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Quantum Optics: single photon
For a long time, it was thought that light could be fully described as a classical electromagnetic wave obeying Maxwell's equations. In the last decades, however, it became possible to produce new states of light with unprecedented properties, impossible to understand in the context of classical electromagnetism, and demanding the use of the quantum optics formalism. This course will introduce many basic tools of the quantum optics formalism, and use them to describe an emblematic example of fully quantum states of light: onephoton wavepackets. These tools will be also used in the follow up of this course, which will present entangled pairs of photons, squeezed states of light, interaction of quantized light with matter, and their applications to quantum technologies. Behave both as a wave and a particle, and/or how to use it in quantum technologies, embark with us!
About the Course
In order to obtain the formalism of quantum optics, one must start from classical Maxwell's equations, and quantize them. One can then use the general formalism of quantum mechanics, with some adaptation. In order to benefit from the course, it is thus necessary to have some knowledge of classical electromagnetism, and of standard quantum mechanics. The course can then be considered as an advanced undergraduate course.
Quantum optics phenomena are fascinating, and the central theme of this course, waveparticle duality for a single particle (here a single photon) was declared by Richard Feynman “the great mystery” of quantum mechanics, in his famous Lectures on physics (volume III, chapter 1). The great physicist then added: ‘We cannot make the mystery go away by “explaining” how it works. We will just tell you how it works.” This is one of the goals of this MOOC, helping you to get a better understanding of quantum optics by learning how it works.
But there is another big interest in quantum optics. It turns out that quantum optics is one of the key components of the quantum technologies, which are a result of the ongoing second quantum revolution. Onephoton sources play a very important role in these quantum technologies.
If you want to understand what is a photon, how it can behave both as a wave and a particle, and/or how to use it in quantum technologies, embark with us!
Course Syllabus

Week 1 : Quantization of free radiation: one mode
 Canonical quantization
 Material harmonic oscillator
 Single mode of radiation
 Canonical quantization of a single mode
 Observables
 Number states, photon
 Vacuum fluctuations
 Homework 1: quantization of a standing wave in a cavity

Week 2 : One photon in a single mode: particlelike behavior
 The semiclassical model of optics
 One photon in a single mode
 Photodetection signals
 Single photodetection signal for one photon
 Double photodetection signal for one photon: a fully quantum behavior
 Quantum optics: a must
 Homework 2: The "coherent states" of light

Week 3 : One photon interferences: Waveparticle duality
 Beamsplitter in quantum optics
 One photon on a beam splitter
 MachZehnder interferometer in classical optics
 Onephoton interference
 Waveparticle duality: a quantum mystery; a consistent formalism
 Homework 3: Field state transformation on a beamsplitter

Week 4 : Multimode quantized radiation: quantum optics in a real laboratory
 Canonical quantization of multimode radiation
 Eigenstates of the Hamiltonian; energy of the vacuum
 Total number of photons
 Linear and angular momentum of a photon
 Field observables; vacuum fluctuations
 Photodetection signals
 Homework 4: Multimode radiation field states: localized single photon state

Week 5 : Onephoton sources in the real world
 Heisenberg formalism; photo detection signals
 Multimode onephoton wave packet
 Spontaneous emission photon
 A detour to Fourier transforms
 Real onephoton sources
 Onephoton sources for what?
 Homework 5: Generation of photon pairs by parametric down conversion

Week 6 : Waveparticle duality for a single photon in the real world
 Anticorrelation on a beam splitter
 Anticorrelation experiments
 Supplementary photons
 Onephoton interference: formalism
 Onephoton interference: experiments
 Waveparticle duality and complementarity
 A fruitful mystery
 Homework 6: Shotnoise limit of interferometry

Week 7 : One photon based quantum technologies
 Quantum Random Number Generator (QRNG)
 Weak light pulses on a beam splitter
 One photon polarization as a qubit
 Quantum cryptography: the BB84 scheme
 The nocloning theorem
 Conclusion and outlook
Suggested Readings
The lectures are selfcontained, but the learners will find useful additional information in the following book: G. Grynberg, Alain Aspect and Claude Fabre: Introduction to quantum optics, Cambridge University Press (2010).