#### 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

We assume that the characteristic size of the interaction region of the effective field of SPP and chromophores

For a metal-dielectric boundary, the relation

is valid, where the parameters

We assume that the pumping volume

where

The parameter

where

The parameter

The dispersive and dissipative corrections

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 (

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

where

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

We use the known dependence [17] of the

where

For chosen model parameters and the QD concentration

However, taking (1a) into account, the choice of the appropriate dielectric host-medium can partially or completely compensate the increase of

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

To analyze the contribution of dissipative effects related to the imaginary part

and the effective field

formed in it.

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

where

By passing to new dimensionless variables

The solution of system (8) can be written in the form

By substituting the expression for

where the amplitude of the decay coefficient is defined as

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

when the central part of SPP pulse is formed, the enhancement of pendulum oscillations is observed;

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 (

The initial conditions in simulation of (10) are chosen equal to

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

#### 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.