Mathematical Methods for Quantum Engineering

1. Basic concepts of linear algebra leading to a rigorous formulation of the postulates of quantum mechanics

2. Schrödinger, Heisenberg and Dirac pictures
3. Introduction to the density operator and the von Neumann equation

4. Simple systems such as the spin 1/2, the harmonic oscillator, and the Jaynes-Cummings model;

5. Time-independent and time-dependent perturbation theory

6. Introduction to three specific mathematical methods illustrated on physical examples encountered in classical/quantum engineering:

i) Single frequency averaging and rotating wave approximation (phase-looked loop, resonant control of a qubit, Kapitsa pendulum and Paul traps)
ii) Euler/Lagrange and Hamilton equations with the classical/quantum correspondence (1D/2D pendulum, LC and Josephson electrical circuits)
iii) Stability of dynamical systems and feedback (damped pendulum, Lyapunov function, transfer functions of first/second-order systems, PID-regulator/cascade, slow/fast systems)