still stirs in some faint wise even the practiced dilettanti of our day. The face upon the cross, with its majestic patience, seemed to shed a blessing down on the company of saints of all ages who were grouped by their representative men at the foot. Saint Dominic, Saint Ambrose, Saint Augustin, Saint Jerome, Saint Francis, and Saint Benedict were depicted as standing before the Great Sacrifice in company with the Twelve Apostles, the two Maries, and the fainting mother of Jesus,—thus expressing the unity of the Church Universal in that great victory of sorrow and glory. The painting was enclosed above by a semicircular bordering composed of medallion heads of the Prophets, and below was a similar medallion border of the principal saints and worthies of the Dominican order. In our day such pictures are visited by tourists with red guide-books in their hands, who survey them in the intervals of careless conversation; but they were painted by the simple artist on his knees, weeping and praying as he worked, and the sight of them was accepted by like simple-hearted Christians as a perpetual sacrament of the eye, by which they received Christ into their souls.
- 76. So absorbed was the father in the contemplation of this picture, that he did not hear the approaching footsteps of the knight and monk. When at last they came so near as almost to touch him, he suddenly looked up, and it became apparent that his eyes were full of tears. He rose, and, pointing with a mute gesture toward the painting, said,— "There is more in that than in all Michel Angelo Buonarotti hath done yet, though he be a God-fearing youth,—more than in all the heathen marbles in Lorenzo's gardens. But sit down with me here. I have to come here often, where I can refresh my courage." The monk and knight seated themselves, the latter with his attention riveted on the remarkable man before him. The head and face of Savonarola are familiar to us by many paintings and medallions, which, however, fail to impart what must have been that effect of his personal presence which so drew all hearts to him in his day. The knight saw a man of middle age, of elastic, well-knit figure, and a flexibility and grace of motion which seemed to make every nerve, even to his finger-ends, vital with the expression of his soul. The close-shaven crown and the plain white Dominican robe gave a severe and statuesque simplicity to the lines of his figure. His head and face, like those of most of the men of genius whom modern Italy has produced, were so strongly cast in the antique mould as to leave no doubt of the identity of modern Italian blood with that of the great men of ancient Italy. His low, broad forehead, prominent Roman nose, well-cut, yet fully outlined lips, and strong, finely moulded jaw and chin, all spoke the old Roman vigor and energy, while the flexible delicacy of all the muscles of his face and figure gave an inexpressible fascination to his appearance. Every emotion and changing thought seemed to flutter and tremble over his countenance as the shadow of leaves over sunny water. His eye had a wonderful dilating power, and when he was excited seemed to shower sparks; and his voice possessed a surprising scale of delicate and melodious inflections, which could take him in a moment
- 77. through the whole range of human feeling, whether playful and tender or denunciatory and terrible. Yet, when in repose among his friends, there was an almost childlike simplicity and artlessness of manner which drew the heart by an irresistible attraction. At this moment it was easy to see by his pale cheek and the furrowed lines of his face that he had been passing through severe struggles; but his mind seemed stayed on some invisible centre, in a solemn and mournful calm. "Come, tell me something of the good works of the Lord in our Italy, brother," he said, with a smile which was almost playful in its brightness. "You have been through all the lowly places of the land, carrying our Lord's bread to the poor, and repairing and beautifying shrines and altars by the noble gift that is in you." "Yes, father," said the monk; "and I have found that there are many sheep of the Lord that feed quietly among the mountains of Italy, and love nothing so much as to hear of the dear Shepherd who laid down His life for them." "Even so, even so," said the Superior, with animation; "and it is the thought of these sweet hearts that comforts me when my soul is among lions. The foundation standeth sure,—the Lord knoweth them that are his." "And it is good and encouraging," said Father Antonio, "to see the zeal of the poor, who will give their last penny for the altar of the Lord, and who flock so to hear the word and take the sacraments. I have had precious seasons of preaching and confessing, and have worked in blessedness many days restoring and beautifying the holy pictures and statues whereby these little ones have been comforted. What with the wranglings of princes and the factions and disturbances in our poor Italy, there be many who suffer in want and loss of all things, so that no refuge remains to them but the altars of our Jesus, and none cares for them but He." "Brother," said the Superior, "there be thousands of flowers fairer than man ever saw that grow up in waste places and in deep dells
- 78. and shades of mountains; but God bears each one in his heart, and delighteth Himself in silence with them: and so doth He with these poor, simple, unknown souls. The True Church is not a flaunting queen who goes boldly forth among men displaying her beauties, but a veiled bride, a dove that is in the cleft of the rocks, whose voice is known only to the Beloved. Ah! when shall the great marriage-feast come, when all shall behold her glorified? I had hoped to see the day here in Italy: but now"— The father stopped, and seemed to lapse into unconscious musing,—his large eye growing fixed and mysterious in its expression. "The brothers have been telling me somewhat of the tribulations you have been through," said Father Antonio, who thought he saw a good opening to introduce the subject nearest his heart. "No more of that!—no more!" said the Superior, turning away his head with an expression of pain and weariness, "rather let us look up. What think you, brother, are all these doing now?" he said, pointing to the saints in the picture. "They are all alive and well, and see clearly through our darkness." Then, rising up, he added, solemnly, "Whatever man may say or do, it is enough for me to feel that my dearest Lord and his blessed Mother and all the holy archangels, the martyrs and prophets and apostles, are with me. The end is coming." "But, dearest father," said Antonio, "think you the Lord will suffer the wicked to prevail?" "It may be for a time," said Savonarola. "As for me, I am in His hands only as an instrument. He is master of the forge and handles the hammer, and when He has done using it He casts it from Him. Thus He did with Jeremiah, whom He permitted to be stoned to death when his preaching mission was accomplished; and thus He may do with this hammer when He has done using it." At this moment a monk rushed into the room with a face expressive of the utmost terror, and called out,—
- 79. "Father, what shall we do? The mob are surrounding the convent! Hark! hear them at the doors!" In truth, a wild, confused roar of mingled shrieks, cries, and blows came in through the open door of the apartment; and the pattering sound of approaching footsteps was heard like showering rain-drops along the cloisters. "Here come Messer Nicolo de' Lapi, and Francesco Valori!" called out a voice. The room was soon filled with a confused crowd, consisting of distinguished Florentine citizens, who had gained admittance through a secret passage, and the excited novices and monks. "The streets outside the convent are packed close with men," cried one of the citizens; "they have stationed guards everywhere to cut off our friends who might come to help us." "I saw them seize a young man who was quietly walking, singing psalms, and slay him on the steps of the Church of the Innocents," said another; "they cried and hooted, 'No more psalm-singing!'" "And there's Arnolfo Battista," said a third;—"he went out to try to speak to them, and they have killed him,—cut him down with their sabres." "Hurry! hurry! barricade the door! arm yourselves!" was the cry from other voices. "Shall we fight, father? Shall we defend ourselves?" cried others, as the monks pressed around their Superior. When the crowd first burst into the room, the face of the Superior flushed, and there was a slight movement of surprise; then he seemed to recollect himself, and murmuring, "I expected this, but not so soon," appeared lost in mental prayer. To the agitated inquiries of his flock, he answered, "No, brothers; the weapons of monks must be spiritual, not carnal." Then lifting on high a crucifix, he said, "Come with me, and let us walk in solemn procession to the altar, singing the praises of our God."
- 80. The monks, with the instinctive habit of obedience, fell into procession behind their leader, whose voice, clear and strong, was heard raising the Psalm, "Quare fremunt gentes:"— "Why do the heathen rage, and the people imagine a vain thing? "The kings of the earth set themselves, and the rulers take counsel together, against the Lord, and against his Anointed, saying,— "Let us break their bands asunder, and cast away their cords from us. "He that sitteth in the heavens shall laugh: the Lord shall have them in derision." As one voice after another took up the chant, the solemn enthusiasm rose and deepened, and all present, whether ecclesiastics or laymen, fell into the procession and joined in the anthem. Amid the wild uproar, the din and clatter of axes, the thunders of heavy battering-implements on the stone walls and portals, came this long-drawn solemn wave of sound, rising and falling,—now drowned in the savage clamors of the mob, and now bursting out clear and full like the voices of God's chosen amid the confusion and struggles of all the generations of this mortal life. White-robed and grand the procession moved on, while the pictured saints and angels on the walls seemed to smile calmly down upon them from a golden twilight. They passed thus into the sacristy, where with all solemnity and composure they arrayed their Father and Superior for the last time in his sacramental robes, and then, still chanting, followed him to the high altar, where all bowed in prayer. And still, whenever there was a pause in the stormy uproar and fiendish clamor, might be heard the clear, plaintive uprising of that strange singing, "O Lord, save thy people, and bless thine heritage!" It needs not to tell in detail what history has told of that tragic night: how the doors at last were forced, and the mob rushed in; how citizens and friends, and many of the monks themselves, their
- 81. instinct of combativeness overcoming their spiritual beliefs, fought valiantly, and used torches and crucifixes for purposes little contemplated when they were made. Fiercest among the combatants was Agostino, who three times drove back the crowd as they were approaching the choir, where Savonarola and his immediate friends were still praying. Father Antonio, too, seized a sword from the hand of a fallen man and laid about him with an impetuosity which would be inexplicable to any who do not know what force there is in gentle natures when the objects of their affections are assailed. The artist monk fought for his master with the blind desperation with which a woman fights over the cradle of her child. All in vain! Past midnight, and the news comes that artillery is planted to blow down the walls of the convent, and the magistracy, who up to this time have lifted not a finger to repress the tumult, send word to Savonarola to surrender himself to them, together with the two most active of his companions, Fra Domenico da Pescia and Fra Silvestro Maruffi, as the only means of averting the destruction of the whole order. They offer him assurances of protection and safe return, which he does not in the least believe: nevertheless, he feels that his hour is come, and gives himself up. His preparations were all made with a solemn method which showed that he felt he was approaching the last act in the drama of life. He called together his flock, scattered and forlorn, and gave them his last words of fatherly advice, encouragement, and comfort, —ending with the remarkable declaration, "A Christian's life consists in doing good and suffering evil." "I go with joy to this marriage- supper," he said, as he left the church for the last sad preparations. He and his doomed friends then confessed and received the sacrament, and after that he surrendered himself into the hands of the men who he felt in his prophetic soul had come to take him to torture and to death. As he gave himself into their hands, he said, "I commend to your care this flock of mine, and these good citizens of Florence who have
- 82. been with us;" and then once more turning to his brethren, said, "Doubt not, my brethren. God will not fail to perfect His work. Whether I live or die, He will aid and console you." At this moment there was a struggle with the attendants in the outer circle of the crowd, and the voice of Father Antonio was heard crying out earnestly, "Do not hold me! I will go with him! I must go with him!" "Son," said Savonarola, "I charge you on your obedience not to come. It is I and Fra Domenico who are to die for the love of Christ." And thus, at the ninth hour of the night, he passed the threshold of San Marco. As he was leaving, a plaintive voice of distress was heard from a young novice who had been peculiarly dear to him, who stretched his hands after him, crying, "Father! father! why do you leave us desolate?" Whereupon he turned back a moment, and said, "God will be your help. If we do not see each other again in this world, we surely shall in heaven." When the party had gone forth, the monks and citizens stood looking into each other's faces, listening with dismay to the howl of wild ferocity that was rising around the departing prisoner. "What shall we do?" was the outcry from many voices. "I know what I shall do," said Agostino. "If any man here will find me a fleet horse, I will start for Milan this very hour; for my uncle is now there on a visit, and he is a counselor of weight with the King of France: we must get the King to interfere." "Good! good! good!" rose from a hundred voices. "I will go with you," said Father Antonio. "I shall have no rest till I do something." "And I," quoth Jacopo Niccolini, "will saddle for you, without delay, two horses of part Arabian blood, swift of foot, and easy, and which will travel day and night without sinking."
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