Program Overview

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Program Overview

The Quantum Optics: Single Photon program is a 7-week course that introduces students to the basics of quantum optics formalism and its application to fully quantum states of light, specifically one-photon wave-packets.

Course Details

• Start Date: 10/28/2019
• Duration: 7 weeks
• Rhythm: 4-5 hours/week
• Prerequisites: Undergraduate knowledge of quantum mechanics and classical electromagnetism.
• Discipline: Physics
• Course Language: English

Presentation

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: one-photon wave-packets.

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.

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: Particle-Like Behavior

• The semi-classical model of optics
• One photon in a single mode
• Photo-detection signals
• Single photo-detection signal for one photon
• Double photo-detection signal for one photon: a fully quantum behavior
• Quantum optics: a must
• Homework 2: The "coherent states" of light

Week 3: One Photon Interferences: Wave-Particle Duality

• Beam-splitter in quantum optics
• One photon on a beam splitter
• Mach-Zehnder interferometer in classical optics
• One-photon interference
• Wave-particle 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
• Photo-detection signals
• Homework 4: Multimode radiation field states: localized single photon state

Week 5: One-Photon Sources in the Real World

• Heisenberg formalism; photo detection signals
• Multimode one-photon wave packet
• Spontaneous emission photon
• A detour to Fourier transforms
• Real one-photon sources
• One-photon sources for what?
• Homework 5: Generation of photon pairs by parametric down conversion

Week 6: Wave-Particle Duality for a Single Photon in the Real World

• Anti-correlation on a beam splitter
• Anti-correlation experiments
• Supplementary photons
• One-photon interference: formalism
• One-photon interference: experiments
• Wave-particle duality and complementarity
• A fruitful mystery
• Homework 6: Shot-noise 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 no-cloning theorem
• Conclusion and outlook

Suggested Readings

The lectures are self-contained, 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).

Teachers

• Aspect Alain
• Brune Michel

Institution

École Polytechnique, is a founding member of Institut Polytechnique de Paris