Analytical and numerical investigation of the two-photon spontaneous emission process near plasmonic nanostructures

The electric dipole approximation is prevalent in the field of light-matter interaction.
However, in nanophotonic structures sustaining highly confined fields, the approximation no longer holds and higher-order processes can become significant.
Among them, two-photon spontaneous emission (TPSE) is a second-order process that involves the simultaneous emission of two photons from a quantum emitter and is a promising alternative to conventional entangled photon pair sources for quantum applications.
However, the study of advanced nanostructures for TPSE is hampered by a lack of efficient numerical and theoretical methods.

First, we develop a general framework that calculates the two-photon emission rate enhancement of a quantum emitter near an arbitrarily shaped nanostructure.
The framework is based on the classical computation of Purcell factors in classical electromagnetic simulations and we consider the interaction up to the electric quadrupolar order.
For a hydrogen atom near a plasmonic silver nanodisk, we demonstrate a substantial enhancement in the photon-pair emission rates by 5 and 11 orders of magnitude for the two-electric dipole (2ED) and two-electric quadrupole (2EQ) transitions, respectively.
Second, our framework also includes the quantum interferences between the multipolar transitions of TPSE, which has never been studied before.
For a hydrogen atom near a plasmonic graphene nanotriangle, we calculate the interference between the 2ED and 2EQ transitions, which can increase the total transition rate by more than $65 \%$.
Third, we do a first step towards the design of innovative two-photon nanoantennas.
We exploit dipolar and quadrupolar modes on one or two silver nanorods to emit photons of different frequencies in separate directions.

In conclusion, we developed a powerful framework based on the classical computation of Purcell factors to design nanoantennas for TPSE that can be exploited for various quantum applications.
It can be used for arbitrarily shaped nanostructures to optimize the efficiency and directionality, amongst others.