Statistical methods in quantum optics: 1 Master equations and Fokker-Planck equations
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| Main Author: | |
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| Format: | Book |
| Language: | English |
| Published: |
Berlin [u.a.]
Springer
2002
|
| Edition: | Corr. 2. print. |
| Series: | Physics and astronomy online library
Texts and monographs in physics |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XXIII, 365 S. graph. Darst. |
| ISBN: | 3540548823 9783540548829 |
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MARC
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| 008 | 021213s2002 d||| |||| 00||| eng d | ||
| 020 | |a 3540548823 |9 3-540-54882-3 | ||
| 020 | |a 9783540548829 |9 978-3-540-54882-9 | ||
| 035 | |a (OCoLC)230765794 | ||
| 035 | |a (DE-599)BVBBV015472344 | ||
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| 100 | 1 | |a Carmichael, Howard |d 1950- |e Verfasser |0 (DE-588)120461315 |4 aut | |
| 245 | 1 | 0 | |a Statistical methods in quantum optics |n 1 |p Master equations and Fokker-Planck equations |c H. J. Carmichael |
| 250 | |a Corr. 2. print. | ||
| 264 | 1 | |a Berlin [u.a.] |b Springer |c 2002 | |
| 300 | |a XXIII, 365 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 0 | |a Physics and astronomy online library | |
| 490 | 0 | |a Texts and monographs in physics | |
| 650 | 4 | |a Quantum optics |x Industrial applications | |
| 650 | 4 | |a Quantum optics |x Statistical methods | |
| 773 | 0 | 8 | |w (DE-604)BV012561429 |g 1 |
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| 999 | |a oai:aleph.bib-bvb.de:BVB01-010110469 | ||
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| _version_ | 1804129704851013632 |
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| adam_text | Table
of
Contents
Volume
1.
Master Equations and
Fokker—
Planck Equations
1.
Dissipation in Quantum Mechanics:
The Master Equation Approach
......................... 1
1.1
Introduction
......................................... 1
1.2
Inadequacy of an Ad Hoc Approach
..................... 2
1.3
System Plus Reservoir Approach
........................ 3
1.3.1
The
Schrödinger
Equation
in Integro-DirFerential Form
...................... 5
1.3.2
Born and Markov Approximations
................ 6
1.3.3
The Markov Approximation
and Reservoir Correlations
....................... 7
1.4
The Damped Harmonic Oscillator
...................... 9
1.4.1
Master Equation
for the Damped Harmonic Oscillator
.............. 9
1.4.2
Some Limitations
............................... 17
1.4.3
Expectation Values and Commutation Relations
.... 18
1.5
Two-Time Averages and the Quantum Regression Formula
. 19
1.5.1
Formal Results
................................. 22
1.5.2
Quantum Regression
for a Complete Set of Operators
.................. 25
1.5.3
Correlation Functions
for the Damped Harmonic Oscillator
.............. 27
2.
Two-Level Atoms and Spontaneous Emission
............ 29
2.1
Two-Level Atom as a Pseudo-Spin System
............... 29
2.2
Spontaneous Emission in the Master Equation Approach
.. 32
2.2.1
Master Equation
for a Radiatively Damped Two-Level Atom
........ 32
2.2.2
The Einstein A Coefficient
....................... 35
2.2.3
Matrix Element Equations. Correlation Functions.
and Spontaneous Emission Spectrum
.............. 36
2.2.4
Phase Destroying Processes
...................... 39
XVI Table of
Contents
2.3
Resonance Fluorescence
............................... 43
2.3.1
The Scattered Field
............................. 45
2.3.2
Master Equation for a Two-Level Atom
Driven by a Classical Field
...................... 48
2.3.3
Optical Bloch Equations and Dressed States
....... 51
2.3.4
The Fluorescence Spectrum
...................... 56
2.3.5
Second-Order Coherence
......................... 60
2.3.6
Photon Antibunching and Squeezing
.............. 65
3.
Quantum—Classical Correspondence
for the Electromagnetic Field I:
The Glauber—
Sudar shan
Ρ
Representation
.............. 75
3.1
The Glauber-Sudarshan
Ρ
Representation
............... 76
3.1.1
Coherent States
................................ 77
3.1.2
Diagonal Representation for the Density Operator
Using Coherent States
........................... 81
3.1.3
Examples: Coherent States, Thermal States,
and Fock States
................................ 83
3.1.4
Fokker-Planck Equation
for the Damped Harmonic Oscillator
.............. 89
3.1.5
Solution of the Fokker-Planck Equation
........... 91
3.2
The Characteristic Function for Normal-Ordered Averages
. 94
3.2.1
Operator Averages and the Characteristic Function
. 95
3.2.2
Derivation of the Fokker-Planck Equation
Using the Characteristic Function
................ 96
4.
Quantum—Classical Correspondence
for the Electromagnetic Field II:
P, Q, and Wigner Representations
...................... 101
4.1
The
Q
and Wigner Representations
..................... 102
4.1.1
Antinormal-Ordered Averages
and the
Q
Representation
....................... 102
4.1.2
The Damped Harmonic Oscillator
in the
Q
Representation
......................... 105
4.1.3
Antinormal-Ordered Averages
Using the
P
Representation
...................... 108
4.1.4
The Wigner Representation
...................... 110
4.2
Fun with Fock States
.................................. 114
4.2.1
Wigner Distribution for a Fock State
.............. 114
4.2.2
Damped Fock State in the
P
Representation
....... 117
4.2.3
Damped Fock State
in the
Q
and Wigner Representations
............. 120
4.3
Two-Time Averages
................................... 123
Table
of Contents
XVII
4.3.1
Quantum-Classical Correspondence
for General Operators
........................... 124
4.3.2
Associated Functions and the Master Equation
..... 129
4.3.3
Normal-Ordered Time-Ordered Averages
in the
Ρ
Representation
......................... 131
4.3.4
More General Two-Time Averages
Using the
Ρ
Representation
...................... 133
4.3.5
Two-Time Averages
Using the
Q
and Wigner Representations
.......... 137
5.
Fokker—
Planck Equations
and Stochastic Differential Equations
................... 147
5.1
One-Dimensional Fokker-Planck Equations
.............. 148
5.1.1
Drift and Diffusion
.............................. 149
5.1.2
Steady-State Solution
........................... 153
5.1.3
Linearization and the System Size Expansion
...... 155
5.1.4
Limitations of the Linearized Treatment
of Fluctuations
................................. 160
5.1.5
The Truncated Kramers-Moyal Expansion
......... 164
5.2
Linear Fokker-Planck Equations
........................ 165
5.2.1
The Green Function
............................ 166
5.2.2
Aloments of Multi-Dimensional Gaussians
.......... 169
5.2.3
Formal Solution for Time-Dependent Averages
..... 171
5.2.4
Equation of Motion for the Covariance Matrix
...... 174
5.2.5
Steady-State Spectrum of Fluctuations
............ 176
5.3
Stochastic Differential Equations
....................... 178
5.3.1
A Comment on Notation
........................ 179
5.3.2
The Wiener Process
............................. 180
5.3.3
Stochastic Differential Equations
................. 183
5.3.4
Ito
and Stratonovich Integrals
.................... 186
5.3.5
Fokker-Planck Equations
and Equivalent Stochastic Differential Equations
.... 190
5.3.6
Multi-Dimensional Ornstein-Uhlenbeck Process
.... 192
6.
Quantum—Classical Correspondence
for Two-Level Atoms
.................................... 195
6.1
Hakend
Representation
and the Damped Two-Level Atom
...................... 195
6.1.1
The Characteristic Function
and Associated Distribution
...................... 196
6.1.2
Some Operator Algebra
......................... 197
6.1.3
Phase-Space Equation of Motion
for the Damped Two-Level Atom
................. 199
XVIII Table of
Contents
6.1.4
A Singular
Solution
to the Phase-Space Equation of Motion
............ 205
6.2
Normal-Ordered Representation
for a Collection of Two-Level Atoms
.................... 211
6.2.1
Collective Atomic Operators
..................... 212
6.2.2
Direct Product States,
Dicke
States,
and Atomic Coherent States
..................... 216
6.2.3
The Characteristic Function
and Associated Distribution
...................... 222
6.2.4
Nonsingular Approximation for the
Ρ
Distribution
. . 223
6.2.5
Two-Time Averages
............................. 226
6.2.6
Other Representations
.......................... 232
6.3
Fokker-Planck Equation
for a Radiatively Damped Two-Level Medium
............ 233
6.3.1
Master Equation
for Independently Damped Two-Level Atoms
....... 233
6.3.2
Closed Dynamics for Normally-Ordered Averages
of Collective Operators
.......................... 236
6.3.3
Operator Averages
Without Quantum Fluctuations
.................. 241
6.3.4
Phase-Space Equation of Motion
for Independently Damped Two-Level Atoms
....... 245
6.3.5
Fokker-Planck Equation:
First-Order Treatment of Quantum Fluctuations
. .. 248
6.3.6
Steady-State Distribution of Inversion
............. 252
7.
The Single-Mode Homogeneously Broadened Laser I:
Preliminaries
........................................... 257
7.1
Laser Theory from Einstein Rate Equations
.............. 258
7.1.1
Rate Equations and Laser Threshold
.............. 258
7.1.2
Spontaneous Emission and Thermal Photons
....... 263
7.1.3
Quantum Fluctuations: A Stochastic Model
........ 268
7.1.4
Two-Level Model and Laser Parameters
........... 276
7.2
Phase-Space Formulation
in the Normal-Ordered Representation
.................. 280
7.2.1
Model and Hamiltonian
......................... 280
7.2.2
Master Equation for the Single-Mode
Homogeneously Broadened Laser
................. 284
7.2.3
The Characteristic Function
and Associated Distribution
...................... 286
7.2.4
Phase-Space Equation of Motion
for the Single-Mode Homogeneously Broadened Laser
287
7.3
The Laser Output Field
............................... 289
Table
of
Contents
XIX
7.3.1
Pree
Field and Source Field
for a Lossy Cavity Mode
......................... 289
7.3.2
Coherently Driven Cavities
...................... 293
7.3.3
Correlations Between the Free Field
and Source Field for Thermal Reservoirs
........... 295
7.3.4
Spectrum of the Free Field plus Source Field
for the Laser Below Threshold
.................... 302
8.
The Single-Mode Homogeneously Broadened Laser II:
Phase-Space Analysis
................................... 305
8.1
Linearization:
Laser Fokker-Planck Equation Below Threshold
.......... 305
8.1.1
System Size Expansion Below Threshold
........... 305
8.1.2
Laser Equations Without Fluctuations
............ 312
8.1.3
Linearized Treatment
of Quantum Fluctuations Below Threshold
......... 316
8.1.4
Adiabatic Elimination
of the Polarization and Laser Linewidth
........... 320
8.2
Laser Fokker-Planck Equation at Threshold
.............. 325
8.2.1
System Size Expansion
and Adiabatic Elimination of Atomic Variables
..... 326
8.2.2
Steady-State Solution
and Threshold Photon Number
................... 329
8.3
Quasi-Linearization:
Laser Fokker-Planck Equation Above Threshold
.......... 331
8.3.1
System Size Expansion Above Threshold
.......... 333
8.3.2
Adiabatic Elimination
........................... 340
8.3.3
Quantum Fluctuations Above Threshold
........... 345
References
.................................................. 349
Index
....................................................... 357
XX
Table
of Contents
Volume
2.
Modern Topics
9.
The Degenerate Parametric Oscillator I: Preliminaries
9.1
Introduction
9.2
Degenerate Parametric Amplification and Squeezed States
9.2.1
Degenerate Parametric Amplification
Without Pump Depletion
9.2.2
Quantum Fluctuations and Squeezed States
9.2.3
The Degenerate Parametric Oscillator
9.2.4
Master Equation
for the Degenerate Parametric Oscillator
9.2.5
Cavity Output Fields
9.3
The Spectrum of Squeezing
9.3.1
Intracavity Field Fluctuations
9.3.2
The Spectrum of Squeezing Defined
9.3.3
Homodyne
Detection:
The Source-Field Spectrum of Squeezing
9.3.4
The Source-Field Spectrum of Squeezing
with Unit Efficiency
9.3.5
Free-Field Contributions
9.3.6
Vacuum Fluctuations
9.3.7
Squeezing in the Wigner Representation:
A Comment on Interpretation
10.
The Degenerate Parametric Oscillator II:
Phase-Space Analysis in the Small-Noise Limit
10.1
Phase-Space Formalism
for the Degenerate Parametric Oscillator
10.1.1
Phase-Space Equation of Motion
in the
Ρ
Representation
10.1.2
Phase-Space Equations of Motion
in the
Q
and Wigner Representations
10.2
Squeezing: Quantum Fluctuations
in the Small-Noise Limit
10.2.1
System Size Expansion Far from Threshold
10.2.2
Quantum Fluctuations Below Threshold
10.2.3
Quantum Fluctuations Above Threshold
10.2.4
Quantum Fluctuations at Threshold
11.
The Positive
P
Representation
11.1
The Positive
P
Representation
11.1.1
The Characteristic Function
and Associated Distribution
Table
of
Contents
XXI
11.1.2
Fokker-Planck Equation
for the Degenerate Parametric Oscillator
11.1.3
Linear Theory of Quantum Fluctuations
11.2
Miscellaneous Topics
11.2.1
Alternative Approaches
to the Linear Theory of Quantum Fluctuations
11.2.2
Dynamical Stability of the Classical Phase Space
11.2.3
Preservation of Conjugacy for Stochastic Averages
12.
The Degenerate Parametric Oscillator III:
Phase-Space Analysis Outside the Small-Noise Limit
12.1
The Degenerate Parametric Oscillator
with Adiabatic Elimination of the Pump
12.1.1
Adiabatic Elimination
in the Stochastic Differential Equations
12.1.2
A Note About
Superoperators
12.1.3
Adiabatic Elimination in the Master Equation
12.1.4
Numerical Simulation
of the Stochastic Differential Equations
12.1.5
Deterministic Dynamics
in the Extended Phase Space
12.1.6
Steady-State Distribution
for the Positive
Ρ
Distribution
12.1.7
Quantum Fluctuations and System Size
12.1.8
Quantum Dynamics Beyond Classical Trajectories
plus Fuzz
12.1.9
Higher-Order Corrections
to the Spectrum of Squeezing at Threshold
12.2
Difficulties with the Positive
Ρ
Representation
12.2.1
Technical Difficulties: Two-Photon Damping
12.2.2
Physical Interpretation:
The Anharmonic Oscillator
13.
Cavity QED I: Simple Calculations
13.1
System Size and Coupling Strength
13.2
Cavity QED in the Perturbative Limit
13.2.1
Cavity-Enhanced Spontaneous Emission
13.2.2
Cavity-Enhanced Resonance Fluorescence
13.2.3
Forwards Photon Scattering
in the Weak-Excitation Limit
13.2.4
A One-Atom Laser
13.3
Nonperturbative Cavity QED
13.3.1
Spontaneous Emission
from a Coupled Atom and Cavity
XXII Table of
Contents
13.3.2
Vacuum
Rabi
Splitting
13.3.3
Vacuum
Rabi
Resonances
in the Two-State Approximation
14.
Many Atoms in a Cavity I: Macroscopic Theory
14.1
Optical Bistability: Steady-State Transmission
of a Nonlinear Fabry-Perot
14.2
The Mean-Field Limit for a Homogeneously Broadened
Two-Level Medium
14.2.1
Steady State
14.2.2
Maxwell-Bloch Equations
14.2.3
Stability of the Steady State
14.3
Relationship Between Macroscopic
and Microscopic Variables
14.4
Cavity QED with Many Atoms
14.4.1
Weak-Probe Transmission Spectra
14.4.2
A Comment on Spatial Effects
15.
Many Atoms in a Cavity II:
Quantum Fluctuations in the Small-Noise Limit
15.1
Microscopic Model
15.1.1
Master Equation for Optical Bistability
15.1.2
Fokker-Planck Equation in the
Ρ
Representation
15.1.3
Fokker-Planck Equation in the
Q
Representation
15.1.4
Fokker-Planck Equation in the Wigner Representation
15.2
Linear Theory of Quantum Fluctuations
15.2.1
System Size Expansion for Optical Bistability
15.2.2
Linearization About the Steady State
15.2.3
Covariance Matrix for Absorptive Bistability
15.2.4
Atom-Atom Correlations
15.2.5
Spectrum of the Transmitted Light
in the Weak-Excitation Limit
15.2.6
Forwards Photon Scattering
in the Weak-Excitation Limit
16.
Cavity QED II: Quantum Fluctuations
16.1
Density Matrix Expansion for the Weak-Excitation Limit
16.1.1
Pure-State Factorization of the Density Operator
for One Atom
16.1.2
Pure-State Factorization of the Density Operator
for Many Atoms
16.1.3
Forwards Photon Scattering for TV Atoms in a Cavity
16.1.4
Corrections to the Small-Noise Approximation
16.1.5
Autibimching of Fluorescence for One Atom in a Cavity
Table
of Contents
XXIII
16.1.6
Spectra of Squeezing
16.2
Spatial Effects
16.3
Beyond Classical Trajectories plus Fuzz :
Spontaneous Dressed-State Polarization
16.3.1
Maxwell-Bloch Equations for Zero System Size
16.3.2
Dressed Jaynes-Cummings Eigenstates
16.3.3
Secular Approximation
in the Basis of Dressed Jaynes-Cummings Eigenstates
16.3.4
Spectrum of the Transmitted Light
in the Weak-Excitation Limit
16.3.5
The s/n Anharmonic Oscillator
16.3.6
Quantum Fluctuations for Strong Excitation
17.
Quantum Trajectories I:
Introduction and Physical Interpretation
17.1
Density Operators and Scattering Records
17.2
Generalizing the Bohr-Einstein Quantum Jump
17.2.1
The Einstein Stochastic Process
17.2.2
Trajectories for Known Initial States
17.2.3
Trajectories for Blind Realizations
17.2.4
Superposition States
17.2.5
Record Probabilities and Norms
17.2.6
Summing Over Records: The Master Equation
17.3
Quantum Trajectories
for the Degenerate Parametric Oscillator
17.3.1
Record Probabilities
17.3.2
Unraveling the Density Operator:
Photon Counting Records
17.3.3
Unraveling the Density Operator:
Homodyne- and Heterodyne-Current Records
17.3.4
Systems. Environments, and Complementarity
17.4
Modeling
Projective
Measurements
18.
Quantum Trajectories II: More Examples
18.1
Photon Scattering in Cavity QED
18.2
Unraveling the Density Operator: Cascaded Systems
18.2.1
Coherent States
18.3
Optical Spectra
18.3.1
Incoherent Spectrum Using a Filter Cavity
18.3.2
Spontaneous Emission from a Three-Level Atom
18.3.3
Incoherent Spectrum Using Heterodyne Detection
References
Index
|
| any_adam_object | 1 |
| author | Carmichael, Howard 1950- |
| author_GND | (DE-588)120461315 |
| author_facet | Carmichael, Howard 1950- |
| author_role | aut |
| author_sort | Carmichael, Howard 1950- |
| author_variant | h c hc |
| building | Verbundindex |
| bvnumber | BV015472344 |
| callnumber-first | Q - Science |
| callnumber-label | QC446 |
| callnumber-raw | QC446.2 |
| callnumber-search | QC446.2 |
| callnumber-sort | QC 3446.2 |
| callnumber-subject | QC - Physics |
| ctrlnum | (OCoLC)230765794 (DE-599)BVBBV015472344 |
| dewey-full | 535 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 535 - Light and related radiation |
| dewey-raw | 535 |
| dewey-search | 535 |
| dewey-sort | 3535 |
| dewey-tens | 530 - Physics |
| discipline | Physik |
| edition | Corr. 2. print. |
| format | Book |
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| id | DE-604.BV015472344 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T19:09:21Z |
| institution | BVB |
| isbn | 3540548823 9783540548829 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-010110469 |
| oclc_num | 230765794 |
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| owner | DE-29T DE-19 DE-BY-UBM DE-384 DE-M347 DE-355 DE-BY-UBR DE-83 |
| owner_facet | DE-29T DE-19 DE-BY-UBM DE-384 DE-M347 DE-355 DE-BY-UBR DE-83 |
| physical | XXIII, 365 S. graph. Darst. |
| publishDate | 2002 |
| publishDateSearch | 2002 |
| publishDateSort | 2002 |
| publisher | Springer |
| record_format | marc |
| series2 | Physics and astronomy online library Texts and monographs in physics |
| spelling | Carmichael, Howard 1950- Verfasser (DE-588)120461315 aut Statistical methods in quantum optics 1 Master equations and Fokker-Planck equations H. J. Carmichael Corr. 2. print. Berlin [u.a.] Springer 2002 XXIII, 365 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Physics and astronomy online library Texts and monographs in physics Quantum optics Industrial applications Quantum optics Statistical methods (DE-604)BV012561429 1 Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=010110469&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Carmichael, Howard 1950- Statistical methods in quantum optics Quantum optics Industrial applications Quantum optics Statistical methods |
| title | Statistical methods in quantum optics |
| title_auth | Statistical methods in quantum optics |
| title_exact_search | Statistical methods in quantum optics |
| title_full | Statistical methods in quantum optics 1 Master equations and Fokker-Planck equations H. J. Carmichael |
| title_fullStr | Statistical methods in quantum optics 1 Master equations and Fokker-Planck equations H. J. Carmichael |
| title_full_unstemmed | Statistical methods in quantum optics 1 Master equations and Fokker-Planck equations H. J. Carmichael |
| title_short | Statistical methods in quantum optics |
| title_sort | statistical methods in quantum optics master equations and fokker planck equations |
| topic | Quantum optics Industrial applications Quantum optics Statistical methods |
| topic_facet | Quantum optics Industrial applications Quantum optics Statistical methods |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=010110469&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV012561429 |
| work_keys_str_mv | AT carmichaelhoward statisticalmethodsinquantumoptics1 |