Explicit effective Hamiltonians for linear quantum-optical networks

Linear optical networks are devices that turn classical incident modes by a linear transformation into outgoing ones. In general, the quantum version of such transformations may mix annihilation and creation operators. We derive a simple formula for an effective Hamiltonian generating the scattering operator of a general linear quantum network. Our formula shows how to compute the effective Hamiltonian from the classical scattering matrix. Simple optical instruments [1] such as beam splitters or parametric amplifiers are characterized by linear input-output relations. The beam splitter transforms the annihilation operator of the incident light modes according to the classical laws of optical interference, i.e., by a linear transformation. The parametric amplifier acts like a phase-conjugating mirror, combining the annihilation operator of one incident mode with the creation operator of the other. Complex optical networks can be constructed from beam-splitters, mirrors and active elements such as parametric amplifiers [2, 3, 4]. Networks are essential to the optical processing of quantum information. Furthermore, linear networks may possess interesting quantum-statistical properties when they are large, for example as representations of the Ising model [6, 5] and as examples of quantum localization [7]. Here we derive a simple formula for the Hamiltonian of an arbitrarily large quantum-optical network. Our result allows to predict, in principle, how the quantum state of the incident light modes is processed. Our theory contains as special cases the previously studied Hamiltonians of symmetric passive networks [4], the theory of the beam splitter [8, 9, 10] and of the parametric amplifier [11, 12].