## Context

The graduate curriculum in quantum optics at Toronto evolves continually in order to try to provide you with the best possible background, both in terms of general "cultural" knowledge of physics and optics, and in terms of the specific methods and techniques which are likely to be of use in your own research. You should choose your courses in consultation with your supervisor, after studying the course descriptions, and with advice from senior graduate students. It may also be wise to sit in on a few courses for the first week or two before making your final choices.

On this page, we provide an approximate idea of the sort of programme of study which we believe makes sense for the typical graduate student in quantum optics. This is intended as a guide to help you through the sometimes strange course nomenclature and number schemes, and not as a list of "requirements." (You should note that revision of official titles and descriptions sometimes lags behind the actual content; when in doubt, speak to the professor teaching the course in question!) After a brief description of the various courses which form a part of the quantum optics programme, we present some ideas of typical schedules.

This page covers quantum optics specifically, and not intended to give a complete overview of the physics graduate curriculum. Most quantum optics students also take courses from beyond their specialty, both for their own education and interest and also sometimes for use in their own research. For instance, you might consider PHY 1850F (Condensed Matter Physics); PHY 1500 (Stat Mech), PHY 2301 (Order Parameters), 1810 (Particle Physics), 2403 (Quantum Field Theory), and/or special topics courses offered periodically. In addition, some optics students take courses in ECE, Chemistry, Math, or Computer Science.

Finally, it is worth pointing out that senior students may **audit** courses even after completing their course requirements for the degree. In particular, this affords an opportunity to be exposed to some Special Topics courses which may not have been offered in your first one or two years in the program.

## Relevant Courses

In each research area the basics of scientific literacy will be different, though a broad background in physics and an analytical approach will be important for all. In Quantum Optics, we feel that you can't consider yourself scientifically literate unless you have the following general background. The list below should help guide you to cover most of what you need for this general literacy; you'll very probably need considerably more depth in the area immediately supporting your proposed PhD research, and your advisor may set additional requirements.

### General courses

The following courses, typically offered in the Fall term, are intended as core preparation for graduate students in all disciplines. While they are not required, a background in **Quantum Mechanics**, **Electromagnetism**, and **Statistical Mechanics** will be assumed by most advanced quantum optics courses, and are essential foundations of the discipline. This combination of QM, E&M, and SM are sometimes referred to as the "M courses".

- PHY 1500 Graduate Statistical Mechanics
- PHY 1510 Graduate Electromagnetism
- PHY 1520 Graduate Quantum Mechanics
- PHY 1530 Graduate Fluid Mechanics
- PHY 1540 Graduate Mathematical Methods in Physics
- PHY 1600 Effective Communication for Professional Physicists

### Core quantum optics courses

- PHY1485 Advanced Classical Optics / Laser Physics

One cannot become an expert in quantum optics without a solid grounding in advanced classical optics. This course builds on a student's knowledge of basic electromagnetic waves (at the level of PHY385) to treat interference, diffraction, coherence, the paraxial wave equation, resonators, and lasers. The discussion of partially coherent light demonstrates how temporal coherence relates to frequency power spectrum (the Wiener-Khintchine theorem), and how transverse coherence relates to the spatial extent of the source (the van Cittert-Zernicke theorem). By the end of the course, students will appreciate several unique properties of lasers: their spectral brightness, the diffraction-limited gaussian beams the can produce, and their high occupation of electromagnetic modes in the visible band. The course aims to provide an intuition for the laboratory performance of optical sources, and the essential background for subsequent courses on light-matter interactions and photon statistics. - PHY2203 Quantum Optics I

This course explores atom-photon interactions on a semi-classical treatment. How does a quantum system respond to a classical drive field? We begin by discussing why an atom driven by an optical field reduces to a dipole interaction Hamiltonian. The atom-photon problem can then be mapped onto the problem of a spin one-half electron in a magnetic field, since both are driven two-level quantum systems. We develop the Bloch equations, Rabi oscillations, and magnetic resonance. Returning to the optical regime, damping is necessary, and thus a treatment using density matrices. Dynamics of the density operator are described by the Optical Bloch Equations, with which one can understand a wide range of current experiments in AMO (atomic, molecular, and optical) physics and solid-state physics. These quantum dynamics are contrasted to classical (Lorentz-model) dynamics, such as quantum saturation. In the context of a diagonalized atom-photon Hamiltonian, we discuss inversion, dressed states and light shifts. Applications of this foundational material include electromagnetically induced transparency, slow light, dark states, and laser cooling. - PHY2204 Quantum Optics II

This course deals with the fully quantum-mechanical description of light and matter. The course uses field quantization and photon states to address at least some of the following, and perhaps a few other topics as well: photon wave functions, energy, momentum, angular momentum, and spin; elementary radiation processes and radiative corrections; Glauber detection theory; simple optical elements from a quantum optics perspective; multipole Hamiltonian in QED; cavity QED; quantum optical processes; entanglement in quantum optics; systems, reservoirs, and decoherence; elementary quantum theory of the laser; mock distribution functions.

### Depth courses

- PHY2202 Atomic and Molecular Physics
*(in the spring term of even-numbered years)*

Atomic physics is not only an important discipline in its own right, but is also essential background for a variety of other areas of physics, ranging from laser physics to condensed matter and plasma physics to molecular spectroscopy and atmospheric physics. Furthermore, the concepts and techniques involved in the application of quantum mechanics to atoms and molecules have broad applicability to the description of other quantum systems and their interactions with the electromagnetic field. It is in this spirit that we will attempt to discuss the fundamentals of a variety of issues in atomic, molecular, and optical physics. - PHY2208 Nonlinear Optics
*(offered in the spring term of odd-numbered years)*

Nonlinear optics is the discipline in physics in which the electric polarization density of the medium is studied as a nonlinear function of the electromagnetic field of the light. Being a wide field of research in electromagnetic wave propagation, nonlinear interaction between light and matter leads to a wide spectrum of phenomena, such as optical frequency conversion, optical solitons, phase conjugation, and Raman scattering. In addition, many of the analytical tools applied in nonlinear optics are of general character, such as the perturbative techniques and symmetry considerations, and can equally well be applied in other disciplines in nonlinear dynamics. The concepts of nonlinear optics are central to most modern research in quantum optics; in particular, there is no experimentally observable consequence of the difference between the classical and quantum theories of light until nonlinearities of some sort are taken into account!

### Special topics

- The course codes PHY2205, 2206, 2108, and 2109 are used for special topics offered on a regular basis and usually focussing on the research specialty of the particular faculty member who is teaching that year. Due to the cutting edge nature of the course, each one is nearly unique. Recent topics include the following:
- Special Topics in Quantum Optics: Fundamental physics using AMO techniques

This is a graduate seminar course on fundamental physics experiments using the techniques of Atomic, Molecular and Optical (AMO) physics. From the Michelson-Morley experiment in the 19th century to present-day searches for dark energy, AMO measurements have constantly probed the boundaries of the laws of physics. Using weekly assigned readings, this course will provide a background in precision AMO techniques, and expose students to the details of some well known (and lesser known) classic experiments in the field. The aim of the course is to critically survey the field of precision AMO physics, and to identify new opportunities for research. - Special Topics in Quantum Optics: Ultra-Cold Atoms

This course discusses the basic concepts important to the creation and study of cold quantum gases. Not just for the specialist, the course covers is a "best of" list of topics from statistical mechanics, atomic physics, hydrodynamics, quantum optics, nonlinear optics, and solid state physics. Students will be expected to write a report on a current topic of study in the field, and the goal of the lectures will be to provide background necessary to understand the current literature. - Special Topics in Quantum Optics: Experimental Quantum Measurement

This is a course intended for any students in Quantum Optics or other disciplines who are interested in modern developments in the**experimental**side of fundamental quantum mechanics, such as (but not limited to) quantum information. It obviously assumes a good working knowledge of quantum mechanics, but new formalism will be introduced as needed, so it should be accessible to first-year as well as second-year graduate students. Much of the mystery of quantum mechanics has been tied up with the famed "quantum measurement problem" but nearly all of us have been trained with a very simplistic view of what quantum measurements really are. It turns out there are many different types of measurement in the real world, and almost never do they correspond to what we get from the QM textbooks. Experimental advances in recent years have brought the study of quantum measurement out of the shameful realm of metaphysics and into the lab. Numerous experimental groups now study effects ranging from "interaction-free measurement" to "quantum non-demolition measurements" to "weak measurements" to "generalized quantum measurements" (POVMs), to "quantum cloning" and "quantum teleportation". Ideas about quantum measurement are central to the new fields of quantum cryptography and quantum computation (especially quantum error correction). There are even two distinct paradigms of quantum computation in which the effects of measurement itself are used to carry out operations, in the place of logic gates built from "real" physical interactions. - Quantum Information Theory (typ as PHY2211)

This is a first course on quantum information and communication theory. Topics covered include: 1. basics of quantum mechanics and quantum information, 2. resource model of quantum information processing 3. entanglement and entanglement distillation protocols, and 4. quantum cryptography and security proofs. - Entanglement Physics (typ as PHY2212)

This course is an introduction to quantum entanglement, its impact upon fundamental aspects of quantum theory, and its role in diverse areas of science and technology. Topics will include: separability and entanglement; measures and witnesses of entanglement; entanglement in two level systems and harmonic oscillators; physics of experimental systems in which entanglement has be created; measurement of entangled states by tomography; dynamics and sudden death of entanglement.

## Typical schedules

Graduate students are required to complete their coursework in the **first two years of graduate studies**. The distribution between the two years is however affected by which MSc Option you choose:

**Option I.**6 courses in the first year, 2 courses in the second year. This is typically chosen by theory students.**Option II:**4 courses in the first year, 4 courses in the second year. This is typically chosen by experimental students.

Many students require some bolstering of their background through the "M" courses (ie, General Courses on E&M, Quantum Mechanics, and Statistical Mechanics) listed above. A student with more advanced preparation, who can skip all of these, should instead diversify into other fields. Non-QO courses should be chosen based on interest and on consultation with your research supervisor, but a few common choices are 1487 QTS, 2315 Adv SM, or 2403 QFT. Scheduling will of course constrain the order in which you take classes: check the Calendar in the summer for courses in the upcoming academic year, since graduate courses sometimes move depending on instructor availability. Even if the scheduling work out, since there are ten courses described above and a doctoral program requires only eight, you will not be able to cover the full QO offering. We recommend auditing those courses you missed in later years.