Control of Quantum Systems

Lecture 1: Recap of time-dependent perturbation theory, Using Fermi’s Golden Rule to calculate transition rates for a system under constant or monochromatic perturbations.

Lecture 2: System coupled to an environment. The case of a qubit or a simple harmonic oscillator. Simple derivation of the Lindblad master equation for a qubit. Understanding the energy relaxation and dephasing times of a qubit, working with Bloch’s equations. Theoretical concept: an elementary derivation of Born-Markov master equations, frequently used to model dissipation.

Lecture 3: Adiabatic elimination. From derivation all the way to concrete examples, such as driven dissipative stabilization of bosonic cat codes.

Lecture 4: The Jaynes-Cummings Hamiltonian of a spin coupled to a photon, as a first model of circuit quantum electrodynamics. Introducing the dispersive approximation using a Schrieffer- Wolff approach, and a first take on dispersive qubit readout in circuit QED. Theoretical principles: systematic approaches to the rotating-wave approximation

Lecture 5: One new topic (maybe adiabatic theorem and Berry’s phase), or more in-depth examples.

Lecture 6: The Haroche photon box (micromaser): ideal model with wave function for dispersive/resonant interaction, Quantum Non-Demolition (QND) measurement of photon, realistic model with density operator and Kraus operators including measurement imperfection and decoherence, open-loop Monte-Carlo simulations.

Lecture 7: Feedback stabilisation of the Haroche photon Box: stabilisation of photon-number state either via measurement-based feedback (classical controller) or via decoherence engineering and autonomous feedback (quantum controller), convergence and robustness based on closed-loop Monte-Carlo simulations.

Lecture 8: Measurement and decoherence models for quantum harmonic oscillator: counting measurements, damping and thermal environment, corresponding stochastic master equation driven by Poisson processes, Lindblad master equation, Wigner function, numerical simulations.