{'fraction_non_alphanumeric': 0.04279749478079332, 'fraction_numerical': 0.01103489412466448, 'mean_word_length': 4.626677852348993, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 1, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Graphene Spaser Description by Rate Equations

In this paper a surface plasmon polariton laser (spaser), which generates surface plasmons in graphene nanoflake, is considered. The peculiarities of spaser, such as strong material dispersion, require revision of basic laser equations. We provide a full derivation of equations of the spaser dynamics starting from the Maxwell-Bloch equations. Optical Bloch equations and rate equations are obtained and the relation of the equation parameters through the physical ones is given. In the case of graphene realization, the numerical parameter values are estimated.

. In general, the difference between the SPP generator ("SPP laser") and spaser is vague, so that they are often identified [45]. On the other hand, these two devices can be considered as opposed limiting cases, respectively, of large SPP cavity and small (nanoscopic) system. Beside a rich perspective in applications, plasmonic generators are interesting by themselves as pioneering devices of quantum plasmonics [46,47].
One of the most promising for plasmonics is graphene. [48][49][50][51][52][53] Graphene is an extremely thin 2D material [54,55], which has high carrier mobility [56]. This material supports plasmons, and control of electron Fermi level achievable by doping, for example, by the use of a gate electrode [57], makes it applicable from optical to THz region. The latter frequency region is of high interest as it contains vibrational transitions of molecules. The use of graphene opens up opportunities to create highly sensitive compact THz devices. In near-IR graphene plasmons, the localization factor reaches higher values while losses are lower than in metal plasmons [58]. In spite of the relatively low absorption in graphene, the loss limit of the plasmon mean free path is still a major obstacle for graphene plasmonics applications. As a result, active graphene plasmonics began to develop [59]. In particular, graphene plasmon generators based on the use of a gain medium have been proposed [60][61][62].
In the present paper, the spaser dynamics equations in the single-mode approximation are derived, and the material dispersion is taken into account. In the case of the graphene nanoflake spaser, the numerical estimation of the equation parameters is given.

II. Optical Bloch equations for the graphene spaser
Let us consider a spaser, which consists of a gain medium (quantum dot) and a surface plasmon cavity. This cavity can be realized as a graphene nanoflake. Gain medium has larger coupling to the plasmonic modes than to propagating electromagnetic modes. Therefore, only nonradiative emission of plasmons will be taken into account.
The gain medium is modeled as a two-level system. In this case, the equations for the density matrix elements are transformed to those for the gain medium dipole moment  and population inversion N . Addition of the Maxwell equation for the electric field  gives the Maxwell-Bloch equations [63][64][65]: The latter two variables are associated with a single particle of the gain medium, which are distributed with spatial density   C r . Angular brackets with the subscript r , ... r , mean averaging over gain particles, located in a physically infinitesimal volume at the point r .
Asterisk means complex conjugation.
Let us assume a case of single-mode spasing regime. In this case, we will describe the field distribution by that in the lasing mode,   E r . Furthermore, let us use a slow amplitude approximation using the gain medium transition frequency 0  as a reference. With these assumptions, the field is written as is a slow amplitude. The spatial field profile is determined by the following equation:  is the permittivity distribution at the transition frequency 0  . The expression in square brackets in (6) can be expanded as i t Therefore, Eq. (6) transforms to the following form:

 
Then, Eq. (8) is integrated over the volume. The expression in curly brackets transforms in the following manner:

 
The latter expression describes the radiation loss, which will be disregarded in comparison with the absorption loss in graphene. Thus, Eq. (8) assumes the form Here, the notation is introduced, the imaginary part of the factor   2 /     is neglected, and the time derivative is resolved as 2 Finally, we introduce the field amplitude     e t e t W   , which does not depend on the normalization of the coordinate function   E r : Introducing the field relaxation time T as and the slow amplitude of the gain medium polarization as one writes the field equation: To obtain an equation for   p t , one should average both parts of Eq. (2) over a physically infinitesimal volume ( ... r operation), then scalarly multiply them by and integrate over the volume. This gives In this approximation, introducing the interaction parameter  by the relation we obtain the equation for the gain medium polarization: According to the above definition, To summarize, the laser dynamics in the single-mode approximation is described by Eqs. (14), (18), (19): with the parameters given by Eqs. (12), (17).
Let us note that a single particle population inversion cannot exceed a unity in the absolute value. Thus, the total population inversion is N D N    .

III. Rate equations
Despite the plasmonic generator based on noble metal, the one based on graphene can possess so small loss, that the relation between time parameters may be the same as in the  (20) and (22) . The equations for n and D are written as

IV. Numerical estimation of the parameters  , 
According to the above derivation, the parameters in Eqs. (26), (27) are defined as

 
In the structure under study,  is equal to the permittivity of the substrate at In the assumption of tangential direction of the gain medium dipole moment, the latter expression transforms to where D  is permittivity of the dielectric substrate. Let  and D  be the imaginary parts of the normal components of surface plasmon wave numbers in vacuum and dielectric, so that . This leads to the following expression for the integral in denominators in (28), (29): Finally, let us note the relation between the tangential and normal electric field components: The expressions for the constants transform to

Conclusions
In conclusion, we have obtained the expressions for the parameters, which are needed to describe the spaser dynamics. The numerical values of these parameters in the case of graphene spaser are estimated.
In this Appendix we provide a derivation of the relation with   e t being a slow amplitude.
A causal time operator  with the kernel   G  is defined as which gives the required relation.