Degenerate parametric down-conversion facilitated by exciton-plasmon polariton states in nonlinear plasmonic cavity

Plasmonics has been in the spotlight in engineering, GramotnevNatPhot:2010 chemistry, EbbesenAcCemRes:2016 ; FeistASCNano:2018 and biology HuangLaserSurgMed:2007 for more than a decade. StockmanOptEx:2011 During this time, fascinating progress in nanofabrication provided new tools to create metal-dielectric, i.e., plasmonic, interfaces with nanometer precision.PanoiuJ.Opt:2018 Even though the surface plasmon (SP) modes in such nanostructures show relatively low characteristic quality factors, large local field enhancements WuACSPhot:2021 make them very attractive for applications. EbbesenAcCemRes:2016 ; JiangChemRev:2017 On the other hand, ultrafast optical detection techniques have offered time resolution reaching a few femtosecond limit. VasaNatPhot:2013 These fabrication and characterization tools tremendously assist us in advancing our knowledge of the light-matter interactions at the nanoscale, where the quantum and classical properties of light and matter merge.TormaRepPrgoPhys:2014

Despite such progress, the nonlinear plasmonics relying on the strong local field enhancement GordonACSPhot:2018 and high nonlinear electron susceptibilities in metal KrasavinLaserPhotRev:2018 is still in its infancy.PanoiuJ.Opt:2018 Several recent reports address nonlinear phenomena such as second/third-harmonic generation GalantyLightSciTech:2018 ; WeissmanAdvOptMat:2017 ; YooOptExp:2019 , four-wave mixing AlmeidaSciRep:2015 ; BlechmanNanoLett:2018 , and optical bistability. WurtzPRL:2006 Moreover, the use of coherent control was reported to study the second-harmonic generation (SHG) by gold nanoparticles combining spatial localization of electromagnetic fields with laser pulse shaping.BaharPRB:2020

Quantum plasmonics interprets the metallic nanostructure as a plasmonic cavity where strong local electric fields enhance the spontaneous decay of organic dyes or semiconductor quantum dots representing quantum emitters (QEs).TameNatPhys:2013 This effect is known as the plasmonic Purcell effect. Advances beyond demonstration of the Purcell effect hinge on the use of 2D plasmonic lattices facilitating delocalized hybrid plasmonic-diffraction surface lattice resonance of relatively high quality factor (low energy losses).CherquiAccChemRes:2019 Use of these structures allowed researchers to reach a strong coupling regime between QE and SP in which the SP-QE quantum exchange rate exceeds plasmonic losses and results in the formation of mixed QE and SP exciton-plasmon polariton stats. RamezaniJOSAB:2019 Current theoretical and experimental studies of the exciton-polariton states focus on plasmonic lasing Pusch_ACSnano:2012 ; Guan_ASCnano:2020 ; Winkler_ACSnano:2020 , polariton Bose Einstein condensation, and polariton lasing. MartikainenPRA:2014 ; ZasterJPCS:2016 ; DeGiorgiPEP:2018 ; HakalaNP:2018 All these studies asume linear SP response with the QEs providing source of nonlinearity.

The strong SP-QE coupling regime has recently become a spotlight in nonlinear plasmonics, promising coherent photonic up/down-conversion and lasing applications. One of the authors contributed to a theoretical study demonstrating that strongly coupled nonlinear SP systems and QEs may exhibit unique second-order optical signals facilitated by mixed exciton-plasmon polariton states.DrobnyhJCP:2020 ; SukharevJCP:2021 Subsequent experimental studies confirmed the predictions. LiNanoLett:2020 In another theory report, we examined the effect of QE gain on the SHG by a plasmonic nanostructure, e.g., 2D plasmonic lattice, exhibiting nonlinear response.Sukharev_JCP:2021 For the QEs strongly coupled with fundamental SP mode, we found a lasing state contributing to efficient photon emission at the second-harmonic frequency. In that report, we also introduced a concept of the nonlinear plasmonic cavity in which nonlinear interaction occurs between quantized fundamental and second-harmonic plasmonic modes. Noteworthy that photonic cavities demonstrate nonlinearity due to embedded QEs only. RibeiroPRA:2021

In this paper, we use the concept of nonlinear SP cavity to investigate a reverse effect of the degenerate parametric down-conversion (DPDC) of excited QEs (excitons) strongly coupled to the second-harmonic of the cavity via near-field interactions. For a localized SP mode, the effects of SHG and parametric down-conversion have been studied experimentally in light of nonlinear photon-plasmon interactions.GrossePRL:2012 ; HeckmannOptExp:2013 The authors of Refs. LootJOpt:2018 ; LootOptik:2018 considered spontaneous parametric downconversion in nonlinear dielectric media interfacing with metal surfaces so that the linear SP response of the metal enhances the process via weak local field coupling. None of these reports considers the effects of QEs.

Therefore, we focus our paper on the case in which QEs forming gain medium exchange energy via the near-filed interactions with the second-harmonic of a nonlinear plasmonic cavity. As demonstrated below, the presence of QEs results in new exciton-plasmon polariton states mixing two-SP quanta with a single exciton state. Considering the strong coupling regime, we predict a non-trivial lasing phase supporting these polariton states as elementary excitations. Our analysis of the polariton power spectra predicts coherent photon emission by QEs around the second harmonic frequency and correlated photon pairs at the fundamental frequency by the SP mode. Given the SP photon emission rate significantly exceeds the same rate for QEs, we predict the photon DPDC effect in such a system.

We organize the paper as follows. Sec. II presents our quantum open system theory of the DPDC, identifying lasing steady states and providing expressions for the exciton-plasmon polariton dispersion and fluctuation auto-correlation functions. Sec. III presents numerical calculations and discussion of the polariton dispersion relations and the auto-correlation functions. Finally, we conclude in Sec. IV.

As illustrated in Fig. 1, we introduce an ensemble of identical two-level QEs with the transition frequency $\omega_{o}$. The QEs occupy sites $n=\overline{1,{\cal N}_{o}}$ and interact with a delocalized SP mode whose fundamental frequency is $\omega_{sp}$. The SP mode is assumed to have second-order nonlinear response allowing for the SHG, i.e., creation of a single SP quantum with frequency $2\omega_{sp}$ or a down-conversion of a single SP quantum with frequency $2\omega_{sp}$ to two quanta each having frequency $\omega_{sp}$. By tuning the QEs around the SP second-harmonic frequency, i.e., $\omega_{o}\sim 2\omega_{sp}$, we allow for the down-conversion of the SP quanta produced by QEs. The interaction between each QE and SP is characterized by the quantum exchange rate $\lambda$ proportional to the SP second-order susceptibility $\chi_{sp}^{(2)}$.

For the introduced system, the Hamiltonian, assuming rotating wave approximation for the SP-QE interaction term, reads

Here, the SP mode is described by Bose creation (annihilation) operator $\hat{a}^{\dagger}$ ($\hat{a}$), and the two-level QEs by spin operators $\hat{s}_{n}^{\pm}=\hat{s}_{n}^{x}\pm i\hat{s}^{y}_{n}$, $s_{n}^{z}$, related to the Pauli matrices as $\hat{s}_{n}^{j}=\frac{1}{2}\hat{\sigma}_{n}^{j}$ where $j=x,y,z$.

Both the SPs and QEs are subject to energy supply originating form incoherent excitation of the QEs and dissipation processes due to environment fluctuations. Thus, the time-evolution of such an open quantum system characterized by an operator set $\hat{\cal O}=\{\hat{a},\hat{s}_{n}^{-},\hat{s}_{n}^{z}\}$ can be described by the Heisenberg equation of motion

where the first term in the right-hand side accounts for the coherent dynamics governed by the Hamiltonian $\hat{\cal H}$. The second term describes SP decay with the rate $2\gamma_{sp}$ and the last three terms account for incoherent pumping of QEs with the rate $\gamma_{\uparrow}$, QE decay with the rate $\gamma_{\downarrow}$ and pure dephasing with the rate $\gamma_{\phi}$. Here, the Lindblad operator describing dissipation process in the Markovian approximation is

The substitution of the Hamiltonian (II) and $\hat{\cal O}=\{\hat{a},\hat{s}_{n}^{-},\hat{s}_{n}^{z}\}$ into Eqs. (II) – (3) results in the following set of coupled Heisenberg equations of motion for the SP and QE degrees of freedom

Eqs. (5) and (6) contain pumping rate dependent QE dephasing, $\gamma_{o}=\gamma_{\downarrow}/2+\gamma_{\uparrow}/2+\gamma_{\phi}$ and population decay, $\gamma_{t}=\gamma_{\uparrow}+\gamma_{\downarrow}$, rates. Eq. (6) also contains the population inversion parameter

whose value varies in the interval $-1\leq d_{o}<0$ ($0<d_{o}\leq 1$) for $\gamma_{\uparrow}<\gamma_{\downarrow}$ ($\gamma_{\uparrow}>\gamma_{\downarrow}$) indicating that QEs are pumped below (above) population inversion threshold.

Eq. (4) has a well familial form of equation of motion for the parametric down-conversion processes Loudon_book:10 . While conventional DPDC occurs due to a coherent optical pump tuned in resonance with the second-harmonic, our model replaces the laser pump with the QE coherences interacting with the counter-rotating SP coherence in the last term of Eq. (4).

In this section, we calculate the polariton dispersion (Eq. (II.2)) as a function of the SP-QE coupling and power spectra of the SP and QE elementary excitations. Let us define the spectra as $S_{sp}=\text{Re}[C_{sp}(\omega)]/{\cal N}_{o}^{2}$ and $S_{qe}=\text{Re}[C_{qe}(\omega)]$ using Eqs. (26) and (27) for the auto-correlation functions. The correlation functions are calculated using both the mean-field (Eq. (12)–(18)) and correlated QE-SP (Eqs. (II.1)–(22)) models allowing for comparison between their predictions. Parameterization of the models involves setting SP and QE energies to $\hbar\omega_{sp}=\hbar\omega_{o}/2=1.95\text{eV}$, QE decay (dephasing) rate to $\hbar\gamma_{\downarrow}=0.014\text{eV}$ ($\hbar\gamma_{\phi}=0.01\text{eV}$), and the SP dephasing rate to $\hbar\gamma_{sp}=0.014\text{eV}$. These parameters are consistent with our previous calculations reported in Ref. Sukharev_JCP:2021 . Finally, for the correlated SP-QE model, Eqs. (II.1)–(22) were solved numerically by setting the number of QEs to ${\cal N}_{o}=5000$.

Figure 2 shows the dispersion of polariton frequency $\Omega=\text{Im}[\Lambda_{\pm}]$ (orange curves) and dissipation rate $\Gamma=\text{Re}[\Lambda_{\pm}]$ (blue curves) as a function of normalized coupling parameter. Probing various values of the incoherent pumping rate and associated (Eq. (7)) inversion parameter, $d_{o}$, reveals several distinct features of polariton states and limitations of the mean-field model. For a relatively high pumping rate, $d_{o}=0.33$, results of both mean-field and correlated SP-QE calculations are the same (see panel a). In this case, the lower and upper polariton frequencies do not split as the system transitions from the normal (i.e., trivial ) steady state to the lasing state. In contrast, the polariton dissipation rate branches split above the critical coupling, and the lower branch reaches zero. This trend is due to the QE gain, which fully compensates for the losses indicating lasing regime. For a lower inversion characterized by $d_{o}=0.46$, panel b shows no changes in the behavior of the mean-field solution (thin blue lines). However, the dissipation rate calculated using the correlated QE-SP model (blue dots) behaves differently. Specifically, the polariton frequency splitting associated with the strong coupling regime occurs below the mean-field threshold ($\sim 0.75\lambda_{c}$). At the same time, the lower polariton dissipation rate never reaches zero, i.e., the QE gain does not fully compensate the losses.

These features indicate deviations in the properties of finite-size QE ensembles from the predictions of the mean-field theory. The latter, indeed, is exact only in the thermodynamic limit. For example, failure of the lower polariton dissipation branch to reach zero is a signature of cross-over behavior between normal and lasing states rather than the lasing phase transition. Furthermore, below the QE inversion (panel c), our mean-field theory is not defined, but the correlated QE-SP calculations produce a satisfactory result. They show a well-known behavior of the polariton branches while crossing a threshold between weak and strong coupling regimes at ${\cal N}_{o}\lambda\approx 0.2\lambda_{c}$. At this point, the polariton frequency (orange dots) experiences the Rabi splitting, and the QE and SP dissipation rates (blue dots) merge into a single rate. The trend of lowering polariton strong coupling threshold, observed in panels b and c, with the pumping rate decrease, is likely related to the dependence of the dissipation rates on the pumping rate (see comments following Eqs. (4) and (6)).

Trends in the variations of the QE power spectra, calculated in the mean-field approximation (Fig. 3a) and using the QE-SP correlated model (Fig. 3b), are consistent with the discussed trends in polariton dispersion (Fig. 2). Calculated above critical coupling at ${\cal N}_{o}\lambda=2.0\lambda_{c}$, both panels of the plot show narrow lasing peaks at the pumping rate twice the decay rate (red curves). Differences appear when the pumping rate gets closer to the inversion threshold. In this situation (blue lines), the mean-field spectrum still has a sharp lasing peak, whereas the correlated QE-SP model predicts a broader peak. The broadening occurs because the lower polariton dissipative branch (blue dots in Fig. 2b) does not reach zero, i.e., the QE gain does not fully compensate for the losses.

Below the inversion threshold, our mean-field model predicts a single broad peak (black curve in panel a) originating from the trivial limit of the fluctuation correlation function (Eq. (33)) with the resonance at $\omega_{o}$. In contrast, the correlated QE-SP model (black line in panel b) predicts two resonances due to the Rabi splitting of the polariton frequency (orange dots in Fig. 2c). The two-SP power spectra in panels c and d of Fig. 3 show the same features as panels a and b. The mean-field model predicts no DPDC effect below inversion (notice the absence of a black curve in pane c). To understand this, recall that the two-SP coherence directly couples to the QE steady state coherence $\bar{s}_{-}$ (see, e.g., Eq. (II)). The latter vanishes below the population inversion threshold and cancels the two-SP coherence.

As illustrated in Fig. 1, the radiative decay of QEs and two-SP states results in the photon emission around the second-harmonic frequency and correlated fundamental-frequency photon-pair production, respectively. The QE and SP spectra are appropriately normalized, so comparing peak values, we see an even quantum number distribution between the QEs and two-SP degrees of freedom. In contrast, the number of photons produced via the radiative decay of QEs and two-SP states depends on their material-specific spontaneous emission rates. This quantity in plasmonic nanostructures can be orders of magnitude higher than in organic chromophores or semiconductor QEs. As a result, we expext that the photon emission spectra should be dominated by the correlated two-photon pairs resulting in the photon DPDC effect.

We have examined the mechanism of DPDC in nonlinear plasmonic cavities in which the incoherently pumped QEs interact with the SP second-harmonic mode via near-field coupling. As demonstrated, such a model allows for a lasing phase transition in the strong coupling regime. Interestingly, the effective coupling rate in our model scales as ${\cal N}_{o}\lambda$ in contrast to conventional scaling of $\sqrt{{\cal N}_{o}}\lambda$ known for the Dicke and Tavis-Cummings models facilitating the strong coupling regime for smaller ensembles of QEs. Comparing the finite-size ensemble of QEs with the model thermodynamic limit, we have found that in the former case, the strong coupling threshold lowers as the QE pumping rate decreases. Furthermore, the finite-size ensemble of QEs pumped below population inversion is predicted to demonstrate the polariton Rabi splitting and the DPDC effect. In the thermodynamic limit, no such effects take place. Our model also predicts that the lasing threshold in finite-size systems occurs for high pumping rate than in the thermodynamic limit.

We expect that 2D plasmonic lattices showing the lattice-plasmon resonances RamezaniJOSAB:2019 ; Sukharev_JCP:2021 could be a good material platform for observation predicted DPDC effect. However, our basic quantum optics model cannot incorporate some material-specific features of the lattice-plasmon resonances. This calls for numerical simulations SukharevJPCM:2017 accounting for the composition and geometry of metal nanoparticles, detailed mechanisms of nonlinearity, and QE electronic structure. In particular, the simulations should address the question of photon emission yield for the correlated two-SP states compared to the emission yield of the QEs. Finally, reported progress on developing new plasmonic nanomaterials, e.g., doped semiconductors, AgrawalChenRev:2018 ; Dolgopolova_NanoHorizons:2022 with SP resonances in the near-infrared, will pave the way for practical applications of the proposed effect in on-chip photonic devices.

This work was performed, in part, at the Center for Integrated Nanotechnologies, an Office of Science User Facility operated for the U.S. Department of Energy (DOE) Office of Science by Los Alamos National Laboratory (Contract 89233218CNA000001) and Sandia National Laboratories (Contract DE-NA-0003525) (user project No. 2021BC0087). A.P. would like to acknowledge support via LDRD exploratory research grant No. 20200407ER. M.S. is grateful for the financial support by the Air Force Office of Scientific Research under Grant No. FA9550-22-1-0175. We would like to thank Professor Joseph Zyss for stimulating discussions.