KnE Energy | VII International Conference on Photonics and Information Optics (PhIO) | pages: 204–214

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1. Introduction

Collective processes in a system of quantum emitters for a long time remain a subject of intensive studying [1] with both theoretical and experimental points of view. New opportunities of the well-known cooperative effects in optics can be associated with the collective behavior of the plasmonic oscillators pumped by a near-field of excited chromophores (semiconductor quantum dots, dye molecules, etc. [2]). The kinematics of individual localized systems ”quantum dot+metal nanoparticle” [3], the core-shell nanocrystals [4] is well described in the framework of spaser theory [2]. However, generated plasmons in such systems are strongly localized and their collective dynamic is restricted to the near-field area of the plasmonic nanoparticles [5]. Suitable interfaces for observing collective processes with surface plasmon-polariton (SPP) can be planar metal/dielectric waveguides [6] which were already implemented in practice [7]. One of approaches to solving the problem of fast damping of plasmons in such systems is connected with use of photonic crystals as a dielectric layer [8]. In this case, the long-range SPPs with a maximum energy of the field in the dielectric region are formed. On the other hand, compensation of damping of plasmons in metal can be realized in the model of the dissipative waveguide spaser with a near-field pumping from the chromophores placed in the dielectric layer near a metal surface.

In this work the approach to choosing specific chromophores and the dielectric host medium to increase the energy transmission efficiency of collective excitations of chromophores to SPPs modes in metal/dielectric waveguide is proposed. Considering that the refractive index of the dielectric host medium is a complex value, we have defined such conditions when the spontaneous emission rate of the chromophores near the metal-dielectric interface [9], as well as the collective optical processes with quantum emitters [10,11] are almost completely suppressed by the influence of the dielectric environment. Using model of the waveguide spaser the selfconsistent system of the equations describing dynamics of excitons and propagated SPP pulses was obtained. It is shown that in the mean-field approximation the self-consistent problem can be reduced to a modified pendulum equation with an additional term of nonlinear losses. A separatrix solution of the nonlinear equation, which corresponds to the formation of the single SPP pulse in waveguide spaser model, was obtained. A model of the waveguide spaser with an ensemble of CdS quantum dots placed in the dielectric layer near the metal surface for the realization of the predicted effects was proposed.

2. Master equation for collective process of SPP generation in wavegide spaser

Consider the model of an interface in Fig. 1a in the form of a metal/dielectric waveguide [12] with two-level chromophores located inside a thin dielectric layer, the transition frequency between the two levels ωa=2πc/λa being resonant with the SPP frequency ωSPP=2πc/λSPP (1-ground and 2-upper levels, respectively). By selecting a dielectric medium with appropriate dispersion characteristics and providing the initial excitation (inversion) of a dense ensemble of chromophores in this model, it is possible to produce the collective decay of excitons.

Figure 1

(a) Formation scheme of SPP pulses in a layered (planar) metal/dielectric waveguide pumped by CdS QDs; (b) dependence of the transition energy on the CdS QD size ( Eg=2.42eV at 0 K for a bulk); (c) parametric plane of the complex refractive index n=nR+inI of a dielectric medium with separatrices Γε=0 for the effective rate of radiative losses of QDs in this medium.

fig-1.jpg

We assume that the characteristic size of the interaction region of the effective field of SPP and chromophores h=Lx=Ly=Lz satisfies the inequality h<<λp and the inequality Lzld is also valid, where ld is the SPP decay length along the z axis. The corresponding Rabi frequency can be written in the form Ω=-Aφμ12ε/ , where A=S/ε0εdSω , ε=Np is the SPP amplitude, Np is the number of SPP modes in the interaction region, μ12 is the transition dipole moment in a chromophore, and φ is the scalar potential of the plasmon field linearly decreasing with distance from the surface, is the Planck's constant. In the case of excitation of a mode of the plasmon field at frequency ω , using the normalization φ2dV=1 [13], the expression for the Rabi frequency can be approximated by the function

Ω=μ12Snεdε0VSnωε=gε.

For a metal-dielectric boundary, the relation

λSPP=Re(εm)+ Re (εd)Re(εm)Re(εd)·λ0

is valid, where the parameters εd and εmω¯=1-ωp2/ω¯2+iγsω¯ are the dielectric permittivities of dielectric (with QD) and metal, respectively. Here, ωp=4πnme2/m0 is the plasma frequency in a metal, m0 and nm are the electron mass and concentration, respectively, γs is the electron collision frequency in the metal, ω¯=2πc/λ0 . We chose the wavelength λ0=387nm for our simulation. The spectral properties of the metal-dielectric interface can be described by use of the Bergman's parameter Sω=Reεd/(εd-εmω) [2].

We assume that the pumping volume V' is a dielectric containing QDs with the characteristic radius a and concentration N>>1021m-3 . Assuming that the refractive index n=nR+inI of the dielectric environment of QDs is a complex quantity, where n=εd and εd is the complex permittivity, expressions for the radiative relaxation rate Γa , the Rabi frequency Ω , and the effective frequency detuning Δa can be written in the form [14]

Γε=ΓanRlRnIlI+2δaΓalI,
Ω0=Ω·lR2+lI2,
Δε=δalRΓa2δanIlR+nRlI+Δa,

where ln=lR+ilI is a complex function for which lR=nR2-nI2/3 , lI=2nRnI/3 ; and δa is a small correction caused by the Lamb shift. It is assumed here that the function ln=El/EM coupling the Lorentz local El and Maxwell EM fields will retain its structure in the case of the near field through which SPP are excited in the scheme in Fig. 1.

The parameter Γa*=1/τR+1/τF is the total rate of radiative (with the time τR=1/Γa ) and nonradiative (with the time τF ) losses for QDs in vacuum. In the semiclassical approximation, the system can be described similarly to the “metal nanoparticle in a dielectric with chromophores” spaser model [2] with the help of equations for elements of the density matrix ρ of a two-level chromophore:

ρ˙12=iΔε+Γε2ρ12+iΩ0*+iξ0uRρ12+ξ0uIρ12n21,
n˙21=2iΩ0ρ12Ω0*ρ214ξ0uIρ122Γε1+n21,

where Δa=2πc1/λa-1/λSPP , n21=ρ22-ρ11 . The Rabi frequency can be written as Ω0=gε·lR2+lI2 , where g=μ12Sn/εdε0VSnω is the coupling constant and ε=Apεdε0VSnω/Sn is the normalized field with the amplitude Ap of the total field produced by the perturbed electron density in a metal and the electromagnetic field component in a dielectric.

The parameter ξ0=Nμ122/3ε0 in (2) determines the addition to the Rabi frequency appearing due to transition from the Maxwell EM to the local field El [14] acting on a chromophore.

The dispersive and dissipative corrections

uR=lRεR+lIεI/εR2+εI2,
uI=lIεRlRεI/εR2+εI2,

respectively, are expressed in terms of the real and imaginary parts of the permittivity of the host-medium [14] in which QDs are placed and have the physical meaning of the additional frequency modulation and the effects of absorption ( uI<0 ) or amplification ( uI>0 ) due to the local field (Fig. 2).

Figure 2

(a) Profiles of SPP pulse amplitude squared obtained by the numerical simulation of system (2)–(3) in the following regimes: (a) neglecting the dissipation parameters Γε=Γa=γp=0s-1 ; (b) the dissipative regime with Γε=Γa=6.3×1011s-1 , γp=4.1×1013s-1 ; (c) the regime of QD radiative decay suppression Γε=0 in the host medium with nR=1.6 and nI=1.23 ; (d) dynamics of the angle θ (solid curve) and coefficients cosθ (dotted curve) and -cosθ (dashed curve) for regime (a). The interaction parameters are g=8.1×1011s-1 , ξ0=6.3×109s-1 . The initial polarization of the medium is ρ12(0)=iθ0=i/Na=0.14i , the normalization parameter is Λ=5.7×1012s-1 for Na=50 .

fig-2.jpg

The equation of motion for the Rabi frequency of SPP, which in the case of the exact plasmon resonance has the form

Ω˙0=itR2ρ12γpΩ0,

where

tR=1gNa=εdε0SnωSnμ122N

determines the characteristic formation time for quantum correlations in Fig. 1a (compare with the optical problem [15] when emitters are located in the field formation region).

Note that the plasmon mode decay rate γp=1/τJ+1/τR is high and determined by the characteristic times τR and τJ of radiative and “joule” losses, respectively. Under conditions 1/τJ30/τR [16], radiative losses can be neglected, while “joule” losses are determined by the collision frequency in a metal, i.e., γpγs , and in problem (3) in the absence of pump, the short-range SPP appear. The self-consistent problem (2)–(3) will be valid only under conditions when the characteristic establishment time tR for correlations between QDs proves to be considerably shorter than τJ . Because tR is inversely proportional to the dipole moment of a chromophore, the relation tR<τJ can be valid for pumping a distributed waveguide spaser by QDs with their giant dipole transition moments.

We use the known dependence [17] of the 1Se1Sh transition energy on the QD diameter DQD=2a (Fig. 1b) for regime of strong confinement

E1S(e)1S(h)=Eg+22π2DQD21me+1mh,

where e is the electron charge, me and mh are the effective electron and hole masses, respectively, in the volume of the QD material with the permittivity εqd and band gap energy Eg [18,19]. The corresponding parameters for CdS are me=0.19m0 , mh=0.8m0 and εqd=9 [20], which gives DQD=1.56nm . Bohr radius of exciton Rex for CdS is 2.5nm [21] therefore strong confinement regime [22] will be observed for the considered QDs, and energy sublevels of conductivity zone will be essentially separated. The dipole moment of the corresponding interband transition in QDs is assumed equal to μ=μ12=5×10-29C·m [2].

For chosen model parameters and the QD concentration N=7×1021m-3 , the characteristic correlation time is tR=124fs and the delay time tD=414fs for the number of chromophores in the interaction region Na=50 (here we assume that Na=Np ). The duration of a formed SPP monopulse is only about 200fs for ideal conditions Γε=Γa=γp=0s-1 in Fig. 2a. For dissipative regime in Fig. 2b we take into account the rate of decay Γε=6.3×1011s-1 for QD near the metal surface [23] and the rate of decay γp=4.1×1013s-1 for plasmon. In this case we also find the possibility for the formation of SPP pulses, since the characteristic time tR is shorter than the characteristic decay times in the system.

However, taking (1a) into account, the choice of the appropriate dielectric host-medium can partially or completely compensate the increase of Γa (Fig. 1c), but it is also obvious that the properties of natural media are strongly restricted. Thus, for silica at the wavelength under study λSPP=192nm , we have nR=1.6 , nI=5×10-7 [24] and Γε=2.43Γa . To completely compensate relaxation processes in (2a) ( Γε0 ), the required combination of dispersion-dissipative parameters should satisfy the condition nRlR-nIlI=0 (be neglecting a small Lamb shift), which is satisfied, for example, for the choice nR=1.6 and nI=1.23 . Such conditions can be fulfilled for an artificial microstructured dielectric material with specified dispersion– dissipative characteristics (the Cole-Cole diagram). They lead to the significant increase in the SPP pulse intensity, while energy transfer from chromophores to radiation proves to be suppressed (see Fig. 2c). In this case, the influence of the local field increases, the absolute values of its parameters increase (corrections uR=0.37 and uI=-0.158 in (2)) and the formation dynamics of SPP pulses changes.

3. Collective dynamics of a waveguide spaser in the mean field approximation

To analyze the contribution of dissipative effects related to the imaginary part uI of the local field correction, we can neglect the corresponding phase effects with uR in (2) and decay in (2)–(3) and to pass in the mean field approximation to a simplified system of self-consistent equations for a medium

ρ˙12=iΩ0*+ξ0uIρ12n21,
n˙21=2iΩ0ρ12Ω0*ρ214ξ0uIρ122

and the effective field

Ω˙0=ig2Naρ12

formed in it.

By passing to the representation for the Rabi frequency and polarization in the form

Ω0=12UeiK0t+U*eiK0t,ρ12=12R·eiK0

where K0=ωSPPt-kSPPz , and assuming that Z=n21 , we can obtain the system of Maxwell-Bloch equations for a spaser taking into account the (dissipative) local response of the QD environment

Z˙=i2UR*U*Rξ0uIR2,
R˙=iUiξ0uIRZ,
U˙=ig2NaR.

By passing to new dimensionless variables δ0=-iUΛ and τ=t·Λ , where Λ=gNa and setting R*=R and δ*=δ , we represent system (7) in the form

Zτ=δ0Rξ0uIΛR2,
Rτ=δ0Z+ξ0uIΛRZ,
δ0τ=R.

The solution of system (8) can be written in the form Z=cosθ and R=sinθ , where θ determines the angle of the so-called Bloch vector with coordinates Z and R and their substitution to (8) gives the equation for the angle

θ˙=δ0+ξ0uIΛsinθ.

By substituting the expression for δ0 from (9) into (8c), we obtain a new variant of the pendulum equation with the nonlinear harmonic losses/decay term

θ¨Kcosθ·θ˙=sinθ,

where the amplitude of the decay coefficient is defined as K=ξ0uI/Λ . In the absence of the loss modulation, when Kcosθ·θ˙=K·θ˙ , Eq. (10) is reduced to the usual nonlinear pendulum equation with losses [25]. Taking the modulation into account under the same conditions K<0 ( εI>0 and uI<0 ), the pendulum experiences the additional decay in intervals

θ0+2πm;π2+2πm,θ3π2+2πm;2π+2πm

responsible for the formation of the leading and trailing edges of SPP pulse (see Fig. 2c), whereas in the interval

θπ2+2πm;3π2+2πm

when the central part of SPP pulse is formed, the enhancement of pendulum oscillations is observed; m=0,1,2... .

In other words, the absorbing dielectric host-medium coherently preserves a part of the QD energy during the formation of the leading edge of the pulse and then coherently returns this energy to SPP pulse during formation of the pulse peak. As a result, taking into account the compensation of the spontaneous relaxation rate of QDs ( Γε=0 ) and nonlinear terms with uI in (5), the increase in the peak pulse intensity is observed with respect to the case when the response of the host-medium is neglected (see Figs. 2a,b).

The initial conditions in simulation of (10) are chosen equal to θ0=1/Na for the initial oscillation angle and

υθ=θtt=0=2coshlnθ04

for the initial velocity of the pendulum.

Equation (10) is a particular case of the Lienard equation and its approximate analytic solution can be expressed in terms of elliptic integrals of the first kind. The numerical solution for the Rabi frequency of SPP pulse field obtained from (10) completely coincides with the results of the direct numerical simulation of system (5)–(6) under conditions of the suppression of spontaneous relaxation in QDs for the chosen values nR=1.6 and nI=1.23 ( K=-0.0147 ) (see Fig. 2a).

4. Conclusions

We have proposed efficient method for the formation of short SPP pulses at the dielectric-metal interface containing QDs. The general conditions for selecting parameters of QDs and a dielectric host-medium are determined which provide the maximal collective energy transfer from a QD ensemble to SPP modes dominating over the radiative relaxation of individual chromophores. To tune the system parameters to the plasmon resonance more accurately, it is useful to employ experimental absorption and fluorescence spectra of giant ensembles of emitters [26,27] in different host medium. The models presented in the paper can be useful for practical implementation of multiqubits entanglements [28] and quantum computations in macroscopic and mesoscopic [29] systems. However, for the realization of the external control in such systems one additionally requires the use of multiwave schemes [30,31] of nonlinear coherent interaction by analogy with optics [32,33]. Further development of our research is related to the investigation of collective spin effects based on the photon echo [34] in plasmonic structures, as well as the possibilities of control such effects [35].

The work was supported by RFBR (17-42-330029) and the Ministry of Education and Science of the Russian Federation in the framework of the state task VlSU 2017 in the field of scientific research.

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