Elements of Quantum Information

by Schleich, Wolfgang P.; Walther, Herbert

• ISBN13: 9783527407255
• ISBN10: 3527407251
• Edition: 1st
• Format: Hardcover
• Copyright: 2007-03-26
• Publisher: Wiley-VCH

Summary

'Elements of Quantum Information' introduces the reader to the fascinating field of quantum information processing, which lives on the interface between computer science, physics, mathematics, and engineering. This interdisciplinary branch of science thrives on the use of quantum mechanics as a resource for high potential modern applications. With its wide coverage of experiments, applications, and specialized topics - all written by renowned experts - 'Elements of Quantum Information' provides an indispensable up-to-date account of the state of the art of this rapidly advancing field and takes the reader straight up to the frontiers of current research. The articles have first appeared as a special issue of the journal 'Fortschritte der Physik/Progress of Physics'. Since then, they have been carefully updated. The book will be an inspiring source of information and insight for anyone researching and specializing in experiments and theory of quantum information.

Author Biography

Wolfgang P. Schleich, born in 1957, is head of the Institute of Quantum Physics at the University of Ulm, Germany, and Adjunct Professor at the University of North Texas in Denton, USA. While working at the Universities of New Mexico, Albuquerque, of Texas at Austin and the Max Planck Institute for Quantum Physics, Garching, Germany, he has collaborated with world leaders in physics such as M.O. Scully, J.A. Wheeler and H. Walther. Professor Schleich has published more than 230 papers on problems of quantum optics, foundations of quantum mechanics and general relativity and is the author of the highly acclaimed textbook Quantum Optics in Phase Space. For his work he has received numerous awards and honors, including the Ernst Abbe Medal of the International Commission for Optics, the Gottfried Wilhelm Leibniz Prize of the German Science Foundation and the Max Planck Award. He is a Fellow of the Institute of Physics, the American Physical Society and the Optical Society of America and has been elected a member of several academies, such as the Leopoldina, the Heidelberg Academy of Science, and the Royal Danish Academy of Sciences and Letters.

Herbert Walther (1935-2006) received his Ph.D. from the University of Heidelberg in 1962. After working at various universities in Germany, France and the United States, Professor Walther accepted a post as Professor of Physics at the University of Munich in 1975, from which he took retirement in 2003. From 1981, Professor Walther also worked for the Max Planck Society. He founded the Max Planck Institute of Quantum Optics in Garching, and headed the Institute as its Director until his retirement. From 1990 to 1996, he acted as the Max Planck Society's Vice President. Professor Walther was a Fellow and member of many professional physics organizations and scientific boards. He was awarded numerous honours and awards, among them the Max Born Prize (1978) and the Humboldt Medal (1998).

Table of Contents

Preface to the Book	p. XVII
Preface to the Journal	p. XIX
List of Contributors	p. XXI
The Deterministic Generation of Photons by Cavity Quantum Electrodynamics	p. 1
Introduction	p. 1
Oscillatory Exchange of Photons Between an Atom and a Cavity Field	p. 1
Experimental Set-up of the One-atom Maser	p. 3
One-atom Maser as a Source of Non-classical Light	p. 5
Review of Experiments on Basic Properties of the One-atom Maser	p. 8
Statistics of Detector Clicks	p. 12
Trapping States	p. 13
Trapping State Stabilization	p. 17
Fock States on Demand	p. 17
Dynamical Preparation of n-photon States in a Cavity	p. 18
The One-atom Maser Spectrum	p. 24
Other Microwave Cavity Experiments	p. 26
Collapse-and-revival of the Rabi Oscillations in an Injected Coherent Field	p. 26
Atom-photon and Atom-atom Entanglement	p. 27
Atom-photon Phase Gate	p. 28
Quantum Nondestructive-measurement of a Photon	p. 28
Wigner-function of a One-photon State	p. 29
Multiparticle Entanglement	p. 29
Schrodinger Cats and Decoherence	p. 29
Cavity QED Experiments in the Visible Spectral Region	p. 30
The One-atom Laser	p. 30
Atoms Pushed by a Few Photons	p. 31
Single-photon Sources	p. 33
Single-atom Laser using an Ion Trap	p. 34
Conclusions and Outlook	p. 38
References	p. 39
Optimization of Segmented Linear Paul Traps and Transport of Stored Particles	p. 45
Introduction	p. 45
Optimization of a Two-layer Microstructured Ion Trap	p. 46
Design Objectives	p. 47
Operating Mode and Modeling of the Segmented Linear Paul Trap	p. 49
Optimization of the Radial Potential	p. 51
Optimization of the Axial Potential	p. 52
Open Loop Control of Ion Transport	p. 54
Non-adiabatic Heating Sources	p. 54
Overview of the Applied Optimization Strategies	p. 55
The Optimal Control Method	p. 55
Optimization Results	p. 58
Ion Heating due to Anharmonic Dispersion	p. 59
Quantum Mechanical Estimate of Non-adiabatic Parametric Heating	p. 59
Improved Initial Guess Function and Ultra-fast Transport	p. 60
Discussion of the Open-loop Result	p. 62
Outlook	p. 64
Appendix	p. 65
References	p. 66
Transport Dynamics of Single Ions in Segmented Microstructured Paul Trap Arrays	p. 69
Introduction	p. 69
Classical Equations of Motion	p. 71
Classical Dynamics of Ion Transport	p. 72
Homogeneous Solution	p. 73
Green's Function and General Solution	p. 74
Adiabatic Limit	p. 75
Quantum and Classical, Dragged Harmonic Oscillators with Constant Frequency	p. 76
The Dragged Quantum Harmonic Oscillator	p. 78
Transport Dynamics in a Well-controlled Regime	p. 81
Two Analytical Examples	p. 82
Near-optimum Transport Functions	p. 86
High-frequency Limit, Adiabatic Transport, and Approximate Trajectories	p. 86
Please supply a short title	p. 87
Determination of Waveforms	p. 87
Potential Fluctuations and Aspect-ratio Rule	p. 90
Conclusions	p. 95
Appendix	p. 96
References	p. 96
Ensemble Quantum Computation and Algorithmic Cooling in Optical Lattices	p. 99
Introduction	p. 99
Physical System	p. 102
Bose-Hubbard Model	p. 102
Initial State Properties	p. 103
Entropy as Figure of Merit	p. 105
Basic Operations	p. 106
Ensemble Quantum Computation	p. 108
Cooling with Filtering	p. 112
Algorithmic Ground State Cooling	p. 114
The Protocol	p. 114
Theoretical Description	p. 115
Conclusion	p. 118
References	p. 119
Quantum Information Processing in Optical Lattices and Magnetic Microtraps	p. 121
Introduction	p. 121
Optical Lattices	p. 122
Preparation of a Qubit Register	p. 122
A Quantum Conveyer Belt for Neutral Atoms	p. 123
Controlled Collisions	p. 124
Magnetic Microtraps	p. 127
Qubit States on the Atom Chip	p. 128
State-dependent Microwave Potentials	p. 132
Qubit Readout in Microtraps	p. 135
Stable fiber Fabry-Perot Cavities	p. 137
FFP Cavity Fabrication and Performance	p. 137
On-chip Atom Detection with a FFP Cavity	p. 138
Single Atom Preparation	p. 141
Conclusion	p. 142
References	p. 142
Two-dimensional Bose-Einstein Condensates in a CO[subscript 2]-laser Optical Lattice	p. 145
Introduction	p. 345
Experimental Setup and Procedure	p. 146
Experimental Results	p. 148
Conclusions	p. 151
References	p. 153
Creating and Probing Long-range Order in Atomic Clouds	p. 155
Introduction	p. 155
Collective Coupling	p. 157
Experimental Setup	p. 158
Ring Cavity	p. 159
Dipole Trap for [superscript 85]Rb	p. 160
Optical Molasses	p. 162
Signatures of Collective Atomic Recoil Lasing	p. 163
Beat Note of Field Modes	p. 163
Spectra of Recoil-induced Resonances	p. 165
Atomic Transport	p. 166
Creating Long-range Order	p. 168
Analytic Treatment for Perfect Bunching	p. 168
Radiation Pressure	p. 170
Phase-locking by Imperfect Mirrors	p. 171
Simulations of Atomic Trajectories with Friction and Diffusion	p. 172
Lasing Threshold	p. 173
Self-synchronization	p. 174
Probing Long-range Order	p. 176
Bragg Scattering	p. 176
Heterodyned Bragg Spectra	p. 178
Measuring the Bragg Scattering Phase	p. 179
Conclusion	p. 180
References	p. 181
Detecting Neutral Atoms on an Atom Chip	p. 185
Introduction	p. 185
Detecting Single Atoms	p. 186
Measuring the Scattered Light: Fluorescence Detection	p. 186
Measuring the Driving Field	p. 187
Absorption on Resonance	p. 187
Refraction	p. 189
Cavities	p. 189
Absorption on Resonance	p. 189
Refraction	p. 190
Many Atoms in a Cavity	p. 190
Concentric Cavity	p. 191
Miniaturization	p. 191
Properties of Fiber Cavities	p. 192
Loss Mechanisms for a Cavity	p. 193
Losses due to the Gap Length	p. 194
Losses due to Transversal Misalignment	p. 195
Losses due to Angular Misalignment	p. 196
Fresnel Reflections	p. 197
Other Fiber Optical Components for the Atom Chip	p. 199
Fluorescence and Absorption Detectors	p. 199
A Single Mode Tapered Lensed Fiber Dipole Trap	p. 199
Integration of Fibers on the Atom Chip	p. 201
Building Fiber Cavities	p. 201
The SU-8 Resist	p. 203
Test of the SU-8 Structure	p. 204
Pilot Test for Atom Detection with Small Waists	p. 205
Dropping Atoms through a Concentric Cavity	p. 205
Detecting Magnetically Guided Atoms	p. 207
Conclusion	p. 208
References	p. 209
High Resolution Rydberg Spectroscopy of Ultracold Rubidium Atoms	p. 211
Introduction	p. 211
Experimental Setup and Cold Atom Preparation	p. 212
Vacuum System and Magneto Optical Trap (MOT)	p. 212
Rydberg Laser System and Rydberg Excitation	p. 215
Detection of the Rydberg Atoms	p. 216
Excitation Sequence	p. 217
Spectroscopy of Rydberg States,
Spatial and State Selective Addressing of Rydberg States	p. 220
Spatial Selective Rydberg Excitation	p. 220
Hyperfine Selective Rydberg Excitation	p. 221
Autler-Townes Splitting	p. 222
Conclusion and Outlook	p. 224
References	p. 224
Prospects of Ultracold Rydberg Gases for Quantum Information Processing	p. 227
Introduction	p. 227
Excitation of Rydberg Atoms from an Ultracold Gas	p. 229
Van-der-Waals Interaction	p. 230
Blockade of Excitation	p. 231
Ionization	p. 232
States with Permanent Electric Dipole Moments	p. 234
Forster Resonances	p. 236
Conclusion	p. 239
References	p. 241
Quantum State Engineering with Spins	p. 243
Introduction	p. 243
Quantum States of Spins	p. 244
Deutsch-Josza Algorithm	p. 246
The Deutsch-Josza Algorithm	p. 246
Implementation of the 3-qubit Deutsch-Josza Algorithm Using Liquid State NMR	p. 247
2,3,4-Trifluoroaniline	p. 247
Preparation of Pseudo-pure States	p. 248
Results on the 3-qubit DJ-algorithm	p. 249
Entanglement of an Electron and Nuclear Spin in [superscript 15]N@C[subscript 60]	p. 251
Spin Quantum Computing in the Solid State: S-bus	p. 253
The S-bus Concept	p. 253
Single Crystal CaF[subscript 2] : Ce[superscript 3+] as an S-bus system	p. 255
Experimental Details	p. 256
3-qubit Pseudo-pure States	p. 258
2-qubit Deutsch-Josza Algorithm	p. 259
Controlling Nuclear Spin Decoherence in CaF[subscript 2] : Ce	p. 260
Summary and Outlook	p. 263
References	p. 263
Improving the Purity of One- and Two-qubit Gates	p. 265
Introduction	p. 265
Quantum Gate with Bit-flip Noise	p. 266
Bloch-Redfield Master Equation	p. 267
Purity Decay	p. 268
Numerical Solution	p. 269
Coherence Stabilization for Single Qubits	p. 270
Dynamical Decoupling by Harmonic Driving	p. 271
Coherent Destruction of Tunneling	p. 272
Coherence Stabilization for a CNOT Gate	p. 275
Heisenberg vs. Ising Coupling	p. 276
Coherence Stabilization by an AC Field	p. 278
Numerical Solution	p. 279
Implementation with Quantum Dots	p. 282
Conclusions	p. 282
Appendix	p. 283
References	p. 284
How to Distill Entanglement from a Finite Amount of Qubits?	p. 287
Introduction	p. 287
Entanglement Distillation	p. 288
The Protocol	p. 289
CNOT Distillation for a Finite Set of Entangled Systems	p. 293
Iterative Distillation	p. 294
Example of the Iterative Distillation for Small Finite Sets	p. 297
Conclusions	p. 299
Appendix	p. 300
References	p. 301
Experimental Quantum Secret Sharing	p. 303
Introduction	p. 303
Theory	p. 504
The GHZ-protocol	p. 304
The [Psi][Characters not reproducible]-protocol	p. 305
The Single Qubit Protocol	p. 306
Security of the Protocols	p. 307
Experiment	p. 309
The [Psi][Characters not reproducible]-protocol	p. 309
The Single-qubit Protocol	p. 310
Conclusion	p. 312
References	p. 314
Free Space Quantum Key Distribution: Towards a Real Life Application	p. 315
Introduction	p. 315
Setup	p. 316
Transmitter Unit	p. 316
Free Space Link	p. 317
Receiver Unit	p. 318
Synchronisation and Automatic Alignment Control	p. 329
Sifting, Error Correction and Privacy Amplification	p. 319
Experimental Results	p. 320
Conclusion	p. 322
References	p. 323
Continuous Variable Entanglement Between Frequency Modes	p. 325
Introduction	p. 325
Sideband Separation	p. 327
Theory	p. 328
Pictorial Description	p. 331
Experiment and Results	p. 331
Conclusion and Discussion	p. 335
References	p. 336
Factorization of Numbers with Physical Systems	p. 339
Introduction	p. 339
Chirping a Two-photon Transition	p. 340
Chirped Laser Pulses	p. 340
Excitation Probability Amplitude	p. 341
Example for Factorization	p. 342
Driving a One-photon Transition	p. 343
Model	p. 344
Floquet Ladder	p. 345
Pulse Train	p. 346
Factorization	p. 347
Factorization with Floquet Ladder	p. 348
Factorization with a Pulse Train	p. 349
NMR-experiment	p. 350
Conclusions	p. 352
References	p. 353
Quantum Algorithms for Number Fields	p. 355
Introduction	p. 355
Outline of the Survey	p. 355
Why Number Fields?	p. 356
Some History of the Subject	p. 356
Geometry of Numbers	p. 357
Number Fields	p. 357
Lattices	p. 358
Integral Elements	p. 359
The Class Number	p. 360
The Regulator	p. 361
Complexity Results	p. 361
Reduction	p. 362
Reduced Ideals	p. 362
Infrastructure	p. 363
Geometric Interpretation of G	p. 364
Results from Analytic Number Theory	p. 366
Distribution of Prime Numbers	p. 366
Class Number Formulas	p. 367
Examples of Minima Distributions	p. 368
Computing the Regulator	p. 370
Real Quadratic Case	p. 370
Hallgren's Algorithm	p. 371
Generalization of the Weak Periodicity Condition	p. 372
Computation of Other Invariants	p. 374
The Principal Ideal Problem	p. 374
Computing the Class Number	p. 375
References	p. 376
Implementation Complexity of Physical Processes as a Natural Extension of Computational Complexity	p. 377
Introduction	p. 377
Similar Complexity Bounds for Different Tasks	p. 379
Relating Control Problems to Hard Computational Problems	p. 385
The Need for a Control-theoretic Foundation of Complexity	p. 388
Hamiltonians that Compute Autonomously	p. 393
References	p. 396
Implementation of Generalized Measurements with Minimal Disturbance on a Quantum Computer	p. 399
Introduction	p. 399
Minimal-disturbing Implementations of POVMs	p. 401
Generalized Measurements of Quantum Systems	p. 401
Positive-operator Valued Measures	p. 402
Orthogonal Measurements	p. 403
Disturbance of a Generalized Measurement	p. 404
Minimal-disturbing Implementation of a POVM	p. 405
Symmetric Matrices and their Structure	p. 406
Representations of Finite Groups	p. 407
Projective Representations	p. 408
Symmetry of a Matrix and Schur's Lemma	p. 410
Symmetric POVMs Define Matrices with Symmetry	p. 411
Implementation of Symmetric POVMs	p. 413
Cyclic and Heisenberg-Weyl Groups	p. 416
Cyclic Groups	p. 426
Heisenberg-Weyl Groups	p. 429
Conclusions and Outlook	p. 423
References	p. 424
Full Counting Statistics of Interacting Electrons	p. 425
Introduction	p. 425
Concepts of FCS	p. 428
Full Counting Statistics in Interacting Quantum Dots	p. 435
FCS of a Set for Intermediate Strength Conductance	p. 437
Non-Markovian Effects: Renormalization and Finite Lifetime Broadening of Charge States	p. 440
Keldysh Action and CGF tn Majorana Representation	p. 442
FCS and Coulomb Interaction in Diffusive Conductors	p. 443
Model and Effective Action	p. 445
"Cold Electron" Regime	p. 447
"Hot Electron" Regime	p. 453
Summary	p. 454
References	p. 455
Quantum Limit of the Carnot Engine	p. 457
Introduction	p. 457
Spin-oscillator Model	p. 458
Basic Definitions	p. 458
Thermodynamic Variables for G	p. 461
Master Equation	p. 462
Lindblad Superoperator	p. 462
Time Slot Operators	p. 463
Machine Cycles	p. 465
Choice of Amplitudes a[Characters not reproducible] and Control Functions [theta][superscript (j)] ([tau])	p. 465
Heat and Work	p. 467
Energy Balance	p. 468
Fluctuations	p. 468
Numerical Results	p. 470
Heat Engine	p. 470
Heat Pump	p. 474
Longtime Limit	p. 475
Quantum Limit and Classical Limit	p. 475
Summary and Conclusions	p. 477
References	p. 479
Colour Plates	p. 481
Index	p. 491
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