## Admission Requirements

B.Sc. level Quantum Mechanics and mathematics. The lecturer assumes that the students are familiar with the content of Quantum Theory.

## Description

The quantum optics course describes the interaction between light and matter at the fundamental quantum level. This is done via second quantization where both the atom and the electromagnetic field becomes quantized. This is a minimal requirement to build an understanding of light-matter interaction and is necessary to describe, control, manipulate and detect simple quantum systems. Throughout the course these quantum system are either comprised of light (photons) or are created by coupling a single atom to a quantized light field.

Many interesting and highly relevant questions can be addressed within the framework of quantum optics because the calculations are relatively simple compared to a more complete description of quantum field theory. This makes quantum optics ideally suited to design experiments that test the foundations of quantum mechanics and that probe the crossover between the microscopic realm of quantum physics to the macroscopic domain of classical physics. The course investigates this crossover by analyzing Schrödinger cat states and their decoherence in quantum phase space.

Throughout the course a strong link is made between theoretical concepts and modern experimental research by discussing the important experiments using scientific articles. Key historical papers are given as background, while the students present a relevant article in small groups. The discussion of these experiments help to illustrate and understand the essential difference between the classical and quantum mechanical description.

The course covers the following subjects and topics:

**Basics:** quantization of the electromagnetic field, field quadratures, quantum measurement, operator ordering theorems

**States of light:** Coherent States of Light, Thermal light states, Photon Number states, Quantum Phase space distributions, Wigner functions, quantum phase operator

**Sources of quantum light:** Squeezing operator, squeezed light, parametric down-conversion, single photon sources

**Correlation functions:** Quantum and classical coherence, Hanbury-Brown and Twiss experiment

**Quantum interference:** quantum description of beamsplitters as unitary transformations, Hong-Ou-Mandel effect, interferometers, homodyne detection, backaction and noise, quantum erasure

**Coupled quantum oscillators:** Jaynes-Cummings model and dressed-states picture of strongly coupled systems, opto-mechanical interaction

**Cavity QED:** Single Rydberg atoms, Purcell effect, Schrödinger cat states, decoherence and quantum jumps

**Applications of entanglement:** Experiments on teleportation, quantum key distribution and violations of Bell’s inequality

## Course objectives

At the end of the course you will be able to:

Apply the formalism of creation and annihilation operators for the electromagnetic field to describe quantum states of light

Name the different quasiprobability distributions and use these to draw a phase-space picture of the various quantum states of light

Calculate and explain the fluctuations and correlations of different quantum states of light

Calculate the photon number distribution of different quantum states of light from the Hamiltonian that describes the interaction that generates the state

Compute and interpret the second-order correlation function of states of light and indicate the boundary between classical and quantum light

Explain Bell’s theorem and experimental tests done with entangled photons

Explain and calculate the contribution of quantum fluctuations in measurements involving light

Formulate decoherence of quantum states using the quantum-jump method

Calculate and explain the Eigenstates of the Jaynes-Cummings Hamiltonian in the dressed-state picture

Explain the concept of Schrödinger cat states and name several different ways of creating such macroscopic quantum states

Describe and calculate the properties of squeezed states

Explain the Hong-Ou-Mandel effect as quantum interference related to which-path-information

Calculate the visibility of quantum interference effects

Explain the quantum erasure effect and the role of which-path information

Give operator expressions for the quantum optical output of arbitrary multiports and interferometers using the input-output formalism (s-matrix) of beamsplitters

Explain and design simple setups used to prepare and manipulate the quantum state of an atomic qubit using the interaction of the Rabi-model

Verify if a given quantum state is pure or mixed and if the quantum state is entangled or not

Describe how entanglement can be generated and tested in experiments that involve spontaneous parametric down-conversion or atomic cascades

The following soft skills will be trained during the course:

Presentation skills by giving a short (~10 min.) presentation about a relevant recent article in the field of experimental quantum optics

Collaboration by preparing the presentation by a small group of 2-3 students

## Timetable

## Mode of instruction

Lectures, tutorials (exercise classes) and 15 min. student presentations on an experimental research article selected by the lecturer.

## Assessment method

Student presentations are graded for each group based on the presentation and questions asked. The grade counts as 1/3 of the final grade.

Written examination, with questions modeled after the exercises from the tutorials. The written exams counts as 2/3 of the final grade.

There is a possibility to retake the exam. The date and format (oral or written examination) of the retake will be decided in consultation.

## Blackboard

Blackboard is used to distribute Course information

To have access to Blackboard you need a ULCN-account.Blackboard UL

## Literature

C. Gerry and P. Knight, Introductory Quantum Optics, Cambridge University Press, Cambridge, UK (2005), ISBN 0 521 52735 X (paperback)

Additional lecture notes and papers will be distributed via blackboard

Suggested additional reading for a more experimental perspective: M.Fox, Quantum Optics: An Introduction, Oxford University Press, Oxford, UK (2001), ISBN 0198566735 (paperback)

## Contact information:

Lecturer: Dr. M. de Dood (Michiel)