# Synthetic quantum systems

### Advanced course given by Marcello Dalmonte

**Term: **0

**Start:** 11-05-2021
**End:** 30-06-2021
**Room:** Online

**Credits: **3

**Schedule:** TBA

## Program

**Idea of the course**: make people familiar with techniques in the field of synthetic quantum systems, in particular cold atom and trapped ion architectures, tackling them from the theoretical quantum optics viewpoint.

**Goals:** gain the basic theoretical tools and methods to understand in some detail experiments in synthetic quantum systems.

**Enabling skills:** being able to propose a novel technique to realize or probe quantum matter.

**Pre-requisites: **master equation, a tiny bit of band theory (Bloch theorem), atomic physics (solution of the Hydrogen atom and angular momentum theory), advanced quantum mechanics (including some basics of scattering). I will not cover laser theory.

**Modules:**

**1) Atom-light interactions - basics and simple modelling**

Lec 1:

- basic discussion on setups and energy scales

- crash course in atomic physics - reviewing the Hydrogen atom structure

- useful extras: Zeeman effect, Rydberg states

Lec 2:

- atom-field interactions: Schroedinger equation in the electric dipole approximation

- atom-field interactions for quantised fields: the Jaynes-Cummings model

- discussion on the rotating-wave approximation: validity?

- basic properties of the JC model: undressed and dressed spectra

Lec 3:

- dynamics of atoms coupled to a single mode: Rabi oscillation, collapse and revivals

- brief discussion of Rydberg and ion experiments

- effects of off-resonant coupling: the AC Stark shift

- coupling to a continuum of states: spontaneous emission

- Fermi golden rule and emission from flat spectrum

- Wigner-Weisskopf theory of spontaneous emission, irreversibility of Hamiltonian dynamics

**2) Cold atoms in optical lattices**

Lec 4:

- dipole trapping: basic ideas

- how to treat spontaneous emission

- coupling to a single state: red and blue detuned lattices

Lec 5:

- from microscopic to Hubbard models

- brief comments on energy scales

- reminder on Bloch theorem

- solution of single wave-function problem, energy bands

- Wannier functions and Hubbard model representation

Lec 6:

- interacting Hamiltonian and Bose-Hubbard models

- basic aspects of Bose-Hubbard models: Superfluid to Mott transition

- review of progresses in Bose-Hubbard model physics

**3) Advanced quantum engineering in atomic systems**

Lec 7:

- interaction tuning: contact interactions (resonances, closed-shell atoms)

- interaction tuning: spin-exchange Hamiltonians

- interaction tuning: magnetic atoms and polar molecules

Lec 8:

- classical gauge potentials: Jaksch-Zoller and Gerbier-Dalibard approaches

- a small exercise: spontaneous emission and excitations to other bands at the single building block level

- synthetic gauge potentials in synthetic dimensions: quantum Hall ribbons

- beyond the Hofstadter butterfly: combining interactions and background potentials

Lec 9:

- atoms interacting via cavity modes: infinite-range (?) interactions

- Rydberg atoms: two-atom interactions, spaghetti, and Rydberg blockade

- Rydberg atoms in optical tweezers and strongly interacting spin systems [frozen regime]

**4) Quantum engineering with trapped ions**

Lec 10 - basics:

- microscopic degrees of freedom at hand: alkaline-earth ions

- trapping ions: Penning and Paul traps

- quantum mechanical models for single-ion traps

- light-matter interactions: how spins talk to collective modes via light

Lec 11 - applications:

- spin models with trapped ions in Paul traps

- quantum computing with trapped ions: Cirac-Zoller and Molmer-Sorensen gates

Lec 12 - entanglement measurements:

- measuring entanglement properties in synthetic quantum systems:

- a brief review of bipartite entanglement measures and witnesses

- overview of measurement methods: tomography, copies, entanglement Hamiltonian engineering

- Renyi entropies from random measurements in trapped ion chains