Elements of quantum information

Cover......Page 1
Half-Title......Page 2
Title: Elements of Quantum Information Edited by Wolfgang P. Schleich and Herbert Walther......Page 4
ISBN: 978-3-527-40725-5......Page 5
Contents......Page 6
Preface to the Book Edition......Page 18
Preface to the Journal Edition......Page 20
List of Contributors......Page 22
1.2 Oscillatory Exchange of Photons Between an Atom and a Cavity Field......Page 34
1.2.1 Experimental Set-up of the One-atom Maser......Page 36
1.2.2 One-atom Maser as a Source of Non-classical Light......Page 38
1.2.3 Review of Experiments on Basic Properties of the One-atomMaser......Page 41
1.2.4 Statistics of Detector Clicks......Page 45
1.2.5 Trapping States......Page 46
1.2.7 Fock States on Demand......Page 50
1.2.8 Dynamical Preparation of n-photon States in a Cavity......Page 51
1.2.9 The One-atom Maser Spectrum......Page 57
1.3.1 Collapse-and-revival of the Rabi Oscillations in an Injected Coherent Field......Page 59
1.3.2 Atom-photon and Atom-atom Entanglement......Page 60
1.3.4 Quantum Nondestructive-measurement of a Photon......Page 61
1.3.7 Schrödinger Cats and Decoherence......Page 62
1.4.1 The One-atom Laser......Page 63
1.4.2 Atoms Pushed by a Few Photons......Page 64
1.4.3 Single-photon Sources......Page 66
1.4.4 Single-atom Laser using an Ion Trap......Page 67
1.5 Conclusions and Outlook......Page 71
References......Page 72
2.1 Introduction......Page 78
2.2 Optimization of a Two-layer Microstructured Ion Trap......Page 79
2.2.1 Design Objectives......Page 80
2.2.2 Operating Mode and Modeling of the Segmented Linear Paul Trap......Page 82
2.2.3 Optimization of the Radial Potential......Page 84
2.2.4 Optimization of the Axial Potential......Page 85
2.3.1 Non-adiabatic Heating Sources......Page 87
2.3.3 The Optimal Control Method......Page 88
2.3.4 Optimization Results......Page 91
2.3.6 Quantum Mechanical Estimate of Non-adiabatic Parametric Heating......Page 92
2.3.7 Improved Initial Guess Function and Ultra-fast Transport......Page 93
2.3.8 Discussion of the Open-loop Result......Page 95
2.4 Outlook......Page 97
References......Page 99
3.1 Introduction......Page 102
3.2 Classical Equations of Motion......Page 104
3.3 Classical Dynamics of Ion Transport......Page 105
3.3.1 Homogeneous Solution......Page 106
3.3.2 Green’s Function and General Solution......Page 107
3.3.3 Adiabatic Limit......Page 108
3.4 Quantum and Classical, Dragged Harmonic Oscillators with Constant Frequency......Page 109
3.5 The Dragged Quantum Harmonic Oscillator......Page 111
3.6 Transport Dynamics in a Well-controlled Regime......Page 114
3.6.1 Two Analytical Examples......Page 115
3.6.3 High-frequency Limit, Adiabatic Transport, and Approximate Trajectories......Page 119
3.7.1 Determination of Waveforms......Page 120
3.7.2 Potential Fluctuations and Aspect-ratio Rule......Page 123
3.8 Conclusions......Page 128
References......Page 129
4.1 Introduction......Page 132
4.2.1 Bose-HubbardModel......Page 135
4.2.2 Initial State Properties......Page 136
4.2.3 Entropy as Figure of Merit......Page 138
4.2.4 Basic Operations......Page 139
4.3 Ensemble Quantum Computation......Page 141
4.4 Cooling with Filtering......Page 145
4.5.1 The Protocol......Page 147
4.5.2 Theoretical Description......Page 148
4.6 Conclusion......Page 151
References......Page 152
5.1 Introduction......Page 154
5.2.1 Preparation of a Qubit Register......Page 155
5.2.2 A Quantum Conveyer Belt for Neutral Atoms......Page 156
5.2.3 Controlled Collisions......Page 157
5.3 Magnetic Microtraps......Page 160
5.3.1 Qubit States on the Atom Chip......Page 161
5.3.2 State-dependent Microwave Potentials......Page 165
5.3.3 Qubit Readout in Microtraps......Page 168
5.3.3.2 FFP Cavity Fabrication and Performance......Page 170
5.3.4 On-chip Atom Detection with a FFP Cavity......Page 171
5.3.5 Single Atom Preparation......Page 174
5.4 Conclusion......Page 175
References......Page 176
6.1 Introduction......Page 178
6.2 Experimental Setup and Procedure......Page 179
6.3 Experimental Results......Page 181
6.4 Conclusions......Page 184
References......Page 186
7.1 Introduction......Page 188
7.2 Collective Coupling......Page 190
7.2.1 Experimental Setup......Page 191
7.2.1.1 Ring Cavity......Page 192
7.2.1.2 Dipole Trap for 85Rb......Page 193
7.2.1.3 Optical Molasses......Page 195
7.2.2.1 Beat Note of Field Modes......Page 196
7.2.2.2 Spectra of Recoil-induced Resonances......Page 198
7.2.2.3 Atomic Transport......Page 199
References......Page 10
7.3.1 Analytic Treatment for Perfect Bunching......Page 201
7.3.1.1 Radiation Pressure......Page 203
7.3.1.2 Phase-locking by Imperfect Mirrors......Page 204
7.3.2 Simulations of Atomic Trajectories with Friction and Diffusion......Page 205
7.3.2.1 Lasing Threshold......Page 206
7.3.2.2 Self-synchronization......Page 207
7.4.1 Bragg Scattering......Page 209
7.4.2 Heterodyned Bragg Spectra......Page 211
7.4.3 Measuring the Bragg Scattering Phase......Page 212
7.5 Conclusion......Page 213
References......Page 214
8.1 Introduction......Page 218
8.2.1 Measuring the Scattered Light: Fluorescence Detection......Page 219
8.2.2.1 Absorption on Resonance......Page 220
8.2.3.1 Absorption on Resonance......Page 222
8.2.3.3 Many Atoms in a Cavity......Page 223
8.2.5 Miniaturization......Page 224
8.3 Properties of Fiber Cavities......Page 225
8.3.1 Loss Mechanisms for a Cavity......Page 226
8.3.2 Losses due to the Gap Length......Page 227
8.3.3 Losses due to TransversalMisalignment......Page 228
8.3.4 Losses due to Angular Misalignment......Page 229
8.3.5 Fresnel Reflections......Page 230
8.4.2 A Single Mode Tapered Lensed Fiber Dipole Trap......Page 232
8.5.1 Building Fiber Cavities......Page 234
8.5.2 The SU-8 Resist......Page 236
8.5.3 Test of the SU-8 Structure......Page 237
8.6.1 Dropping Atoms through a Concentric Cavity......Page 238
8.6.2 Detecting Magnetically Guided Atoms......Page 240
8.7 Conclusion......Page 241
References......Page 242
9.1 Introduction......Page 244
9.2.1 Vacuum System and Magneto Optical Trap (MOT)......Page 245
9.2.2 Rydberg Laser System and Rydberg Excitation......Page 248
9.2.3 Detection of the Rydberg Atoms......Page 249
9.2.4 Excitation Sequence......Page 250
9.3 Spectroscopy of Rydberg States, |mj| Splitting of the Rydberg States......Page 252
9.4.1 Spatial Selective Rydberg Excitation......Page 253
9.4.2 Hyperfine Selective Rydberg Excitation......Page 254
9.5 Autler-Townes Splitting......Page 255
References......Page 257
10.1 Introduction......Page 260
10.2 Excitation of Rydberg Atoms from an Ultracold Gas......Page 262
10.3 Van-der-Waals Interaction......Page 263
10.3.1 Blockade of Excitation......Page 264
10.3.2 Ionization......Page 265
10.4 States with Permanent Electric Dipole Moments......Page 267
10.5 Förster Resonances......Page 269
10.6 Conclusion......Page 272
References......Page 274
11.1 Introduction......Page 276
11.1.1 Quantum States of Spins......Page 277
11.2.1 The Deutsch-Josza Algorithm......Page 279
11.2.2.1 2,3,4-Trifluoroaniline......Page 280
11.2.2.2 Preparation of Pseudo-pure States......Page 281
11.2.2.3 Results on the 3-qubit DJ-algorithm......Page 282
11.3 Entanglement of an Electron and Nuclear Spin in 15N@C60......Page 284
11.4.1 The S-bus Concept......Page 286
11.4.2 Single Crystal CaF2 : Ce3+ as an S-bus system......Page 288
11.4.3 Experimental Details......Page 289
11.4.4 3-qubit Pseudo-pure States......Page 291
11.4.5 2-qubit Deutsch-Josza Algorithm......Page 292
11.4.5.1 Controlling Nuclear Spin Decoherence in CaF2 : Ce......Page 293
References......Page 296
12.1 Introduction......Page 298
12.2 Quantum Gate with Bit-flip Noise......Page 299
12.2.1 Bloch-Redfield Master Equation......Page 300
12.2.2 Purity Decay......Page 301
12.2.3 Numerical Solution......Page 302
12.3 Coherence Stabilization for Single Qubits......Page 303
12.3.1 Dynamical Decoupling by Harmonic Driving......Page 304
12.3.2 Coherent Destruction of Tunneling......Page 305
12.4 Coherence Stabilization for a CNOT Gate......Page 308
12.4.1 Heisenberg vs. Ising Coupling......Page 309
12.4.2 Coherence Stabilization by an AC Field......Page 311
12.4.3 Numerical Solution......Page 312
12.5 Conclusions......Page 315
A Appendix......Page 316
References......Page 317
13.1 Introduction......Page 320
13.2 Entanglement Distillation......Page 321
13.2.1 The Protocol......Page 322
13.3 CNOT Distillation for a Finite Set of Entangled Systems......Page 326
13.3.1 Iterative Distillation......Page 327
13.4 Example of the Iterative Distillation for Small Finite Sets......Page 330
13.5 Conclusions......Page 332
A Appendix......Page 333
References......Page 334
14.1 Introduction......Page 336
14.2.1 The GHZ-protocol......Page 337
14.2.2 The |Ψ−4>-protocol......Page 338
14.2.3 The Single Qubit Protocol......Page 339
14.2.4 Security of the Protocols......Page 340
14.3.1 The |Ψ−4>-protocol......Page 342
14.3.2 The Single-qubit Protocol......Page 343
14.4 Conclusion......Page 345
References......Page 347
15.1 Introduction......Page 348
15.2.1 Transmitter Unit......Page 349
15.2.2 Free Space Link......Page 350
15.2.3 Receiver Unit......Page 351
15.2.5 Sifting, Error Correction and Privacy Amplification......Page 352
15.2.6 Experimental Results......Page 353
15.3 Conclusion......Page 355
References......Page 356
16.1 Introduction......Page 358
16.2 Sideband Separation......Page 360
16.2.1 Theory......Page 361
16.3 Experiment and Results......Page 364
16.4 Conclusion and Discussion......Page 368
References......Page 369
17.1 Introduction......Page 372
17.2.1 Chirped Laser Pulses......Page 373
17.2.2 Excitation Probability Amplitude......Page 374
17.2.3 Example for Factorization......Page 375
17.3 Driving a One-photon Transition......Page 376
17.3.1 Model......Page 377
17.3.2 Floquet Ladder......Page 378
17.3.3 Pulse Train......Page 379
17.4 Factorization......Page 380
17.4.1 Factorization with Floquet Ladder......Page 381
17.4.2 Factorization with a Pulse Train......Page 382
17.5 NMR-experiment......Page 383
17.6 Conclusions......Page 385
References......Page 386
18.1.1 Outline of the Survey......Page 388
18.1.3 Some History of the Subject......Page 389
18.2.1 Number Fields......Page 390
18.2.2 Lattices......Page 391
18.2.3 Integral Elements......Page 392
18.2.4 The Class Number......Page 393
18.2.6 Complexity Results......Page 394
18.3.1 Reduced Ideals......Page 395
18.3.2 Infrastructure......Page 396
18.3.3 Geometric Interpretation of G......Page 397
18.4.1 Distribution of Prime Numbers......Page 399
18.4.2 Class Number Formulas......Page 400
18.5 Examples of Minima Distributions......Page 401
18.6.1 Real Quadratic Case......Page 403
18.6.2 Hallgren’s Algorithm......Page 404
18.6.3 Generalization of the Weak Periodicity Condition......Page 405
18.7.1 The Principal Ideal Problem......Page 407
18.7.2 Computing the Class Number......Page 408
References......Page 409
19.1 Introduction......Page 410
19.2 Similar Complexity Bounds for Different Tasks......Page 412
19.3 Relating Control Problems to Hard Computational Problems......Page 418
19.4 The Need for a Control-theoretic Foundation of Complexity......Page 421
19.5 Hamiltonians that Compute Autonomously......Page 426
References......Page 429
20.1 Introduction......Page 432
20.2.1 Generalized Measurements of Quantum Systems......Page 434
20.2.2 Positive-operator Valued Measures......Page 435
20.2.3 Orthogonal Measurements......Page 436
20.2.4 Disturbance of a Generalized Measurement......Page 437
20.2.5 Minimal-disturbing Implementation of a POVM......Page 438
20.3 Symmetric Matrices and their Structure......Page 439
20.3.1 Representations of Finite Groups......Page 440
20.3.2 Projective Representations......Page 441
20.3.3 Symmetry of a Matrix and Schur’s Lemma......Page 443
20.3.4 Symmetric POVMs Define Matrices with Symmetry......Page 444
20.4 Implementation of Symmetric POVMs......Page 446
20.5.1 Cyclic Groups......Page 449
20.5.2 Heisenberg-Weyl Groups......Page 452
20.6 Conclusions and Outlook......Page 456
References......Page 457
21.1 Introduction......Page 458
21.2 Concepts of FCS......Page 461
21.3 Full Counting Statistics in Interacting Quantum Dots......Page 468
21.3.1 FCS of a Set for Intermediate Strength Conductance......Page 470
21.3.2 Non-Markovian Effects: Renormalization and Finite Lifetime Broadening of Charge States......Page 473
21.3.3 Keldysh Action and CGF in Majorana Representation......Page 475
21.4 FCS and Coulomb Interaction in Diffusive Conductors......Page 476
21.4.1 Model and Effective Action......Page 478
21.4.2 “Cold Electron” Regime......Page 480
21.4.3 “Hot Electron” Regime......Page 486
21.5 Summary......Page 487
References......Page 488
22.1 Introduction......Page 490
22.2.1 Basic Definitions......Page 491
22.2.2 Thermodynamic Variables for G......Page 494
22.3.1 Lindblad Superoperator......Page 495
22.3.2 Time Slot Operators......Page 496
22.4.1 Choice of Amplitudes a(j)± and Control Functions θ(j)(τ)......Page 498
22.4.2 Heat and Work......Page 500
22.4.4 Fluctuations......Page 501
22.5.1 Heat Engine......Page 503
22.5.2 Heat Pump......Page 507
22.5.4 Quantum Limit and Classical Limit......Page 508
22.6 Summary and Conclusions......Page 510
References......Page 512
Appendix: Colour Plates......Page 514
Index......Page 524