Introduction to Quantum Optics
Héctor Manuel Moya-Cessa Francisco Soto-Eguibar
202 pages, 9x6 inches
October 2011 Paperback
ISBN 978-1-58949-061-1



Quantum optics is a topic that has recently acquired a great deal of attention, not only for its theoretical and experimental contributions to the understanding of the quantum world, but also for the perspectives of its use in many sophisticated applications. Among them, quantum optical devices are particularly promising tools for quantum information processing applications.

This book intends to teach graduate and postgraduate students several methods used in quantum optics. Therefore, it is mainly about doing calculations. Throughout the book we have emphasized the hows over the whys. Solution to the time dependent harmonic oscillator by applying invariant methods and superoperator techniques for the solution of the Master Equation are extensively used.

undergraduate students, graduate students, teachers, researchers, and all these who are interested in optics



 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 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
5.1 Differential forms
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