Topics of the Lectures 

Lecture 1 - 27/09/2024 

Introduction to the course. The photoelectric effect, revisited. Semi-classical approach to light-matter interaction. Time-dependent pertubation theory. The Fermi's Golden Rule.  

REFERENCES: Introductory slides (click here!),  Lamb and the photoelectric effect  (click here!), GK: Ch. 4 (4.1-4.2).  

Lecture 2 - 02/10/2024 

Again on the photoelectric effect using the semiclassical approach.  Photon-counting statistics with a coherent beam of light with constant intentsity: "quantum" and "semi-classical" approaches. Poissonian distribution. Semi-classical theory of photodetection. 

REFERENCES: FOX: Ch. 5 (5.1,  5.2, 5.3, 5,4) -  LOU: Ch. 3 (3.9)

Lecture 3 - 04/10/2024 

Photocounting statistics of a fluctuating field: the Mandel formula. Classification of light by photon statistics: Poissonian, Super-Poissonian, and sub-Poissonian light. Discharge lamps and a model for chaotic light sources.  Stochastic processes: definition of stationary and ergodic processes. Intensity fluctuations of chaotic light: calculation of the probability distribution and its variance. 

REFERENCES: FOX: Ch. 5 (5.5, 5.6, 5.8) -  LOU: Ch. 3 (3.6, 3.9)

Lecture 4 - 9/10/2024 

Mandel formula applied to a chaotic light source: from Bose-Einsterin to Poissonian distribution. Further details on the Mandel formula and its connection to the second order correlation function:  Mandel Q paramenter. Thermal light and and the super-Poissonian carachter of balck-body radiation. Degradation of photon statistics by optical losses. Experiments demonstrating the existence of sub-poissonian sources of light: Frank-Hertz experiment in space-charge limited-current regime; Experiments performed with a beam of single atoms. 

REFERENCES: FOX: Ch. 5 (5.5, 5.6) - LOU: Ch. 3 (3.9)  - Undergradutate experiment: photon-counting statistics  (click here!) - Paper on the Franck-Hertz experiment (click here!) - Paper on the "single" atom experiment (click here!)

Lecture 5 - 11/10/2024 

Degree of second order coherence. Hanbury-Brown and Twiss interferometer. Connection between the first order and second order correlation functions. Calculation of the first order correlation function for  chaotic light sources. Chaotic light sources with different broadening mechanisms: Collision broadening, lifetime broadening, Doppler broadening. Calculation of the Mandel Q parameter for chaotic light sources with collision broadening. 

REFERENCES: FOX: Ch. 2 (2.2,  2.3) - Ch. 6 (6.1, 6.2, 6.3) -  LOU: Ch. 3 (3.3,3.4,3.5,3.7, 3.8). On the calculation of the Q parameter for cahotic light (up to equation  4, click here!)

Lecture 6 - 16/10/2024 

Calculations of the bounds of the second order correlation function (for any classical light source). New definition of light sources according to the value of the second order  correlation function at zero time delay: Photon bunching and antibunching.  Experimental demonstrations of photon antibunching. Kimble experiment with "single atoms". Experiments with single "artificial atoms".  Triggered single photon sources. The differences between the two classifications of non-classical light sources: photon antibunching vs. sub-poissonian photon statistics. Short introduction on single photon sources based on quantum dots.

REFERENCES: FOX: Ch. 6 (6.1, 6.2, 6.3, 6.4, 6.5, 6.6, 6.7) -  LOU: Ch. 3 (3.7, 3.9).  Kimble experiment (click here!) - Michler experiment (click here!) - Photon Antibunching vs. sub-poissonian photon statistics (up to equation  4, click here!). Slides presented during the lecture (click here!)  

Lecture 7 - 18/10/2024 

How to quantize the electromagnetic field. Hamilton's equations. Potential theory for the classical electromagnetic field.  Coulomb gauge and the Helmotz theorem. The quantum harmonic oscillator: review of the basic concepts and formulas using the raising and lowering operators.

REFERENCES:  LOU: Ch. 4 (4.1, 4.2, 4.3, 4.4) 

Lecture 8 - 23/10/2024 

Quantization of the electromagnetic field.  Fock states. Multi-mode and single-mode states. Fock state or photon number states. Discusssion about the concept of "photon". The quadrature operators in the classical and quantum pictures. Quantum noise. Vaccum fluctuations. Fock state as highly non-classical states of light.

REFERENCES:  LOU: Ch. 4 (4.4), Ch. 5 (5.1, 5.2) 

Lecture 9 - 25/10/2024 

 Coherent or quasi-classical states.  Discussion of the main properties of coherent states: expectiation value of the electric field and noise for a coherent state. Phase-space pticture of coherent states. Displacement operator.  Generation of coherent states.

REFERENCES:  LOU: Ch. 5 (5.3), GK: Ch. 3 (3.1-3.6) 

Lecture 10 - 30/10/2024

Quadrature squeezing and squeezed states. Squeeze operator. Squeezed vaccum state. Decomposition into Fock states. Statistical properties of the squeezed vacuum state.

REFERENCES: LOU: Ch. 5 (5.5, 5.6), GK: Ch. 7 ( 7.1).

Lecture 11 - 06/11/2024 

Squeezed coherent state: Definition and main properties.  Amplitude- and phase-squeezed coherent states: From super-poissonian to sub-poissonian statistics. Application of squeezed states in gravitational-wave interferomenters (only qualitative discussion). Pure states and statistical mixtures. Introduction to the density operator: main properties.

REFERENCES: LOU: Ch. 4 (4.6), GK:  Ch.7 (7.1). Paper on the "Measurement of the quantum states of squeezed light", click here!  Paper on the gravitational-wave interferometer: click here. 

Lecture 12 - 08/11/2024 

Thermal states and chaotic light states: Probability distributions and density operators. Photon statistics and representation  of thermal states and chaotic light states in the phase-space picture.  Classical theory of the beam splitter.  Quantum mechanics of the beam splitter. Input-output relations. Introduction to the practical experience.

REFERENCES:  LOU: Ch 3 (3.2), Ch. 5 (5.4, 5.7,), GK:Ch. 2 (2.5) , Ch. 6 (6.1, 6.2) 

Lecture 13 - 12/11/2024 

EXPERIENCE:  Estimating the second order correlation function for light from a quantum dot and a laser.

REFERENCES:  Slides (click here)

Lecture 13/11/2024  - CANCELLED

Lecture 14 - 15/11/2024 

Quantum mechanics of the beam splitter. Input-output relations. Single-photon input: the appearance of entanglement. Coherence state at the input port of the beam splitter. Discussion and summary of the the properties of the different excitations of the electromagnetic field studied in the previous lectures. The photon intensity operator. Quantum degree of second order coherence. Second order correlation function "before and after" the beam splitter.  Calculation of the second order correlation function for the the different excitations studied in the previous lectures (only single mode).  

REFERENCES:  LOU: Ch4 (4.11, 4.12), Ch. 5 (5.7, 5.8, 5.10), GK:Ch. 5 (5.2, 5.4) , Ch. 6 (6.1, 6.2) 

Lecture 15 - 22/11/2024  (AL)

Seminar: Quantum Key Distribution.

Lecture 16- 27/11/2024  

Two single photon states at the input ports of a beam splitter: The Hong-Ou-Mandel effect. The classical Mach- Zender interferometer: input-output relations. The quantum Mach-Zender interferometer. Interferometry with a single photon.  The Grangier experiment. Interaction-free measurements. Balanced homodyne detection.

REFERENCES:  LOU: Ch. 5 (5.8), GK: Ch. 6 (6.2, 6.3, 6.4) - Ch 7 (7.3). Paper on the Hong-Ou-Mandel effect, click here! Paper on the Grangier experiment, click here!

Lecture 17- 29/11/2024  (FB)

Seminar: Quantum Teleportation

Lecture 18 - 04/12/2024 

Multimode and continuous-mode quantum optics. Formalism for the continuous-mode theory.  Continuous-mode photon number states. Photon wave packets.  Calculation of the intensity at the detector.  Mach-Zender interferometry with continuos-mode single-photon states: differences with resepct to the case of single-mode states.

REFERENCES:  LOU: Ch. 6(6.1, 6.2, 6.3, 6.6-6.8,.11).

Lecture 19 - 06/12/2024 

More insights on the Hong-Ou-Mandel: Time-resolved two-photon interference. Introduction to the light-matter (atom) interaction. The interaction Hamiltonian in the semi-classical approach.  The "length" gauge. The case of a weak interaction: time-dependent perturbation theory. Light-atom interaction: the two-level atom. The case of strong interaction: The Rabi model with no detuning. 

REFERENCES:  LOU: Ch. 2(2.1, 2.2). GK: Ch. 4 (4.1, 4,2, 4.4). FOX: Ch. 9 (9.5.1) . Time-resolved two-photon interference, click here! Quantum beat of two single photons,  click here!

Lecture 20 - 11/12/2024 

Again on the Rabi model: the case with detuning. The rotating frame of the atom and of the field.  Optical Bloch equations. Spontaneous emission. Damping of the Rabi oscillations. Experimental observation of Rabi oscillations (experiments of Gibbs).

REFERENCES:  LOU: Ch. 2(2.7, 2.8). See also Ch. 5.2, 5.3 of D.A. Steck "Quantum and Atom Optics". Paper on the Gibbs experiments, click here.

Lecture 21 - 13/12/2024 

Full quantum treatment of light-matter interaction. Second quantization of the atomic Hamiltonian. Rotating wave approximation. Full Hamiltonian in the Schoedinger and in the Dirac (interaction) picture. Matrix elements, absorption, emission and the appearance of spontaneous emission. The Jaynes-Cummings model.  Quantized Rabi oscillations. Vacuum Rabi oscillations. Interaction between a two-level system and a coherent state. Collapse and revival of Rabi oscillations. Experimental demonstration of quantized Rabi Oscillations as well as collapese and revival or Rabi oscillations. 

REFERENCES:  LOU: 4.9, 4.10,  GK: Ch. 4 (4.5) Ch. 10 (10.3). See also Ch. 6.1 and 6.2 of "Quantum Optics", by M-S. Zubairy and M. O. Scully,  Cambridge University Press.  Paper on the experimental demonstration of quantum Rabi Oscillations, click here.

Lecture 22- 18/12/2024 

Cavity quantum electrodynamics: distinction between strong and weak coupling regimes. Damping of vacuum Rabi oscillations. Weisskopf-Wigner theroy of spontaneous emission in free space. Calculation of the photon decay rate. Calculation of the single photon "wavefunction" for the case of spontaneous emission.

REFERENCES:  See Ch. 6.3 of "Quantum Optics", by M-S. Zubairy and M. O. Scully,  Cambridge University Press and Ch. 11.1, 11.2, 11.3, 11.4 of D.A. Steck "Quantum and Atom Optics". 

Lecture 23 - 20/12/2024

Calculation of the transition rate for an atom in free-space using perturbation theory.  Fundamental properties of Fabry-Perot cavities and resonant modes.  Weak coupling regime: Calculation of the transition rate for an atom in a single-mode cavity. The Purcell effect. Experimental demonstration of the Purcell effect using "artificial atoms" in semiconductor micro cavities.  

REFERENCES:  FOX: Ch. 10 (10.1, 10.2),  Paper on the experimental demonstration of the Purcell effect, click here!