The fourth lesson focused on describing the evolution of an open system based on a differential equation (pilot equation). We began by discussing the conditions for the existence of the pilot equation, introducing the Markov approximation and analyzing its physical interpretation. We then showed that if a pilot equation existed, it could always be put into the so-called Lindblad form, which follows naturally from the Kraus form of super-operators of quantum transformations, introduced in the previous lesson. This form of the pilot equation is expressed using a small number of Lindblad operators. The infinitesimal evolution described by the pilot equation can then be seen as a generalized unread measurement of the system, and the Lindblad operators can be interpreted as the operators describing the quantum jumps of the system interacting with its environment during such "measurements".
The Lindblad form of the pilot equation led us directly to the problem of Monte Carlo simulation of quantum trajectories. It is always possible to describe the system's evolution as a generalized measurement whose results, obtained by a random draw, determine the system's quantum jumps and define stochastic trajectories for its state. The theory makes it possible to calculate the evolution of the system between two quantum jumps and its evolution during a jump, whereas the occurrence of a jump can only be statistically predicted. The density operator is found as an average on quantum trajectories. Simple examples of the calculation of stochastic trajectories were presented, concerning the spontaneous emission of a two-level atom and the decoherence of a qubit.
