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Tables of Contents for Fluctuations Quantiques=Quantum Fluctuations
Chapter/Section Title
Page #
Page Count
Lecturers
ix
2
Participants
xi
6
Preface (French)
xvii
4
Preface (English)
xxi
 
Course 1. Quantum Fluctuations in Optical Systems
1
78
P. L. Knight
1. Introduction
5
1
2. Quantum interference and the construction of nonclassical states
6
29
2.1. Nonclassical light, quadratures and phase-sensitive observables
6
8
2.2. Phase-space and quasi-probabilities
14
2
2.3. Quantum noise as a stochastic process
16
3
2.4. Marginal distributions and reconstruction of Wigner functions
19
1
2.5. Joint measurements of p, q in phase space
20
7
2.6. Phase-space sampling entropies
27
1
2.7. Mutual information and entropic uncertainty
28
2
2.8. Building nonclassical states by quantum interference
30
5
3. Dissipation versus coherent evolution: decoherence
35
2
4. Unravelling the master equation
37
5
5. Quantum jumps
42
7
5.1. Intrinsic or extrinsic jumps
42
1
5.2. Observation of quantum jumps
43
6
6. Quantum state diffusion
49
8
7. Comparison of ensemble averaged results
57
2
8. Entanglement in quantum optics
59
16
8.1. Entanglement
59
2
8.2. The Schmidt decomposition
61
4
8.3. Two-mode squeezed state
65
2
8.4. Entangled fields and atoms
67
3
8.5. Jaynes-Cummings model
70
3
8.6. Perspective on Schmidt decomposition
73
1
8.7. Conditional measurements and entangled photon pairs
73
2
9. Conclusions
75
1
10. References
75
4
Course 2. Quantum Noise in Quantum Optics: the Stochastic Schrodinger Equation
79
58
P. Zoller
C. W. Gardiner
1. Introduction
83
2
2. An introductory example: Mollow's pure state analysis of resonant light scattering
85
3
2.0.1. Hierarchy of equations
85
2
2.0.2. Interpretation
87
1
3. Wave-function quantum stochastic differential equations
88
11
3.1. The model
88
2
3.1.1. Approximations
88
2
3.2. Quantum stochastic calculus
90
3
3.2.1. Quantum stochastic integration
90
1
3.2.2. Example and discussion
91
2
3.2.3. Comparison of Ito and Stratonovich stochastic differential equations (SDE)
93
1
3.3. Quantum stochastic Schrodinger, density matrix and Heisenberg equations
93
3
3.3.1. Equation of motion for the stochastic density operator
95
1
3.3.2. Remarks
96
1
3.4. Number processes and photon counting
96
1
3.5. Input and output
97
2
4. Counting and diffusion processes: stochastic Schrodinger equation and wave function simulation
99
16
4.1. Photon counting and exclusive probability densities
99
4
4.1.1. Mandel's counting formula
99
1
4.1.2. The characteristic functional and system averages
100
2
4.1.3. Generalization to many channels
102
1
4.1.4. Examples of the use of the characteristic functional
103
1
4.2. Conditional dynamics and a posteriori states
103
2
4.3. Stochastic Schrodinger equation and wave function simulation for counting processes
105
4
4.3.1. Stochastic Schrodinger equation
105
1
4.3.2. Equation of motion for the stochastic density matrix
106
1
4.3.3. Wave function simulation
107
2
4.4. Simulation of stationary two-time correlation functions
109
2
4.5. Diffusion processes and homodyne detection
111
1
4.6. Stochastic Schrodinger equation and wave function simulation for diffusion processes
112
3
5. Application and illustrations
115
18
5.1. State preparation by observation of quantum jumps in an icon trap
115
4
5.1.1. Quantum jumps in three-level atoms
116
1
5.1.2. Fock states in the Jaynes-Cummings model
117
2
5.2. Wave function simulations of laser cooling: applications to quantized optical molasses
119
9
5.2.1. Quantized atomic motion in optical molasses
121
4
5.2.2. Localization by spontaneous emission
125
3
5.3. Quantum computing, quantum noise and continuous observation
128
5
5.3.1. Realization of the elements of a quantum computer
129
2
5.3.2. Errors and their correction
131
2
Appendix A. An exercise in Ito calculus
133
1
References
134
3
Course 3. The Quantum Optics of Dielectrics
137
44
S. M. Barnett
1. Introduction
141
1
2. The problem of propagation
142
2
3. Hopfield theory
144
10
3.1. Dispersion only
144
6
3.2. Dispersion and absorption losses
150
4
4. Propagation of non-classical states
154
4
5. Lifshitz theory
158
11
5.1. Infinite homogeneous dielectric
160
3
5.2. Dielectric interfaces
163
4
5.3. Cavity modes
167
2
6. Spontaneous emission in dielectrics
169
7
6.1. General considerations
169
1
6.2. Dispersive absorbing medium
170
2
6.3. Local field corrections
172
2
6.4. Longitudinal field effects
174
2
7. Summary
176
1
References
177
4
Course 4. Quantum Fluctuations in Light Beams
181
34
C. Fabre
1. Introduction
185
1
2. Noise in optical measurements
186
2
2.1. Autocorrelation function and noise spectral density
186
1
2.2. Noise variances
187
1
2.3. Usual noises
188
1
3. Quantum description of light beams
188
6
3.1. Light beam in classical optics
188
1
3.2. Definition of the field envelope operator
189
2
3.3. Direct photodetection of light intensity
191
1
3.4. Quantum calculation of photocurrent noise spectral densities
192
2
4. Input-output relations
194
2
4.1. Free propagation
195
1
4.2. The beamsplitter
195
1
5. Quadrature components of a light beam
196
4
5.1. Balanced homodyne detection
196
1
5.2. Heisenberg relations for the noise spectral density of the quadrature components
197
1
5.3. Standard quantum limit
198
1
5.4. Intensity and phase noise for an intense beam
199
1
6. Linear input-output transformations for light beams
200
2
6.1. Linear losses
200
1
6.2. Linear gain
201
1
6.3. Non ideal photodetector
202
1
7. Generation of light beams with "nonclassical" fluctuations
202
7
7.1. Linear processing of light beams
203
1
7.2. Parametric processes in a second-order nonlinear medium
203
3
7.3. Other X(2) processes
206
1
7.4. X(3) processes
206
2
7.5. Linearization method for quantum fluctuations
208
1
8. Enhancement of nonclassical effects by resonant cavities
209
3
8.1. Single port optical cavities
209
1
8.2. Intracavity X(2) effects
210
1
8.3. Intracavity X(3) effects
211
1
9. Conclusion
212
1
References
212
3
Course 5. Sub-Poisson Photon Statistics
215
52
L. Davidovich
1. Introduction
219
3
2. Squeezing, sub-Poisson photon statistics, and anti-bunching
222
10
2.1. Quadratures of the electromagnetic field
222
1
2.2. Phase-space representations
223
1
2.3. Squeezed states
224
2
2.4. Number-phase squeezing
226
3
2.5. Relation between photon anti-bunching and sub-Poisson statistics
229
3
3. Sources of quantum noise in lasers
232
13
3.1. The generalized master equation
232
2
3.2. Steady state and photon-number dispersion
234
3
3.3. Lasers
237
2
3.4. Micromasers
239
3
3.5. Poissonian-pumped micromaser: the quantum steady state
242
1
3.6. Trapping states
243
2
3.7. Regularly pumped micromaser
245
1
4. Heisenberg-Langevin approach
246
16
4.1. Heisenberg-Langevin equations
248
8
4.1.1. Macroscopic operators
251
3
4.1.2. Equivalent c-number equations
254
2
4.2. Semiclassical theory and steady state
256
1
4.3. Dynamics of fluctuations
257
1
4.4. Photon-number variance
258
1
4.5. Spectra of the output field
259
1
4.6. Quantum noise compression
260
2
5. Conclusions
262
1
References
263
4
Course 6. Quantum Tomography of Nonclassical Light
267
20
S. F. Pereira
G. Breitenbach
S. Schiller
J. Mlynek
1. Introduction
271
1
2. Generation of squeezed light
272
5
2.1. Experimental set-up
273
2
2.2. Experimental procedure
275
2
3. Measurements of squeezing
277
4
3.1. The squeezing spectrum
277
1
3.2. Squeezing and fluctuation statistics at a fixed frequency
278
3
4. Characterization of the squeezed state
281
3
4.1. Determination of the Wigner function
282
1
4.2. Determination of the density matrix and photon number distribution
282
2
5. Conclusions
284
2
References
286
1
Course 7. Sensitivity in Quantum Measurements
287
22
V. B. Braginsky
1. Introduction
291
1
2. Isolation of the object from the heat bath
292
4
2.1. Microwave resonators
292
2
2.2. Optical resonators
294
1
2.3. Mechanical oscillators
294
1
2.4. Free mass
295
1
2.5. Fluctuations and friction produced by electromagnetic vacuum
295
1
3. Quantum non-demolition measurements
296
6
3.1. Standard quantum limits and quantum behavior of macroscopic objects
296
1
3.2. Definition of the Quantum-Non-Demolition measurements
297
1
3.3. Achievements in the area of QND measurements
299
1
3.4. Prospects of QND measurements in the near future
299
1
3.5. Measurements of phase of quantum oscillator
300
2
4. Quantum limitations in gravitational experiments
302
3
4.1. Sensitivity of laser gravitational antenna today and in the near future
302
3
4.2. Is it possible to measure the quantization of gravitational interaction?
305
1
5. Achievable resolution in symmetry tests and the search for new essences
305
2
References
307
2
Course 8. Cavity Quantum Electrodynamics
309
28
J. M. Raimond
S. Haroche
1. The atom-cavity system
314
2
1.1. Orders of magnitude
314
1
1.2. The dressed atom picture
315
1
2. Resonant atom-field interaction
316
4
2.1. Quantum Rabi nutation: a test of field amplitude quantization in the cavity
316
3
2.2. Resonant atom-field entanglement
319
1
3. Non-resonant atom-field entanglement
320
9
3.1. Phase entanglement
320
2
3.2. An experiment on the dispersive interaction
322
2
3.3. Quantum non demolition intensity measurement
324
1
3.4. Phase Schrodinger cats
325
1
3.5. Quantum switches
326
1
3.6. Quantum gates
327
1
3.7. Teleportation
328
1
4. Mie resonances of dielectric microspheres
329
5
5. Conclusion
334
1
References
335
2
Course 9. Quantum Non-Demolition Measurements in Optics
337
14
Ph. Grangier
1. Introduction
341
1
2. Quantum non-demolition measurements in optics
342
1
2.1. Different regimes for QND measurement in optics
342
1
2.2. From QND devices to quantum duplicators
343
1
3. QND criteria
343
4
3.1. Presentation
343
1
3.2. Quantum state preparation ability
344
1
3.3. Input-output criteria
345
1
3.4. Discussion and standard quantum limits
346
1
4. Experimental implementations
347
1
5. Conclusion
348
1
References
349
2
Course 10. Quantum Fluctuations in Electrical Circuits
351
36
M. H. Devoret
1. Introduction
355
4
2. Hamiltonian description of the classical dynamics of electrical circuits
359
11
2.1. Non-dissipative circuits
359
7
2.1.1. Branch variables
359
3
2.1.2. The degrees of freedom of a circuit
362
1
2.1.3. Lagrangian of a circuit
363
1
2.1.4. Node charges: the conjugate momenta of node fluxes
364
1
2.1.5. Hamiltonian of a circuit
365
1
2.1.6. Mechanical analogy
365
1
2.1.7. Fields to circuits, circuits to fields
366
1
2.2. Circuits with linear dissipative elements
366
4
2.2.1. The Caldeira-Leggett model
366
3
2.2.2. Voltage and current sources
369
1
2.2.3. The classical fluctuation-dissipation theorem
369
1
3. Quantum mechanics of linear dissipative circuits
370
9
3.1. Quantum description of electrical circuits
370
1
3.2. Useful relations
371
1
3.3. The quantum LC oscillator
372
2
3.4. The quantum fluctuation dissipation theorem
374
1
3.5. Interpretation of the quantum spectral density
375
1
3.6. Quantum fluctuations in the damped LC oscillator
375
3
3.7. Low temperature limit
378
1
4. Quantum fluctuations in superconducting tunnel junction circuits
379
6
4.1. Energy operator for a Josephson element
379
1
4.2. The phase difference operator
380
1
4.3. Macroscopic quantum tunneling
381
2
4.4. Influence of dissipation on macroscopic quantum tunneling
383
1
4.5. Zero-voltage conductance of small Josephson junctions
384
1
4.6. Circuits with islands
385
1
References
385
2
Course 11. Spectral Fluctuations in Disordered Metals
387
44
G. Montambaux
1. Introduction, the relevant scales
391
2
2. Metal as a quantum chaotic system
393
5
3. Level correlations in a disordered metal
398
12
3.1. A brief reminder about weak-localization
400
2
3.2. Semi-classical description of energy levels correlations
402
3
3.3. Breakdown of time-reversal symmetry, Aharonov-Bohm flux
405
5
4. Persistent currents
410
9
4.1. Introduction
410
1
4.2. The typical current
411
2
4.3. The average current
413
6
4.3.1. The canonical current
414
1
4.3.2. Electron-electron interactions
415
2
4.3.3. Local versus global fluctuations of the DOS
417
2
4.4. Temperature dependence
419
1
5. Conductance and spectrum
419
3
5.1. The Thouless formula and its extensions
419
2
5.2. Universal Conductance Fluctuations
421
1
6. Parametric correlations
422
4
6.1. Curvatures distribution
422
2
6.2. Parametric correlations
424
2
7. Conclusion
426
1
References
427
4
Course 12. Quantum Fluctuations and Nonlinear Optical Patterns
431
36
L. A. Lugiato
A. Gatti
H. Wiedemann
1. Introduction
435
1
2. Pattern formation in optical systems
436
7
3. The quantum model for the degenerate Optical Parametric Oscillator (OPO)
443
2
4. Quantum effects in the OPO above threshold
445
6
5. Spatial structure of squeezed states
451
7
6. Quantum images in the OPO below threshold
458
6
7. Conclusions
464
1
References
465
2
Course 13. Quantum and Wave Signatures of Chaos
467
32
F. Haake
1. Introduction
471
1
2. Periodically kicked tops
472
3
3. Some quantum distinctions between regular and chaotic behaviour
475
12
3.1. Weak vs. strong sensitivity to perturbations
475
1
3.2. Near-periodic vs. erratic sequences of recurrences
476
3
3.3. Uninhibited vs. avoided crossings of nodal lines in two dimensional systems
479
1
3.4. Level clustering vs. level repulsion
479
8
4. Two dimensional microwave billiards
487
9
References
496
3
Course 14. Quantum Fluctuations and Inertia
499
42
M. T. Jaekel
S. Reynaud
1. Introduction
503
2
2. Quantum fields with boundaries
505
7
2.1. Radiation pressure of quantum fluctuations
505
5
2.2. Quantum fluctuations of radiation pressure
510
2
3. Quantum Brownian motion
512
9
3.1. Linear response formalism in Quantum Field Theory
513
4
3.2. Quantum Langevin equation
517
3
3.3. Quantum fluctuations of position
520
1
4. Quantum limits in space-time
521
8
4.1. Quantum limits
522
3
4.2. Ultimate quantum noise
525
1
4.3. Space-time fluctuations
526
3
5. Quantum fluctuations and accelerated motion
529
9
5.1. Inertia of quantum fluctuations
529
4
5.2. Quantum fluctuations of mass
533
2
5.3. Acceleration and vacuum fluctuations
535
3
6. Conclusion
538
1
References
538
3
Course 15. Vacuum Fluctuations and Cosmology
541
22
L. P. Grishchuk
1. Electrodynamics and gravidynamics
545
4
2. Amplification of classical cosmological perturbations
549
5
3. Quantized cosmological perturbations and squeezing
554
4
4. Statistics of the microwave background anisotropies caused by squeezed cosmological perturbations
558
3
References
561
2
Seminar 1. The Hermitian Optical Phase Operator
563
14
S. M. Barnett
D. T. Pegg
1. The nature of phase
564
1
2. Quantum optical phase
565
7
3. Phase measurement
572
1
4. Summary
573
1
References
574
3
Seminar 2. Dissipative Quantum Systems
577
8
G. -L. Ingold
1. Introduction
578
1
2. Damped harmonic oscillator
578
2
3. Dissipation in quantum mechanics
580
1
4. Path integrals and effective action
581
1
5. Decay of a metastable state
582
2
References
584
1
Seminar 3. Sonoluminescence as Quantum Vacuum Radiation
585
10
C. Eberlein
1. Introduction - What is sonoluminescence?
586
1
2. Static and dynamic vacuum effects
587
1
3. Moving dielectric interfaces
588
2
4. Two-photon emission
590
1
5. Application to sonoluminescence
591
2
References
593
2
Seminar 4. Fundamental Physics Experiments in Space
595
 
Y. Jafry
1. Introduction
596
1
2. LISA
596
5
3. STEP
601
3
4. Ultimate limits
604
1
References
605