Lecture (30 hrs)
1. General information about the subject – a review (basic phenomena and processes, photons).
Orders of magnitude and units of physical quantities characterizing atoms, optical fields and interaction between them. Estimating
the number of photons in a laser beam of given power and frequency.
Examples of quantum behaviour: photon detection after its passing through a Mach-Zehnder interferometer, nondemolition
measurement.
Photon statistics: counting photons, statistics of the number of photons in a coherent beam, Poisson distribution and "Poissonian,
super-Poissonian and sub-Poissonian light".
2. Quantum theory of radiation
Maxwell's equations (ME) for electric and magnetic fields, electromagnetic potentials, gauge
ME in Fourier space (r,t) -> (kn, t), longitudinal and transverse components of electric and magnetic fields ("electrostatics" and
"radiation")
Polarization and radiation modes
Dynamics of transverse fields, expression through normal variables
Vector and scalar electromagnetic potentials in Fourier space, longitudinal and transverse components of the vector potential,
gauge, evolution equations
Coulomb gauge
Energy of radiation field, expression through longitudinal and transverse components of the electric field and vector potential in
the Fourier space, analogy to a set of uncoupled harmonic oscillators
Quantization rules, creation and ahhinilation operators, Hamiltonian and momentum operator, number operator, eigenstates and
eigenvalues of the Hamiltonian and momentum operator, radiation modes, photons.
3. Quantum states of radiation
Vacuum state and its basic properties.
Single-mode states, Fock (number) and coherent (quasiclassical, Glauber) states, their basic properties and interpretation.
Multimode states.
Single- and multimode single-photon states.
Beam splitter and its classical and quantum model. Input and output states. Single-photon experiments, Hong – Ou – Mandel (HOM)
effect.
Quadrature operators for radiation fields (definition, commutation rules, Heisenberg relations).
Squeezed states of radiation (definitnion, properties, generation scheme in a parametric process).
4. Interaction of electromagnetic fields with atomic systems
Time-dependent perturbation theory, transition amplitude and probability, transitions among discrete states and from discrete to
continuous spectrum.
Interaction of atomic systems with classical electromagnetic fields (interaction Hamiltonian, electric – dipole and magnetic –
dipole interaction, absorption and stimulated emission).
"Exactly solvable" models:Rabi model, Weisskopf-Wigner model..
Remarks on more complicated cases: more levels, more fields.
Exercises (30 hrs)
5. Harmonic oscillator. A review of the subject (a, a+ and n = a+a operators, their basic properties and algebra, eigenstates and eigenvalues of
harmonic oscillator Hamiltonian). Coherent states of harmonic oscillator and their properties: definition, decomposition in the Fock basis,
Poissonian statistics of excitation numbers, graphical representation, temporal evolution, Heisenberg relations, quasiorthogonality and
completeness, displacement operator for generation operator of coherent states from vacuum state.
6. A few operator relations. Functions of operators, commutation relations involving functions of operators, derivative of an operator,
Glauber and Baker-Hausdorff formulas, displacement and squeezing operators.
7. Spin ½ dynamics in a magnetic field as a prototype two-level system. Magnetic resonance, classical and quantum description, Bloch,
Schroedinger and von Neumann equations, evolution of expectation value of magnetization.
8. Optical Bloch equations
