The fifth lesson applied the previous results to the study of the evolution of a mode of the electromagnetic field coupled to an environment, an essential problem in quantum optics. We established the Lindblad form of the pilot equation for a field mode coupled to an environment in thermodynamic equilibrium. We studied the evolution of a Fock state of well-defined photon number at T = 0K and T > 0K , describing the corresponding stochastic trajectories and their mean. We then looked at the evolution of a coherent state coupled to an environment at T = 0K : we showed that there was a unique trajectory and no entanglement with the environment. We sought to explain these classical properties of the coherent state and showed that they were linked to paradoxical properties of the evolution of this state, which only changes between two quantum jumps and remains invariant when it loses a photon. Finally, we have shown that fictitious "homodyne measurements" in the field environment can be associated with different but equivalent Lindblad forms of the pilot equation, enabling us to understand the relaxation of a field in a cavity in a number of complementary ways.
We then generalized coherent states to a general class of states that evolve without becoming entangled with their environment, the so-called pointer states. These states play an essential role in any decoherence process described by a pilot equation. They are in fact eigenstates of quantum jump operators, when these operators satisfy certain switching properties between themselves and with the system Hamiltonian. We also defined approximate pointer states that "slowly" intricate with the environment. We gave various examples and concluded the lesson by showing the importance of the notion of pointer state in measurement theory.
