Quantum Optics
A.Y. 2024/2025
Learning objectives
The course aims to provide the basic notions and the analytical methods for the quantum
description of the radiation field, and its interaction with matter. The principal
quantum states of radiation are discussed, and measurement and interferometry processes
are described in detail. In particular, fundamental topics of quantum mechanics applied to interaction with single atoms and with circuits based on superconductors are examined
in depth. The course also includes a discussion on the relevant technological applications
of quantum optics in its more recent developments, such as atomic fountain clocks and the
practical use of squeezed states.
description of the radiation field, and its interaction with matter. The principal
quantum states of radiation are discussed, and measurement and interferometry processes
are described in detail. In particular, fundamental topics of quantum mechanics applied to interaction with single atoms and with circuits based on superconductors are examined
in depth. The course also includes a discussion on the relevant technological applications
of quantum optics in its more recent developments, such as atomic fountain clocks and the
practical use of squeezed states.
Expected learning outcomes
At the end of the course the student is expected to acquire the following knowledge:
1) The student will be able to discuss the quantization of the radiation field starting from classical electrodynamics;
2) He will be able to characterize the main observables and the most relevant states of the field
of radiation, classic and nonclassical;
3) He will know the basic elements of the quantum theory of coherence and of the radiation detection.
4) He will know how to describe the generation and manipulation of nonclassical states via parametric processes, with particular regard to the properties of squeezing and entanglement;
5) He will be able to discuss the dynamics of the radiation field as an open quantum system;
6) He will be able to discuss the models and the dynamics of the interaction of the quantized field
with atoms on two levels;
7) He will know how to describe various optical-quantum systems in fundamental experiments and in applications to quantum information.
1) The student will be able to discuss the quantization of the radiation field starting from classical electrodynamics;
2) He will be able to characterize the main observables and the most relevant states of the field
of radiation, classic and nonclassical;
3) He will know the basic elements of the quantum theory of coherence and of the radiation detection.
4) He will know how to describe the generation and manipulation of nonclassical states via parametric processes, with particular regard to the properties of squeezing and entanglement;
5) He will be able to discuss the dynamics of the radiation field as an open quantum system;
6) He will be able to discuss the models and the dynamics of the interaction of the quantized field
with atoms on two levels;
7) He will know how to describe various optical-quantum systems in fundamental experiments and in applications to quantum information.
Lesson period: First semester
Assessment methods: Esame
Assessment result: voto verbalizzato in trentesimi
Single course
This course can be attended as a single course.
Course syllabus and organization
Single session
Responsible
Lesson period
First semester
Course syllabus
1) Quantization of the Electromagnetic Field and Basic Concepts. Single- and multi-mode quantization. Quantum harmonic oscillator, Fock space, photons, quadratures, and field operators. Thermal radiation and density operator. The vacuum of the quantized radiation field and its physical effects.
2) Quantum States of the Electromagnetic Field. Coherent states and their properties. Displacement operator and BCH formulas. Generation of coherent states from classical currents. Single- and two-mode squeezed states and their properties. Single- and two-mode squeezing operators. Rotating frames and interaction picture. Generation of squeezed states.
3) Quantum Phase Space. Glauber formula (Fourier-Weyl relation). Moment-generating functions. Quasi-probability distributions (Glauber-Sudarshan, Husimi, Wigner) and their properties. Optical equivalence theorem. Dynamics in phase space. Non-classical states.
4) Radiation-Matter Interaction. Minimal coupling and electric dipole Hamiltonian. Interaction with a classical field via the perturbative approach, Fermi's golden rule. Two-level system and semi-classical Rabi model. Interaction with a quantized field via the perturbative approach, spontaneous and stimulated emission, absorption of radiation, Einstein coefficients. Two-level system and Jaynes-Cummings model: dressed states, vacuum Rabi oscillations, collapses and revivals, experimental demonstrations, entangled states, and state swapping. Dispersive Jaynes-Cummings interaction and Schrödinger cat states.
5) Quantum Interferometry. Quantum mechanics of the beam splitter. Effective Hamiltonian and field evolution. Mixing of two photons and fluorescence from a single atom. Mach-Zehnder interferometer. Interaction-free measurements. Interferometry with coherent states.
6) Photon Detection and Photon Sources. Photocurrents, intensity detectors, on/off detectors, and photon number detectors. Homodyne detector. Single-photon sources.
7) Further Technological Applications. Atomic clocks. Interferometry with squeezed states. Interferometry with entangled states. Continuous-variable quantum computing, GKP bosonic codes.
2) Quantum States of the Electromagnetic Field. Coherent states and their properties. Displacement operator and BCH formulas. Generation of coherent states from classical currents. Single- and two-mode squeezed states and their properties. Single- and two-mode squeezing operators. Rotating frames and interaction picture. Generation of squeezed states.
3) Quantum Phase Space. Glauber formula (Fourier-Weyl relation). Moment-generating functions. Quasi-probability distributions (Glauber-Sudarshan, Husimi, Wigner) and their properties. Optical equivalence theorem. Dynamics in phase space. Non-classical states.
4) Radiation-Matter Interaction. Minimal coupling and electric dipole Hamiltonian. Interaction with a classical field via the perturbative approach, Fermi's golden rule. Two-level system and semi-classical Rabi model. Interaction with a quantized field via the perturbative approach, spontaneous and stimulated emission, absorption of radiation, Einstein coefficients. Two-level system and Jaynes-Cummings model: dressed states, vacuum Rabi oscillations, collapses and revivals, experimental demonstrations, entangled states, and state swapping. Dispersive Jaynes-Cummings interaction and Schrödinger cat states.
5) Quantum Interferometry. Quantum mechanics of the beam splitter. Effective Hamiltonian and field evolution. Mixing of two photons and fluorescence from a single atom. Mach-Zehnder interferometer. Interaction-free measurements. Interferometry with coherent states.
6) Photon Detection and Photon Sources. Photocurrents, intensity detectors, on/off detectors, and photon number detectors. Homodyne detector. Single-photon sources.
7) Further Technological Applications. Atomic clocks. Interferometry with squeezed states. Interferometry with entangled states. Continuous-variable quantum computing, GKP bosonic codes.
Prerequisites for admission
Fundamental concepts of: a) non relativistic quantum mechanics, in particular for the description of atomic energy levels; b) classical electromagnetic field and electromagnetic waves in vacuum; c) basic optical instruments
Teaching methods
The teaching is provided through lectures and classroom discussions. Attendance is strongly recommended.
Teaching Resources
The following textbooks are recommended:
1) C.G.Gerry and P.L.Knight, "Introductory Quantum Optics", Cambridge University Press 2005
2) G.Grynberg, A.Aspect and C.Fabre ''Introduction to Quantum Optics'', Cambridge University Press
3) U Leonhardt, "Essential Quantum Optics - From Quantum Measurements to Black Holes", Cambridge University Press 2010
4) P. Kok and B. Lovett, "Introduction to Optical Quantum Information Processing", Cambridge University Press 2010
1) C.G.Gerry and P.L.Knight, "Introductory Quantum Optics", Cambridge University Press 2005
2) G.Grynberg, A.Aspect and C.Fabre ''Introduction to Quantum Optics'', Cambridge University Press
3) U Leonhardt, "Essential Quantum Optics - From Quantum Measurements to Black Holes", Cambridge University Press 2010
4) P. Kok and B. Lovett, "Introduction to Optical Quantum Information Processing", Cambridge University Press 2010
Assessment methods and Criteria
The exam consists of an oral interview covering the topics discussed in the course. Prior to the interview, students may be required to solve some exercises. During the exam, which lasts at least 1 hour, both the student's knowledge and critical skills in the quantum description of the radiation field and its interaction with atoms will be assessed, including knowledge of fundamental experiments. The final mark is expressed on a scale of thirty.
Professor(s)