Interference between multipolar two-photon transitions in quantum emitters near plasmonic nanostructures

Discover Nano Research Interference between multipolar two‑photon transitions in quantum emitters near plasmonic nanostructures S. Smeets1 · B. Maes1 · G. Rosolen1 Received: 16 May 2024 / Accepted: 11 September 2024 © The Author(s) 2024  OPEN Abstract In the vicinity of plasmonic nanostructures that support highly confined light fields, spontaneous emission processes, such as two-photon spontaneous emission (TPSE), exhibit higher-order multipolar emission pathways beyond the dipolar one. These multipolar emission channels occur simultaneously and can interfere with each other. We develop a novel framework that computes these interference effects for TPSE of a quantum emitter close to an arbitrary nanostructure. The model is based on the computation of Purcell factors that can be calculated with conventional electromagnetic simulations, which avoids complex analytic calculations for the environment. For a transition of a hydrogen-like emitter close to a graphene nanotriangle, we demonstrate a breakdown of the dipolar selection rule in the TPSE process. This breakdown is due to a huge enhancement of the two-electric dipole (2ED) and of the two-electric quadrupole (2EQ) transitions. We observe an important interference between these multipolar transitions, as it increases the total rate by 67 %. In the end, our framework is a complete tool to design emitters and nanostructures for TPSE, where the exploitation of previously ignored interference effects provides an additional degree of freedom, for example to boost desired transitions and to supress undesirable ones. Keywords Two-photon spontaneous emission · Interference · Dipole approximation breakdown · Plasmonic nanostructure · Framework · Purcell factor 1 Introduction Two-photon spontaneous emission (TPSE) is a broadband second-order process in the field of light-matter interaction that involves the simultaneous emission of two photons from a quantum emitter. Enhancing and tailoring this process is of great interest, as it promises several applications [1, 2], for example as an alternative to conventional entangled photon pair sources using the parametric down-conversion process in nonlinear crystals [3] for quantum applications [4–7]. Since its prediction in 1931 [8], this second-order process has been investigated in many systems, such as in atoms [9–11], molecules [12], quantum dots [13–15], semiconductors [4, 16–18], epsilon-near-zero-materials [19], plasmonic nanostructures [2, 20, 21], and cosmic strings [22]. Despite the interest in controlling the TPSE process [23], it typically occurs 8 to 10 orders of magnitude slower than the competing spontaneous emission of a single photon [24]. However, electromagnetic interactions with the surrounding environment are known to modify spontaneous emission rates of quantum emitters: the Purcell effect [25, * S. Smeets, ; B. Maes, ; G. Rosolen, | 1Micro‑ and Nanophotonic Materials Group, Research Institute for Materials Science and Engineering, University of Mons, 20 Place du Parc, 7000 Mons, Belgium. Discover Nano (2024) 19:155 | https://doi.org/10.1186/s11671-024-04111-8 Vol.:(0123456789) Research Discover Nano (2024) 19:155 | https://doi.org/10.1186/s11671-024-04111-8 26]. Specifically, when the electromagnetic field is confined at the nanometer scale, such as in plasmonic nanostructures [27–36], there is a light emission enhancement by several orders of magnitude [24, 37]. Moreover, higher-order transitions can be enhanced and even outperform the electric dipole single-photon transition, leading to multiquanta and multipolar transitions [24, 27, 36]. For example, a local breakdown of the dipole selection rule was calculated for a hydrogen-like emitter near graphene nanoislands, where the one-photon quadrupolar transition rate becomes 100 times larger than the dipolar one [36]. Allowed transitions between two given states of a quantum emitter are dictated by selection rules [38]. Between two states of an atom, there is only one possible decay channel for the emission of a single photon (e.g., electric dipole, magnetic dipole, electric quadrupole, etc.) [38], and therefore no interference. However in the TPSE process, the second-order transition is mediated by virtual intermediate states, allowing multiple multipolar emission channels to exist simultaneously [39]. For example, in vacuum and for a 2s → 1s transition in hydrogen, the two-electric dipole (2ED), two-magnetic dipole (2MD), and two-electric quadrupole (2EQ) multipolar emission channels occur simultaneously, however, with the 2MD and 2EQ transition being twelve and thirteen orders of magnitude slower than the 2ED one, respectively [39]. Unlike in vacuum, inside a photonic environment the multipolar field distributions are modified and interference between multipolar emission channels can occur, leading to an increase or decrease of the total transition rate [40]. Thereby, if in a system (atom plus environment) several multipolar two-photon emission channels are of the same order magnitude and are responsible for the emitter transition rate, it is necessary to take into account the interference effects among the dominant channels [40]. Note that in the case of molecules and asymmetric quantum dots, multiple multipolar singlephoton emission channels can occur simultaneously, and therefore can interfere. For example, interference effects were studied between the one-photon magnetic dipole and electric quadrupole transitions in molecules [37] and between one-photon electric dipole, magnetic dipole and electric quadrupole transitions in quantum dots [41], but never between two-photon multipolar transitions to our knowledge. In this paper we study the interference effects between the two-photon multipolar emission channels of an atom near a plasmonic nanostructure. For this purpose we add the interference terms to our framework [42] that computes the TPSE rate of a quantum emitter at any position, close to an arbitrary structure, and beyond the dipolar approximation by considering interactions up to the electric quadrupolar order. The framework relies on the analytical calculation of the emitter contribution and on the classical computation of one-photon Purcell factors by modeling classical point emitters in electromagnetic simulations for the environment contribution, which allows the consideration of complex geometries without available analytical models. In comparison with the formula derived by Muniz et al. [43] to calculate the 2ED transition rate, for example near plasmonic nanostructures [20] and in cosmic strings [22], the formula we derive contains additional terms that are linked to the off-diagonal components of the Green’s function, and it is therefore more general [42]. Note that our framework is based on Fermi’s golden rule and is therefore limited to the weak-coupling regime [42]. Furthermore, for the extreme case of large emitters (larger than 1 nm) placed very close to a nanostructure (≈ (...truncated)