Photoemissive sources and quantum stochastic calculus

Abstract

Just at the beginning of quantum stochastic calculus (QSC), Hudson and Parthasarathy proposed a quantum stochastic Schrodinger equation linked to dilations of quantum dynamical semigroups [7, 8, 11]. Such an equation has found applications in physics, mainly in quantum optics, but not in its full generality [6, 1, 2]. It has been used to give, at least approximately, the dynamics of photoemissive sources such as an atom absorbing and emitting light or matter in an optical cavity, which exchanges light with the surrounding free space. But in these cases the possibility of introducing the gauge (or number) process in the dynamical equation has not been considered. In this paper we want to show, in the case of the simplest photoemissive source, namely a two–level atom stimulated by a laser, how the full Hudson– Parthasarathy equation allows to describe in a consistent way not only absorption and emission, but also the scattering of the light by the atom. Let us recall the Hudson–Parthasarathy equation; this is just to fix our notations, while for the proper mathematical definitions and the rules of QSC we refer to the book by Parthasarathy [11]. We denote by F := F(X ) the Boson Fock space over the Hilbert space X := Z ⊗ L(R+) ≃ L(R+;Z), where Z is another separable complex Hilbert space. Let {ei, i ≥ 1} be a c.o.n.s. in Z and let us denote by Ai(t), A†i (t), Λij(t) the annihilation, creation and gauge processes associated with such a c.o.n.s. We denote by E(h), h ∈ X , the exponential vectors in F with normalization ‖E(h)‖2 = exp{‖h‖2}; E(0) is the Fock vacuum. We shall also use the Boson Fock spaces Ft := F ( L([0, t];Z) ) and F t := F ( L2((t,∞);Z) ) , for which we have F = Ft ⊗F , and the Weyl operators