Advances in atomic, molecular, and optical physics, volume 64

EDITORS ENNIO ARIMONDO University of Pisa Pisa, Italy CHUN C LIN University of Wisconsin Madison, Madison, WI, USA SUSANNE F YELIN University of Connecticut Storrs, CT, USA EDITORIAL BOARD P.H BUCKSBAUM SLAC Menlo Park, California C JOACHAIN Universite Libre de Bruxelles, Brussels, Belgium J.T.M WALRAVEN University of Amsterdam Amsterdam, The Netherlands Academic Press is an imprint of Elsevier 225 Wyman Street, Waltham, MA 02451, USA 525 B Street, Suite 1800, San Diego, CA 92101–4495, USA The Boulevard, Langford Lane, Kidlington, Oxford OX5 1GB, UK 125 London Wall, London, EC2Y 5AS, UK Radarweg 29, PO Box 211, 1000 AE Amsterdam, The Netherlands First edition 2015 Copyright © 2015 Elsevier Inc All rights reserved No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or any information storage and retrieval system, without permission in writing from the publisher Details on how to seek 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law, neither the Publisher nor the authors, contributors, or editors, assume any liability for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions, or ideas contained in the material herein ISBN: 978-0-12-802127-9 ISSN: 1049-250X For information on all Academic Press publications visit our website at store.elsevier.com CONTRIBUTORS Tunna Baruah (153) Department of Physics, University of Texas El Paso, El Paso, Texas, USA Scott D Bergeson (223) Department of Physics and Astronomy, Brigham Young University, Provo, Utah, USA Giovanni Borghi (105) Theory and Simulations of Materials (THEOS), and National Center for Computational  cole Polytechnique Fe´de´rale de Design and Discovery of Novel Materials (MARVEL), E Lausanne, Lausanne, Switzerland Carlo Maria Canali (29) Department of Physics and Electrical Engineering, Linnæus University, Kalmar, Sweden Yiwen Chu (273) Department of Applied Physics, Yale University, New Haven, Connecticut, USA Ismaila Dabo (105) Department of Materials Science and Engineering, Materials Research Institute, and Penn State Institutes of Energy and the Environment, The Pennsylvania State University, Pennsylvania, USA Phuong Mai Dinh (87) CNRS, and Universite´ de Toulouse, UPS, Laboratoire de Physique The´orique (IRSAMC), Toulouse Ce´dex, France David Gelbwaser-Klimovsky (329) Department of Chemical Physics, Weizmann Institute of Science, Rehovot, Israel Nikitas Gidopoulos (129) Department of Physics, Durham University, Durham, United Kingdom Koblar Alan Jackson (15) Physics Department and Science of Advanced Materials Program, Central Michigan University, Mt Pleasant, Michigan, USA Nathan Daniel Keilbart (105) Department of Materials Science and Engineering, Materials Research Institute, and Penn State Institutes of Energy and the Environment, The Pennsylvania State University, Pennsylvania, USA Stephan Kuămmel (143) Theoretische Physik IV, Universitaăt Bayreuth, Bayreuth, Germany Gershon Kurizki (329) Department of Chemical Physics, Weizmann Institute of Science, Rehovot, Israel Nektarios N.N Lathiotakis (129) Theoretical and Physical Chemistry Institute, National Hellenic Research Foundation, Athens, Greece, and Max Planck Institute of Microstructure Physics, Halle (Saale), Germany ix x Contributors Mikhail Lukin (273) Department of Physics, Harvard University, Cambridge, Massachusetts, USA Michael S Murillo (223) New Mexico Consortium Los Alamos, New Mexico, USA Ngoc Linh Nguyen (105) Theory and Simulations of Materials (THEOS), and National Center for Computational  cole Polytechnique Fe´de´rale de Design and Discovery of Novel Materials (MARVEL), E Lausanne, Lausanne, Switzerland Wolfgang Niedenzu (329) Department of Chemical Physics, Weizmann Institute of Science, Rehovot, Israel Mark R Pederson (1, 29, 153) Department of Chemistry, Johns Hopkins University, Baltimore, Maryland, USA John P Perdew (1) Department of Physics, and Department of Chemistry, Temple University, Philadelphia, Pennsylvania, USA Anna Pertsova (29) Department of Physics and Electrical Engineering, Linnæus University, Kalmar, Sweden Nicolas Poilvert (105) Department of Materials Science and Engineering, Materials Research Institute, and Penn State Institutes of Energy and the Environment, The Pennsylvania State University, Pennsylvania, USA Paul-Gerhard Reinhard (87) Institut fuăr Theoretische Physik, Universitaăt Erlangen, Erlangen, Germany Ivan Rungger (29) School of Physics, AMBER and CRANN Institute, Trinity College, Dublin, Ireland Adrienn Ruzsinszky (1) Department of Physics, Temple University, Philadelphia, Pennsylvania, USA Stefano Sanvito (29) School of Physics, AMBER and CRANN Institute, Trinity College, Dublin, Ireland Swati Singh (273) ITAMP, Harvard-Smithsonian Center for Astrophysics, Cambridge, Massachusetts, and Department of Physics, University of Connecticut, Storrs, Connecticut, USA Duncan G Steel (181) Departments of EECS and Physics, University of Michigan, Ann Arbor, Michigan, USA Jianwei Sun (1) Department of Physics, Temple University, Philadelphia, Pennsylvania, USA Eric Suraud (87) CNRS; Universite´ de Toulouse, UPS, Laboratoire de Physique The´orique (IRSAMC), Toulouse Ce´dex, France, and Physics Department, University at Buffalo, The State University New York, Buffalo, New York, USA Contributors xi Marc Vincendon (87) CNRS, and Universite´ de Toulouse, UPS, Laboratoire de Physique The´orique (IRSAMC), Toulouse Ce´dex, France Kevin Wang (105) Department of Materials Science and Engineering, Materials Research Institute, and Penn State Institutes of Energy and the Environment, The Pennsylvania State University, Pennsylvania, USA Susanne Yelin (273) Department of Physics, University of Connecticut, Storrs, Connecticut, and Department of Physics, Harvard University, Cambridge, Massachusetts, USA PREFACE A large part of this volume is on the subject of self-interaction corrections (SICs) to the density functional theory (DFT) In the Hartree–Fock formalism for a many-electron system, the self-interaction of the Coulomb repulsion is offset by a similar term in the exchange component, but the cancelation is incomplete when semi-local approximations to DFT, such as the local-density approximation (LDA) and generalized gradient approximation (GGA), are adopted The residual self-interaction is troublesome in many applications of the LDA-DFT For example, it produces, for the case of a neutral atom, a one-electron potential with an exponential tail instead of the correct (1/r) asymptotic form leading to serious problems in the calculated energies Of the numerous attempts to address this deficiency, the SIC scheme proposed by Perdew and Zunger in 1981 (PZ SIC) has received a great deal of attention Early applications of PZ SIC to simple atoms and molecules demonstrated significant improvements over the uncorrected LDA calculations However, one serious drawback is that the PZ SIC Hamiltonian generally depends on the individual orbital densities in contrast to the fundamental view that the energy of the entire system is dictated only by the total density and should be invariant under a unitary transformation of the orbitals A consequence of this is the necessity of introducing two sets of orbitals (referred to as the canonical and localized orbitals by Pederson et al.) which greatly increases the computational labor As researchers moved on to the more complex systems and simulation of dynamic processes, the need for an effective means to handle the problems of self-interactions becomes more pressing during the past 10 years One of us (C.C.L.) is fortunate to have enjoyed a long and fruitful association with Dr Mark R Pederson who was an early explorer of the applications of the PZ SIC to molecules while working on his doctoral dissertation and is actively involved in the current surge of research efforts to deal with the problems of self-interactions With the invaluable advice and assistance from Dr Pederson, we have compiled the first eight chapters of this volume which provide different viewpoints on the SIC along with discussions of both past and current work as well as some indications on where the future might lead Chapter introduces the PZ SIC in general terms and compares the various degrees of success (also the lack of it) in different kinds of calculations Applications of the SIC to study the electronic structure of substitutional xiii xiv Preface impurity atoms in ionic crystals constitute the main theme of the second chapter Addressed in Chapter is the broad area of spin-dependent phenomena in nanostructures Specifically, an important question is how well one can predict the behaviors of a few spins in a given environment by means of first-principles theoretical treatments The computational challenge of such analyses is discussed pointing toward possible future directions for improving the predictive power of the DFT-based methods As indicated in the preceding paragraph, adaptation of the PZ SIC to the LDA resulted in an iteration scheme involving two sets of orbitals The use of such a twoset scheme is discussed by the authors of Chapter They introduced an average-density SIC procedure which drastically simplifies the computational work In Chapter 5, a new way to incorporate the PZ SIC to the LDA is presented The self-interaction-corrected functional employed here is still dependent on the individual orbitals but is made to conform to the Koopmans theorem Successful applications of such Koopmans-compliant functionals as presented in this chapter are indicative of the potential power of this approach As mentioned earlier, one manifestation of the selfinteraction errors is that for a neutral system the LDA fails to reproduce the (1/r) behavior of the potential seen by an electron at a large distance Using an effective local potential may provide an alternative avenue to reduce the self-interaction errors, and construction of optimal local potentials for this purpose is the main theme of Chapter In contrast to works based on PZ SIC that result in orbital-specific potentials, Chapter presents SIC with one global multiplicative potential along with a discussion of the relations of this approach to the method of optimized effective potential Extension to the time-dependent formulation is also discussed In spite of its success in many areas, the PZ SIC has been criticized for the undesirable orbital dependence in the functional which spoiled the invariance under a unitary transformation of the orbitals inherent in the general DFT, not to mention the complications, both conceptual and computational, resulting from this orbital dependence Concluding this sequence of articles on self-interactions, Chapter reviews the recent work on recasting the PZ SIC in terms of the Fermi orbitals which restores the unitary invariance This step puts a constraint on the original PZ SIC formalism, but bypasses the need for the localization equations and the two-set procedure Results of the new SIC calculations and future outlook are also discussed therein A quarter century after quantum coherence effects such as dark states and electromagnetically induced transparency (EIT) have become mainstream, they are finally applied in solid-state systems in a wider context To see such Preface xv effects in typical semiconductor environments is at the same time among the most desired and the most difficult Steel shows in Chapter recent advances with quantum dot exciton artificial atoms and successes, and shows among other things typical EIT and dark-state signatures In Chapter 10, Murillo and Bergeson present a new kind of plasmas formed by photoionization of laser-cooled atoms These ultracold neutral plasmas are strongly coupled systems and are particularly well suited to study many-body interactions in atomic and molecular processes like thermalization, three-body recombination, and collisional ionization The authors begin with an introduction to the strong coupling parameter as an index of classification and then focus the discussion on generating strongly coupled plasmas using calcium atoms in a magneto-optical trap Molecular dynamics simulations provide insight into electron screening Techniques such as multiple ionization to higher ionization states, Rydberg atom dynamics, and direct laser cooling of the ions for producing strongly coupled plasmas are also discussed In parallel with the great progresses associated to the tools developed by atomic, molecular, and optical physics, in the last few years an important trend was established by the solid-state community: apply those sophisticated tools to solid systems where the complexity is not too large and instead descriptions in terms of few atom-like objects can be applied This approach was exemplified in Chapter Along a similar vein, the contribution in Chapter 11 by Singh, Chu, Lukin, and Yelin targets the control of the nuclear spins modifying the optical excitation of a single electronic spin for the case of nitrogen-vacancy color centers in diamond Owing to impressive technological advances, it is today possible to monitor a single-color center that represents a single atomic-like system This center interacts with the nuclear spin of the surrounding crystals, between tens and hundreds of the 13C isotope within the diamond The control of those nuclear spins is essential for the application of the color center electronic spin qubit, for instance, for quantum information The authors of the present contribution present the control achieved by applying the coherent population trapping approach originally developed for the laser cooling of atoms and ions That method is successfully applied to the cooling and the realtime projective measurement of the nuclear spin environment surrounding the electronic spin In a combination of quantum optics and thermodynamics, GelbwaserKlimovsky, Niedenzu, and Kurizki asked the question in Chapter 12 whether quantum mechanics can allow to violate any of the laws of xvi Preface thermodynamics While this article does not claim to answer this wide and impactful question fully, it turns out that the structure of the system-plusbath of a quantum mechanical heat engine allows in certain aspects to improve on the classical Carnot limit In their article, the authors review the questions how partially and fully quantum systems behave, systems that are driven steady state or periodically modulated, Markovian and nonMarkovian systems, and systems that are stripped down to the qubit stage The editors would like to thank all the contributing authors for their contributions and for their cooperation in assembling this volume They are especially grateful to Dr Mark R Pederson for his help in organizing the first eight chapters Sincere appreciation is also extended to Ms Helene Kabes at Elsevier for her untiring assistance throughout the preparation of this volume ENNIO ARIMONDO CHUN C LIN SUSANNE F YELIN CHAPTER ONE Paradox of Self-Interaction Correction: How Can Anything So Right Be So Wrong? John P Perdew*,†, Adrienn Ruzsinszky*, Jianwei Sun*, Mark R Pederson{,1 *Department of Physics, Temple University, Philadelphia, Pennsylvania, USA † Department of Chemistry, Temple University, Philadelphia, Pennsylvania, USA { Department of Chemistry, Johns Hopkins University, Baltimore, Maryland, USA Corresponding author: e-mail address: mpeder10@jhu.edu Contents Introduction What Is Right About PZ SIC? What Is Wrong About PZ SIC? SIC: How Can Anything So Right Be So Wrong? (Conclusions) Acknowledgments Appendix Do Complex Orbitals Resolve the Paradox of SIC? References 10 11 12 Abstract Popular local, semilocal, and hybrid density functional approximations to the exchangecorrelation energy of a many-electron ground state make a one-electron self-interaction error which can be removed by its orbital-by-orbital subtraction from the total energy, as proposed by Perdew and Zunger in 1981 This makes the functional exact for all one-electron ground states, but it does much more as well: It greatly improves the description of negative ions, the dissociation curves of radical molecules and of all heteronuclear molecules, the barrier heights for chemical reactions, charge-transfer energies, etc PZ SIC even led to the later discovery of an exact property, the derivative discontinuity of the energy It is also used to understand strong correlation, which is beyond the reach of semilocal approximations The paradox of SIC is that equilibrium properties of molecules and solids, including atomization energies and equilibrium geometries, are at best only slightly improved and more typically worsened by it, especially as we pass from local to semilocal and hybrid functionals which by themselves provide a ladder of increasing accuracy for these equilibrium properties The reason for this puzzling ambivalence remains unknown In this speculative chapter, we suggest that the problem arises because the uncorrected functionals provide an inadequate description of compact but noded one-electron orbital densities We suggest that a meta-generalized gradient approximation designed to satisfy a tight lower bound on Advances in Atomic, Molecular, and Optical Physics, Volume 64 ISSN 1049-250X http://dx.doi.org/10.1016/bs.aamop.2015.06.004 # 2015 Elsevier Inc All rights reserved 260 Michael S Murillo and Scott D Bergeson @v + v rv ẳ utị2 rn @t n (32) where u2 ¼ kBTe(t)/mi In this isothermal model, the electron temperature is assumed to be constant in space and much larger than the ion temperature The self similar expansion that solves these equations is described by the velocity equation v0r r, tị ẳ u20 rt 20 + ve2 t2 (33) and also the plasma density, as given in Eq (26) (see also Robicheaux and Hanson, 2002) We add the optical force δf ¼ h/2λτ to the ion momentum equation This produces an additional radial acceleration given by δa ¼ δf : mi (34) The new radial ion momentum equation is @vr @ @n + vr ẳ utị2 + δa: @t @r n @r (35) When the ion trapping is weak, we can linearize the continuity and radial momentum equations about the self similar expansion to obtain à @n1 @  + r ðn0 v1r + n1 v0r ¼ @t r @r (36) and ! @v1r @v1r @v0r Àn1 @n0 @n1 @n0 + v0r + v1r ẳ u2 tị + utị2 Þ1 + δa: @t @r @r n0 @r n0 @r n0 @r (37) For the sake of simplicity, we assume that the optical force is directed radially inward In the MOT configuration with the laser frequency tuned below the resonance, we model the force as depending linearly on the radial coordinate of the ions We also assume that it operates on all velocities below a certain cutoff velocity The acceleration due to this force is of the form δa ¼ CrHðvc À vr Þ: (38) Ultracold Neutral Plasmas Well into the Strongly Coupled Regime 261 Figure 16 A calculation of optical cooling and trapping of ions in a MOT The left panel shows the ion density distribution with (red (black in the print version)) and without (blue (black in the print version) dashed) the optical force after 10 μs of cooling The right panel shows the difference between these two distributions Ions that normally would expand to larger radii are trapped by the MOT laser beams, giving rise to the “negative” density in the right hand panel This calculation used σ ¼ 0.3 mm, Te ¼ 20 K, 106 ions, C ¼ (h/2λτ)(r/3.5 mm), and vc ¼ 100 MHz See text for an explanation of the calculation Because the force itself is small, we replace vr in the Heaviside function by its value in the unperturbed expansion These equations are solved on a radial grid by upwind differencing in r and t The results of a calculation for 106 MOT ions is shown in Fig 16 Two conclusions are immediately obvious One is that the trapping efficiency is somewhat low That can be improved by increasing the initial size of the MOT before ionization, a discussed in Section 9.2 The other, and perhaps more important, is that an ion MOT is expected to be successful on this time scale The fraction of ions trapped in this calculation is 0.5%, the same order of magnitude as estimated in Section 9.3 9.5 Velocity-Changing Collisions One potential issue for laser-cooling the ions comes from velocity-changing collisions The optical force in a MOT is strongest when atoms move away 262 Michael S Murillo and Scott D Bergeson from the trap center At the trap “edge,” the gradient in the magnetic field shifts the atomic energy levels into resonance with red-detuned laser beams The atoms scatter light more efficiently and the photon momentum is transferred to the atoms, pushing them back toward the center of the trap Velocity-changing collisions can occur over relatively long distances In the plasma they are mediated by the Coulomb interaction One can imagine a situation in which an ion near the edge of the plasma begins to scatter photons with high efficiency Normally this ion would slow down However, this slower momentum could be transferred to nearby ions either nearer or farther from the trap center, and the higher momentum from nearby hotter ions could be transferred to the one near the trap edge Therefore an ion that should have been cooled can escape The velocity-changing collision rate is approximately equal to the ion plasma frequency Again, for calcium ions under our typical conditions of n ¼ 1010cmÀ3, the ion plasma frequency is  107sÀ1 The Ca+ photon scattering rate is half of the natural linewidth, or about 11 MHz These two rates are comparable, and this might complicate laser-cooling efforts However, it is important to remember that the collision rate scales with density as n1/2 By moving toward lower density, it is possible to lower the collision rate until it is well below the photon scattering rate 9.6 Preliminary Experiments Trapping Ca+ We have completed some preliminary laser-cooling studies using the ions in our Ca+ plasmas For these initial studies, we created the plasma by photoionizing the laser-cooled Ca atoms in a MOT We tuned the ion MOT laser wavelength to the 397 nm s ! p transition While scanning the laser frequency across the Ca+ absorption profile, we collected laser-induced fluorescence from the plasma as a function of time and probe laser frequency This generates a fluorescence map, similar to the one shown in Fig As mentioned previously, we analyze this data by fitting a Voigt lineshape to the data at a given time The fluorescence signal in Fig is nearly symmetric, but not entirely so A careful analysis shows that the center of the fluorescence signal shifts to lower values at later times The rms frequency at a particular time is calculated as vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi #" # u" X u X (39) frms tị ẳ t fi F i tị F i tị , i i Ultracold Neutral Plasmas Well into the Strongly Coupled Regime 263 Figure 17 The rms frequency of the fluorescence signal plotted in Fig A steady decrease of the rms frequency is observed as a result of the ion MOT laser beams, suggesting that the ion MOT laser beams are confining the ions This data is preliminary It is suggestive but not definitive where F i ðtÞ is the fluorescence signal at a particular time t and the index i runs over the range of measured frequencies The shift of the rms frequency from zero to a lower and constant value indicates that Ca+ ions are being trapped by the laser fields (see Fig 17) The optical pumping problem is significant in Ca+ A typical ion expansion velocity in our system is 20 m/s In the MOT configuration, the laser beams change the ion momentum radially, toward the center of the trap For calcium, the recoil velocity is 2.5 cm/s The ions need to scatter 800 photons to come to rest Unfortunately, the upper state of the transition has a branching fraction to a dark metastable 2D3/2 state of 0.06435 (Ramm et al., 2013) After an ion scatters 15 photons, it is optically pumped over into the dark state This minuscule amount of cooling makes it impossible to trap the ions without a repumper We optically pump the ions out of the 2D3/2 state into the 2P3/2 state But unfortunately for us, this state has a weak branch (BF ¼ 0.0587) to the 2D5/2 state (Gerritsma et al., 2008) With this repumper (at 850 nm), ions scatter 240 photons before being optically pumped into the metastable 2D5/2 state 264 Michael S Murillo and Scott D Bergeson This abbreviated laser-cooling traps ions moving less than m/s—a small fraction of the total population Optically pumping out of the metastable D5/2 state should be possible using a pair of lasers, chopped in time so that they are not both on at the same time This will make it possible to empty the metastable levels while avoiding coherences due to the λ-level scheme 10 DUAL-SPECIES PLASMAS The last innovation that we will discuss in this chapter is our work toward setting up a dual-species ultracold neutral Ca+ and Yb+ plasma The double MOT/plasma experiment opens completely new opportunities to study equilibration, sympathetic cooling, ion acoustic wave damping, and other topics While dual-species MOTs and optical traps are common for neutral atoms, they appear not to have been used for UNP experiments previously A dual species plasma experiment would open exiting new possibilities in plasma physics For example, when very strongly coupled neutral plasmas are generated, either by laser-cooling the ions, adiabatic expansion, working through the Rydberg blockade, or exciting to higher ionization states, it will be important to create new methods for manipulating and probing the strongly-coupled system The double MOT/plasma is ideal for this The two ion species will interact strongly by the Coulomb force, yet each of the ion systems can be addressed separately using spectroscopy The temperature equilibration problem in a two-temperature plasma is unresolved in fusion-class plasmas In a fusion event, an α-particle and neutron are generated The neutron passes out of the system without interacting Unfortunately, the α-particle deposits all of its energy into the electron system In order to get that energy back into the ion system where it can facilitate more fusion events, the electrons and the nuclei have to thermalize The rate at which this occurs is as yet unknown and it is a potentially important problem in fusion science The two-temperature plasma problem is also important for RF heating in fusion plasmas (Tuccillo et al., 2014) Initial work in the dual-plasma experiment will focus on ionizing the Ca and Yb MOT atoms at different times, and studying the way in which the second plasma perturbs the first 11 CONCLUSION In this document, we have attempted to motivate some of our work in UNPs These plasmas operate in a similar Γ–κ range as many high energy Ultracold Neutral Plasmas Well into the Strongly Coupled Regime 265 density plasmas UNPs can be prepared with well-defined initial conditions They can be diagnosed with great precision using laser spectroscopy in nearly real-time These qualities make UNPs ideas systems in which to gain deeper understanding of the physics of “dense” plasma systems ACKNOWLEDGMENTS This work is supported in part by the National Science Foundation (Grant No PHY0969856) and the Air Force (Grant No FA9950-12-1-0308) We thank Ross Spencer for his help in the calculations for laser-cooling the ions REFERENCES Anderegg, F., Dubin, D., O’Neil, T., Driscoll, C., 2009 Measurement of correlationenhanced collision rates Phys Rev Lett 102, 185001, URL http://link.aps.org/doi/ 10.1103/PhysRevLett.102.185001 Baalrud, S.D., 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Population Trapping, Nuclear Spin Cooling, and Lévy Flights in Solid-State Atom-Like Systems Swati Singh*,†, Yiwen Chu{, Mikhail Lukin}, Susanne Yelin†,},1 *ITAMP, Harvard-Smithsonian Center for Astrophysics, Cambridge, Massachusetts, USA † Department of Physics, University of Connecticut, Storrs, Connecticut, USA { Department of Applied Physics, Yale University, New Haven, Connecticut, USA } Department of Physics, Harvard University, Cambridge, Massachusetts, USA Corresponding author: e-mail address: syelin@physics.harvard.edu Contents Introduction Physical System and Experiments: Overview 2.1 Coherent Population Trapping with NV Centers 2.2 Model: Laser–NV Interaction 2.3 Hyperfine Interaction 2.4 Nuclear Bath Dynamics 2.5 Mechanism for Nuclear Spin Cooling Simulating Spin Bath Cooling 3.1 Random Walk Model for Nuclear Spin Diffusion 3.2 MC Simulations for Nuclear Configurations 3.3 Prospects of Lévy Flights in 13C Spin Bath Diffusion 3.4 Measuring the Laser Cooled Spin Bath Photon Statistics 4.1 Photon Random Walk 4.2 Coupled Nuclear Spin-Photon Random Walks 4.3 Photon Statistics Dominated by Nuclear Dynamics Conclusion Acknowledgments Appendix A NV–Laser Interaction Details Appendix B Details of Hyperfine Interaction B.1 Secular Part—Energy Shifts B.2 Nonsecular Part—Nuclear Bath Interactions Appendix C Simulating a Realistic 13C Spin Bath C.1 Concentration Study C.2 Nuclear Diffusion and Lévy Statistics References Advances in Atomic, Molecular, and Optical Physics, Volume 64 ISSN 1049-250X http://dx.doi.org/10.1016/bs.aamop.2015.05.001 274 277 278 281 283 286 287 290 291 292 294 296 297 298 301 303 306 307 307 309 310 312 314 314 319 324 # 2015 Elsevier Inc All rights reserved 273 274 Swati Singh et al Abstract We describe and analyze a method for controlling nuclear spin environment of atomlike quantum emitters in the solid state The method makes use of laser manipulation of an electronic spin transition via coherent population trapping Specifically, we present a detailed description of the nuclear spin dynamics and its interplay with the optical excitation of the electronic spin of nitrogen-vacancy color centers in diamond We introduce a simple model of this process that allows us to study both optimal cooling parameters for nuclear spins and optimal information transfer between the optical measurement of the electron and the nuclear bath dynamics This allows us to investigate the statistical properties of the nuclear spin bath Potential applications to quantum information processing and quantum metrology are possible INTRODUCTION Intense recent interest in solid-state quantum emitters is being driven by their unique potential applications to nanoscale sensors (Balasubramanian et al., 2008; Kucsko et al., 2013; Maletinsky et al., 2012; Maze et al., 2008; Sage et al., 2013) and realization of novel quantum information platforms (Bernien et al., 2013; Neumann et al., 2008) These atom-like solid-state defects combine the tunability and precision of atomic systems, along with the robust and scalable infrastructure provided by solid-state devices (Weber et al., 2010) Many defects in solid-state materials, such as quantum dots in semiconductors (Hennessy et al., 2007) and color centers in diamond (Wrachtrup and Jelezko, 2006) or silicon (Baranov et al., 2011) have emerged as competitive platforms for such quantum based technologies However, the practical performance of these systems is limited by strong interactions of solid-state defects with their local environment For example, the electronic spin coherence properties of atom-like systems are limited by the random Overhauser field, which is the effective field created by the interaction of the random surrounding nuclear spins (Awschalom et al., 2013) It originates due to the interaction of the electron spin with the surrounding nuclear spins consisting of both Fermi contact and dipole–dipole interactions, which cannot be turned off entirely using dynamical decoupling techniques The goal of nuclear spin cooling is to eliminate or considerably suppress the electronic spin decoherence due to surrounding nuclei Within this context, nuclear spin cooling can take place when one or several electron Coherent Population Trapping, Nuclear Spin Cooling, and Lévy Flights 275 spins, polarized and controlled by fine-tuned lasers and other external fields, interact with all surrounding nuclear spins This can amount to about 104–105 nuclear spins of all the surrounding atoms in typical semiconductor quantum dots or to as few as the 10–100 spins of the 13C atoms in an otherwise spinless 12C diamond lattice The quantum dynamics of a solid-state defect center can be described by a central spin model, in which the (central) electron spin can be controlled directly, while the environment of nuclear spins is only indirectly accessible via the electron (Prokof’ev and Stamp, 2000) The spin interactions existing in this system encompass three distinct time scales: (i) the interaction of the electron spin with electromagnetic fields, in the range of 1–100 MHz; (ii) the hyperfine and dipole–dipole interaction between electron and nuclear spins (of the order of 0.01–10s of MHz), which are responsible for the Overhauser field; and (iii) the dipole–dipole interaction among the nuclear spins, of the order of mostly a few kHz One specific approach involves the optical technique known as “coherent population trapping” (CPT) The essence of this techniques, widely known in atomic systems, is that the atomic population is trapped in a superposition of electronic states, decoupled from the laser fields, known as the “dark state.” Over the past two decades, CPT has been employed for laser cooling of neutral atoms (Aspect et al., 1988) and ions (Roos et al., 2000), creation of ultra-cold molecules (Ni et al., 2008), optical magnetometry (Budker and Romalis, 2007; Scully and Fleischhauer, 1992), and atomic clocks (Vanier, 2005), as well as for slowing and stopping light pulses (Fleischhauer et al., 2005) The same technique has been applied to solid-state systems including the nitrogen-vacancy (NV) center in diamond (Santori et al., 2006), and individual quantum dots (Xu et al., 2009) where interesting dynamics between the quantum dot and its nuclear environment have been observed In particular, this quantum optical technique can be used for cooling, measurement and manipulation of artificial atom-like solid-state systems and their local environment (Giedke et al., 2006; Issler et al., 2010; Stepanenko et al., 2006; Xu et al., 2009) Specifically, the method can be applied to cooling, real-time projective measurement and control of the nuclear spin environment surrounding the electronic spin qubit associated with individual NV centers in diamond (Batalov et al., 2009; Hanson et al., 2006) The NV center has a long-lived spin triplet as its electronic ground state (Manson et al., 2006), whose ms ¼ Ỉ1,0 sublevels are denoted as jỈ1i and j0i in the following In pure ... experience and knowledge in evaluating and using any information, methods, compounds, or experiments described herein In using such information or methods they should be mindful of their own safety and. .. corresponding delocalized Bloch functions In general, finding the exact WF is a difficult problem; however, Heaton and Lin (1984) and Erwin and Lin (1988) described a method for obtaining simple... uncorrected DFT Jackson and Lin addressed two systems, NaCl:Cu+ and LiCl:Ag+ ( Jackson and Lin, 1988, 1990) These calculations are described in the following sections Erwin and Lin also treated a similar