Controlling quantum-dot light absorption and emission by a surface-plasmon field

The possibility for controlling the probe-field optical gain and absorption switching and photon conversion by a surface-plasmon-polariton near field is explored for a quantum dot above the surface of a metal. In contrast to the linear response in the weak-coupling regime, the calculated spectra show an induced optical gain and a triply-split spontaneous emission peak resulting from the interference between the surface-plasmon field and the probe or self-emitted light field in such a strongly-coupled nonlinear system. Our result on the control of the mediated photon-photon interaction, very similar to the `gate' control in an optical transistor, may be experimentally observable and applied to ultra-fast intrachip/interchip optical interconnects, improvement in the performance of fiber-optic communication networks and developments of optical digital computers and quantum communications.


I. INTRODUCTION
such as spatially selective illumination of individual neuron cells to locally control neuron firing activities in optogenetics and neuroscience and near-field high-resolution imaging beyond the optical diffraction limit as well.
Theoretically, a big hurdle also exists for studying photon-photon interactions in the strong-coupling regime mainly due to intractable numerical computation for systems with very strong nonlinearity. The obstacle of nonlinearity in such a system means that any perturbative theories, e.g. using bare electron states or linear response theory, [5] become inadequate for describing both field and electron dynamics in this system. The presence of an induced polarization, regarded as a source term to the Maxwell equations, [15,16] from photo-excited electrons makes it impossible for us to solve the field equations by simply using finite-element analysis [17] or finite-difference-time-domain methods [18]. Although the semiconductor-Bloch equations [19] and density-matrix equations [5,20], derived from manybody theory, are able to accurately capture the nonlinear optical response of electrons, the inclusion of pair scattering effects on both energy relaxation and optical dephasing precludes an analytical approach for seeking solution of these equations. As a result, there exists only very few theoretical studies [21], which heavily depend on computer simulation, that focus on simplified one-dimensional strongly-coupled microcavity systems, in contrast to the threedimensional structure and self-consistent approach presented in this paper.
Physically, not only the high-quality microcavities [22] but also the intense surfaceplasmon near fields [23,24] could be employed for reaching the strong-coupling goal in semiconductors. In this paper we solve the self-consistent equations for strongly-coupled electromagnetic-field dynamics and electron quantum kinetics in a quantum dot above the surface of a thick metallic film, which has not been fully explored so far from either a theoretical or experimental point of view. This is done based on finding an analytical solution to Green's function [25,26] for a quantum dot coupled to a semi-infinite metallic material system, which makes it easy to calculate the effect of the induced polarization field as a source term to the Maxwell equations. In our formalism, the strong light-electron interaction is reflected in the photon-dressed electronic states with a Rabi gap and in the feedback from the induced optical polarization of dressed electrons to the incident light. The formalism derived in this paper goes beyond the weak-coupling limit and deals with a much more realistic structure in the strong-coupling limit for the development of a surface-plasmon polariton laser with a very low threshold pumping. Our results clearly demonstrate the ability to control probe-field optical gain and absorption switching and photon conversion by a surface-plasmon field with temperature-driven frequency detuning in such a nonlinear system led by dressed electron states, very similar to the 'gate' control in an optical transistor. These conclusions should be experimentally observable [27,28]. On the other hand, our numerical results also provide an example for demonstrating the so-called quantum plasmonics, [29] where the nature of surface-plasmon polaritons and the nature of quantum-confined electrons are hybridized through near-field coupling.
In Sec. II, we will introduce our physics model and derive self-consistent equations for determining the coupled scattering dynamics of a surface-plasmon field and the quantum kinetics of electrons in quantum dots. Section III is devoted to a full discussion of our numerical results, including scattering and optical absorption of surface-plasmon-polariton field by quantum dots, spontaneous emission and nonlinear optical response of dressed electron states. Some concluding remarks are given in Sec. IV.

II. MODEL AND THEORY
Our model system, as shown in Fig. 1, consists of a semi-infinite metallic materialwith a semiconductor quantum dot above its surface. A surface-plasmon-polariton (SPP) field is locally excited through a surface grating by normally-incident light. This propagating SPP field further excites an interband electron-hole (e-h) plasma in the quantum dot. The induced optical-polarization field of the photo-excited e-h plasma is strongly coupled to the SPP field to produce split degenerate e-h plasma and SPP modes with an anticrossing gap.
Part of the brief description for our self-consistent formalism was reported earlier. [16] In order to let readers follow up easily with the details of our model and formalism, we present here the full derivation of the Maxwell-Bloch numerical approach for an SPP field coupled to a photo-excited e-h plasma in the quantum dot.

A. General Formalism
The Maxwell's equation for a semi-infinite non-magnetic medium in position-frequency space can be written as [25] ∇×∇×E(r; ω) − ǫ b (x 3 ; ω) ω 2 c 2 E(r; ω) = ω 2 ǫ 0 c 2 P loc (r; ω) , where E(r; ω) is the electric component of an electromagnetic field, H(r; ω) = − i ωµ 0 ∇×E(r; ω) is the magnetic component of the electromagnetic field, r = (x 1 , x 2 , x 3 ) is a three-dimensional position vector, ω is the angular frequency of the incident light, ǫ 0 , µ 0 and c are the permittivity, permeability and speed of light in vacuum, P loc (r; ω) is an off-surface local polarization field generated by optical transitions of electrons in a quantum dot, and the position-dependent dielectric function is Here, ǫ d is for the semi-infinite dielectric material in the region x 3 > 0, while ǫ M (ω) represents the semi-infinite metallic material in the region x 3 < 0. For the Maxwell's equation in Eq. (1), we introduce the Green's function G µν (r, r ′ ; ω) satisfying the following equation where

B. Solving Green's Function
For a semi-infinite medium, the Green's function can be formally expressed by its Fourier where we have introduced the notations for the two-dimensional vectors r = (x 1 , x 2 ) and After a rotational transformation [25] is performed in k -space, i.e., where the rotational matrix is selected as we acquire an equivalent version of Eq. (7) To get the solution of Eq. (10), we need to employ both the finite-value boundary condition at x ′ 3 = ±∞ and the continuity boundary condition at the x 3 = 0 interface. This leads to the following five non-zero f µν (k , ω|x 3 , x ′ 3 ) elements [25,26] where sgn(x) is the sign function, Im[p d (k , ω)] ≥ 0 and Im[p(k , ω)] ≥ 0. In addition, from these non-zero f µν (k , ω|x 3 , x ′ 3 ) functions, we obtain which can be substituted into Eq. (6) to calculate the Green's function G µν (r, r ′ ; ω) in position space.

C. Local Polarization Field
In order to find the explicit field dependence in P loc (r; ω), we now turn to the study of electron dynamics in a quantum dot. Here, the optical-polarization field P loc (r; ω) plays a unique role on bridging the classical Maxwell's equations for electromagnetic fields to the quantum-mechanical Schrödinger equation for electrons. The electron dynamics in photoexcited quantum dots can be described quantitatively by the so-called semiconductor Bloch equations [30][31][32]. These generalize the well-known optical Bloch equations in two aspects including the incorporation of electron scattering with impurities, phonons and other electrons as well as many-body effects on dephasing in the photo-induced optical coherence.
The physical system considered in this paper is illustrated in Fig. 2, where we assume two levels for electrons and holes, respectively, in a quantum dot. These two energy levels of both electrons and holes are efficiently coupled by phonon scattering at high temperatures.
Additionally, the lowest electron and hole energy levels are optically coupled to each other by an incident SPP field to form the dressed states of excitons. The SPP-controlled optical properties of quantum-dot excitons can either probed by a plane-wave field or seen from the spontaneous emission of excitons.
For photo-excited spin-degenerated electrons in the conduction band, the semiconductor Bloch equations with ℓ = 1, 2, · · · are given by where R sp is the spontaneous emission rate and n e ℓ represents the electron level population. In Eq. (19), the term marked 'rel' is the non-radiative energy relaxation for n e ℓ , and the Y j ℓ , M eh ℓ,j , and V eh ℓ,j;j,ℓ terms are given later in the text.
Similarly, for spin-degenerate holes in the valence band, the semiconductor Bloch equations with j = 1, 2, · · · are found to be where n h j stands for the hole energy level population. Again, the non-radiative energy relaxation for n h j is incorporated in Eq. (20). Moreover, we know from Eqs. (19) and (20) that where N e (t) and N h (t) are the total number of photo-excited electrons and holes, respectively, in the quantum dot at time t.
Finally, for spin-averaged e-h plasmas, the induced interband optical coherence, which is introduced in Eqs. (19) and (20), with j = 1, 2, · · · and ℓ = 1, 2, · · · satisfies the following equations, wherehγ 0 =hγ eh +hγ ext is the total energy-level broadening due to both the finite carrier lifetime and the loss of an external evanescent field, ω is the frequency of the external field, and ε e ℓ (ω) and ε h j (ω) are the kinetic energies of dressed single electrons and holes, respectively (see Appendix A with α = 1). In Eq. (22), the diagonal dephasing (γ 0 ) of Y j ℓ , the renormalization of interband Rabi coupling (Y j ℓ V eh ℓ,j;j,ℓ ), the renormalization of electron and hole energies (third and fourth terms on the right-hand side), as well as the exciton binding energy, are all taken into consideration. Since the e-h plasmas are independent of spin index in this case, they can be excited by both left-circularly and right-circularly polarized light. The off-diagonal dephasing of Y j ℓ has been neglected due to low carrier density.
The steady-state solution to Eq. (22), i.e. under the condition of dY j ℓ /dt = 0, is found to be where the photon and Coulomb renormalized interband energy-level separationhΩ eh ℓ,j (ω|t) is given byh The steady-state solution in Eq. (23) can be substituted into Eqs. (19) and (20) above.
The Coulomb interaction matrix elements introduced in Eqs. (19), (20) and (22) are defined as where the static screening length 1/q s at temperatures (k B T ≫ E F ) is determined from S is the cross-sectional area of a quantum dot, T is the lattice temperature, ψ e ℓ (r) and ψ h j (r) are the envelope wave-functions of electrons and holes in a quantum dot (see Appendix A), where L 0 is the thickness of a disk-like quantum dot. In addition, the matrix elements employed in Eqs. (19), (20) and (22) for the Rabi coupling between photo-excited carriers and an evanescent external field E(r; t) = θ(t) E(r; ω) e −iωt are given by where θ(x) is a unit step function, the static interband dipole moment d c,v (see Appendix u c (r) and u v (r) are the Bloch functions associated with conduction and valence bands at the Γ-point in the first Brillouin zone of the host semiconductor, and the effective electric field coupled to the quantum dot is The Boltzmann-type scattering term [33] for non-radiative electron energy relaxation in Eq. (19) is where the microscopic scattering-in and scattering-out rates are calculated as Here,the primed summations in Eqs. (35) and (36) 20) is where the scattering-in and scattering-out rates are and again the primed summations in Eqs. (38) and (39) exclude the terms satisfying either The coupling between the longitudinal-optical phonons and electrons or holes in Eqs. (35), (36), (38) and (39) are calculated as where ǫ ∞ and ǫ s are the high-frequency and static dielectric constants of the host polar semiconductor.
By generalizing the Kubo-Martin-Schwinger relation, [20] the time-dependent spontaneous emission rate, R sp (t), introduced in Eqs. (19) and (20), can be expressed as where is the density-of-states of spontaneously-emitted photons in vacuum, m 0 is the free electron mass, m * e is the effective mass of electrons, and the Coulomb renormalization of the energy bandgap E c (t) is calculated as In Eq. (44), the first two terms are associated with the Hartree-Fock energies of electrons and holes, while the rest of the terms are related to the exciton binding energy.
Finally, the photo-induced interband optical polarization P loc (r; ω), which is related to the induced interband optical coherence, by dressed electrons in the quantum dot is given where d c,v = d c,vêd represents the interband dipole moment [see Eq. (32)],ê d is the unit vector of the dipole moment, and |ξ(r)| 2 comes from the confinement of a quantum dot.

D. Self-Consistent Field Equation
Since the wavelength of the incident light is much larger than the size of a quantum dot, we can treat the quantum dot, which is excited resonantly by the incident light, as a point dipole at r = r 0 = (0, 0, z 0 ), i.e. we can assume P loc (r ′ ; ω) = P loc (ω) δ(r ′ − r 0 ) in Eq. (4) to neglect its geometry effect. This greatly simplifies the calculation and gives rise to where where the quantum-dot level populations n e ℓ (t) and n h j (t) depend nonlinearly on E(r 0 ; ω) in the strong-coupling regime.
If the electromagnetic field is not very strong, we can neglect the pumping effect. In this linear-response regime, we can write down the electron and hole populations in a thermalequilibrium state [without solving Eqs. (19) and (20)] where f 0 (x) is the Fermi function, µ e and µ h are the chemical potentials of electrons and holes, respectively, determined by Eq. (21). As a result of Eqs. (50) and (51), we get from where, according to Eq. (6), we have The solution E(r 0 ; ω) of the linear-matrix equation in Eq. (52) can be substituted into Eq. (49) to find the spatial distribution of the electromagnetic field E(r; ω) at all positions other than r = r 0 , i.e., In order to find the coupled e-h plasma and plasmon dispersion relation ω = Ω ex−pl (k ), we perform Fourier transforms to both E(r; ω) and E (0) (r; ω) in Eq. (46) with respect to r . This leads to After setting Here, the zero determinant of the coefficient matrix in Eq. (57) determines the coupled e-h plasma and plasmon dispersion relation ω = Ω ex−pl (k ). We emphasize that the assumption of thermal-equilibrium states for electrons and holes is just for obtaining analytical expressions. Therefore, some qualitative conclusions can be drawn for guidance from these analytical solutions. Our numerical results, however, are based on the non-thermal-equilibrium steady states calculated after solving self-consistently the coupled Maxwell-Bloch equations.
By assuming an incident SPP field within the x 1 x 2 -plane, we can write x 3 directions, E sp is the field amplitude, ω sp is the field frequency, θ 0 is the angle of the incident SPP field with respect to the x 1 direction, D 0 = {−x g , −y g } is the position vector of the surface grating, and the two wave numbers are with Re[k 0 (ω sp )] ≥ 0 and Re[β 3 (k 0 , ω sp )] ≥ 0. Here, the in-plane wave number k 0 is produced by the surface-grating diffraction of the p-polarized normally-incident light, which in turn determines the resonant frequency ω of the SPP mode. Equation (59) stands for the full dispersion relation of the SPP field, including both radiative and non-radiative parts. From Eq. (58), it is easy to find its Fourier transformed expression

E. Quantum-Dot Absorption
On the basis of the above electromagnetic field E(r 0 ; ω) at the quantum dot, we are able to compute the time-resolved nonlinear interband absorption coefficient of dressed electrons in a quantum dot for the SPP field. [34] In this case, we find where α spp (ω; t) is the complex Lorentz function given by and the scaled refractive index function n spp (ω; t) can be calculated by In Eqs. (63) and (64) F. Probing Quantum-Dot Dressed States We are also able to compute the time-resolved linear interband absorption coefficient of electrons, dressed by the SPP field, for a weak probe field (not the strong SPP field) on the basis of the above calculated electromagnetic field E(r 0 ; ω) at the quantum dot. [34] Assuming a spatially-uniform probe field E p (t) = θ(t − τ ) E p e −iωpt with τ being the delay time, the probe-field absorption coefficient β abs (ω p ; t) of the lowest dressed state is given by Eq. (62) with the replacements of ω, n spp , and α spp by ω p , n pf , and α pf , respectively, where Here, using Eq. (24) we havē andh Moreover, the time-resolved photoluminescence spectrum P em (ω ′ ; t) of dressed electrons in the quantum dot is proportional to  We know that a decrease in temperature T gives rise to an increase in the crystal bandgap E G . On the other hand, the localization of an SPP field (i.e. an exponential decay of the field strength on either side of a metallic surface) is greatly enhanced when the SPP frequency ω sp approaches that of a surface plasmon. As a result, the field at the quantum dot is expected to decrease as T is reduced. This gives rise to a higher absorption coefficient for a lower temperature, as shown in the upper-left panel of Fig. 4. Interestingly, although the suppressed absorption coefficient can be seen from β 0 (ω sp ) for high SPP-field amplitudes, as shown by Eq. (63), from the upper-right panel of this figure we find the resonant peak athω sp = E G + ε 1,e + ε 1,h initially increases with T but then decreases with T at room temperature. This subtle difference demonstrates the effect of reduced phonon absorption at T = 77 K on the resonant scattering field by the factor 1−n e (t)−n h (t) in Eq. (49). Moreover, the strong effect of the suppressed optical-phonon absorption between two electron energy levels at 77 K is clearly demonstrated in the lower panels of Fig. 4, where the level occupation n 2,e becomes negligible at T = 77 K in comparison with that at T = 300 K.
The electron thermal dynamics due to phonon absorption has been demonstrated in Fig. 4 for various temperatures. In Fig. 5, we present the electron dynamics resulting from the optical dephasing, due to the finite lifetime of electrons, at different energy-level broadenings hγ 0 . Ashγ 0 is increased from 3 meV to 7 meV, the dip in β 0 (ω sp ) at resonance is suppressed, leading to a single peak with a reduced strength and an increased width, as shown in the upper-left panel of the figure. This increase in the resonant absorption is further accompanied by an enhanced resonant peak for the scattering field in the upper-right panel of this figure. As expected, the energy-level occupations athγ 0 = 7 meV become much broader than those athγ 0 = 3 meV, as displayed in the lower two panels of the figure.
We further notice that the effective bandgap E G + ε 1,e + ε 1,h also depends on the size L x of a quantum dot due to the quantization effect, and the effective bandgap will increase with decreasing L x . The size effect from such an L x dependence is displayed in Fig. 6.
From the upper-left panel of Fig. 6, we find that the peak of β 0 (ω sp ) is enhanced as L x is reduced. This phenomenon is connected to the increased localization of the SPP field at L x = 170Åas the SPP frequency approaches the saturation part of its dispersion. Moreover, the dip in β 0 (ω sp ) is lifted somewhat uniformly at the same time due to decreased n e 1 (t) from the enhanced Coulomb and phonon scattering at L x = 170Å. Here, β 0 (ω sp ) is proportional to the population factor 1 − n e 1 (t) − n h 1 (t), as can be seen from Eq. (63). Besides the slightlyreduced resonant peak strength of the scattering field for L x = 170Å (also resulting from the enhanced carrier scattering), |E tot − E sp | keeps the same peak position, as shown in the upper-right panel of the figure. In this case, |E tot −E sp | at the dot approaches a nonzero value at resonance, as can be seen from Eq. (55), and tends to zero rapidly away from resonance.
Additionally, n 2,h is reduced for L x = 170Å, as can be found from a comparison between the two lower panels of the figure. This is attributed to the reduced phonon absorption between two hole energy levels.
In Figs. 4 and 6, we vary the localization of an SPP field by changing the effective bandgap.
Since the frequency of the surface plasmon (saturated dispersion part) is proportional to the factor of 1/ √ 1 + ǫ d , a smaller value of ǫ d implies a higher surface-plasmon frequency or a reduced localization of the SPP field. We verify the change in the SPP localization by observing the upper two panels of Fig. 7, where the absorption peak, as well as the resonant scattering-field peak, become much stronger as ǫ d is increased from 8 to 12 due to the reduction of saturated absorption for a lower field strength at the quantum dot.
Furthermore, from the two lower panels of this figure we also observe, via the jumps in the population curves, an enhanced Rabi-split energy gap in the electron dressed states as ǫ d is reduced from 12 to 10 due to the enhanced field strength at the quantum dot.
In the presence of the localization of an SPP field, we can move a quantum dot closer to a metallic surface to gain a higher field at the quantum dot. The upper-left panel of  Fig. 3.

B. Results for the dressed states of electrons
In the second part of the numerical calculations, besides the parameters given in the first subsection, we have fixed L x = 210Å,hγ 0 = 3 meV, z 0 = 610Å and ǫ d = 12. Other parameters, including T , E sp and ∆hω sp , will be directly indicated in the figures. Additionally, ∆hω sp is given with respect to the energy gap at T = 300 K.
From the left panel of Fig. 9 we find a strong absorption (positive) peak and a weak gain (negative) peak for the probe-field absorption coefficient β abs (ω p ) due to a quantum coherence effect from the electron states being dressed by an SPP field. In the strongcoupling regime, the dispersion of the quantum-dot e-h plasmas (dot-like branch) and SPPs (photon-like branch) form an anticrossing gap, where a higher-energy dot-like branch at a negative frequency detuning switches to a photon-like branch for a positive detuning. The positive peak is associated with the absorption of a probe-field photon by a quantum-dot e-h plasma, while the negative peak relates to the process with absorption of two photons from an SPP field and emission of one probe-field photon. The absorption peak is significantly reduced by saturation at E sp = 1000 kV/cm, and the gain peak is suppressed by a smaller Rabi-coupling frequency at E sp = 250 kV/cm (see the inset of the left panel). In addition, we observe from the right panel of Fig. 9 that two Rabi-splitting-induced side emission peaks for the spontaneous emission P em (ω) become weaker and closer to the strong central peak as  other. Using a laser pulse to launch a pulsed SPP field, we are able to study the dynamics of phonon scattering (narrow pulse) as well as the dynamics of e-h pair radiative recombination (wide pulse), separately. Dynamically, phonon scattering becomes effective only after a characteristic time (around 1 ps), its effect can be seen from a significant increase of n 2,e in our system. Figure 12  Technically, detecting dynamics of photo-excited e-h plasmas by using another timedelayed weak probe field is much more feasible, as shown in Fig. 13. From the left panel of this figure, we find that β abs (ω p ) starts with a pair of positive absorption and negative gain peaks due to a very strong photon dressing effect for the delayed times τ d = 60 and 120 fs.
This is changed to a strong absorption peak plus a very weak gain peak at τ d = 240 fs. At the end, β abs (ω p ) becomes independent of τ d , indicating that a linear optical-response regime has been reached. On the other hand, from the right panel of this figure, we see that the central peak of P em (ω) is gradually built up with increasing τ d due to enhanced n 1,e and n 1,h around resonance, while two side peaks become weakened and disappear at the same time due to weakened Rabi oscillations. Interestingly, we also find that the central peak of P em (ω) slightly decreases at τ d = 1 ps, which agrees with the observed start of significant phonon absorption seen in the lower-left panel of Fig. 12.
In order to explore the dynamics of e-h pair radiative recombination in our system, a wide pulse with a full-pulse width around 300 ps is required, as displayed in Fig. 14. From the upper-middle panel of this figure, we find that β 0 (ω sp ) starts with a resonant dip due to a strong photon dressing effect, then shifts to a sole peak at ∆hω sp = 0 as τ 0 ≥ 400 ps where a steady state is almost reached in the linear-response regime. Accordingly, the level populations n 1,e and n 2,e in the lower two panels show a transition from an initial nonresonant behavior to a final resonant behavior. This is accompanied by dramatically reduced level populations due to the start of a radiative recombination process for photo-excited e-h pairs.
Recombination dynamics for e-h plasmas can also be demonstrated clearly by the timedelayed probe-field absorption as well as by the time-resolved spontaneous emission, as shown in Fig. 15. As presented in the left panel of this figure, we find that the initial weak absorption and gain peaks (see the inset) in β abs (ω p ) occur at τ d = 200 ps and are replaced by a strong single absorption peak due to a suppressed photon dressing effect and phase-space blocking. On the other hand, from the right panel of the same figure, we see that the initial central peak in P em (ω) is increased very rapidly due to accumulation of photo-excited e-h pairs and accompanied by the reduction of two side peaks resulting from the weakened Rabi oscillations. Importantly, the very-strong central peak in P em (ω) is significantly reduced at τ d = 200 ps, indicating the start of a radiative-recombination process for photo-excited e-h plasmas. This recombination process is continuously enhanced with the increasing delay time τ d and suppresses the central peak in P em (ω) after τ d ≥ 400 ps due to draining out the photo-generated electrons and holes at the same time.

IV. CONCLUSIONS AND REMARKS
In conclusion, we have demonstrated the possibility of using a SPP field to control the optical gain and absorption of another passing light beam due to their strong nonlinear field coupling mediated by electrons in the quantum dot. We have also predicted the coherent conversion of a surface-plasmon-field photon to a spontaneously-emitted free-space photon, which is simultaneously accompanied by another pair of blue-and red-shifted photons.
Although we studied only the coupling of a SPP field to a single quantum dot in this paper for the simplest case, our formalism can be generalized easily to include many quantum dots. We have employed a box-type potential with hard walls for a quantum dot, which is given by where the position vector r = (x 1 , x 2 , x 3 ), L 1 , L 2 and L 3 are the widths of the potential in the x 1 , x 2 and x 3 directions, respectively. The Schrödinger equation for a single electron or hole in a quantum dot is written as where the effective mass m * is m * e for electrons or m * h for holes. The eigenstate wave-function associated with Eq. (A2) is found to be ψ n 1 ,n 2 ,n 3 (r) = 2 L 1 sin which is same for both electrons and holes, and the eigenstate energy associated with Eq. (A2) is where the quantum numbers n 1 , n 2 , n 3 = 1, 2, · · ·.
Based on the calculated wave-functions in Eq. (A3), the form factors introduced in Eqs. (11) and (12) can be obtained from F e n 1 ,n 2 ,n 3 ; n ′ 1 ,n ′ 2 ,n ′ where the wave vector q = (q 1 , q 2 , q 3 ) and we have defined the following notation for j = 1, 2, 3 Moreover,the overlap of the electron and hole wave-functions in this model can be easily calculated as The interband dipole moment d c,v = d c,vêd at the isotropic Γ-point, which is defined in Eq. (32), can be calculated according to the Kane approximation [35,36] Furthermore, the direction of the dipole momentê d is determined by the quantum-dot energy levels in resonance with the photon energyhω. in the adjacent quantum dot. As a result, the optical-polarization field of the photo-excited e-h plasma is strongly coupled to the propagating SPP field to form split plasma-SPP modes with an anticrossing gap. Also, a probe-field is used for studying the photon dressing effect.         for zero SPP-field detuning indicates the result is multiplied by a factor of 100.     Here, E sp = 500 kV/cm is taken, and the other parameters are the same as those in Fig. 3