Two-plasmon spontaneous emission from a nonlocal epsilon-near-zero material

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-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, comparable 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. Plasmonics involves the engineering of light-matter interactions and has range of possible uses in the study of quantum phenomenon, but efficient plasmonic sources are required. The authors report a method to achieve efficient two-plasmon spontaneous emission using an epsilon-near-zero material with highly-confined surface plasmon polaritons to simultaneously serve as a two-plasmon emitter for emission acceleration.

P lasmonics is a burgeoning field of research that exploits the light-matter interaction in metallic nanostructures 1,2 . Recently, there has been a growing interest in quantum plasmonic devices 1,2 for demonstrating wave-particle duality 3 , quantum interference 4,5 , and quantum entanglement 6,7 , using surface plasmon polaritons (SPPs). Quantum plasmonic circuitry requires the development of efficient quantum plasmonic sources 1 . The generation of single-or entangled-SPP can be done either by using the combination of external quantum sources and SPP couplers [6][7][8] , or by the emission of SPPs from quantum sources, such as colloidal quantum dots [9][10][11] , that is directly coupled to a plasmonic cavity. The former approach may suffer from the limited efficiency of the external source and the SPP coupler. For the latter approach, the spatial separation between the emitter and the plasmonic cavity may pose a fundamental limit on the enhancement of the emission rate 9,10 . It remains a grand challenge to generate single-or entangled-SPPs directly from the plasmonic metal.
Two-photon emission (TPE) refers to the simultaneous emission of two photons during a radiative transition process [12][13][14][15][16][17][18] . Recent researches suggest TPE as a promising approach to generate entangled photon pairs in semiconductors, as it emits two photons with intrinsic energy conservation and time concurrence 19,20 . Compared to spontaneous parametric downconversion, a prevailing method to generate entangled photons, TPE does not have the restriction on phase matching and can have 3-order-of-magnitude higher occurrence probability [21][22][23] . TPE can also occur in a wide temperature range, unlike semiconductor quantum dots that strictly require cryogenic operation [24][25][26] . In addition, while the typical one-photon emission (OPE) process emits photons with energy above the material bandgap, the TPE spectrum can be extremely broad starting from near-zero frequencies. This indicates that TPE has the potential to provide emission and gain spectrum with an ultra-broad bandwidth not restricted by the material bandgap 19 . However, as a second-order quantum transition, the TPE process has an emission rate typically 5-10 orders-of-magnitude lower than the OPE rate, thus has typically been considered a "forbidden" process. In semiconductors, the low TPE rate is mainly due to the mismatch between the characteristic emitter size and the light wavelength 21,27 , 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 may be selectively enhanced by the Purcell effect. Previous researchers have demonstrated enhancements of the TPE rate in semiconductors by coupling emitters to a plasmonic bowtie nanoantenna array 14 . However, the experimentally estimated TPE intensity is enhanced by 3 orders of magnitude, still much lower than the OPE rate. This is likely because the optical field is 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 alternative approaches to enhance the TPE rate by placing an atomic emitter near a single-layer graphene that supports SPPs 28 , or a polar dielectric film that supports surface phonon polaritons (SPhPs) 29 . More recently, Y. Muniz et al. proposed a similar approach by placing an atomic emitter near thin plasmonic nanostructures 18 . However, a spatial separation between emitters and the field maximum still pose a limit to the overall TPE rate. An integrated light-emitting scheme that is straightforward for the experimental implementation is yet identified. Furthermore, the nonlocal effect, which may significantly change the optical response of ultrathin films supporting SPPs or SPhPs, needs to be carefully considered when dealing with polariton propagation with a large wavevector. Rivera et al. considered the nonlocal correction of TPE near graphene using the zero-temperature random phase approximation 28 . Gonçalves et al. considered the nonlocal correction of TPE near a jellium metal using the dparameter framework 30 .
Here, we propose a scheme to achieve efficient two-plasmon spontaneous emission (2PSE), by employing a degenerately doped semiconductor thin film that simultaneously serves as the light-emitting medium and the plasmonic cavity. Doped semiconductors are known to support SPP modes in the mid-and farinfrared 31 . In this work, we design the epsilon-near-zero (ENZ) [32][33][34][35][36][37][38][39] frequency ω ENZ of the semiconductor film to be around half of its OPE frequency, thus allowing the selective enhancement of the 2PSE rate. Using degenerately doped InSb as a prototype material, a remarkable Purcell factor up to 1.33 × 10 5 is obtained due to the high field confinement and the slow-light effect [40][41][42] near its ω ENZ . The nonlocal effect that governs the optical response of the thin film near its ω ENZ is considered using a hydrodynamic model. We show that the 2PSE lifetime can be reduced from tens of millisecond to tens of nanoseconds, making the 2PSE rate comparable to the OPE rate. In addition, the 2PSE spectral peak can be flexibly tuned by varying the doping density of the InSb film.

Results
Model. Our proposed structure consists of a 50-nm-thick degenerately doped InSb film sandwiched between two AlSb layers, as depicted in Fig. 1a. The bottom AlSb layer can serve as a buffer layer between GaAs and InSb 43 . It can also serve as a a SPP Pump beam x y Fig. 1 Schematic of two-plasmon spontaneous emission from a degenerately doped InSb thin film. a Schematics of the proposed structure that supports surface plasmon polaritons (SPPs). b Schematics of the onephoton emission process in a semiconductor with a degenerate doping level. The dashed line marks the Fermi level E F . Yellow arrows represent the thermalization of photoexcited carriers. ω 0 represents the angular frequency of the photon emitted during one-photon emission. ω ENZ is the angular frequency where the permittivity of InSb approaches zero. k is the electron wavevector. c Schematics of the two-plasmon spontaneous emission process in a semiconductor with a degenerate doping level. ω 1 and ω 2 represent the angular frequencies of the two plasmons emitted in the spontaneous emission process, respectively.
wide-bandgap barrier to trap the photoexcited carriers in InSb 44 . This layered structure supports SPPs at both AlSb-InSb interfaces at frequencies below ω ENZ of doped InSb. With the deep-subwavelength thickness of the InSb layer, SPPs at both interfaces couple and split into the long range-SPP (LR-SPP) and the short range-SPP (SR-SPP) 45 . Meanwhile, the doped InSb layer may also serve as the light-emitting material through the radiative transition from its conduction band to the valence band (see Supplementary Note 1 for the model of the electronic band structure). With its narrow band gap of 0.17 eV after doping, one may design the OPE frequency of the doped InSb thin film to be about twice its ω ENZ , to selectively enhance the 2PSE rate. Figure 1b, c shows the schematics of the OPE and 2PSE process in the doped InSb thin film. In the initial calculation, we choose a carrier concentration N e of 6 × 10 18 cm −3 for InSb and a photoexcited carrier density of 1 × 10 14 cm −3 .
The photoexcited carrier density is set to be much smaller than the carrier concentration, such that the electron temperature remains unchanged, and only the linear permittivity of InSb needs to be considered.
In the local response approximation (LRA), the permittivity of InSb ε is independent of the wavevector of light, and is described by the Drude model as, where ε ∞ is the high-frequency permittivity, ω is the angular frequency, ω p is the plasma frequency, and γ is the Drude scattering rate. Following the Gauss's law ε∇·E = 0, the condition of transverse waves ∇·E = 0 persists unless ε = 0. Therefore, SPPs are conventionally treated as transverse waves except at ω ENZ . The permittivity of transverse waves ε T (ω) is thus equal to the local permittivity ε(ω), and is plotted in Fig. 2a, with an ENZ energy of 0.105 eV. Figure 2b illustrates the dispersion relation q-Re(ω q ) of the layered structure with LRA, where q is the wavevector along the x-y plane. The density of states (DOS) for SPP modes at a given ω is inversely proportional to the slope of the dispersion curve, i.e., dq/dRe(ω q ) 46 . The dispersion of LR-SPP modes in the thin film, also known as ENZ modes 47 , is extremely flat, giving rise to a large DOS near ω ENZ . Figure 2c plots the field confinement factor as a function of q in LRA. Here, the field confinement factor L q /λ q is related to the modal length L q and the modal wavelength λ q with L q defined as, The dispersion curve is extended to infinity in LRA, with a singularity of DOS and field confinement at large wavevectors. To resolve the singularity issue, which leads to an unphysical, diverging Purcell enhancement, we further consider the nonlocal effect in the InSb thin film 38 . In the nonlocal response approximation (NLRA), SPPs are coupled modes of transverse electromagnetic excitations and longitudinal plasmons. Longitudinal plasmons are collective electron oscillations and hold a distinct dielectric response compared to transverse electromagnetic excitations. Its permittivity ε L can be described by a hydrodynamic model, and is related to the wavevector of longitudinal plasmons k L as 45,48,49 , The k-dependence of permittivity corresponds to the nonlocal effect, and ξ 2 is the nonlocal parameter. Here, we adopt ξ 2 ¼ Á v 2 F with v F the Fermi velocity, which is derived from a viscoelastic fluid model for semiconductor films and agrees well with experiments 49,50 . When k 2 L ¼ ðω 2 þ iγω À ω 2 p =ε 1 Þ=ξ 2 , ε L is equal to zero and ∇·E = 0 is not required, thereby yielding longitudinal plasmons in the InSb film. Figure 2d depicts the dispersion relation of SPP with NLRA. In the low-q region, dispersion relations with LRA and NLRA are almost identical, which suggests that LRA and NLRA only differ for large in-plane wavevectors (see Methods and Supplementary Note 2 for details). Compared to the dispersion relation with LRA, the nonlocal effect leads to a blue-shift of the SPP modal frequency and a cut-off in-plane wavevector since SPPs only exist below ω ENZ . The singularity of DOS for SPP modes is consequently eliminated. In addition, the cut-off wavevector at ω ENZ is still below ω/v F . Therefore, the Landau damping effect can be ignored 51,52 . As shown in Fig. 2e, field confinement quickly drops for extremely large in-plane wavevectors, due to the field screening caused by longitudinal waves 45 . Nevertheless, with the nonlocal effect considered, SPP modes in the InSb film can Here, ℏ is the reduced Planck constant, ω P is the plasma frequency, γ is the Drude damping rate. b-c Dispersion relations (b) and field confinement factor L q /λ q (c) of the the longrange surface plasmon polariton (LR-SPP) and the short-range surface plasmon polariton (SR-SPP) modes under the local-response assumption, respectively. Here, q is the in-plane momentum of the SPP. ℏω q is the modal energy, L q is the modal length, and λ q is the modal wavelength. The gray dashed line represents the epsilon-near-zero (ENZ) energy. d-e Dispersion relations (d) and field confinement factor (e) of the LR-SPP and SR-SPP modes under the nonlocal-response assumption, respectively. still retain deep subwavelength confinement, which is critical for the enhancement of the 2PSE rate.
Calculated Purcell enhancement and 2PSE rate. We quantize SPPs using the modal expansion method tailored for a nonlocal plasmonic material, with the material loss included in the complex modal frequency 46 , the material dispersion considered in the Brillouin energy density formula 53,54 , and the material nonlocality incorporated in Maxwell's equations 45,48,55 . Our modified modal expansion method can prevent a divergent decay rate caused by the unphysically flat dispersion relation and by the breakdown of the dipole approximation when the field is highly concentrated near the emitter. Based on the modal expansion method, we can express the Purcell factor as an integration of the q-resolved Purcell factor f, The explicit expression of f(q, ω, z) is given in Supplementary Note 3.
Electron-hole pairs may form either in-plane dipoles that can be excited by the in-plane electric field E x-y or out-of-plane dipoles that can be excited by the out-of-plane electric field E z . Figure 3a-d depicts f(q, ω) contributed by E x-y and E z , respectively. The spectral broadening of f(q, ω) is induced by the imaginary part of the modal frequency 56 . The spectral broadening indicates non-radiative channels related to material losses, giving rise to a high Purcell factor at low frequencies as shown in Fig. 3e, f. This feature is also observed in the Green's function method 16,55 . An enhanced non-radiative decay rate in lossy materials may affect the Purcell enhancement of emitters, which is important in realizing our proposal to enhance the 2PSE process. However, in this work, the enhanced non-radiative decay at low frequencies barely contributes to the 2PSE process. The non-radiative decay related to material losses near the ENZ frequency can be neglected due to the high Q-factor of the SPP mode (~43). We further exclude the non-radiative decay caused by Landau damping by introducing a wavevector cut-off where the modal frequency reaches the ENZ frequency.
At the center of the InSb layer (z = 25 nm), due to the parity, E x-y of LR-SPPs equals to zero, while E z of SR-SPPs equals to zero. LR-SPPs have a weak spectral broadening, high peak value and flat dispersion compared to SR-SPPs. Out-of-plane dipoles therefore obtain a Purcell factor F up to 1.33 × 10 5 with a narrow spectral bandwidth, while the Purcell effect for in-plane dipoles is much broader with a lower peak value. At the edge of the InSb film (z = 3 nm), LR-SPPs and SR-SPPs can excite both in-plane and out-of-plane dipoles. For in-plane dipoles, the Purcell enhancement is stronger and narrower due to the contribution of LR-SPPs. However, for out-of-plane dipoles, the peak Purcell enhancement drops due to the field screening effect near the edge 45,51 .
The OPE and 2PSE rates tailored by the Purcell effect can be formulated as 14,29 , where R 0 OPE (ω 0 ) and R 0 2PSE (ω 1 , ω 2 ) represent the OPE and 2PSE rates in the vacuum state, respectively (see Supplementary Note 4 for details). The valance band of InSb splits into subbands due to the confinement in the z-direction. Here, we focus on the transition from the conduction (C) band to the heavy-hole (HH) band. There are two 2PSE paths: C n -HH n and C n+/−1 -HH n , where n is the quantum number of the subband. 2PSE includes one interband transition and one intersubband transition. In the C n -HH n transition, both transitions are excited by E x-y , as shown in Fig. 3g. In the C n+/−1 -HH n transition, the interband transition is excited by E x-y and the intersubband transition is excited by E z , as shown in Fig. 3h. The q-resolved Purcell factors f as a function of the in-plane momentum q and the energy ℏω for both in-plane (||) and out-of-plane (⊥) dipole orientations at z = 3 nm. c-d The q-resolved Purcell factors f as a function of q and ℏω for both dipole orientations at z = 25 nm. e-f The Purcell factor F as a function of ℏω for both dipole orientations. The gray dashed line represents the epsilon-near-zero (ENZ) energy. g Two-plasmon spontaneous emission through the C n -HH n transition. Here, C represents the conduction subband and HH represents the heavy-hole subband. k || is the the in-plane electron wavevector. The blue arrow represents the transition excited by the in-plane polarization E x-y . h Two-plasmon spontaneous emission through the C n+/-1 -HH n transition. The blue arrow represents the transition excited by E x-y . The red arrow represents the transition excited by the out-of-plane polarization E z . i The one-photon emission (OPE) and the two-plasmon spontaneous emission (2PSE) rates R (ℏω) in the InSb film averaged over the z-position at an operation temperature T = 77 K. Figure 3i shows the resulting OPE and 2PSE spectra in the InSb film averaged in the z-direction at 77 K. The calculated spectral rate of 2PSE in the InSb film close to its ENZ frequency is comparable to its OPE rate. The ratio of the z-averaged lifetimes between OPE and 2PSE increases from 5.8 × 10 −8 in bulk InSb to 0.27, with a 2PSE lifetime of only 2.5 ns. This enhancement is robust at lower temperatures, with a ratio of 0.15 at 30 K and a ratio of 0.03 at 4 K. The current design requires low temperature operation of the two-plasmon emitter, which may be a concern for certain applications.

Discussion
Moreover, while the spectrum of OPE in a semiconductor is generally fixed by the material bandgap, the spectrum of 2PSE in the proposed structure can be flexibly engineered. We further investigate the dependence of the 2PSE spectrum on the carrier density in the InSb film. ω ENZ of InSb can be largely tuned by the carrier concentration N e . As shown in Fig. 4, with an increasing N e from 4 × 10 18 cm −3 to 1 × 10 19 cm −3 , the spectral peak of 2PSE in InSb can blue-shift by 34%. Besides, the emission bandwidth can be flexibly tuned from 13 to 33 meV. The tuning of N e can be done by chemical doping or by applying a static electric field. Alternatively, the SPP dispersion can be actively tuned by including additional tunable materials, such as phasechanging vanadium dioxide in the vicinity of the plasmonic cavity 57 .
In conclusion, we theoretically show that ultrafast and tunable 2PSE can be realized in a nonlocal ENZ film, by spatially and spectrally matching 2PSE with highly confined SPP modes. Similar concepts can be extended to other semiconductors, 2D materials 58,59 , and superconductors supporting Josephson plasmons 60 . The emitted SPP pair have intrinsic time concurrence and energy conservation, which predicts a time-energy entanglement or correlation 61 . TPE also shows the polarization entanglement when eliminating the degeneracy of the heavyhole and light-hole valence band at the band edge 19 . One of the potential advantages of an entangled light source emitting in the 8-12 μm wavelength range is that it's within an atmospheric window which is desirable for free-space quantum communication [62][63][64] . The single-photon detection in midinfrared can be realized using detectors based on either quantum wells 65,66 or superconducting devices [67][68][69] . The efficient SPP emission can be used for realizing plasmonic amplification and increasing plasmon coherence, two crucial elements in plasmonic circuitry 70,71 and nano-lasing 72 . The generated SPPs can be efficiently coupled to the free space using nanoantennas (see Supplementary Note 5 for details) 73 . In addition, the carrier can potentially be injected by electrical means, with possibilities for further on-chip integration.

Methods
Model of the material permittivity. The linear optical properties of doped semiconductors are mainly governed by the inter-band transition, intra-band transition, and optical phonon absorption 74 . Here we use the same optical properties for both bulk InSb and InSb with a finite thickness. Within the OPE and 2PSE spectra, the inter-band absorption below the Fermi energy is mostly prohibited by Pauli blocking, due to the degenerate doping level in InSb. Meanwhile, since the optical phonon frequency of InSb 75 is relatively far away from ω ENZ , the optical phonon absorption is also ignored here. The Drude model 76 can be written as ε = ε ∞ -ω p 2 /(ω 2 + iγω). The plasma frequency ω p is given by ω p 2 = N e e 2 /(ε 0 m c ), where m c is the effective mass of conduction-band electrons. For a given N e , we can obtain γ = e/(m c μ e ) and v F = ℏ(3π 2 N e ) 1/3 /m c , where e is the electron charge, μ e is the electron mobility 5 . In this work, we assume μ e to be 11000 cm 2 V −1 s −1[ 77 .
Field distribution and dispersion relation. The nonlocality in the metallic film is described by a semiclassical hydrodynamical model 45,[48][49][50] . Here we incorporate longitudinal waves into Maxwell's equations by introducing longitudinal components into the polarization of free electrons. The field distribution can therefore be solved analytically by assuming that transverse waves exponentially decay near the doped InSb/AlSb interface and that longitudinal waves only exist within the InSb layer.

Data availability
The datasets generated during and/or analyzed during the current study are available from the corresponding author on reasonable request.

Code availability
The codes used during the current study are available from the corresponding author on reasonable request.