This chapter surveys several different mathematical methods for time-dependent change of quantum states using quantum field theory. The Bloch sphere method is introduced, which can be used to show the physics discussed in Chapter 3, that electronic transitions, or “jumps,” are not instantaneous.

Appendix G: interaction between a monochromatic field and two-level atom. The problem is treated first in the case of a classical field and a quantum two-level system (semiclassical approach): It is characterised by a rotation of the Bloch vector (Rabi ocillation) and allows us to generate any qubit state by applying a field of well-controlled duration and amplitude. One then includes spontaneous emission to the model, and finally obtains the set of Bloch equations that are used in many different problems of light–matter interaction. One then considers the full quantum case of cavity quantum electrodynamics (CQED), where the field is single mode and fully quantum: this is the Jaynes–Cummings Hamiltonian approach, which is fully solvable when one negelcts spontaneous emission: quantum oscillations and revivals are predicted. Damping is then introduced in the model, and two regimes of strong and weak couplings are predicted in this case.

Typically, we are interested in a small system (a few particles, or some region in space), entangled with its surroundings. Exact equations (Dyson or Nakajima–Zwanzig) for the reduced system dynamics are readily derived using suitable projection operators. They are rarely solvable, however, since full account is taken for mutual influences between the system and its environment. Nevertheless, for weak mutual coupling, environment-induced dynamics in the (small) system can be much slower than system-induced dynamics in the (large) environment. This justifies the Born–Markov approximation, leading to closed equations for the system. In Hilbert space, we demonstrate the emergence of irreversible dynamics and exponential decay for pure sates. In Liouville space, we derive the Redfield equation. Invoking the secular (or, rotating-wave) approximation, we derive Pauli’s master equations, which properly account for relaxation to equilibrium. Rates of spontaneous emission, coherence transfer (Bloch equations), and pure dephasing are derived and analyzed for a dissipative qubit.