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Two-Plasmon Spontaneous Emission from a Nonlocal Epsilon-Near-Zero Material ✉ ✉ Futai Hu1, Liu Li1, Yuan Liu1, Yuan Meng 1, Mali Gong1,2 & Yuanmu Yang1

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Two-Plasmon Spontaneous Emission from a Nonlocal Epsilon-Near-Zero Material ✉ ✉ Futai Hu1, Liu Li1, Yuan Liu1, Yuan Meng 1, Mali Gong1,2 & Yuanmu Yang1 ARTICLE https://doi.org/10.1038/s42005-021-00586-4 OPEN Two-plasmon spontaneous emission from a nonlocal epsilon-near-zero material ✉ ✉ Futai Hu1, Liu Li1, Yuan Liu1, Yuan Meng 1, Mali Gong1,2 & Yuanmu Yang1 Plasmonic cavities can provide deep subwavelength light confinement, opening up new avenues for enhancing the spontaneous emission process towards both classical and quantum optical applications. Conventionally, light cannot be directly emitted from the plasmonic metal itself. Here, we explore the large field confinement and slow-light effect near the epsilon-near-zero (ENZ) frequency of the light-emitting material itself, to greatly enhance the “forbidden” two-plasmon spontaneous emission (2PSE) process. Using degenerately- 1234567890():,; doped InSb as the plasmonic material and emitter simultaneously, we theoretically show that the 2PSE lifetime can be reduced from tens of milliseconds to several nanoseconds, com- parable to the one-photon emission rate. Furthermore, we show that the optical nonlocality may largely govern the optical response of the ultrathin ENZ film. Efficient 2PSE from a doped semiconductor film may provide a pathway towards on-chip entangled light sources, with an emission wavelength and bandwidth widely tunable in the mid-infrared. 1 State Key Laboratory of Precision Measurement Technology and Instruments, Department of Precision Instrument, Tsinghua University, Beijing, China. ✉ 2 State Key Laboratory of Tribology, Department of Mechanical Engineering, Tsinghua University, Beijing, China. email: [email protected]; [email protected] COMMUNICATIONS PHYSICS | (2021) 4:84 | https://doi.org/10.1038/s42005-021-00586-4 | www.nature.com/commsphys 1 ARTICLE COMMUNICATIONS PHYSICS | https://doi.org/10.1038/s42005-021-00586-4 lasmonics is a burgeoning field of research that exploits the correction of TPE near graphene using the zero-temperature Plight-matter interaction in metallic nanostructures1,2. random phase approximation28. Gonçalves et al. considered the Recently, there has been a growing interest in quantum nonlocal correction of TPE near a jellium metal using the d- plasmonic devices1,2 for demonstrating wave-particle duality3, parameter framework30. quantum interference4,5, and quantum entanglement6,7, using Here, we propose a scheme to achieve efficient two-plasmon surface plasmon polaritons (SPPs). Quantum plasmonic circuitry spontaneous emission (2PSE), by employing a degenerately requires the development of efficient quantum plasmonic doped semiconductor thin film that simultaneously serves as the sources1. The generation of single- or entangled-SPP can be done light-emitting medium and the plasmonic cavity. Doped semi- either by using the combination of external quantum sources and conductors are known to support SPP modes in the mid- and far- SPP couplers6–8, or by the emission of SPPs from quantum infrared31. In this work, we design the epsilon-near-zero 9–11 32–39 sources, such as colloidal quantum dots , that is directly (ENZ) frequency ωENZ of the semiconductor film to be coupled to a plasmonic cavity. The former approach may suffer around half of its OPE frequency, thus allowing the selective from the limited efficiency of the external source and the SPP enhancement of the 2PSE rate. Using degenerately doped InSb as coupler. For the latter approach, the spatial separation between a prototype material, a remarkable Purcell factor up to 1.33 × 105 the emitter and the plasmonic cavity may pose a fundamental is obtained due to the high field confinement and the slow-light 9,10 40–42 ω limit on the enhancement of the emission rate . It remains a effect near its ENZ. The nonlocal effect that governs the grand challenge to generate single- or entangled-SPPs directly optical response of the thin film near its ωENZ is considered using from the plasmonic metal. a hydrodynamic model. We show that the 2PSE lifetime can be Two-photon emission (TPE) refers to the simultaneous emis- reduced from tens of millisecond to tens of nanoseconds, making sion of two photons during a radiative transition process12–18. the 2PSE rate comparable to the OPE rate. In addition, the 2PSE Recent researches suggest TPE as a promising approach to gen- spectral peak can be flexibly tuned by varying the doping density erate entangled photon pairs in semiconductors, as it emits two of the InSb film. photons with intrinsic energy conservation and time concurrence19,20. Compared to spontaneous parametric down- conversion, a prevailing method to generate entangled photons, Results Model TPE does not have the restriction on phase matching and can . Our proposed structure consists of a 50-nm-thick – fi have 3-order-of-magnitude higher occurrence probability21 23. degenerately doped InSb lm sandwiched between two AlSb TPE can also occur in a wide temperature range, unlike semi- layers, as depicted in Fig. 1a. The bottom AlSb layer can serve as 43 conductor quantum dots that strictly require cryogenic a buffer layer between GaAs and InSb . It can also serve as a operation24–26. In addition, while the typical one-photon emis- sion (OPE) process emits photons with energy above the material a bandgap, the TPE spectrum can be extremely broad starting from Pump beam near-zero frequencies. This indicates that TPE has the potential to provide emission and gain spectrum with an ultra-broad band- z width not restricted by the material bandgap19. However, as a (nm) second-order quantum transition, the TPE process has an emis- SPP sion rate typically 5–10 orders-of-magnitude lower than the OPE 50 AlSb “ ” 0 y InSb rate, thus has typically been considered a forbidden process. In AlSb semiconductors, the low TPE rate is mainly due to the mismatch x between the characteristic emitter size and the light GaAs wavelength21,27, as well as the discrepancy between the electron velocity and the group velocity of light. Due to the spectral separation of OPE and TPE, the TPE rate bc may be selectively enhanced by the Purcell effect. Previous researchers have demonstrated enhancements of the TPE rate in E E semiconductors by coupling emitters to a plasmonic bowtie F F nanoantenna array14. However, the experimentally estimated ω ≈ω TPE intensity is enhanced by 3 orders of magnitude, still much ω ω 1 ENZ fi 0≈2 ENZ ω ω lower than the OPE rate. This is likely because the optical eld is 2≈ ENZ only enhanced in the vicinity of the antenna tip. In addition, the spontaneous emission enhancement near semiconductor emitters is limited by the relatively low quality- (Q-) factor of the antenna resonance. Recently, Rivera et al. theoretically proposed alter- k k native approaches to enhance the TPE rate by placing an atomic emitter near a single-layer graphene that supports SPPs28,ora Fig. 1 Schematic of two-plasmon spontaneous emission from a polar dielectric film that supports surface phonon polaritons degenerately doped InSb thin film. a Schematics of the proposed structure (SPhPs)29. More recently, Y. Muniz et al. proposed a similar that supports surface plasmon polaritons (SPPs). b Schematics of the one- approach by placing an atomic emitter near thin plasmonic photon emission process in a semiconductor with a degenerate doping 18 nanostructures . However, a spatial separation between emitters level. The dashed line marks the Fermi level EF. Yellow arrows represent and the field maximum still pose a limit to the overall TPE rate. the thermalization of photoexcited carriers. ω0 represents the angular An integrated light-emitting scheme that is straightforward for frequency of the photon emitted during one-photon emission. ωENZ is the the experimental implementation is yet identified. Furthermore, angular frequency where the permittivity of InSb approaches zero. k is the nonlocal effect, which may significantly change the optical the electron wavevector. c Schematics of the two-plasmon spontaneous response of ultrathin films supporting SPPs or SPhPs, needs to be emission process in a semiconductor with a degenerate doping level. ω1 and carefully considered when dealing with polariton propagation ω2 represent the angular frequencies of the two plasmons emitted in the with a large wavevector. Rivera et al. considered the nonlocal spontaneous emission process, respectively. 2 COMMUNICATIONS PHYSICS | (2021) 4:84 | https://doi.org/10.1038/s42005-021-00586-4 | www.nature.com/commsphys COMMUNICATIONS PHYSICS | https://doi.org/10.1038/s42005-021-00586-4 ARTICLE λ fi wide-bandgap barrier to trap the photoexcited carriers in modal wavelength q with Lq de ned as, 44 – R InSb . This layered structure supports SPPs at both AlSb InSb ½∂ðω εÞ=∂ω j ð Þj2 ω dzRe q q E z interfaces at frequencies below ENZ of doped InSb. With the Lq ¼ noð Þ 2 2 deep-subwavelength thickness of the InSb layer, SPPs at both max Re½∂ðωqεÞ=∂ωqjEðzÞj interfaces couple and split into the long range-SPP (LR-SPP) and the short range-SPP (SR-SPP)45.Meanwhile,thedoped InSb layer may also serve as the light-emitting material through The dispersion curve is extended to infinity in LRA, with a the radiative transition from its conduction band to the valence singularity of DOS and field confinement at large wavevectors. To band (see Supplementary Note 1 for the model of the electronic resolve the singularity issue, which leads to an unphysical, band structure). With its narrow band gap of 0.17 eV after diverging Purcell enhancement, we further consider the nonlocal doping, one may design the OPE frequency of the doped InSb effect in the InSb thin film38. In the nonlocal response fi ω thin lm to be about twice its ENZ, to selectively enhance the approximation (NLRA), SPPs are coupled modes of transverse 2PSE rate. Figure 1b, c shows the schematics of the OPE and electromagnetic excitations and longitudinal plasmons. Long- 2PSE process in the doped InSb thin film. In the initial calcu- itudinal plasmons are collective electron oscillations and hold a 18 −3 lation, we choose a carrier concentration Ne of 6 × 10 cm distinct dielectric response compared to transverse electromag- 14 −3 for InSb and a photoexcited carrier density of 1 × 10 cm .
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