Topics of the Lectures

LOU: Loudon, The quantum theory of light, Oxford University Press;

GK: Gerry and Knight, Introductory quantum optics, Cambridge University Press;

FOX: Fox, Quantum optics, an introduction, Oxford University Press;

Lecture - 03/10/2025 - Cancelled due to the strike

Lecture 1 - 08/10/2025

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 - 09/10/2025

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 - 10/10/2025

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.

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

Lecture 4 - 15/10/2025

Stochastic processes: definition of stationary and ergodic processes. Intensity fluctuations of chaotic light: calculation of the probability distribution and its variance. Mandel formula applied to a chaotic light source: from Bose-Einsterin to Poissonian distribution.

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

Lecture 5 - 16/10/2025

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 6 - 17/10/2025

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.

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).

Lecture 7 - 22/10/2025

Degree of second order coherence for chaotic light sources (collision broadening and Doppler broadening). Calculation of the Mandel Q parameter for chaotic light sources with collision broadening. 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".

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!).

Lecture 8 - 23/10/2025

Photon bunching and antibunching. Experimental demonstrations of photon antibunching. Kimble experiment with "single atoms". Experiments with single "artificial atoms". Triggered single photon sources. Introduction on single photon sources based on quantum dots.

REFERENCES: FOX: Kimble experiment (click here!) - Michler experiment (click here!) Slides presented during the lecture (click here!)

Lecture 9 - 24/10/2025

How to quantize the electromagnetic field. Hamilton's equations. Potential theory for the classical electromagnetic field. Coulomb gauge and the Helmotz theorem.

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

Lecture 10 - 29/10/2025

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 11 - 30/10/2025

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.

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

Lecture 12 - 31/10/2025

Displacement operator. How to generate coherent states. Quadrature squeezing and squeezed states. Squeeze operator. Squeezed vaccum state.

REFERENCES: LOU: Ch. 5 (5.3-5.5), GK: Ch. 3 (3.1-3.6)

Lecture 13 - 5/11/2025

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 14- 6/11/2025

Squeezed coherent state: Definition and main properties. Amplitude- and phase-squeezed coherent states: From super-poissonian to sub-poissonian statistics. 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!

Lecture 15- 7/11/2025

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. Application of squeezed states in gravitational-wave interferomenters (only qualitative discussion). Classical theory of the beam splitter.

REFERENCES: LOU: Ch 3 (3.2), Ch. 5 (5.4) GK: Ch.2 (2.5). Paper on the gravitational-wave interferometer: click here.

Lecture 16 - 11/11/2025

EXPERIENCE: "Hanbury-Brown and Twiss experiment: Estimating the degree of second-order coherence for a single-photon source and a laser"

GROUPS FOR THE PRACTICAL EXPERIENCE. click here

REFERENCES: Slides (click here)

Lecture 17 - 12/11/2025

Quantum mechanics of the beam splitter. Input-output relations. Single-photon input: the appearance of entanglement. The photon intensity operator. Quantum degree of second order coherence. 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 18 - 13/11/2025

Second order correlation function "before and after" the beam splitter. More details about the appearance of entanglement when illuminating the beam splitter with single photon states. Two single photon states at the input ports of a beam splitter: The Hong-Ou-Mandel effect.

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!

Lecture 19 - 14/11/2025

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 Grangier experiment, click here!

Lecture 20 - 19/11/2025 (AL)

Research Seminar: Quantum Teleportation

Lecture 21- 20/11/2025

Multimode and continuous-mode quantum optics. Formalism for the continuous-mode theory. Continuous-mode photon number states. Photon wave packets.

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

Lecture 22 - 21/11/2025

Photon wave packets. Calculation of the intensity at the detector. First and second order correlation function for multimode states. Single photon "wavefunction". 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 23 - 26/11/2025

More insights on the Hong-Ou-Mandel: Time-resolved two-photon interference.

REFERENCES: Time-resolved two-photon interference, click here! Quantum beat of two single photons, click here!

Lecture 24- 27/11/2025

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.

REFERENCES: LOU: Ch. 2(2.1, 2.2). GK: Ch. 4 (4.1, 4,2, 4.4). FOX: Ch. 9 (9.5.1) .

Lecture 25 - 28/11/2025

The Rabi model with and without detuning. The rotating frame of the atom and of the field. Optical Bloch equations with and without spontaneous emission.

REFERENCES: LOU: Ch. 2(2.7, 2.8). See also Ch. 5.2, 5.3 of D.A. Steck "Quantum and Atom Optics".

Lecture 26 - 3/12/2025

Again on the optical Bloch equations in the presence of damping. Experimental observation of Rabi oscillations (experiments of Gibbs). Steady-state solutions of the optical Bloch equations. Resonance fluorescence spectroscopy.

REFERENCES: LOU: Ch. 2(2.7, 2.8). FOX: Ch. 9 (9.5.1, 9.5.2, 9.5.3). See also Ch. 5.2, 5.3 of D.A. Steck "Quantum and Atom Optics". Paper on the Gibbs experiments, click here.

Lecture 27- 4/12/2025

Bloch Sphere and Bloch vector. Dynamics of the two-level system on the Bloch sphere.

REFERENCES: FOX: Ch. 9 (9.6). See also Ch. 5.4, 5.5.1, 5.5.2, 5.5.3, 5.6, 5.7,1 of D.A. Steck "Quantum and Atom Optics".

Lecture 28 - 5/12/2025

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.

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.

Lecture 29 - 10/12/2025

Paper on the experimental demonstration of quantum Rabi Oscillations, click here.