Synthetic Quantum Systems

by M. Dalmonte (2 credits, type C)

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.


  1. ) Atom–light interactions — basics and simple modelling
      Lecture 1
    • basic discussion on setups and energy scales
    • crash course in atomic physics
    • reviewing the Hydrogen atom structure
    • useful extras: Zeeman effect, Rydberg states
      Lecture 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
      Lecture 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
      Lecture 4
    • dipole trapping: basic ideas
    • how to treat spontaneous emission
    • coupling to a single state: red and blue detuned lattices
      Lecture 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
      Lecture 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
      Lecture 7
    • interaction tuning: contact interactions (resonances, closed-shell atoms)
    • interaction tuning: spin-exchange Hamiltonians
    • interaction tuning: magnetic atoms and polar molecules
      Lecture 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
      Lecture 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
      Lecture 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
      Lecture 11 (applications)
    • spin models with trapped ions in Paul traps
    • quantum computing with trapped ions: Cirac-Zoller and Molmer-Sorensen gates
      Lecture 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

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