Preface
1 Operator
algebra and the harmonic oscillator
1.1
Introduction
1.2 von Neumann equation
1.3 Baker-Hausdorff formula
1.4 Quantum mechanical harmonic oscillator
1.4.1
Ladder operators
1.4.2 Fock states
1.4.3 Coherent states
1.4.4 Displaced number states
1.4.5 Phase states
1.5.
Ordering of ladder operators
1.5.1 Normal ordering
1..5.1.1
Lemma 1
1.5.2 Anti-normal ordering
1.5.2.1 Lemma 2
1.5.3 Coherent states
1.5.4 Fock states
2 Quasiprobability distribution functions
2.1 Introduction
2.2 Wigner function
2.2.1 Properties of the Wigner function
2.2.2 Obtaining expectation values from the
Wigner function
2.2.3 Symmetric averages
2.2.4 Series representation of the Wigner
function
2.3 Glauber-Sudarshan P-function
2.4 Husimi Q-function
2.5 Relations between quasiprobabilities
2.5.1
Differential forms
2.5.2 Integral
forms
2.6 The Wigner function as a took to
calculate divergent (or not) series
2.7 Number-phase Wigner function
27.1 Coherent state
2.7.2 A special superposition of number
states
3 Time Dependent Harmonic Oscillator
3.1 Time dependent harmonic Hamiltonian
3.1.1 Minimum uncertainty states
3.1.2 Step function
3.2 More states of the filed
3.2.1 Squeezed states
3.2.2 Schrodinger cat states
3.2.3 Thermal distribution
4 (Tow-level) Atom-field interaction
4.1 Semi classical interaction
4.2 Quantum interaction
4.2.1 Atomic inversion
4.3 Dispersive interaction
4.4 Mixing classical and quantum
interactions
4.5 Slow atom interacting with a quantized filed
5 A real cavity: Master equation
5.1 Cavity losses at zero temperature
5.1.1 Coherent states
5.1.2 Number states
5.1.3 Cat states
5.2 Master equation at finite temperature
6 Pure
states and statistical mixtures
6.1 Entropy
6.2 Purity
6.3 Entropy and purity in the atom-field
interaction
6.4 Some properties of reduced density matrices
6.4.1 A proof by induction
6.4.2 Atomic entropy operator
6.4.3 Field entropy operator
6.4.4 Entropy operator from orthonormal
states
6.5 Entropy of the damped oscillator: Cat
states |
7 Reconstruction
of quasi probability distribution functions
7.1 Reconstruction in an ideal cavity
7.1.1 Direct measurement of the Wigner function
7.1.2 Fresnel approach
7.2 Reconstruction in a lossy cavity
7.3 Quasi probabilities and losses
7.3 Single Form Multiple Canvas Approach
7.4 Measuring field properties
7.4.1 Squeezing
7.4.2 Phase properties
8
Ion-laser interaction
8.1 Paul trap
8.1.1 The quandrupolar potential of the
trap
8.1.2 Oscillating potential of the trap
8.1.3 Motion in the Paul trap
8.1.4 Approximated solution to the Mathieu
equation
8.2 Ion-laser interaction in a trap with a
frequency independent of time
8.2.1 Interaction out of resonance and low
intensity
8.3 Ion-laser interaction in a trap with a
frequency dependent of time
8.3.1 Linearization of the system
8.4 Adding vibrational quanta
8.5 Filtering specific superpositions of number
states
9 Nonliear
coherent states for the Susskind-Glogower
operators
9.1 Approximated displacemnet operator
9.2 Exact solution for the displacement
operator
9.3 Susskind-Glogover coherent states
analysis
9.3.1 The Husimi Q-function
9.3.2 Photon number distribution
9.3.3 Mandel Q-parameter
9.4 Eigenfunctions of the Susskind-Glogover
Hamiltonian
9.4.1 Solution for |0> as initial condition
9.4.2 Solution for |m> as initial condition
9.5 Time-dependent Susskind-Glogover coherent
states analysis
9.5.1 Q function
9.5.2 Photon number distribution
9.5.3 Mandel Q-Parameter
9.6 Classical quantum analogies
Appendix A
Master equation
A.1 Kerr Medium
A.2 Master equation describing phase
sensitive processes
Appendix B
Methods to solve the Jaynes-Cummings model
B.1 A naive method
B.2 A traditional method
Appendix C
Interaction of quantized fields
C.1 Two fields interacting: beam splitters
C.2 Generalization to n modes
C.3 A particular interaction
C.4 Coherent states as initial fields
Appendix D
Quantum phase
D.1 Turski's operator
D.2 A formulism for phase
D.2.1 Coherent states
D.3 radially integrated Wigner function
Appendix E
Sums of the Bessel functions of the first kind
of integer order
Bibliography
Index |