Abstract
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 slowlight effect near the epsilonnearzero (ENZ) frequency of the lightemitting material itself, to greatly enhance the “forbidden” twoplasmon spontaneous emission (2PSE) process. Using degeneratelydoped 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 onephoton 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 onchip entangled light sources, with an emission wavelength and bandwidth widely tunable in the midinfrared.
Introduction
Plasmonics is a burgeoning field of research that exploits the lightmatter interaction in metallic nanostructures^{1,2}. Recently, there has been a growing interest in quantum plasmonic devices^{1,2} for demonstrating waveparticle 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 entangledSPP 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 entangledSPPs directly from the plasmonic metal.
Twophoton 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 3orderofmagnitude 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 onephoton emission (OPE) process emits photons with energy above the material bandgap, the TPE spectrum can be extremely broad starting from nearzero frequencies. This indicates that TPE has the potential to provide emission and gain spectrum with an ultrabroad bandwidth not restricted by the material bandgap^{19}. However, as a secondorder quantum transition, the TPE process has an emission rate typically 5–10 ordersofmagnitude 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 singlelayer 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 lightemitting 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 zerotemperature 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 twoplasmon spontaneous emission (2PSE), by employing a degenerately doped semiconductor thin film that simultaneously serves as the lightemitting 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 epsilonnearzero (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 slowlight 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 50nmthick 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 widebandgap 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 deepsubwavelength thickness of the InSb layer, SPPs at both interfaces couple and split into the long rangeSPP (LRSPP) and the short rangeSPP (SRSPP)^{45}. Meanwhile, the doped InSb layer may also serve as the lightemitting 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 highfrequency 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 qRe(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 LRSPP 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 kdependence of permittivity corresponds to the nonlocal effect, and ξ^{2} is the nonlocal parameter. Here, we adopt \(\xi ^2 = \left( {\frac{3}{5}  i\frac{4}{{15}}\frac{\gamma }{\omega }} \right)v_{\mathrm{F}}^2\) 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 \(k_{\mathrm{L}}^2 = (\omega ^2 + i\gamma \omega  \omega _{\mathrm{p}}^2/\varepsilon _\infty )/\xi ^2\), 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 lowq region, dispersion relations with LRA and NLRA are almost identical, which suggests that LRA and NLRA only differ for large inplane wavevectors (see Methods and Supplementary Note 2 for details). Compared to the dispersion relation with LRA, the nonlocal effect leads to a blueshift of the SPP modal frequency and a cutoff inplane wavevector since SPPs only exist below ω_{ENZ}. The singularity of DOS for SPP modes is consequently eliminated. In addition, the cutoff 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 inplane wavevectors, due to the field screening caused by longitudinal waves^{45}. Nevertheless, with the nonlocal effect considered, SPP modes in the InSb film can 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 qresolved Purcell factor f,
The explicit expression of f(q, ω, z) is given in Supplementary Note 3.
Electron–hole pairs may form either inplane dipoles that can be excited by the inplane electric field E_{xy} or outofplane dipoles that can be excited by the outofplane electric field E_{z}. Figure 3a–d depicts f(q, ω) contributed by E_{xy} 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 nonradiative 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 nonradiative 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 nonradiative decay at low frequencies barely contributes to the 2PSE process. The nonradiative decay related to material losses near the ENZ frequency can be neglected due to the high Qfactor of the SPP mode (~43). We further exclude the nonradiative decay caused by Landau damping by introducing a wavevector cutoff where the modal frequency reaches the ENZ frequency.
At the center of the InSb layer (z = 25 nm), due to the parity, E_{xy} of LRSPPs equals to zero, while E_{z} of SRSPPs equals to zero. LRSPPs have a weak spectral broadening, high peak value and flat dispersion compared to SRSPPs. Outofplane dipoles therefore obtain a Purcell factor F up to 1.33 × 10^{5} with a narrow spectral bandwidth, while the Purcell effect for inplane dipoles is much broader with a lower peak value. At the edge of the InSb film (z = 3 nm), LRSPPs and SRSPPs can excite both inplane and outofplane dipoles. For inplane dipoles, the Purcell enhancement is stronger and narrower due to the contribution of LRSPPs. However, for outofplane 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 zdirection. Here, we focus on the transition from the conduction (C) band to the heavyhole (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_{xy}, as shown in Fig. 3g. In the C_{n+/−1}HH_{n} transition, the interband transition is excited by E_{xy} and the intersubband transition is excited by E_{z}, as shown in Fig. 3h.
Figure 3i shows the resulting OPE and 2PSE spectra in the InSb film averaged in the zdirection 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 zaveraged 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 twoplasmon 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 blueshift 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 timeenergy entanglement or correlation^{61}. TPE also shows the polarization entanglement when eliminating the degeneracy of the heavyhole and lighthole 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 freespace quantum communication^{62,63,64}. The singlephoton 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 nanolasing^{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 onchip integration.
Methods
Model of the material permittivity
The linear optical properties of doped semiconductors are mainly governed by the interband transition, intraband 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 interband 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 conductionband 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.
References
Tame, M. S. et al. Quantum plasmonics. Nat. Phys. 9, 329–340 (2013).
Bozhevolnyi, S. I. & Khurgin, J. B. The case for quantum plasmonics. Nat. Photonics 11, 398–400 (2017).
Kolesov, R. et al. Waveparticle duality of single surface plasmon polaritons. Nat. Phys. 5, 470–474 (2009).
Di Martino, G. et al. Observation of quantum interference in the plasmonic HongOuMandel effect. Phys. Rev. A 1, 034004 (2014).
Fakonas, J. S., Lee, H., Kelaita, Y. A. & Atwater, H. A. Twoplasmon quantum interference. Nat. Photonics 8, 317–320 (2014).
Altewischer, E., Van Exter, M. & Woerdman, J. Plasmonassisted transmission of entangled photons. Nature 418, 304–306 (2002).
Fasel, S., Halder, M., Gisin, N. & Zbinden, H. Quantum superposition and entanglement of mesoscopic plasmons. N. J. Phys. 8, 13 (2006).
Tsakmakidis, K. L., Boyd, R. W., Yablonovitch, E. & Zhang, X. Large spontaneousemission enhancements in metallic nanostructures: towards LEDs faster than lasers. Opt. Express 24, 17916–17927 (2016).
Fedutik, Y., Temnov, V., Schöps, O., Woggon, U. & Artemyev, M. Excitonplasmonphoton conversion in plasmonic nanostructures. Phys. Rev. Lett. 99, 136802 (2007).
Falk, A. L. et al. Nearfield electrical detection of optical plasmons and singleplasmon sources. Nat. Phys. 5, 475–479 (2009).
Tamada, A. et al. Single plasmon generation in an InAs/GaAs quantum dot in a transferprinted plasmonic microring resonator. ACS Photonics 6, 1106–1110 (2019).
He, L.s & Feng, X.l Twophoton emission spectrum of a twolevel atom in an ideal cavity. Phys. Rev. A 49, 4009 (1994).
Lissandrin, F., Saleh, B. E., Sergienko, A. V. & Teich, M. C. Quantum theory of entangledphoton photoemission. Phys. Rev. B 69, 165317 (2004).
Nevet, A. et al. Plasmonic nanoantennas for broadband enhancement of twophoton emission from semiconductors. Nano Lett. 10, 1848–1852 (2010).
Lin, Z. & Vučković, J. Enhanced twophoton processes in single quantum dots inside photonic crystal nanocavities. Phys. Rev. B 81, 035301 (2010).
Poddubny, A. N., Ginzburg, P., Belov, P. A., Zayats, A. V. & Kivshar, Y. S. Tailoring and enhancing spontaneous twophoton emission using resonant plasmonic nanostructures. Phys. Rev. A 86, 5 (2012).
Melzer, S., Ruppert, C., Bristow, A. D. & Betz, M. Stimulated twophoton emission in bulk CdSe. Opt. Lett. 43, 5066–5069 (2018).
Muniz, Y., Manjavacas, A., Farina, C., Dalvit, D. & KortKamp, W. Twophoton spontaneous emission in atomically thin plasmonic nanostructures. Phys. Rev. Lett. 125, 033601 (2020).
Hayat, A., Nevet, A., Ginzburg, P. & Orenstein, M. Applications of twophoton processes in semiconductor photonic devices: invited review. Semicond. Sci. Technol. 26, 18 (2011).
Ota, Y., Iwamoto, S., Kumagai, N. & Arakawa, Y. Spontaneous twophoton emission from a single quantum dot. Phys. Rev. Lett. 107, 233602 (2011).
Hayat, A., Ginzburg, P. & Orenstein, M. Observation of twophoton emission from semiconductors. Nat. Photonics 2, 238–241 (2008).
Kwiat, P. G. et al. New highintensity source of polarizationentangled photon pairs. Phys. Rev. Lett. 75, 4337 (1995).
Hayat, A., Ginzburg, P. & Orenstein, M. Highrate entanglement source via twophoton emission from semiconductor quantum wells. Phys. Rev. B 76, 4 (2007).
Huber, D. et al. Straintunable GaAs quantum dot: a nearly dephasingfree source of entangled photon pairs on demand. Phys. Rev. Lett. 121, 033902 (2018).
Chen, Y., Zopf, M., Keil, R., Ding, F. & Schmidt, O. G. Highlyefficient extraction of entangled photons from quantum dots using a broadband optical antenna. Nat. Commun. 9, 2994 (2018).
Liu, J. et al. A solidstate source of strongly entangled photon pairs with high brightness and indistinguishability. Nat. Nanotechnol. 14, 586–593 (2019).
Hayat, A., Ginzburg, P. & Orenstein, M. Measurement and model of the infrared twophoton emission spectrum of GaAs. Phys. Rev. Lett. 103, 023601 (2009).
Rivera, N., Kaminer, I., Zhen, B., Joannopoulos, J. D. & Soljačić, M. Shrinking light to allow forbidden transitions on the atomic scale. Science 353, 263–269 (2016).
Rivera, N., Rosolen, G., Joannopoulos, J. D., Kaminer, I. & Soljacic, M. Making twophoton processes dominate onephoton processes using midIR phonon polaritons. Proc. Natl Acad. Sci. U.S.A. 114, 13607–13612 (2017).
Gonçalves, P. et al. Plasmon–emitter interactions at the nanoscale. Nat. Commun. 11, 1–13 (2020).
Wagner, M. et al. Ultrafast dynamics of surface plasmons in InAs by timeresolved infrared nanospectroscopy. Nano Lett. 14, 4529–4534 (2014).
Alu, A., Silveirinha, M. G., Salandrino, A. & Engheta, N. Epsilonnearzero metamaterials and electromagnetic sources: tailoring the radiation phase pattern. Phys. Rev. B 75, 155410 (2007).
Capretti, A., Wang, Y., Engheta, N. & Dal Negro, L. Comparative study of secondharmonic generation from epsilonnearzero indium tin oxide and titanium nitride nanolayers excited in the nearinfrared spectral range. ACS Photonics 2, 1584–1591 (2015).
Liberal, I. & Engheta, N. Nearzero refractive index photonics. Nat. Photonics 11, 149–158 (2017).
Reshef, O., De Leon, I., Alam, M. Z. & Boyd, R. W. Nonlinear optical effects in epsilonnearzero media. Nat. Rev. Mater. 4, 535–551 (2019).
Khurgin, J. B. et al. Adiabatic frequency shifting in epsilonnearzero materials: the role of group velocity. Optica 7, 226–231 (2020).
Kinsey, N., DeVault, C., Boltasseva, A. & Shalaev, V. M. Nearzeroindex materials for photonics. Nat. Rev. Mater. 4, 742–760 (2019).
Yang, Y. et al. Femtosecond optical polarization switching using a cadmium oxidebased perfect absorber. Nat. Photonics 11, 390 (2017).
Yang, Y. et al. Highharmonic generation from an epsilonnearzero material. Nat. Phys. 15, 1022–1026 (2019).
Tsakmakidis, K. L., Hess, O., Boyd, R. W. & Zhang, X. Ultraslow waves on the nanoscale. Science 358, eaan5196 (2017).
Boyd, R. W. Material slow light and structural slow light: similarities and differences for nonlinear optics. JOSA B 28, A38–A44 (2011).
Khurgin, J. B. & Tucker, R. S. Slow Light: Science and Applications (CRC Press, 2018).
Kang, S. S. et al. Highquality 100 nm thick InSb films grown on GaAs (001) substrates with an InxAl1–xSb continuously graded buffer layer. lACS Omega 3, 14562–14566 (2018).
Liu, C., Li, Y. & Zeng, Y. Progress in antimonide based IIIV compound semiconductors and devices. Engineering 2, 617 (2010).
Moreau, A., Ciraci, C. & Smith, D. R. Impact of nonlocal response on metallodielectric multilayers and optical patch antennas. Phys. Rev. B 87, 045401 (2013).
Archambault, A., Teperik, T. V., Marquier, F. & Greffet, J.J. Surface plasmon Fourier optics. Phys. Rev. B 79, 195414 (2009).
Campione, S., Brener, I. & Marquier, F. Theory of epsilonnearzero modes in ultrathin films. Phys. Rev. B 91, 121408 (2015).
Raza, S., Christensen, T., Wubs, M., Bozhevolnyi, S. I. & Mortensen, N. A. Nonlocal response in thinfilm waveguides: loss versus nonlocality and breaking of complementarity. Phys. Rev. B 88, 115401 (2013).
De Ceglia, D. et al. Viscoelastic optical nonlocality of lowloss epsilonnearzero nanofilms. Sci. Rep. 8, 9335 (2018).
Halevi, P. Hydrodynamic model for the degenerate freeelectron gas: generalization to arbitrary frequencies. Phys. Rev. B 51, 7497 (1995).
Enoch, S. & Bonod, N. Plasmonics: From Basics to Advanced Topics. Vol. 167 (Springer, 2012).
Khurgin, J. B. Ultimate limit of field confinement by surface plasmon polaritons. Faraday Discuss. 178, 109–122 (2015).
Rivera, N. & Kaminer, I. Light–matter interactions with photonic quasiparticles. Nat. Rev. Phys. 2, 538–561 (2020).
Jackson, J. D. Classical electrodynamics. 3rd edn (John Wiley & Sons, 1999).
Ferreira, B. A., Amorim, B., Chaves, A. & Peres, N. Quantization of graphene plasmons. Phys. Rev. A 101, 033817 (2020).
Iwase, H., Englund, D. & Vučković, J. Analysis of the Purcell effect in photonic and plasmonic crystals with losses. Opt. Express 18, 16546–16560 (2010).
Folland, T. G. et al. Reconfigurable infrared hyperbolic metasurfaces using phase change materials. Nat. Commun. 9, 4371 (2018).
Rana, F., Strait, J. H., Wang, H. & Manolatou, C. Ultrafast carrier recombination and generation rates for plasmon emission and absorption in graphene. Phys. Rev. B 84, 045437 (2011).
Kaminer, I. et al. Efficient plasmonic emission by the quantum Čerenkov effect from hot carriers in graphene. Nat. Commun. 7, ncomms11880 (2016).
Rajasekaran, S. et al. Parametric amplification of a superconducting plasma wave. Nat. Phys. 12, 1012 (2016).
Van Driel, H. M. Semiconductor optics: on the path to entanglement. Nat. Photonics 2, 212 (2008).
Manor, H. & Arnon, S. Performance of an optical wireless communication system as a function of wavelength. Appl. Opt. 42, 4285–4294 (2003).
Temporão, G. et al. Midinfrared singlephoton counting. Opt. Lett. 31, 1094–1096 (2006).
Corrigan, P., Martini, R., Whittaker, E. A. & Bethea, C. Quantum cascade lasers and the Kruse model in free space optical communication. Opt. Express 17, 4355–4359 (2009).
Ueda, T., An, Z., Hirakawa, K. & Komiyama, S. Chargesensitive infrared phototransistors: characterization by an allcryogenic spectrometer. J. Appl. Phys. 103, 093109 (2008).
Ueda, T., An, Z., Hirakawa, K. & Komiyama, S. Narrow Gap Semiconductors 2007 (Springer, 2008).
Verma, V. et al. in Advanced Photon Counting Techniques XIII. 109780N (International Society for Optics and Photonics).
Korneev, A., Korneeva, Y., Florya, I., Voronov, B. & Goltsman, G. NbN nanowire superconducting singlephoton detector for midinfrared. Phys. Procedia 36, 72–76 (2012).
Santavicca, D. et al. Energy resolution of terahertz singlephotonsensitive bolometric detectors. Appl. Phys. Lett. 96, 083505 (2010).
Basov, D. N. & Fogler, M. M. Quantum materials: the quest for ultrafast plasmonics. Nat. Nanotechnol. 12, 187 (2017).
Heeres, R. W., Kouwenhoven, L. P. & Zwiller, V. Quantum interference in plasmonic circuits. Nat. Nanotechnol. 8, 719 (2013).
Ma, R.M. & Oulton, R. F. Applications of nanolasers. Nat. Nanotechnol. 14, 12–22 (2019).
Bogdanov, S. I. et al. Ultrafast quantum photonics enabled by coupling plasmonic nanocavities to strongly radiative antennas. Optica 7, 463–469 (2020).
Amirtharaj, P. M. & Seiler, D. G. Devices, Measurements, and Properties 2nd edn, Vol. 2 (McGrawHill Professional, 1994).
Lockwood, D., Yu, G. & Rowell, N. Optical phonon frequencies and damping in AlAs, GaP, GaAs, InP, InAs and InSb studied by oblique incidence infrared spectroscopy. Solid State Commun. 136, 404–409 (2005).
Hofmann, P. Solid State Physics: an Introduction (John Wiley & Sons, 2015).
Litwin‐Staszewska, E., Szymańska, W. & Piotrzkowski, R. The electron mobility and thermoelectric power in InSb at atmospheric and hydrostatic pressures. Phys. Status Solidi 106, 551–559 (1981).
Acknowledgements
This work was supported by the startup funding provided to Y.Y. by Tsinghua University and by the National Natural Science Foundation of China (Grant 61975251).
Author information
Authors and Affiliations
Contributions
Y.Y. and F.H. conceived the idea; F.H. designed the structure and finished most calculation; L.L. provided the theoretical analysis for the polarization entanglement; Y.L. helped the calculation of the dispersion relation. Y.M. provided insight into the nonlocal effect. All authors analyzed the numerical data. F.H. and Y.Y. wrote the manuscript. Y.Y. and M.G. supervised the project.
Corresponding authors
Ethics declarations
Competing interests
The authors declare no competing interests.
Additional information
Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Supplementary information
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Hu, F., Li, L., Liu, Y. et al. Twoplasmon spontaneous emission from a nonlocal epsilonnearzero material. Commun Phys 4, 84 (2021). https://doi.org/10.1038/s42005021005864
Received:
Accepted:
Published:
DOI: https://doi.org/10.1038/s42005021005864
This article is cited by

Optical metawaveguides for integrated photonics and beyond
Light: Science & Applications (2021)
Comments
By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.