Quantum Hardware
Advanced Course given by Marcello Dalmonte
Term
3

Program

Marcello Dalmonte and Rosario Fazio will hold the course.

Introduction and overall goal

The idea of these lectures is to cover relevant topics in the field of synthetic
quantum systems. The course is structured in two modules: the first on atom
and ion platforms, and the second on solid state platforms.
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. Basic knowledge of quantum resource theory
(entanglement) and many-body theory is a plus, but not required.

Schedule of the lectures

Crash course on quantum optics:

Lect 1: warm-up
• basic discussion on setups and energy scales;
• review of Hydrogen atom structure;
• useful extras: Zeeman effect, Rydberg states.

Lect 2: light-matter interactions
• atom-field interactions: Schroedinger equation in the electric dipole
approximation;
• atom-field interactions for quantised fields: the Jaynes-Cummings
model;
• basic properties of the JC model: undressed and dressed spectra.

Lect 3: building blocks
• dynamics of atoms coupled to a single mode: Rabi oscillations, collapse and revivals;
• intermezzo – how to perform a ’gate’;
• effects of off-resonant coupling: the AC Stark shift;
• coupling to a continuum of states: theory of spontaneous emission;
• Fermi golden rule and emission from flat spectrum;
• Wigner-Weisskopf theory of spontaneous emission, and irreversibility
of Hamiltonian dynamics.

Atoms in optical lattices

Lect 4: from microscopics to atomic band structures
• dipole trapping: basic ideas, spontaneous emission;
• coupling to a single state: red and blue detuned lattices;
• brief comments on energy scales;
• reminder on Bloch theorem;
• solution of single wave-function problem, energy bands

Lect 5: atom Hubbard model
• Wannier functions and Hubbard model representation
• interacting Hamiltonian and Bose-Hubbard models
• basic aspects of Bose-Hubbard models: Superfluid to Mott transition
• review of modern applications: SU(N) fermions, mesoscopic Hubbard
models

Lect 6: Rydberg atom basics
• two-atom interactions, spaghetti, and Rydberg blockade
• Rydberg atoms in optical tweezers and strongly interacting spin systems
• Rydberg dressing

Lect 7: strongly correlated quantum matter with Rydberg atom arrays –
some examples
• frustrated magnetism and phase transitions
• Schwinger model dynamics
• [opt] topological matter: SSH model and deconfined lattice gauge
theories

Probing synthetic quantum matter – from entanglement to complexity

Lect 8: measuring and witnessing entanglement
• a brief review of quantum correlations, and bipartite entanglement
measures and witnesses
• state tomography
• Renyi entropies from random measurements in trapped ion chains
• [opt] entanglement Hamiltonian tomography

Lect 9: Kolmogorov complexity
• brief intro to algorithmic complexity
• wave function snapshots and network interpretation
• scale-free and Erdos-Renyi networks in quantum simulators

Quantum information processing with superconducting networks

Lect 10: Recap of basic notions of superconductivity
• BCS Theory and the superconducting order parameter
• Josephson effect

Lect 11: Quantum effects in superconducting nanocircuits
• From small Josephson junction to superconducting qubits
• circuit-QED

Lect 12: Sources of decoherence

Lect 13: Many-Body effects
• Josephson arrays
• ciruit-QED arrays

Lect 14 15: Topological quantum computation
• Majorana modes in hybrid systems
• Basics of TQC with superconducting systems