![1.2 Elements of Quantum Information Theory 9
equivalent. Yet, the correlations of separable, nonzero discord states are still of clas-
sical origin as they can be characterized by a probability distribution [see discussion
preceding Eq. (1.2)].
In the light of the above discussion, we conclude that states of nonzero discord are
correlated quantum states which show quantum properties (quantum noise, measure-
ment disturbance, nonzero commutators) under the influence of local measurements.
The nature of their correlations is, however, only quantum when they are also entan-
gled. A more detailed discussion will be given in Sect.3.1.
1.2.2 Role of Correlated Quantum States in Quantum
Information Theory
Much effort is spent to define, characterize, and quantify different notions of corre-
lations in quantum states including quantum entanglement and discord (Horodecki
et al. 2009; Modi et al. 2012), usually motivated by referring to their role as a resource
for certain quantum information tasks. But which quantity is actually relevant? This
question cannot be answered in general. Instead, a clear answer can be given only
in few special cases. Even then, it depends very much on the task one aspires to
accomplish.
For simplicity, let us first restrict to pure states. In this case, all of the different
concepts of correlations collapse into the same notion and the set of zero-discord
states coincides with the set of separable states; they even coincide with completely
uncorrelated product states. Furthermore, any entangled pure state can violate a Bell
inequality (Bell 1964), which is useful, for instance, to establish certified, secure
quantum key distribution5
(Ekert 1991). Pure state entanglement is rather well char-
acterized by now (Mintert et al. 2005; Horodecki et al. 2009), even though this already
represents a very intricate task once multipartite states are considered.
The picture becomes considerably more complex when we consider mixed states.
Not every mixed entangled state is able to violate a Bell inequality (Popescu 1994),
hence, not all mixed entangled states serve to distribute a certified, secure quan-
tum key. Yet, other tasks, such as quantum teleportation, can be accomplished if
only some nonzero entanglement is available (Vidal and Werner 2002). So clearly,
it is not always necessary to ask for the strongest incarnations of correlations in
quantum states. This suggests a hierarchical structure of correlated quantum states,
with states that violate a Bell inequality on the top. Entangled states form a subset
thereof, and an even weaker class of correlated quantum states are those with nonzero
discord. The remaining set, the classical (zero discord) states, also includes the com-
pletely uncorrelated product states. The set of zero-discord states are actually very
5Note that the first proposal for a quantum key distribution protocol did not make use of quantum
correlations (Bennett and Brassard 1984). Instead, it uses the destructive nature of the quantum
measurement and the no-cloning theorem (Wootters and Zurek 1982; Dieks 1982) to rule out the
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- 4. SpringerTheses Recognizing Outstanding Ph.D. Research Dynamicsand Characterizationof Composite QuantumSystems Manuel Gessner
- 5. Springer Theses Recognizing Outstanding Ph.D. Research
- 6. Aims and Scope The series “Springer Theses” brings together a selection of the very best Ph.D. theses from around the world and across the physical sciences. Nominated and endorsed by two recognized specialists, each published volume has been selected for its scientific excellence and the high impact of its contents for the pertinent field of research. For greater accessibility to non-specialists, the published versions include an extended introduction, as well as a foreword by the student’s supervisor explaining the special relevance of the work for the field. As a whole, the series will provide a valuable resource both for newcomers to the research fields described, and for other scientists seeking detailed background information on special questions. Finally, it provides an accredited documentation of the valuable contributions made by today’s younger generation of scientists. Theses are accepted into the series by invited nomination only and must fulfill all of the following criteria • They must be written in good English. • The topic should fall within the confines of Chemistry, Physics, Earth Sciences, Engineering and related interdisciplinary fields such as Materials, Nanoscience, Chemical Engineering, Complex Systems and Biophysics. • The work reported in the thesis must represent a significant scientific advance. • If the thesis includes previously published material, permission to reproduce this must be gained from the respective copyright holder. • They must have been examined and passed during the 12 months prior to nomination. • Each thesis should include a foreword by the supervisor outlining the signifi- cance of its content. • The theses should have a clearly defined structure including an introduction accessible to scientists not expert in that particular field. More information about this series at http://www.springer.com/series/8790
- 7. Manuel Gessner Dynamics and Characterization of Composite Quantum Systems Doctoral Thesis accepted by Albert Ludwigs University of Freiburg, Germany 123
- 8. Author Dr. Manuel Gessner Quantum Science and Technology in Arcetri (QSTAR) European Laboratory for Non-Linear Spectroscopy (LENS) Florence Italy Supervisor Prof. Andreas Buchleitner Institute of Physics Albert Ludwig University of Freiburg Freiburg im Breisgau Germany ISSN 2190-5053 ISSN 2190-5061 (electronic) Springer Theses ISBN 978-3-319-44458-1 ISBN 978-3-319-44459-8 (eBook) DOI 10.1007/978-3-319-44459-8 Library of Congress Control Number: 2016949575 © Springer International Publishing AG 2017 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. Printed on acid-free paper This Springer imprint is published by Springer Nature The registered company is Springer International Publishing AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland
- 9. Supervisor’s Foreword Formidable progress in the control and assembly of single constituents of matter has been achieved during the last two decades in quantum optics labs around the world. Experiments with trapped ions and atoms, with tailored states of light, and with metamaterials provide versatile platforms to study and optimize essential building blocks of future quantum computing architectures, to explore the observable dynamical and spectral features of fundamental models of condensed matter theory, and to monitor the emergence of robust macroscopic properties from the micro- scopic dynamics of an increasing number of coupled degrees of freedom. The complexity of such highly engineered, composite quantum systems rapidly increases with the number of interwoven degrees of freedom, and novel tools are required to characterize, monitor and control their dynamical behaviour, in exper- iment and theory, with scalable overhead. The present treatise by Manuel Gessner blends advanced theoretical tools from the theory of open quantum systems, quantum optics, quantum chaos, quantum information theory, and quantum statistics, to provide a fresh perspective on the characterisation and control of “complex” quantum systems such as those now within reach in quantum optics laboratories. Often with specific experimental implementations in mind, this text not only offers a pedagogical introduction to the broader scientific context (together with the relevant literature) and the necessary theoretical machinery, but also elaborates on several innovative applications. Readers with an interest in the efficient probing of collective and spectral properties of composite quantum systems, in their open-system entanglement dynamics, in many-particle dephasing phenomena and in the impact thereon of the particles’ mutual (in)distinguishability will find this text both informative and inspiring. Newcomers to the field will appreciate being able to follow the essential physics issues from their theoretical foundations to cutting-edge research. Freiburg im Breisgau, Germany Prof. Andreas Buchleitner June 2016 v
- 10. Abstract Due to experimental developments over the past decades on quantum optical and atomic systems, a wealth of composite quantum systems of variable sizes has become accessible under rather well-controlled conditions. Typical systems of tens of trapped ions or thousands of cold neutral atoms are usually too large to fully measure all of their constituents’ microscopic quantum properties, but not large enough to be described completely in terms of thermodynamic quantities. This challenging intermediate regime of controllable quantum few- to many-body sys- tems is particularly interesting, since it combines a variety of different phenomena and applications, ranging from quantum information theory to solid-state physics. The effective characterization of these systems requires flexible and experi- mentally feasible observables, complemented by efficient theoretical methods and models. In this dissertation we employ concepts from the fields of open quantum systems, quantum information theory, quantum many-body theory and physical chemistry, to construct dynamical approaches for the study of various aspects of correlations, and to describe spectral and dynamical features of complex, interacting quantum systems. Some of the developed theoretical ideas are complemented by experimental realizations with trapped ions or photons. To facilitate the scalable analysis of bipartite correlation properties in the context of quantum information theory, we introduce a method which allows to detect and estimate discord-type correlations when only one of the two correlated subsystems can be measured. The method makes use of the influence of the correlations on the local subsystem dynamics, which illustrates the fundamental role of initial corre- lations for the theory of open quantum systems. We present an experimental realization with a single trapped ion, as well as the description of a photonic experiment. Further theoretical studies are presented, including the application to a spin-chain model, which relates the dynamical single-spin signature of the ground-state quantum correlations to a quantum phase transition. Having established this local detection technique for quantum discord, whose information about the state’s correlations is limited, we introduce the correlation rank to assess the degree of total correlations of bipartite quantum states. This allows us to identify strongly correlated states which cannot be generated with local vii
- 11. operations. Classically correlated noise processes, however, are able to generate strongly correlated, but separable quantum states. This is confirmed in a trapped-ion experiment, where such noise processes occur naturally, and represent one of the dominant sources of error. We further develop a fully analytical description of the generated ensemble-average dynamics, allowing us to de derive conditions that ensure the robustness of entanglement in bipartite and multi-particle systems. Information-theoretic quantifiers of the correlation properties between the con- stituents no longer represent suitable observables for increasingly complex com- posite quantum systems. Hence, we develop a multi-configurational mean-field approach in order to understand the dynamical features and, with it, the role of the energy spectrum in the vicinity of the quantum phase transition in a quantum magnet. Specifically, we study a spin-chain model with variable-range interactions in a transverse field, which can be realized in trapped-ion quantum simulators. The obtained semiclassical model allows for an analytical analysis of the excitation spectrum, whose predictions are exact in the limit of very strong or vanishing external magnetic fields. Bifurcations of a series of excited-state energy landscapes below a threshold value of the external magnetic field reflect the quantum phase transition from the paramagnetic phase to the (anti-)ferromagnetic phase in the entire excitation spectrum—and not just in the ground state. To develop a set of experimentally accessible, suitable observables, able to cope with complex dynamics in quantum optical systems, we develop a general frame- work based on ideas from nonlinear spectroscopy. Sequences of phase-coherent laser pulses allow us to extract multi-time correlation functions, which may be combined with single-site addressability to achieve spatial resolution. We propose specific schemes to realize the elementary steps with existing trapped-ion tech- nology, and discuss a variety of applications based on second-order and fourth-order signals. The obtained multi-dimensional spectra are particularly suited to extract information about the system’s environmental influences, the relevant transport mechanisms, and particle–particle interactions. The theoretical description of interacting many-body systems becomes particu- larly hard when the quantum statistics is explicitly taken into account. Generalizing concepts from open-system theory to the case of identical particles, we study the dynamics of a subsystem of interacting bosons. We obtain a hierarchical expansion of the coherent subsystem evolution, which can be truncated by a mean-field ansatz. When applied to a dilute Bose–Einstein condensate, we recover the Gross– Pitaevskii equation. Based on a perturbative second-order expansion in the inter- action strength, we establish first steps towards a microscopic derivation of a master-equation description that is able to account for interaction-induced decoherence. viii Abstract
- 12. Acknowledgments My sincere gratitude goes to Andreas Buchleitner for advising and supporting me and my work on this dissertation, for carefully reading the manuscript, as well as for the great experience I had in the ninth floor of the Institute of Physics in Freiburg. I would further like to thank Hartmut Häffner for organizing my stay in Berkeley, and his group, especially Michael Ramm and Thaned (Hong) Pruttivarasin, for a great time, including, but not limited to the lab. Just like the experiments in Berkeley, many parts of this dissertation are the result of collaborations. In particular, I would like to thank Heinz-Peter Breuer for continuous advice and a long-standing collaboration on open quantum systems, Frank Schlawin for the stimulating joint work on nonlinear spectroscopy, and Edoardo Carnio, for his talented work on collective dephasing during his Master’s thesis. I would like to thank Elsi-Mari Laine and Jyrki Piilo for hosting me in Turku, as well as Tobias Brandes and Victor Bastidas for the time in Berlin. I would also like to thank Shaul Mukamel for interesting discussions and collaborations. Moreover, my gratitude goes to Christian Roos and his team, in particular, Ben Lanyon, for the collaboration. I also thank Chuan-Feng Li and his group in Hefei. Moreover, I thank everyone in the ninth floor for the good atmosphere. Special thanks go to Stefan Fischer and Frank Schlawin with whom I enjoyed uncountable lunch and coffee breaks. I also thank Ugo Marzolino and Mattia Walschaers for frequent discussions. Further thanks go to Gislinde Bühler and Susanne Bergmann for a lot of organizational support. I thank the German National Academic Foundation (Studienstiftung des deut- schen Volkes) for supporting my work on this thesis and providing me with travel funds. Last but not least I thank my parents, my sisters and Patricia. Freiburg im Breisgau, Germany Manuel Gessner June 2015 ix
- 13. Contents 1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Recent Advances and Current Challenges: A Brief Overview . . . . 1 1.2 Elements of Quantum Information Theory . . . . . . . . . . . . . . . . . . . 4 1.2.1 Correlations in Composite Quantum Systems: Entanglement and Discord . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.2.2 Role of Correlated Quantum States in Quantum Information Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.3 Controllable Quantum Systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.3.1 Trapped Ions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.3.2 Cold Gases of Neutral Atoms in Optical Lattices . . . . . . . . 25 1.3.3 Photons and (Non-)linear Optics . . . . . . . . . . . . . . . . . . . . . 28 1.3.4 Other Systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 1.4 The Certification of Large-Scale Quantum Devices . . . . . . . . . . . . 31 1.5 Nonlinear Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 1.6 Theoretical Description of Composite Quantum Systems . . . . . . . . 37 1.6.1 Semiclassical Approximations and Mean-Field Theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 1.6.2 Complex Systems, Spectral Analysis, and Random Matrix Theory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 1.6.3 Open Quantum Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 1.6.4 Identical Particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 1.7 Quantum Phase Transitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 1.8 Scope and Structure of This Dissertation . . . . . . . . . . . . . . . . . . . . 52 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 2 Local Detection of Correlations in Composite Quantum Systems . . . 69 2.1 The Local Detection Protocol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 2.1.1 Local Witness for Bipartite Quantum Discord. . . . . . . . . . . 70 2.1.2 Local Bound for the Minimum Entanglement Potential . . . 72 2.1.3 Efficacy of the Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 xi
- 14. 2.2 Trapped-Ion Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 2.2.1 Resonant Laser-Ion Interactions . . . . . . . . . . . . . . . . . . . . . 77 2.2.2 The Effect of Small Detunings . . . . . . . . . . . . . . . . . . . . . . 82 2.2.3 The Local Detection Protocol for the First Blue Sideband. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 2.2.4 Generalization to Arbitrary Sidebands. . . . . . . . . . . . . . . . . 94 2.2.5 Extension of the Experimental Technique . . . . . . . . . . . . . . 97 2.3 Photonic Experiment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 2.3.1 The Pre-initial State . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 2.3.2 Preparation of the Initial State. . . . . . . . . . . . . . . . . . . . . . . 99 2.3.3 Local Dephasing Operation. . . . . . . . . . . . . . . . . . . . . . . . . 100 2.3.4 Reduced Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 2.3.5 Total Trace Distance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 2.3.6 Open-System Evolution Depending on Initial Correlations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 2.3.7 Local Trace Distance. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 2.4 Theoretical Studies of Further Examples . . . . . . . . . . . . . . . . . . . . 109 2.4.1 Atom-Photon Correlations During Spontaneous Decay . . . . 109 2.4.2 Many-Mode Extension of the Trapped-Ion Experiment: A Proposal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 2.4.3 Quantum Phase Transition in a Transverse-Field Ising Chain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 2.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 3 From Local Operations to Collective Dephasing: Behavior of Correlated Quantum States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 3.1 Creation of Quantum Discord by Local Operations . . . . . . . . . . . . 129 3.2 Correlation Rank: Schmidt Decomposition for Mixed States . . . . . 131 3.3 Trapped-Ion Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 3.3.1 Local Amplitude Damping . . . . . . . . . . . . . . . . . . . . . . . . . 134 3.3.2 Collective Dephasing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 3.4 General Dynamics of Collective Dephasing . . . . . . . . . . . . . . . . . . 139 3.4.1 Kraus Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 3.4.2 Ensemble-Average Evolution: Interpretation and Non-Markovian Effects. . . . . . . . . . . . . . . . . . . . . . . . . 142 3.4.3 Robustness of Bipartite Entanglement. . . . . . . . . . . . . . . . . 143 3.4.4 Time-Invariant States: Multipartite Werner States . . . . . . . . 144 3.4.5 Robustness of Multipartite Entanglement . . . . . . . . . . . . . . 145 3.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 xii Contents
- 15. 4 Quantum Phase Transition in a Family of Quantum Magnets . . . . . 151 4.1 Variable-Range Quantum Magnets: From Ising to Lipkin–Meshkov–Glick . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 4.1.1 A One-Parameter Family of Models . . . . . . . . . . . . . . . . . . 151 4.1.2 Special Case: Nearest-Neighbor Ising Model . . . . . . . . . . . 153 4.1.3 Special Case: Fully-Connected Lipkin–Meshkov–Glick Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 4.2 Single-Spin Signatures of a Quantum Phase Transition . . . . . . . . . 154 4.2.1 Distribution of Dephasing-Induced Excitations . . . . . . . . . . 155 4.3 Spectral Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158 4.3.1 Density of States. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160 4.3.2 Level Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162 4.4 Semiclassical Mean-Field Description. . . . . . . . . . . . . . . . . . . . . . . 164 4.4.1 Semiclassical Approximations and Quantum Phase Transitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164 4.4.2 Spin Coherent States. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166 4.4.3 Multidimensional Numerical Search for Critical Points. . . . 168 4.4.4 Analytical Critical Points from a Set of Single-Parameter Energy Landscapes . . . . . . . . . . . . . . . 173 4.4.5 Performance of the Semiclassical Ansatz . . . . . . . . . . . . . . 181 4.4.6 Scaling of Highest and Lowest Eigenvalues . . . . . . . . . . . . 187 4.4.7 Distribution of Critical Fields . . . . . . . . . . . . . . . . . . . . . . . 191 4.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197 5 Multidimensional Nonlinear Spectroscopy of Controllable Quantum Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201 5.1 Introduction: Nonlinear Spectroscopy and Controllable Quantum Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201 5.2 Phase-Coherent Two-Pulse Measurements of Atomic Vapor . . . . . 202 5.2.1 Atomic Vapor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203 5.2.2 Light-Matter Interactions. . . . . . . . . . . . . . . . . . . . . . . . . . . 204 5.2.3 Fluorescence Measurements . . . . . . . . . . . . . . . . . . . . . . . . 205 5.2.4 Single Quantum Coherence. . . . . . . . . . . . . . . . . . . . . . . . . 206 5.2.5 Phase Cycling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209 5.2.6 Second-Order Quantum Coherence . . . . . . . . . . . . . . . . . . . 210 5.2.7 Higher-Order Quantum Coherence . . . . . . . . . . . . . . . . . . . 216 5.2.8 Coupling to Vacuum Modes . . . . . . . . . . . . . . . . . . . . . . . . 217 5.2.9 Interpretation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220 5.3 Diagrammatic Description of Nonlinear Spectroscopic Experiments of Controllable Quantum Systems . . . . . . . . . . . . . . . 220 5.3.1 Basic Elements for the Design of Nonlinear Measurement Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221 5.3.2 Externally Induced Excitations and De-Excitations . . . . . . . 222 Contents xiii
- 16. 5.3.3 Time Evolution and Readout. . . . . . . . . . . . . . . . . . . . . . . . 223 5.3.4 Design of a Pulse-Sequence . . . . . . . . . . . . . . . . . . . . . . . . 224 5.4 Excitation and Readout Schemes for Trapped Ions. . . . . . . . . . . . . 225 5.4.1 Electronic Degree of Freedom: Dynamics. . . . . . . . . . . . . . 226 5.4.2 Electronic Degree of Freedom: Excitation and Readout . . . 226 5.4.3 Vibrational Degree of Freedom: Dynamics . . . . . . . . . . . . . 227 5.4.4 Vibrational Degree of Freedom: Excitation . . . . . . . . . . . . . 228 5.4.5 Vibrational Degree of Freedom: Readout . . . . . . . . . . . . . . 233 5.5 Nonlinear Signals and Applications for Trapped-Ion Systems . . . . 234 5.5.1 Two-Pulse Sequence: Single Quantum Coherence . . . . . . . 236 5.5.2 Four-Pulse Sequence: Double Quantum Coherence. . . . . . . 247 5.5.3 Four-Pulse Sequence: Photon Echo. . . . . . . . . . . . . . . . . . . 250 5.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254 6 Open Quantum Systems of Identical Particles . . . . . . . . . . . . . . . . . . 257 6.1 Introduction: Identical Particles. . . . . . . . . . . . . . . . . . . . . . . . . . . . 257 6.1.1 Symmetrized States. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258 6.1.2 The Single Particle Subspace . . . . . . . . . . . . . . . . . . . . . . . 260 6.1.3 N-Particle Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262 6.1.4 The Single-Particle Density Operator and Partial Trace. . . . 262 6.2 General Formalism for Symmetrized States . . . . . . . . . . . . . . . . . . 263 6.3 Density Matrices and Expectation Values. . . . . . . . . . . . . . . . . . . . 267 6.3.1 N-Particle Density Operator . . . . . . . . . . . . . . . . . . . . . . . . 267 6.3.2 Matrix Elements and Traces in a Larger Hilbert Space . . . . 268 6.3.3 M-Particle Reduced Density Operator . . . . . . . . . . . . . . . . . 269 6.3.4 Many-Body Hamiltonian. . . . . . . . . . . . . . . . . . . . . . . . . . . 271 6.3.5 Bosonic Product States . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271 6.3.6 Field Operator Representation. . . . . . . . . . . . . . . . . . . . . . . 273 6.4 Coherent Dynamics of a Subgroup of Interacting Bosons. . . . . . . . 276 6.4.1 Time Evolution of Many-Body Quantum Systems . . . . . . . 276 6.4.2 The Gross–Pitaevskii Equation . . . . . . . . . . . . . . . . . . . . . . 278 6.4.3 Hierarchical Expansion of the Reduced Bosonic Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 280 6.4.4 From the Hierarchical Expansion to Gross–Pitaevskii . . . . . 287 6.5 Second-Order Master Equation for Identical Particles: Incoherent Effects. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288 6.5.1 Single-Particle Subdynamics of a Two-Particle Bosonic System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289 6.5.2 General Ansatz: M-Particle Subdynamics of an N-Particle Bosonic System. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294 6.5.3 Operator Structures Within the Pure Product State Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297 6.5.4 General Bosonic Master Equation Under the Pure Product State Assumption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302 xiv Contents
- 17. 6.5.5 Single-Particle Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . 303 6.5.6 Nonlinear Redfield-Type Equation . . . . . . . . . . . . . . . . . . . 304 6.5.7 Mean-Field Approximation . . . . . . . . . . . . . . . . . . . . . . . . . 305 6.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309 7 Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314 Appendix A: Derivation of the Bosonic Master Eq. (6.151) . . . . . . . . . . . 315 Appendix B: Representation of Double-Commutator Terms in a Larger Basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331 Appendix C: Transformation Properties of Interaction-Picture Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335 Curriculum Vitae . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345 Contents xv
- 18. Chapter 1 Introduction Abstract In this chapter, we provide the background of different conceptual approaches to quantum many-body systems, as they will become relevant in the course of this dissertation. We first introduce basic ideas of quantum information the- ory, with an emphasis on the role of correlated quantum states. This is followed by an overview over a selection of existing technologies, allowing for high-precision exper- iments on quantum mechanical systems. Experiments on trapped ions are described in particular detail, as they represent a recurring theme throughout this dissertation. Interesting applications, such as quantum simulations, lead us to the discussion of reliable tools to diagnose increasingly larger quantum systems. In this context, we discuss a very powerful probing technique from a different background: Nonlinear spectroscopy. We further review some commonly employed theoretical methods to describe complex quantum systems, including concepts of random matrix theory, the theory of open quantum systems, and the description of interacting, identical particles. Finally, the phenomenology and description of quantum phase transitions is discussed. 1.1 Recent Advances and Current Challenges: A Brief Overview While the fundamental laws of quantum mechanics were developed in the last century (Cohen-Tannoudji et al. 1977a,b; Sakurai and Napolitano 1994), recent experimental developments have allowed to probe, and confirm, their validity on the single particle level under well-controlled conditions (Phillips 1998; Cohen-Tannoudji 1998; Chu 1998; Wineland 2013; Haroche 2013). The theoretical and experimental character- ization of composite quantum systems based on the knowledge of the basic laws is, however, rather challenging due to the rapidly increasing number of degrees of freedom. In the extreme limit of large-scale ensembles of ∼1023 particles, suitable approximations and tools from the well-established fields of solid-state physics or statistical mechanics allow for the efficient description of macroscopic observables (Landau and Lifshitz 1969; Mahan 2000). A remaining challenge to both theory and experiment is the intermediate regime of few- to many-body composite quan- © Springer International Publishing AG 2017 M. Gessner, Dynamics and Characterization of Composite Quantum Systems, Springer Theses, DOI 10.1007/978-3-319-44459-8_1 1
- 19. 2 1 Introduction tum systems in the presence of interactions, which shall be the subject of the present dissertation. In recent years, this regime has become available, for example, in exper- iments on trapped atomic particles (Bloch et al. 2008; Häffner et al. 2008; Schneider et al. 2012; Blatt and Roos 2012). Quantum optical experiments, involving, for instance, trapped ions, cold atoms, or photons, represent a flexible testbed for studying quantum systems of variable sizes, ranging from elementary, small-scale systems of few degrees of freedom (Leibfried et al. 2003a; Kok et al. 2007; Häffner et al. 2008), to the many-body dynamics of a Bose–Einstein condensate (Pitaevskii and Stringari 2003; Bloch et al. 2008). As the system size changes, the experimentally accessible observables that characterize the system’s properties, as well as the theoretical methods which predict their behavior, need to be adjusted accordingly. Let us first provide a very brief overview of differ- ent manifestations of controllable, composite quantum systems, their predominant characteristics, their relevant observables, and the emerging challenges. As long as one limits to effectively very low-dimensional quantum systems, the full collection of microscopic quantum properties can be measured by quantum state tomography (Paris and Řeháček 2004) in the experiment, as well as numerically or even analytically handled in theory. Prominent examples are the, by now experimen- tally well-controlled, elementary building blocks of quantum information theory: Systems of few two-level systems can be encoded, for instance, into the electronic states of trapped ions (Häffner et al. 2008), or the polarization degrees of freedom of individual photons (Kok et al. 2007). In this context, correlation properties are frequently at the center of the theoretical analysis, since they often can identify a boarder between “quantum” and “classical” features, and, in some cases, also play an important role for applications of quantum information theory (Horodecki et al. 2009; Modi et al. 2012). As the number or the dimension of the constituents of composite quantum systems increases, it is no longer reasonable to measure the entirety of microscopic properties experimentally—feasible full state tomography is limited to systems of roughly no more than ten two-level systems (Häffner et al. 2005a). Thus, the electronic levels of a collection of ten to twenty trapped ions already represent a system that is beyond the reach of standard tools of quantum information theory. Similarly, the system rapidly escapes the easily manageable parameter range if other degrees of freedom, such as the ions’ motion in the trap potential, are taken into account. Characterization of the correlation properties in this intermediate, few-particle regime, is significantly complicated by the emerging multipartite nature (Mintert et al. 2005; Gühne and Tóth 2009), and, experimentally, the preparation of specific quantum states and their efficient isolation from external noise sources becomes more and more challenging (Schindler et al. 2013). When the isolation of the system from its environment is no longer possible, an open-system description that takes decoherence into account is required (Breuer and Petruccione 2002). In fact, we may also consider the bipar- tite setting of system and environment as another instance of a composite quantum system. The larger number of particles also leads to the emergence of new phenomena, such as the onset of collective effects with macroscopically observable signatures;
- 20. 1.1 Recent Advances and Current Challenges: A Brief Overview 3 for instance, a measurable magnetization generated by the long-range order of the atom’s spins, determined by their mutual interaction. These macroscopic properties may further depend on externally controllable parameters, for example, the strength of an external magnetic field. Tuning these parameters, thus, allows to employ con- trollable quantum systems for the observation of quantum phase transitions, that is, the macroscopic change of a system’s properties induced by the variation of an external parameter (Sondhi et al. 1997; Sachdev 1999; Vojta 2003). When a system transitions from one phase into another at zero temperature, its ground state under- goes a non-analytic change. Despite indications that this transition also affects the excitation spectrum (Emary and Brandes 2003a), the relation between the quantum phase transition and the excited states is not yet well understood in general. Collec- tive effects may further arise due to the interaction of an ensemble of particles with a common environment (Dicke 1954), which, on the one hand, can lead to enhanced decoherence, but, on the other hand, given a detailed understanding of the resulting dynamics, allows for the efficient control of available parameters, such that coherent evolution can be protected (Palma et al. 1996). The signature of phase transitions typically becomes more and more pronounced as the number of particles increases. Cold, trapped ensembles of neutral atoms repre- sent a platform for composite, controllable quantum systems (Bloch et al. 2008), with yet another parameter regime, characterized by a significantly higher particle num- ber (∼103 −106 ) and density than that of trapped-ion systems. Moreover, appropriate design of the potential landscape via optical lattices can render cold-atom systems formally equivalent to solid-state systems (Jaksch et al. 1998). Cold atoms, therefore, allow for the studies of a variety of physical phenomena, including many-particle dynamics from experimentally controlled conditions (Greiner et al. 2002) to truly complex settings (Moore et al. 1995; Raizen 1999; Oberthaler et al. 1999; Hensinger et al. 2001; Madroñero et al. 2006; Modugno 2010), and the emergence of a semi- classical limit (Smerzi et al. 1997; Wimberger et al. 2003; Hiller et al. 2009). The high particle density implies that the quantum statistics, induced by the particles’ indistinguishability, needs to be taken explicitly into account. Typical experimen- tally accessible observables that characterize the properties of quantum many-body systems of identical particles are single-particle observables, such as the average momentum. In the presence of interactions their theoretical description is severely complicated by the particles’ indistinguishability, and a microscopic theory of the effective dynamics and decoherence (Buchleitner and Kolovsky 2003; Meinert et al. 2014) of a subset of identical particles is presently not available. Hence, composite quantum systems, as represented by state-of-the-art quantum optical experiments, span a large range of system sizes and physical phenomena. Their characterization requires the identification of appropriate observables, as well as the development of theoretical models and tools for their efficient description. The systems mentioned above often allow for a surprisingly flexible experimental frame- work: Despite the large number of degrees of freedom, systems of trapped atomic particles allow for a remarkable level of quantum control, for instance, by providing laser access to individual constituents (Häffner et al. 2008; Weitenberg et al. 2011). Moreover, the interactions between the particles—to some extent—can be controlled
- 21. 4 1 Introduction externally, and the typical microsecond time scales of the associated evolution allow for the convenient time-resolved measurements of observables, employing nanosec- ond laser pulses. This rather convenient experimental access to the quantum dynam- ics stands in contrast to its theoretical description. Interacting composite quantum systems between tens and thousands of particles represent the most challenging para- meter regimes: Full diagonalization is no longer plausible, efficient numerical tools are often limited to the treatment of weakly correlated states, and have troubles to predict the dynamical features of systems with more and more complex interactions (Schollwöck 2005), and, yet, the system is not large enough to be described purely on the level of thermodynamic observables. Therefore, this combination of high- dimensional Hilbert spaces, complex interactions, and high level of experimental access, provides an unprecedented, highly versatile, yet challenging setting. In the present dissertation we employ methods from fields of rather distinct back- grounds to develop experimentally accessible observables and suitable theoretical approaches that are able to treat a broad range of composite quantum systems of very different sizes and phenomena, as outlined above. The theoretical ideas are further- more combined with experimental realizations based on trapped ions and photons. In particular, Chaps.2 and 3 are founded on the fields of quantum information the- ory and open quantum systems. Specifically, we introduce a dynamical observable for correlation properties of quantum states, when tomographic access to a part of the full system is feasible in Chap.2, and analyze the generation and protection of correlations under a collective dephasing process in Chap.3. We move towards more complex systems in Chap.4, where we develop a semiclassical mean-field approach to describe the excitation spectrum in the context of a quantum phase transition. Chapter 5 deals with the development of a rather general toolbox to experimentally probe multi-time correlation functions of controllable quantum many-body systems based on ideas from nonlinear spectroscopy, a formalism originally developed for physical chemistry. Finally, in Chap.6, combining the open-system perspective with the indistinguishability of quantum particles in many-body systems, we microscop- ically describe the dynamics of a subsystem of interacting, identical particles, from coherent, mean-field dynamics towards the description of incoherent effects. In the following we introduce the different basic concepts in further detail, and we will also conclude with a more detailed view on the scope and structure of the thesis in Sect.1.8. 1.2 Elements of Quantum Information Theory The experimental control of objects behaving according to the laws of quantum mechanics has become common practice in many laboratories around the world. Today, there exists a large list of controllable quantum systems, ranging from individ- ual atoms, ions and photons via molecules to nano- or mesoscopic solid-state devices (see also Sect.1.3). Besides the possibility to explore a wide range of interesting
- 22. 1.2 Elements of Quantum Information Theory 5 physical phenomena, this motivates physicists to study the opportunities provided by quantum mechanics in order to achieve tasks beyond the reach of systems follow- ing the laws of classical mechanics. In classical information theory, an elementary unit of abstract information is encoded into a bit, which can take binary values such as 0 and 1. Quantum mechanics allows quantum objects, such as atoms, to realize arbitrary superposi- tions of ground and excited state, | = α|0 + β|1, with complex parameters α, β which only have to satisfy the normalization condition |α|2 + |β|2 = 1. Hence, two-level systems which are used as elementary quantum bits (qubits) in quantum information science can carry significantly more information1 than their classical counterparts (Nielsen and Chuang 2000; Hayashi 2006). The enormous information content of quantum systems is further manifested when many-particle systems are considered, and a Hilbert space whose dimension grows exponentially with the num- ber of particles is required to describe the many-body quantum state. Quantum theory permits arbitrary coherent superpositions of many-particle states, which includes so- called entangled states. Such states contain correlations which are beyond the reach of classical physics (Bell 1964), and underline the futility of attempts to grasp cer- tain quantum phenomena with an intuition based on classical physics (Einstein et al. 1935; Bell 1964; Englert 2013). Notwithstanding, experimental tests confirmed these predictions of quantum theory (Aspect et al. 1982a,b), which indicates the novel opportunities that are expected to open up when concepts from classical information theory are extended to the quantum realm. For instance, the non-classical correlations of quantum states can be harnessed to develop secure key distribution protocols (Ekert 1991; Acín et al. 2006), in which information is encoded into the quantum states of particle pairs. Each of the parties then receives one of the two strongly correlated particles and performs measurements on it. By sharing the settings of their detectors and part of their measurement data publicly, the two parties can establish a secret key (based on the data kept private) and at the same time confirm that their data is correlated in a way only achievable with undisturbed quantum states, which excludes the influence of a third-party eaves- dropper. This protocol, quantum key distribution, is an important example of a field called quantum communication, which strives to make efficient use of the possibili- ties quantum mechanics offers. Another example is given by quantum teleportation (Bennett et al. 1993): By exchanging a combination of quantum information and classical information, it is possible to transfer the quantum state of one particle to another, possibly in a remote laboratory. Entangled quantum states enable to per- form this task deterministically in a single run, without even knowing the teleported quantum state (Bennett et al. 1993). Quantum teleportation has been realized exper- imentally first using photons (Bouwmeester et al. 1997; Boschi et al. 1998) and later with atoms (Barrett et al. 2004; Riebe et al. 2004). 1The quantum state of a qubit is described by a unit vector in a two-dimensional complex vector space, while a classical bit only contains binary information (Nielsen and Chuang 2000). Even though a quantum measurement of a qubit only yields a classical bit of information, the full infor- mation of the complex coefficients is relevant to describe the quantum evolution of the qubit.
- 23. 6 1 Introduction Besides quantum communication, quantum information theory promises to improve the power of computational algorithms. If it was possible to build computers which work according to the laws of quantum mechanics, the additional resources provided by coherent superpositions of quantum states could be used to achieve tasks which otherwise are believed to be unfeasible (Nielsen and Chuang 2000; Hayashi 2006). An important problem, potentially suitable to be treated efficiently with a quantum computer, is given by the factorization of large integer numbers into their prime factors, which may be achieved based on Shor’s algorithm (Shor 1994). This algorithms’ runtime scales polynomially as a function of the number of classical bits required to represent the integer at question. For classical computers, prime fac- torization is a very hard task—the runtime of the fastest existing algorithm scales exponentially with the number of bits, which implies that quantum computers have the potential to provide an exponential speed-up over classical computers (Nielsen and Chuang 2000). The potential implications are immense since the security of stan- dard cryptography methods is based on the assumption that factoring large numbers is computationally intractable. One prerequisite for the realization of a quantum computer is the ability to control and engineer quantum mechanical systems. Assuming that such level of experimen- tal control was available, one could engineer a quantum system which mimics the dynamics of another interacting many-body system. By measuring the controllable quantum system at hand, one could then infer the properties of the quantum system in question. Without using considerable approximations, predicting the dynamics of large quantum many-body systems is also intractable for classical computers as the dimension of the system, and with it the required computational (classical) mem- ory, grows exponentially with the number of particles in the system. This approach, referred to as quantum simulation (Feynman 1982), is currently being pursued by experimental groups working on a variety of different platforms (see Sect.1.3)—the reliability and efficient certification of potential quantum simulations is, however, still debated (see Sect.1.4). Entangled states form the basis of many applications of quantum information theory. In the next section, we review the formal definition of quantum entanglement and also introduce the concept of quantum discord, a weaker form of correlations in quantum states which emerges in the context of local measurements of composite quantum systems. Often it turns out difficult to clearly identify the resource which empowers the quantum speedup in the particular protocol. Later, in Sect.1.2.2, we discuss a selection of examples which rely on entanglement, and others which do not necessarily require entangled states, but instead are enabled by mixed states with nonzero discord. 1.2.1 Correlations in Composite Quantum Systems: Entanglement and Discord We now briefly review the definition of separable and entangled states. For further details and pedagogical introductions we refer to the extensive literature on the topic
- 24. 1.2 Elements of Quantum Information Theory 7 (Nielsen and Chuang 2000; Mintert et al. 2005; Amico et al. 2008; Horodecki et al. 2009; Tichy 2011). Formally, quantum states are represented by normalized, positive semi-definite operators (density operators) ρ on a Hilbert space H: ρ ≥ 1, Trρ = 1 (Cohen-Tannoudji et al. 1977a,b), where Tr denotes the trace operation. A composite bipartite system, describing for instance two distinguishable particles or two degrees of freedom, is represented by quantum states on the Hilbert space H = HA ⊗ HB formed by the tensor product of the Hilbert spaces HA and HB of the individual subsystems. We introduce the notion of correlated quantum states by first defining what is considered a completely uncorrelated state: A product state ρP = ρA ⊗ ρB (1.1) describes a situation where the two subsystems A and B are statistically completely independent,sinceprobabilitiesforthemeasurementoutcomesofindependentexper- iments will factorize (Werner 1989). Quantum states may also exhibit purely classical correlations, which can be seen from the following example: Imagine a device, able to prepare the respective subsystems arbitrarily within the two sets of quantum states {ρi A} and {ρi B} (Werner 1989). If the device is connected to a classical random num- ber generator in a way that with probability pi it prepares system A in state ρi A and system B in state ρi B, we describe the resulting composite quantum system with the state ρS = i pi ρi A ⊗ ρi B. (1.2) This state clearly contains some correlations, since the subsystem probabilities do not factorize. However, the example shows that the correlations can be attributed to a classical probability distribution. A state of this form is called separable, and conversely, if a state cannot be represented in this form, it is called entangled. This can be generalized to characterize multiparticle entanglement in a hierarchical order ranging from bipartite to genuine multipartite entangled states (Mintert et al. 2005; Levi and Mintert 2013); which we will discuss in Sect.3.4.5. The definition (1.2) characterizes the most general set of bipartite states whose correlations are of classical nature. A different approach to defining the boarder between classical and quantum states is based on quantum measurements. The mea- surement uncertainty of two incompatible observables is most prominently deter- mined by nonzero commutators (Heisenberg 1927). However, quantum mechanics may already predict a finite variance for the measurement of a single observable. For example, if a system is prepared in a pure quantum state |, an observable M can be measured with zero variance if and only if | is an eigenstate of M. If the sys- tem is described by a mixed state ρ, the resulting variance must be attributed to two different origins: One is the lack of knowledge expressed by the statistical mixture in the construction of ρ, the other is the remaining intrinsic quantum uncertainty. One may use the commutator of ρ and M to quantify the quantum contribution to
- 25. 8 1 Introduction the uncertainty (Wigner and Yanase 1963; Luo 2003; Girolami et al. 2013) which is consistent with the well-known special case for pure states (Luo 2003). This implies certain consequences for correlated quantum states in a bipartite scenario, as considered before. To see this we apply the above reasoning to local measurements on subsystem A. It is possible to show that there exists a local observ- able2 MA ⊗ IB which commutes with ρ if and only if ρ has the form (Girolami et al. 2013) ρ = i pi |ii| ⊗ ρi B, (1.3) where {|i} represents an orthonormal basis of HA. States of this form are called states of zero discord (Henderson and Vedral 2001; Ollivier and Zurek 2001) and the lack of quantum uncertainty motivates to call them classical states3 (Modi et al. 2012). One may also consider a non-selective measurement of MA ⊗IB—a measurement where the outcome was forgotten4 : In this case the final state is described by an incoherent mixture of the eigenstates of MA, ρ f = i (|ii| ⊗ IB)ρ(|ii| ⊗ IB). (1.4) An equivalent definition of zero-discord states may be given as follows: There exists a basis {|i} (which can be interpreted as the eigenbasis of an observable), such that ρ f = ρ if and only if ρ is of the form (1.3). This is another indicator for classicality since invariance under measurements is typically not granted in quantum mechanics. NoticefromcomparingEqs.(1.2)and(1.3)thatstatesofzerodiscordformasubset of separable states: A separable state has zero discord only if the decomposition of the form (1.2) can be expressed in terms of a local orthonormal basis {|i}, instead of arbitrary density matrices ρi A. What does the definition (1.3) tell us about the correlations of nonzero discord states? First of all we note that nonzero discord states are certainly correlated since product states have always zero discord (Li and Luo 2008). Of course the set of nonzero discord states includes the entire set of entangled states, which are quantum correlated—in fact, when restricting to pure states, discord and entanglement are 2We assume that the spectrum of MA is non-degenerate, since otherwise a given measurement outcome cannot be uniquely identified with a quantum state, thus, the observable carries limited information. 3Note that, due to the asymmetry of this definition, one should always specify which of the two subsystems is being measured. 4The post-measurement state after a non-selective measurement is described by a mixture ρ = i pi i of all measurement projectors i , where the overlap of the initial state ρ with these projectors determines the probability distribution, pi = Tr{i ρ}.
- 26. 1.2 Elements of Quantum Information Theory 9 equivalent. Yet, the correlations of separable, nonzero discord states are still of clas- sical origin as they can be characterized by a probability distribution [see discussion preceding Eq. (1.2)]. In the light of the above discussion, we conclude that states of nonzero discord are correlated quantum states which show quantum properties (quantum noise, measure- ment disturbance, nonzero commutators) under the influence of local measurements. The nature of their correlations is, however, only quantum when they are also entan- gled. A more detailed discussion will be given in Sect.3.1. 1.2.2 Role of Correlated Quantum States in Quantum Information Theory Much effort is spent to define, characterize, and quantify different notions of corre- lations in quantum states including quantum entanglement and discord (Horodecki et al. 2009; Modi et al. 2012), usually motivated by referring to their role as a resource for certain quantum information tasks. But which quantity is actually relevant? This question cannot be answered in general. Instead, a clear answer can be given only in few special cases. Even then, it depends very much on the task one aspires to accomplish. For simplicity, let us first restrict to pure states. In this case, all of the different concepts of correlations collapse into the same notion and the set of zero-discord states coincides with the set of separable states; they even coincide with completely uncorrelated product states. Furthermore, any entangled pure state can violate a Bell inequality (Bell 1964), which is useful, for instance, to establish certified, secure quantum key distribution5 (Ekert 1991). Pure state entanglement is rather well char- acterized by now (Mintert et al. 2005; Horodecki et al. 2009), even though this already represents a very intricate task once multipartite states are considered. The picture becomes considerably more complex when we consider mixed states. Not every mixed entangled state is able to violate a Bell inequality (Popescu 1994), hence, not all mixed entangled states serve to distribute a certified, secure quan- tum key. Yet, other tasks, such as quantum teleportation, can be accomplished if only some nonzero entanglement is available (Vidal and Werner 2002). So clearly, it is not always necessary to ask for the strongest incarnations of correlations in quantum states. This suggests a hierarchical structure of correlated quantum states, with states that violate a Bell inequality on the top. Entangled states form a subset thereof, and an even weaker class of correlated quantum states are those with nonzero discord. The remaining set, the classical (zero discord) states, also includes the com- pletely uncorrelated product states. The set of zero-discord states are actually very 5Note that the first proposal for a quantum key distribution protocol did not make use of quantum correlations (Bennett and Brassard 1984). Instead, it uses the destructive nature of the quantum measurement and the no-cloning theorem (Wootters and Zurek 1982; Dieks 1982) to rule out the presence of a third party.
- 27. 10 1 Introduction sparse—they form a set of Lebesgue measure zero in the full set of quantum states (Ferraro et al. 2010), which demonstrates that quantum features are quite generic to correlated states of composite systems. Most algorithms for quantum computations employ entangled states at some point (Shor 1994; Grover 1997). Some particular implementation schemes even make explicit use of large entangled states (Briegel and Raussendorf 2001) as their initial resource (Raussendorf and Briegel 2001). Are there interesting applications of quan- tum information theory which do not necessarily require entanglement, and if so, is discord useful for certain tasks? This question has been intensively investigated throughout the last decade, since, due to their ubiquity, nonzero discord states are eas- ier to produce than entangled states. The debate was initiated by a quantum algorithm which efficiently determines the trace of large unitary matrices (Knill and Laflamme 1998), with fixed efficiency independent of the size of the unitary, whereas on classi- cal computers the required resources increase exponentially (Datta et al. 2005). The algorithm only requires one pure qubit, while the rest of the quantum state can be highly mixed, leading to vanishing entanglement in the system (Knill and Laflamme 1998; Lanyon et al. 2008). However, despite efforts (Datta et al. 2008), it has not been possible to clearly identify the resource for the quantum speed-up in this system, and doubts prevail about the necessity of discord in this context (Dakić et al. 2010). There exist other examples where the identification of discord as a necessary element is clearer. Consider, for example, a three-partite setup consisting of the two parties A and B, as well as a carrier particle C. Starting from an initially separable state, A and B can establish quantum entanglement only by exchanging the particle C, which, remarkably, is never entangled with either one of the two parties (Cubitt et al. 2003). Nevertheless, there is a quantum cost for this entanglement distribution quantified by a measure of quantum discord (Streltsov et al. 2012; Chuan et al. 2012), which was confirmed in a recent series of experiments (Fedrizzi et al. 2013; Vollmer et al. 2013; Peuntinger et al. 2013). In a similar scenario, two parties A and B share an initially separable state. A local measurement on B induces a minimal amount of entanglement between the measurement apparatus and the bipartite state of A and B, which can be identified by the quantum discord of the state before the measurement (Streltsov et al. 2011; Piani et al. 2011). Also this entanglement activation protocol has been realized in a photonic experiment (Adesso et al. 2014). An example from quantum metrology is given when we consider the measurement of an unknown phase-shift induced in one arm of an interferometer. The maximum obtainable information per measurement about the phase, again, is bounded by a measureof theinitial state’s quantumdiscordbetweenbotharms of theinterferometer (Girolami et al. 2014). Essentially, all of these applications make use of the property of nonzero discord states to contain a mixture of states which are non-orthogonal in at least one of the subsystems. Under the influence of local measurements, these non-orthogonal states lead to non-trivial disturbance of the total state which may be harnessed in different ways. These examples show that quantum discord can be the figure of merit whenever local measurements and operations play a decisive role. This is often the case in mul-
- 28. 1.2 Elements of Quantum Information Theory 11 tipartite scenarios, where a large quantum state is shared by many parties while each of them can only manipulate their part of the state—for instance in quantum com- munication protocols, which explains its relevance for quantum information theory. Another particularly natural setting for such a situation is an open quantum system; that is, a quantum system which is in contact with an inaccessible environment. Thus, the experimentalist can only control the open system with local operations. In this dissertation, we often try to control and measure composite quantum systems by local operations, thus, quantum discord will naturally emerge in various situations. Chapter 2, for instance, is dedicated to a method which detects and estimates the quantum discord in a bipartite setting, when only local access to one of the two parties is available. As we have seen discussing the previous examples, there exists a remarkable selection of quantum systems which can be controlled with high efficiency. These allow us to develop theoretical ideas in rather abstract terms, considering mostly ideal quantum mechanical systems. In the next section we will review a selection of important experimental platforms for controllable quantum systems. 1.3 Controllable Quantum Systems The past century has seen tremendous progress with regard to the understanding and manipulation of particles on the smallest scales. Years after the basic foundations of quantum mechanics had been developed, it was still believed that experiments with individual particles belong to the realm of thought experiments, and can never become an experimental reality (Schrödinger 1952). When electromagnetic traps for charged particles (Penning 1936; Paul et al. 1958; Paul 1990) were first constructed, such experiments became indeed possible, and soon led to the observation of indi- vidual electrons (Wineland et al. 1973) and ionized atoms (Neuhauser et al. 1980). This eventually evolved into the nowadays highly successful research field devoted to controlling and manipulating the quantum states of trapped ions, which will be discussed in further detail in Sect.1.3.1. Another important development towards the control of quantum mechanical sys- tems in a laboratory was laser cooling (Phillips 1998; Chu 1998; Cohen-Tannoudji 1998), which is not only important for ion trapping, but also renders trapping of neutral atomic ensembles possible. This enabled, for instance, the experimental gen- eration of a Bose–Einstein condensate from ultra-cold atoms (Davis et al. 1995; Anderson et al. 1995). Today, a large number of experimental groups work with cold trapped atoms, often in combination with optical lattices to observe the atom’s dynamics in specific potential landscapes. This research field will be discussed in Sect.1.3.2. Apart from the quantum states of atomic particles, it is also important to be able to control the quantum state of light. The previous two research fields use light fields to manipulate the quantum states of atoms. The converse approach is followed by cavity quantum electrodynamics experiments, where atoms are sent through high-
- 29. 12 1 Introduction finesse cavities to probe and manipulate the quantum state of the light mode inside the cavity (Meschede et al. 1985; Benson et al. 1994; Haroche 2013). Alternatively, the quantum state of individual photons or photon-pairs can be controlled by sending the photons through specifically designed arrays of optical instruments, and analyzed via single-photon detectors. We will briefly discuss such experiments in Sect.1.3.3. These examples represent a selection of quantum systems which can be controlled with high precision in today’s experiments, and which are most relevant for the present dissertation. A brief overview of further systems can be found in Sect.1.3.4. 1.3.1 Trapped Ions Trapped ions represent one of the most advanced platforms for quantum control at the single particle level (Wineland 2013). In this section, we summarize some of the essential experimental aspects and introduce some of the key tools which will be needed also for applications later in this dissertation. We will further briefly introduce the experimental setup where part of the research for this dissertation was carried out. 1.3.1.1 Paul Traps Due to their positive charge, ions can be trapped using electromagnetic fields. A static field, however, is unable to provide a potential minimum in three dimensions, since, according to Maxwell’s equations, the potential has to satisfy the Laplace equation = 0 in the charge-free center of the trap (Jackson 1999). Thus, a Paul trap uses a combination of static and radio-frequency modulated electric fields to confine charged particles in three dimensions (Paul 1990). Figure1.1a displays the typical design of a three-dimensional linear Paul trap, where a two-dimensional quadrupole potential generated by the radio-frequency electrodes in the x and y directions (in V sin(Ωrf t) ˜ x y z Ωrf x y y x (a) (b) (c) Fig. 1.1 a The radio-frequency electrodes of a linear Paul trap generate a time-dependent quadru- pole potential with a saddle point (b), which, upon averaging over the periodic radio-frequency oscillations, can be approximated by a static harmonic potential (c). Subfigures (b) and (c) are adapted from Littich (2011)
- 30. 1.3 Controllable Quantum Systems 13 trapped-ion literature, these directions are referred to as radial directions). The time- dependent potential can be pictured as a saddle point whose edges flop up and down with the radio frequency rf (Leibfried et al. 2003a). The time-averaged effective potential can be approximated by a static harmonic potential when the driving fre- quency is sufficiently strong compared to the frequency of the ion’s secular motion (Leibfried et al. 2003a). This approximation is particularly well justified when the ion is close to the center of the trap potential. This is the case when the ion’s kinetic energy is low (Leibfried et al. 2003a). Small displacements from the trap center, however, can lead to an additional driving force at the radio frequency called micro- motion. This can cause unwanted heating of the ions. Static electric fields are used to shift the ion’s equilibrium position striving to compensate this effect. Micromotion becomes more relevant when many ions are placed into the same trap potential since it gets increasingly difficult to maintain all ions close to the trap center. Along the z-axis or the axial direction, two positively charged direct current (dc) endcaps (not shown in the figure) generate a static potential to prevent the ions from escaping into the third dimension. Typical orders of magnitude for the radial trap frequencies are νx,y ≈ 2π × (1 − 10 MHz), while the axial frequencies are usually one order of magnitude smaller, νz ≈ 2π × (0.1 − 1) MHz (Leibfried et al. 2003a; Schindler et al. 2013). 1.3.1.2 The Berkeley Setup Figure1.2 shows a Paul trap used in the group of Hartmut Häffner at the University of California, Berkeley, where part of the work on this dissertation was done during a 10months stay from September 2012 to June 2013. The trap consists of three-way Fig. 1.2 Paul trap used in the experiments carried out at Berkeley, and a fluorescence image of a one-dimensional linear ion chain
- 31. 14 1 Introduction segmented electrodes of which the center segment is driven with an out-of-phase radio-frequency voltage while the two outer segments are used as dc-endcaps. The out-of-phase drive supplies opposing electrodes with the radio-frequency voltage with an 180◦ phase shift. This doubles the radial trap frequency at the same drive amplitude and reduces micromotion along the axial direction, since the potential can- cels along the trap axis (Pruttivarasin 2014; Ramm 2014). The setup which was used for the experiment discussed in Sect.2.2, was designed and assembled by Thaned Pruttivarasin and Michael Ramm under the supervision of Hartmut Häffner. In the following we briefly summarize the key elements of the setup. Details can be found in the dissertations (Pruttivarasin 2014; Ramm 2014). To load ions into the trap, an ionization laser pointing at the center of the trap ionizes a beam of thermal atoms emitted from an oven on demand. Interactions with red-detuned lasers Doppler-cool the ions which allows them to fall into the trap potential generated by the electrodes of the Paul trap. The fluorescence light which is scattered in the course of the Doppler cooling process is used to monitor the ions in the trap. Turning off the oven and the photon-ionization laser after the fluorescence light of a certain number of ions is observed allows to steer the number of trapped ions reliably. The Paul trap is placed in an ultra-high-vacuum chamber with a pressure below 5×10−9 Pa to minimize collisions of the ions with background atoms (Ramm 2014). The potential generated by such Paul traps can be deep enough to trap atoms at room temperature and, once trapped, the ions can be kept for many hours in the trap; under good conditions the lifetime is only limited by the time the experimental equipment can be continuously supported. The cooling and quantum state manipulation of ions requires elaborate laser tech- nology which was developed for trapped-ion systems to perform high-precision spec- troscopy in the context of atomic frequency standards (Berkeland et al. 1998; Young et al. 1999; Wineland 2013). Before being sent into the trap, the laser light is guided through acousto-optic modulators (AOM), which generate acoustic standing waves to modulate the incoming laser light and, thus, can be used to fine-tune the laser detuning and may induce a controllable phase shift. The Berkeley group has developed their own python-software to conveniently control the AOM settings to change beam intensities, detunings and phases, and to run frequently used standard pulse sequences and algorithms automatically and on demand (Ramm 2014). The computer con- nects to a field-programmable gate array (FPGA), which controls custom-designed direct digital synthesis (DDS) boards to provide the control voltages to the AOMs (Pruttivarasin 2014). Apart from AOMs the computer also controls the voltages of the trap electrodes and is provided with the photon counts from the photomulti- plier tube, which is used to detect the ion’s fluorescence. The fluorescence light is guided through the imaging system which at the same time provides laser access to the trap, and eventually is also collected by a CCD (charge coupled device) camera (Pruttivarasin 2014). The fluorescence of individual ions can be resolved spatially, allowing one to read out the ions’ populations independently (Ramm 2014).
- 32. 1.3 Controllable Quantum Systems 15 1.3.1.3 Motion of Trapped Ions in Linear Chains When several ions are trapped in the same harmonic potential, the ions are well- separated due to strong Coulomb repulsions. The full potential consists in the global harmonic trap potential and the Coulomb repulsion between them6 : V = N i=1 1 2 m ν2 x x2 i + ν2 y y2 i + ν2 z z2 i + N i, j=1 (i j) e2 4π0 1 (xi − xj )2 + (yi − yj )2 + (zi − z j )2 , (1.5) where N is the total number of ions, m is the ion mass, and ri = (xi , yi , zi ) the ith ion’s position, respectively, e the electron charge and 0 the dielectric constant of the vacuum. Given the ratio of trap frequencies assumed above (νx,y νz), their equilibrium positions form a linear chain along the axial (z) direction (James 1998). The separation of neighboring ions is typically on the order of 10 μm (Schindler et al. 2013). Assuming that the ions remain close to their equilibrium positions, that is, the ions’ displacement is much smaller than the inter-ion distance, the Coulomb potential is determined by the ions’ equilibrium positions along the axial direction (James 1998). By expanding the potential around the ions’ equilibrium positions to second order, for example in xi , we obtain a Hamiltonian for the description in terms of quantized local phonon modes in x-direction (and analogously for the y-direction) given by H = N i=1 ω0 i a† i ai + N i, j=1 (i j) ti j a† i aj + a† j ai , (1.6) where a† i creates a local phonon at site i (Porras and Cirac 2004a). The average inter-ion distance is given by the length scale l3 0 = e2 /(mν2 z ), with the axial trap frequency νz (James 1998). The Hamiltonian (1.6) is a valid approximation if the parameter β0 := e2 /(l3 0mν2 x ) = ν2 z /ν2 x 1, assuming that the radial trap frequencies are comparable, νx ≈ νy (Porras and Cirac 2004a), which is the case for a linear trap architecture as described above. The second-order expansion yields for the local trap frequencies and the coupling matrix (Porras and Cirac 2004a) ω0 i /νx = 1 − β0 2 j=i 1 |u0 i − u0 j |3 , (1.7) ti j /νx = β0 2 1 |u0 i − u0 j |3 , 6Here and in the following, we assume single-ionized ions.
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- 36. Fig. 6. Sethoconus capreolus, n. sp., × 400 1291 Lateral view. Fig. 7. Lampromitra quadricuspis, n. sp., × 400 1214 Apical view. Fig. 8. Lampromitra furcata, n. sp., × 400 1215 The collar septum after removal of the cephalis. Fig. 9. Lampromitra dendrocorona, n. sp., × 400 1216 Apical view. PLATE 59. Legion NASSELLARIA. Order CYRTOIDEA.
- 37. Families T r i p o c y r t i d a , P o d o c y r t i d a et P h o r m o c y r t i d a . PLATE 59. Tripocyrtida, Podocyrtida et Phormocyrtida. Diam. Page. Fig. 1. Lampromitra huxleyi, n. sp., × 400 1215 Fig. 2. Amphiplecta callistoma, n. sp., × 400 1224 Fig. 3. Corocalyptra agnesæ, n. sp., × 400 1323 Fig. 4. Corocalyptra emmæ, n. sp., × 400 1323 The shell encloses the trilobate central capsule, with the trilobate nucleus. Fig. 5. Clathrocyclas × 400 1390
- 38. cassiopejæ, n. sp., Fig. 6. Clathrocyclas alcmenæ, n. sp., × 400 1388 Fig. 7. Clathrocyclas latonæ, n. sp., × 400 1389 Apical view. Fig. 8. Diplocylas bicorona, n. sp., × 400 1392 Fig. 9. Clathrocyclas ionis, n. sp., × 400 1389 Fig. 10. Corocalyptra elisabethæ, n. sp., × 400 1323 Oblique apical view of the shell, with the quadrilobate central capsule enclosed. Fig. 11. Clathrocyclas europæ, n. sp., × 400 1388
- 39. Apical view of the shell, after removal of the cephalis. Fig. 12. Clathrocyclas europæ, n. sp., × 400 1388 Central capsule, seen from above, with the quadrilobate nucleus. Fig. 13. Clathrocyclas danaës, n. sp., × 300 1388 Vertical section through the cephalis and the quadrilobate central capsule, with the quadrilobate nucleus. Fig. 14. Clathrocyclas danaës, n. sp., × 300 1388 Apical view of the shell.
- 40. PLATE 60. Legion NASSELLARIA. Order CYRTOIDEA. Family T r i p o c y r t i d a . PLATE 60. Tripocyrtida. Diam. Page. Fig. 1. Dictyophimus cienkowskii, n. sp. (vel Lamprotripus squarrosus), × 300 1200 Shell seen from the side. Fig. 2. Dictyophimus bütschlii, n. sp. (vel Lamprotripus horridus), × 300 1201 Fig. 3. Dictyophimus hertwigii, n. sp. × 400 1201
- 41. (vel Lamprotripus spinosus), The cephalis of the shell includes the central capsule, with three lobes depending in the pyramidal thorax. Fig. 4. Dictyophimus platycephalus, n. sp., × 400 1198 Central capsule with four thoracic lobes, each of which contains an oil-globule; kidney-shaped nucleus in the cephalic lobe. Fig. 5. Dictyophimus platycephalus, n. sp., × 400 1198 Shell seen from the side. Fig. 6. Dictyophimus brandtii, n. sp., × 300 1198
- 42. Shell seen from the base, with the four large pores of the collar septum, two minor jugular and two major cardinal pores. Fig. 7. Lampromitra coronata, n. sp., × 400 1214 Shell seen from below, with the quadrilobate central capsule. Fig. 7a. A portion of the shell-margin, × 800 1214 Fig. 8. Lampromitra arborescens, n. sp., × 400 1216 Shell from above. Fig. 8a. The collar septum with the four crossed rods of the cortina, × 400 1216
- 43. Fig. 9. Tripocyrtis plectaniscus, n. sp., × 400 1202 Fig. 10. Tripocyrtis plagoniscus, n. sp., × 400 1201 PLATE 61. Legion NASSELLARIA. Order CYRTOIDEA. Family T r i p o c y r t i d a .
- 44. PLATE 61. Tripocyrtida. Diam. Page. Fig. 1. Dictyophimus cortina, n. sp., × 400 1197 Fig. 2. Lychnocanium pudicum, n. sp., × 200 1230 Fig. 3. Dictyophimus longipes, n. sp., × 400 1197 Fig. 4. Lychnocanium clavigerum, n. sp., × 300 1230 Fig. 5. Dictyophimus lasanum, n. sp., × 300 1197 Fig. 6. Lychnocanium favosum, n. sp., × 300 1225 Fig. 7. Lychnocanium lanterna, n. sp., × 300 1224 Fig. 8. Dictyophimus × 300 1196
- 45. plectaniscus, n. sp., Apical view. Fig. 9. Dictyophimus plectaniscus, n. sp., × 300 1196 Lateral view. Fig. 10. Lychnocanium fenestratum, n. sp., × 400 1228 Fig. 11. Lychnocanium pyriforme, n. sp., × 300 1225 Fig. 12. Lychnocanium fortipes, n. sp., × 300 1227 Fig. 13. Lychnocanium tuberosum, n. sp., × 300 1227 Fig. 14. Lychnocanium nodosum, n. sp., × 300 1225 Fig. 15. Lychnocanium × 400 1228
- 46. sigmopodium, n. sp., Fig. 16. Dictyophimus pyramis, n. sp., × 300 1196 Fig. 17. Dictyophimus triserratus, n. sp., × 300 1200 PLATE 62. Legion NASSELLARIA. Order CYRTOIDEA. Families A n t h o c y r t i d a , S e t h o c y r t i d a et P h o r m o c y r t i d a .
- 47. PLATE 62. Anthocyrtida, Sethocyrtida et Phormocyrtida. Diam. Page. Fig. 1. Dictyocephalus australis, n. sp., × 300 1306 Fig. 2. Dictyocephalus mediterraneus, n. sp., × 300 1307 Fig. 3. Sethamphora costata, n. sp. (vel Dictyocephalus costatus), × 300 1251 Fig. 4. Dictyocephalus amphora, n. sp., × 400 1305 Fig. 5. Cycladophora (?) favosa, n. sp. (an Dictyocephalus?), × 400 1380 Fig. 6. Cycladophora (?) favosa, n. sp. (an Dictyocephalus?), × 400 1380
- 48. A variety with obliterated ribs (?). Fig. 7. Dictyocephalus globiceps, n. sp., × 400 1308 Fig. 8. Sethocorys achillis, n. sp., × 400 1301 Fig. 9. Sethocyrtis oxycephalis, n. sp., × 400 1299 Fig. 10. Sethocorys odysseus, n. sp., × 400 1302 Fig. 11. Sethocyrtis agamemnonis, n. sp., × 300 1300 Seen from above (apical view). Fig. 11A. Sethocyrtis agamemnonis, n. sp., × 300 1300 Seen from above, after removal of the cephalis.
- 49. Fig. 12. Anthocyrtium pyrum, n. sp., × 400 1276 Fig. 13. Anthocyrtis ovata, n. sp., × 300 1272 Fig. 14. Anthocyrtium chrysanthemum, n. sp × 400 1272 Fig. 15. Anthocyrtidium ligularia, n. sp., × 400 1278 Fig. 16. Anthocyrtidium cineraria, n. sp., × 400 1278 Fig. 17. Anthocyrtium campanula, n. sp., × 400 1274 Fig. 18. Anthocyrtium doronicum, n. sp., × 300 1276 Fig. 19. Anthocyrtium flosculus, n. sp., × 300 1277
- 50. Fig. 20. Anthocyrtium adonis, n. sp., × 300 1273 Fig. 21. Sethoconus anthocyrtis, n. sp. (vel Anthocyrtium sethoconium), × 300 1296 PLATE 63. Legion NASSELLARIA. Order CYRTOIDEA. Family T r i p o c y r t i d a .
- 51. PLATE 63. Tripocyrtida. Diam. Page. Fig. 1. Callimitra carolotæ, n. sp., × 400 1217 Lateral view. Fig. 2. Callimitra annæ, n. sp., × 400 1217 Dorsal view. Fig. 3. Callimitra emmæ, n. sp., × 300 1218 Lateral view. Fig. 4. Callimitra emmæ, n. sp., × 400 1218 Cephalis alone, with the enclosed four- lobed central capsule, and the internal four divergent beams; surrounded by some scattered xanthellæ. Fig. 5. Callimitra agnesæ, n. sp., × 400 1217 Dorsal view.
- 52. Fig. 6. Callimitra elisabethæ, n. sp., × 400 1218 Lateral view. Fig. 7. Callimitra carolotæ, n. sp., × 200 1217 Seen from above (from the apical pole). Fig. 8. Callimitra carolotæ, n. sp., × 200 1217 Seen from below (from the basal pole). PLATE 64. Legion NASSELLARIA. Order CYRTOIDEA. Families T r i p o c y r t i d a et P o d o c y r t i d a .
- 53. PLATE 64. Tripocyrtida et Podocyrtida. Diam. Page. Fig. 1. Clathrocanium sphærocephalum, n. sp., × 600 1211 Fig. 2. Clathrocanium diadema, n. sp., × 600 1212 Fig. 3. Clathrocanium triomma, n. sp., × 600 1211 Fig. 4. Clathrocanium reginæ, n. sp., × 600 1212 Fig. 5. Clathrolychnus araneosus, n. sp., × 600 1240 Fig. 6. Clathrolychnus periplectus, n. sp., × 600 1241 Fig. 7. Pteropilium clathrocanium, n. sp., × 400 1327
- 54. Fig. 8. Clathrocorys murrayi, n. sp., × 600 1219 Fig. 9. Clathrocorys giltschii, n. sp., × 600 1220 Fig. 10. Clathrocorys teuscheri, n. sp., × 600 1220 PLATE 65. Legion NASSELLARIA. Order CYRTOIDEA. Family P h o r m o c y r t i d a .
- 55. PLATE 65. Phormocyrtida. Diam. Page. Fig. 1. Alacorys friderici, n. sp. (vel Hexalacorys friderici), × 400 1372 The central capsule, enclosed in the fenestrated shell, exhibits in its lower half four large club-shaped lobes, each of which includes in its upper part a large oil-globule. The uppermost, undivided part of the capsule includes the nucleus, which protrudes four small nuclear lobes through the four holes of the cortinar septum into the thorax. Numerous long pseudopodia
- 56. arise from the granular sarcomatrix, which the capsule surrounds, and pass through the pores of the siliceous shell. Fig. 2. Alacorys guilelmi, n. sp. (vel Hexalacorys guilelmi), × 300 1372 Fig. 3. Alacorys bismarckii, n. sp. (vel Pentalacorys bismarckii), × 200 1372 Fig. 4. Alacorys lutheri, n. sp. (vel Tetralacorys lutheri), × 400 1370 Fig. 5. Cycladophora goetheana, n. sp. (vel Lampterium goetheanum), × 300 1376 PLATE 66.
- 57. Legion N A S S E L L A R I A . Order CYRTOIDEA. Family T h e o c y r t i d a . PLATE 66. Theocyrtida. Diam. Page. Fig. 1. Tricolocapsa theophrasti, n. sp., × 400 1432 Fig. 2. Tricolocapsa schleidenii, n. sp., × 300 1433 Fig. 3. Tricolocapsa dioscoridis, n. sp., × 300 1432 Fig. 4. Tricolocapsa decandollei, n. sp., × 300 1433 Fig. 5. Tricolocapsa × 400 1432
- 58. linnæi, n. sp., Fig. 6. Theocapsa aristotelis, n. sp., × 300 1427 Fig. 7. Theocapsa mülleri, n. sp., × 400 1431 Fig. 8. Theocapsa democriti, n. sp., × 400 1427 Fig. 9. Theocapsa forskalii, n. sp., × 400 1429 Fig. 10. Theocapsa cuvieri, n. sp., × 400 1430 Fig. 11. Theocapsa wottonis, n. sp., × 400 1428 Fig. 12. Theocapsa darwinii, n. sp., × 300 1431 Fig. 13. Theocapsa linnæi, n. sp., × 400 1429
- 59. Fig. 14. Theocapsa wolffii, n. sp., × 400 1429 Fig. 15. Theocapsa malpighii, n. sp., × 400 1428 Fig. 16. Theocapsa lamarckii, n. sp., × 400 1430 Fig. 17. Tricolocampe amphizona, n. sp., × 400 1413 Fig. 18. Theocampe collaris, n. sp., × 300 1425 Fig. 19. Tricolocampe polyzona, n. sp., × 400 1412 Fig. 20. Tricolocampe stenozona, n. sp., × 400 1413 Fig. 21. Tricolocampe cylindrica, n. sp., × 300 1412
- 60. Fig. 22. Tricolocampe urnula, n. sp., × 400 1422 Fig. 23. Theocampe stenostoma, n. sp., × 300 1423 Fig. 24. Theocampe costata, n. sp., × 300 1424 Fig. 25. Theocampe sphærothorax, n. sp., × 300 1424 PLATE 67. Legion NASSELLARIA. Order CYRTOIDEA. Family P o d o c y r t i d a .
- 61. PLATE 67. Podocyrtida. Diam. Page. Fig. 1. Lithornithium falco, n. sp., × 400 1355 Fig. 2. Lithornithium fringilla, n. sp., × 400 1355 Fig. 3. Lithornithium ciconia, n. sp., × 400 1354 Fig. 4. Lithornithium trochilus, n. sp., × 400 1355 Fig. 5. Theopera fusiformis, n. sp., × 400 1357 Fig. 6. Theopera chytropus, n. sp., × 400 1358 Fig. 7. Theopera prismatica, n. sp., × 300 1357 Fig. 8. Theopera cortina, n. sp., × 400 1358
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