Simulating a quantum system exposed to some explicitly time-dependent influence differs from that of quantum systems without time dependence in the Hamiltonian. In the latter case, one can, as in Chapter 2, study the full time evolution by means of a relatively simple time-evolution operator, whereas small time steps must be imposed to study the more dynamic case in which also the Hamiltonian changes in time. The first examples of such address the comparatively simple cases of one and two spin-½ particles exposed to magnetic fields. In this context, the rotating wave approximation is introduced. Later, the spatial wave function of a one-dimensional model of an atom exposed to a laser pulse is simulated. To this end, so-called Magnus propagators are used. It is also outlined how the same problem may be recast as an ordinary differential equation by expanding the wave function in the so-called spectral basis consisting of the eigenstates of the time-independent part of the Hamiltonian. The time evolution in this context may be found by more standard methods for ordinary differential equations. Also, the two-particle case if briefly addressed before what is called the adiabatic theorem is introduced. Its validity is checked by implementing a specific, dynamical system.

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.