Tensor product space for studying the interaction of bipartite states of light with nanostructures

Pairs of entangled photons are important for applications in quantum nanophotonics, where their theoretical description must accommodate their bipartite character. Such character is shared at the other end of the intensity range by, for example, the two degenerate instances of the pump field involved in second-harmonic generation. The description and numerical simulation of the interaction of nanophotonic structures with bipartite states of light are challenging regardless of their intensity and have important technological applications. To address such a challenge, we develop here a theoretical and computational framework for studying the interaction of material structures with bipartite states of light. The theory of the framework rests on the symmetrized tensor product space of two copies of an electromagnetic Hilbert space. For the computational side, the convenient $T$-matrix method is extended to the tensor product space. When the response of the object to one part of the state is independent of the other part, the $T$ matrix for bipartite states is a simple function of the typical $T$ matrix of the single Hilbert space. Such a separable material response is relevant, for example, in the interaction of entangled biphoton states with nanostructures. Nonseparable operators are identified as adequate objects to fully integrate nonlinear effects such as sum-frequency generation or parametric down-conversion. As an example of application, we derive selection rules for second-order nonlinear processes in objects with rotational and/or mirror symmetries and verify them numerically in two different ${\mathrm{MoS}}_{2}$ clusters.