Mahler· WeberruB Quantum Networks

Springer-Verlag Berlin Heidelberg GmbH

Gunter Mahler Volker A. WeberruB

Quantum Networks Dynamics of Open Nanostructures

Second, Revised and Enlarged Edition With 166 Figures

i Springer

Professor Dr. rer. nat. Giinter Mahler 1. Institut ffu Theoretische Physik und Synergetik Abteilung Festkorperspektroskopie Universitlit Stuttgart Pfaffenwaldring 57 D-70569 Stuttgart, Germany

Dr. rer. nat. Volker A. WeberruB Y.A.W. scientific consultation Im Lehenbach 18 D-73650 Winterbach, Germany

Working out of the text, graphics, parts of the numerical calculations, softbook, and production of the camera-ready manuscript by Y.A.W. scientific consultation.

Ubrary of Congress Cataloging-in-Publication Data

Mahler, Gunter. Quantum networks: dynamics of open nanostructures 1 Gunter Mahler, Volker A. Weberruss. - 2nd rev. and en!. ed. p. cm.lncludes bibliographical references and index.

1. Nanostructures. 2. Quantum theory. I. Weberruss, Volker Amim. II. Title. QCI76.8.N35M34 1998 530.4'I-dc21 98-12039 CIP

This work is subject to copyright. Ali rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on micro­film or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, '965, in its current version, and permission for use must always be obtained from Springer -Verlag. Violations are liable for prosecution under the German Copyright Law.

© Springer-Verlag Berlin Heidelberg '995, 1998 Originally published by Springer-Verlag Berlin Heidelberg New York in 1998 Softcover reprint of the hardcover 2nd edition '998

The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that sum names are exempt from the relevant protective laws and regulations and therefore free for general use.

Typesetting: Camera ready by the authors using a Springer TEX macro package Cover design: design & production GmbH, Heidelberg

56/3144 - 5 43 2 1 o - Printed on acid-free paper

ISBN 978-3-642-08350-1 ISBN 978-3-662-03669-3 (eBook)DOI 10.1007/978-3-662-03669-3

Preface to the Second Edition

The first edition of this book has enjoyed a positive reaction from the physics community, beyond our expectation. The concept of quantum networks as used in this book is widely accepted in the recent literature; it is clear now that it underlies a current trend in modern quantum physics.

We welcome the opportunity to present a revised and enlarged version. A number of printing errors have been removed, many paragraphs have been re-organized for improved clarity. The basic outline and the scope of the book remains unchanged, though: we still focus on the physics or" quantum networks rather than on more spectacular (but also more speculative) sub­jects such as quantum computation and quantum information. Even without these futuristic applications, the field of quantum networks is quite demand­ing indeed: in the ideal case, one tries to keep complete control of a finite state system, the number of independent state parameters of which increases exponentially with the number of subsystems. The traditional density matrix theory has typically been concerned with the reduction to few macroscopic parameters (such as thermal states) or other few parameter states, relating, for example, to single modes.

We have completely revised the layout of the book to improve its peda­gogical approach. Graphic elements have been added in order to enhance the transparency of the representation:

• important formulas now have grey shaded backgrounds; • grey lines differentiate examples and proofs from the text; • theorems now have titles; • the terminology has been updated.

In this way, we hope to have achieved a representation applicable to general discussions as well as to special cases in a self-contained way.

Some readers may not find entirely what they expect in such a monograph. Nevertheless, we hope we have found a satisfactory compromise.

Stuttgart and Winterbach January 1998

Gunter Mahler Volker A. Weberrufl

Preface to the First Edition

This book grew out of lectures on density matrix theory given by one of us (G. Mahler) at the Universitat Stuttgart in the academic years 1989/90 and 1993.

Basic properties of the density matrix are covered in many books and from a number of different points of view; however, we felt that an extensive treatment on coupled few-level quantum objects is missing. This may not be too surprising as the necessity to understand such systems has emerged only in the 1980s. In Stuttgart the main motivation has originally been a special research project on molecular electronics, funded by the Deutsche Forschungsgemeinschaft.

However, there are many other areas like photon optics, atom optics, cavity electrodynamics, and combinations thereof, which typically can be mapped onto the same mathematical framework: the density matrix theory cast into an SU (n) lattice description.

Though there are powerful alternatives, we think that this approach is particularly useful to "see quantum dynamics at work": quantum dynamics is often believed to be "counter-intuitive" (which may simply mean that we have no or the wrong intuition). In fact, from experiments and detailed mod­elling of individual quantum objects one almost gets the opposite impression. Though part of the game is non-deterministic, the type of events, the alter­natives, are controlled by an amazingly strict "logic", which derives from the embedding of the quantum object into a classical environment. The measure­ment protocol ("information dynamics") feeds back into the system dynam­ics. The resulting visualization of quantum dynamics in terms of "clockwork of pointers", moving and jumping, disappearing and reappearing, may pro­vide us with some kind of experience, which we are used to getting for free in the classical world.

This book is intended to attract not only specialists, but also students try­ing to gain some working knowledge of quantum mechanics. For this purpose all calculations are given fairly explicitly; furthermore, they do not require more than a decent understanding of vector analysis and vector algebra. A quantum-mechanical background will be provided in the introduction.

We have focussed on the discussion of so-called nanostructures, as these seem to be of great importance for technical products. In order to write a

VIII Preface

book not only aimed at a scientifically interested public, we have added an extensive introductory part dealing with technical aspects of nanotechnology. Moreover, this part is to convince those not interested in technology that even the most complicated formulae and trains of thought presented in this book control objects which are becoming - though often invisible for us - a matter of course for all our lives: electronic devices such as micro-chips.

Stuttgart and Winterbach January 1995

Gunter Mahler Volker A. Weberrufl

Acknowledgements

The authors thank Dipl. Phys. Claus Granzow, Dipl. Phys. Holger Hofmann, Dr. Matthias Keller, Dipl. Phys. Alexander Otte, Dr. Jurgen Schlienz and Dipl. Phys. Rainer Wawer (Institut fUr Theoretische Physik I, Universitat Stuttgart) for many valuable discussions and for supplying us with numeri­cal data and figures. We have profited a lot from conversations with Prof. Thomas Beth (Karlsruhe) and Prof. Howard Carmichael (Eugene, USA). Careful checking of equations by Dipl. Phys. Ilki Kim is gratefully acknowl­edged. Furthermore, it is a pleasure to thank Dr. Heinz Schweizer, Dipl. Phys. Uwe Griesinger, and Dipl. Phys. Renate Bergmann (4. Physikalisches Institut, Universitat Stuttgart) for making available measurement data on nanostruc­tures and for valuable discussions. We thank Prof. Dr. Wolfgang Eisenmenger and Dr. Bruno Gompf (1. Physikalisches Institut, Universitat Stuttgart) for the kindly released video scanning tunnelling microscope images of molecular structures. It is a pleasure for the authors to thank Springer-Verlag, espe­cially Dr. Hans J. K6lsch, Dr. Victoria Wicks, Gertrud Dimler, Jacqueline Lenz, and Gisela Schmitt, for their excellent cooperation. This cooperation has guaranteed the rapid and smooth passing of the project. Last but not least, we would like to thank Dorothee Klink for helpful proofreading and for translations of parts of the text.

Table of Contents

1. Introduction.............................................. 1 1.1 Motivation............................................ 1 1.2 Localized Nuclear Spins. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.3 Confined Electrons: N anostructures . . . . . . . . . . . . . . . . . . . . . . . 7

1.3.1 Fabrication...................................... 9 1.3.2 Characterization Methods . . . . . . . . . . . . . . . . . . . . . . . .. 11 1.3.3 From Structure to Dynamics. . . . . . . . . . . . . . . . . . . . . .. 13 1.3.4 Granular Superconductors: Confined Cooper Pairs. . .. 16

1.4 Confined Photons: Cavity Electrodynamics . . . . . . . . . . . . . . .. 17 1.4.1 Mirror Gaps. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 17 1.4.2 Ring Cavities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 17 1.4.3 Box Cavities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 18

1.5 Confined Ions: Electrodynamic Traps ..................... 18 1.5.1 "Point" Traps... .. .... ... ..... .. .... ...... .. .. . .. 18 1.5.2 Linear Traps. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 20

1.6 Applications: Present and Future. . . . . . . . . . . . . . . . . . . . . . . .. 20 1. 7 Fundamentals.......................................... 22

1. 7.1 Operators in Hilbert Space . . . . . . . . . . . . . . . . . . . . . . .. 22 1.7.2 Aspects of Group Theory. . . . . . . . . . . . . . . . . . . . . . . . .. 24 1.7.3 Application to Quantum Systems ................... 26

2. Quantum Statics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 33 2.1 Introduction........................................... 33 2.2 Quantum-Mechanical Systems. . . . . . . . . . . . . . . . . . . . . . . . . . .. 34

2.2.1 Transition Operators.. . . . . . . . .. . . . . . . . . . . . . . . . . . .. 34 2.2.2 Angular Momentum Operators. . . . . . . . . . . . . . . . . . . .. 36 2.2.3 SU(n) Algebra.. . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . .. 41 2.2.4 Unitary Operators.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 53 2.2.5 Unitary Transformations . . . . . . . . . . . . . . . . . . . . . . . . .. 57 2.2.6 Raising and Lowering Operators. . . . . . . . . . . . . . . . . . .. 62 2.2.7 Discrete Hamilton Models . . . . . . . . . . . . . . . . . . . . . . . .. 65

2.3 The Density Operator. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 73 2.3.1 Fundamental Properties.. . . . . . . . . . . . . . . . . .. . . . . . .. 73 2.3.2 The Coherence Vector ............................ 78

XII Table of Contents

2.3.3 State Models in SU(n) . . . . . . . . . . . . . . . . . . . . . . . . . . .. 84 2.3.4 Entropy......................................... 88 2.3.5 The Canonical Statistical Operator. . . . . . . . . . . . . . . .. 93 2.3.6 Ensemble Measurements .......................... 98 2.3.7 Axiomatic Measurement Theory .................... 110

2.4 Composite Systems: Two Nodes .......................... 114 2.4.1 Product Space ................................... 114 2.4.2 Hamilton Models: Pair Interactions ................. 121 2.4.3 Coupling Between Higher-Dimensional Subsystems ... 129 2.4.4 The Density Operator ............................. 133 2.4.5 Projections and Entanglement ..................... 148

2.5 Composite Systems: Three Nodes ......................... 163 2.5.1 Product Spaces .................................. 163 2.5.2 Hamilton Models in SU(nd 0 SU(n2) 0 SU(n3) ..... 167 2.5.3 The Density Operator ............................. 167 2.5.4 Projections and Entanglement ..................... 176

2.6 N-Node Systems ....................................... 181 2.6.1 The Hamilton Operator and the Density Operator .... 181 2.6.2 The Physics of Entanglement ...................... 182

2.7 Summary .............................................. 185

3. Quantum Dynamics ...................................... 187 3.1 Introduction ........................................... 187 3.2 Unitary Dynamics ...................................... 187

3.2.1 The Liouville Equation ............................ 187 3.2.2 The Dynamics of the Coherence Vector ............. 193 3.2.3 A Hamilton Model with Periodic Time Dependence ... 197 3.2.4 The Heisenberg Picture ........................... 215 3.2.5 Network Dynamics ............................... 221 3.2.6 Temporal Non-Locality and Quantum Parallelism .... 237

3.3 Dynamics of Open Systems .............................. 241 3.3.1 Open Systems ................................... 241 3.3.2 The Markovian Master Equation ................... 245 3.3.3 The Quantum Dynamical Semigroup ................ 257 3.3.4 Damping Channels ............................... 261 3.3.5 The Damped Bloch Equations in SU(2) ............. 273 3.3.6 The Damped Bloch Equations in SU(3) ............. 284 3.3.7 Open Networks .................................. 294

3.4 Incoherent Networks .................................... 316 3.4.1 The Pauli Master Equation ........................ 316 3.4.2 The SU(2) Chain ................................ 317 3.4.3 Stability of States and the Ising Limit ............... 320

3.5 Summary .............................................. 321

Table of Contents XIII

4. Quantum Stochastics ..................................... 323 4.1 Introduction ........................................... 323

4.1.1 Quantum Noise and Langevin Equations ............ 324 4.1.2 Self-Reduction ................................... 324 4.1.3 Stochastics as a Source of Information .............. 325

4.2 Continuous Measurement ................................ 326 4.2.1 Basics .......................................... 326 4.2.2 Simple Systems (N = 1) ........................... 327 4.2.3 Applications ..................................... 337

4.3 Partly Coherent Networks ............................... 348 4.3.1 The Stochastic Algorithm ......................... 348 4.3.2 Different Trajectories

with the Same Ensemble Behaviour. . . . . . . . . . . . . . . . . 349 4.3.3 Reacting and Non-Reacting Environments ........... 351 4.3.4 Quantum Stochastics of a SU(2) 0 SU(2) Network ... 355 4.3.5 Non-Local Damping: Superradiance ................. 356 4.3.6 A Driven 3-Node System:

Relaxation into Entanglement. . . . . . . . . . . . . . . . . . . . . . 360 4.3.7 Decoherence ..................................... 362

4.4 Incoherent Networks .................................... 365 4.4.1 Random Walks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365 4.4.2 A Single 2-Level Node: Random Telegraph Signals .... 369 4.4.3 An Interacting 2-Level Pair: Classical Correlations .... 370 4.4.4 A Single 3-Level Node: Random Telegraph Signals .... 373

4.5 Summary .............................................. 376

5. Summary ................................................. 377 5.1 The Background ..................... , ................. 377 5.2 Key Topics ............................................ 379

5.2.1 Quantum Networks and Nanostructures ............. 379 5.2.2 Coherence and Correlation ........................ 379 5.2.3 Closed and Open Systems ......................... 379 5.2.4 Network Equations ............................... 380 5.2.5 Stochastic Unravelling ............................ 380 5.2.6 The Measurement Record ......................... 381

References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383

Index ......................................................... 391

About the Authors ........................................... 403

List of Symbols

In this book, operators are indicated by /\ (e. g. A). The symbol * denotes conjugate complex quantities, and t denotes adjoint quantities. For vectors of physical quantities (or of operators), bold italic letters are used (e. g. A or A, respectively), where the components are indicated by indices (e. g. Ai or Ai are components of A or A). In the case of matrices (or tensors), letters without serifs are used (e. g. A). Greek letters are exceptions: in this case, matrices (tensors) are indicated by parentheses (e. g. (ilij». Components of SU (n) vectors (matrices) are specified by script letters (e. g. Ai or Aj, respectively). SU(n) vectors and matrices are also indicated by parentheses (e. g. (A) or (Aj)). The SU (n) vector r with components ri represents an exception. The corresponding SU(n) Q9 SU(n) matrix is written as (rij ).

For the Heisenberg picture the symbol (H) is used as an upper index (e. g. the operator A(H) is an operator in the Heisenberg picture). The inter­action picture is indicated by the additional index (i) (e. g. the operator A (i) represents an operator in interaction picture).

Blackboard bold letters are used if a vector space is considered, e. g., a Hilbert space is written in the form lHl.

The main symbols are presented in the following list. The Greek symbols are at the end of the list.

i a+,a A A,Ai IAi) , Ii) A, (iIAlj) = Aij (A) = (7fJIAI7fJ) = 11 (A),A (Aj) ,Aj A(H) (t) (A~H) (t»), A~H) (t) A(i)(t) C(n, q)

Unit operator Creation and annihilation operator Operator Vector of eigenvalues, eigenvalue Eigenstates (discrete) Matrix, matrix element Expectation value SU (n) vector, element of SU (n) Q9 SU (n) matrix, element of Heisenberg operator SU (n) vector (Heisenberg picture), element of Operator in interaction picture Trace of pq in SU(n)

XVI

6 CF, CR

d dijk D1212

ijkl e ei E Ei

List of Symbols

Eij = Ei - E j fijk F 9 98 Q)

Gij(T) Gij(W) Ii = h/27f if

(Hi)

(Hij)

if(H)(t) if(i)(t)

IHI

Ii) I 1m k,kB

k k K, Kij(l, 2), KI} Kijk(l, 2, 3), KIN k ii t = (ix, i y , L) ii £, mo,me

Casimir operator Coulomb coupling constants Dipole operator Anti-structure constants of SU (n) 2-node interaction matrix Electron charge Unit vector Electric field vector Eigenvalues of the Hamilton operator Transition energy Structure constants Free energy Coupling to the electromagnetic field Lande 9 factor Super operator 2-time-correlation function Spectral density Planck's constant Hamilton operator Vector representation of a Hamilton operator in SU(n) Matrix representation of a Hamilton operator in SU(n) 0 SU(n) Hamilton operator (Heisenberg picture) Hamilton operator (interaction picture) Hilbert space Imaginary unit (i = A) Basis state Intensity Imaginary part Proportional constant, Boltzmann constant (do not confuse the index B with the index B used in the context of operators of a "bath"!) Wave vector k = k modn (mod = modulus) 2-node correlation tensor, matrix element of 3-node correlation tensor, matrix element of

Statistical operator Angular momentum operator, component i Vector operator of angular momenta Environment operator (within Lindblad form) Lindblad super operator Particle mass, electron mass

Mi Mij(p" v), Mf/

nl-'

N

Pi P,Pi Fij = Ii) (jl P = (Px,Py,Pz) Qjk, Qjkl Q r r,R

Rei> R,R(v)

R, k;jkl

R(i)(t) Re s = n2 - 1 S t T T1 , T2 Tr Trl-' U12, V12, WI

U12, V12, WI

U U, Uab U,Uij V, V(i)(t)

(3

List of Symbols XVII

Magnetic moment, component of 2-node correlation tensor proper (covariance), matrix element of Dimension of Hilbert space (subsystem p,) Number of subsystems in networks Probability for state i Polarization vector, component of Transition operator Momentum operator Cluster operators (N = 2,3) Transformation matrix Rank of a group Position vector, position vector of centre of mass (COM) Rotation matrix Reduced density operator, reduced density operator of subsystem v Matrix representation of reduced density operator, element of Relaxation matrix, element of Reduced density operator (interaction picture) Real part Number of generating operators in SU(n) Entropy Time coordinate Absolute temperature, time period Longitudinal and transverse relaxation time Trace Trace operation in subsystem p, Generating operators in SU(2) Expectation values of generating operator in SU(2) Internal energy Unitary operator, unitary basis operator Unitary matrix, matrix elements of Interaction operator, interaction operator in the interaction picture Euclidean vector space Transition probability from state j into state i Partition function

Angle, coefficient in a superposition, phase, index Angle, coefficient in a superposition, index

XVIII List of Symbols

(31 = 1/ (kBT) (3v (v = 1, ... ,b) "I r/}kl r i = Hdh rij = Hij/h 0= Wij - W

{ I: i = j Oij = 0: i =1= j

V tlE = Ei+1 - Ei EO

Er

E, E

(Eijk) , Eijk (rJi) ,rJi rJi rJF

v

/-L /-LB (~ij) '~ij(v),~ir p (Pij) ,Pij p(H)

p(i)(t)

if! 1>, ¢I/¢ 11P) , (11P)* = (1P1

11P(i) (t») lJt

Thermodynamic parameter, T = temperature Lagrangian parameter Damping parameter (off-diagonal) Damping parameters in the relaxation matrix Element of the SU (n) vector r Element of the SU(n) 0 SU(n) matrix (rij ) Detuning parameter of laser with frequency W

Kronecker delta

Nabla operator Energy level spacing Dielectric constant Relative dielectric constant Electric field, vector of E tensor, elements of Damping vector, component of Alternative notation for components rJi(V) Measure for the indeterminacy of the experiment F Coherence vector (generalized Bloch vector), component of Basis operators of SU (n) Subsystem index, common index Subsystem index, common index Bohr magneton Damping matrix, component of Density operator Density matrix, matrix element Density operator (Heisenberg picture) Density operator (interaction picture) Spin operator (for n = 2) Vector of spin operators Pauli matrices with components (ai)jk Creation and annihilation operator (of spin states) System Correlation time Subsystem index with v = 1,2 Phase Angle of rotation, unit vector of rotation General wave function (vector in Hilbert space) General wave function (interaction picture) Scalar field

.p

W = 21rlto

Wij = (Ei - Ej ) In w (Dij) , Dij(V), Dr;

List of Symbols XIX

Column matrix Circular frequency (to = time period), driver frequency Transition frequency Vector of rotation Rotation matrix for subsystem v, matrix elements of Rabi frequency