Quantum theory of plasmon

Since very early works on plasma oscillations in solids, it was known that in collective excitations (fluctuations of the charge density) of the electron gas there exists the resonance appearing as a quasiparticle of a special type called the plasmon. The elaboration of the quantum theory of plasmon in the framework of the canonical formalism is the purpose of the present work. We start from the establishment of the Lagrangian of the system of itinerant electrons in metal and the definition of the generalized coordinates and velocities of this system. Then we determine the expression of the Hamiltonian and perform the quantization procedure in the canonical formalism. By means of this rigorous method we can derive the expressions of the Hamiltonians of the interactions of plasmon with photon and all quasiparticles in solid from the first principles.

In many of the above-mentioned fundamental research works on plasmonic processes there was the need to use the Hamiltonians of the interactions between plasmon and other elementary excitations in matter. All those interaction Hamiltonians were introduced in a phenomenological manner. In order to exactly derive the Hamiltonians of the interactions of plasmon from first principles it is necessary to elaborate a rigorous procedure for quantizing the collective excitations of electron gas. This task will be performed in the present work. The main ideas were outlined in our previous publication [30].

Lagrangian of the system of itinerant electrons in metal
Consider a simple model of metal consisting of a gas of itinerant electrons freely moving inside metal over a background of ions with homogeneous distribution of the positive charge. Denote n t r ( , ) the electron density (number of electrons per unit volume) and n 0 its mean value (averaged over space and time). The average charge density −en 0 of electron, −e being the electron charge, compensates the average positive charge of the ions in the background, and the fluctuating charge density in the metal is According to the Coulomb law the effective charge distribution (1) creates a time-dependent electrical field with the potential From formula (2) there follows the Poisson equation 2 The mutual interaction between effective charge densities at two different regions in the space gives rise to the potential energy of the electron gas which can be also written in the form As a consequence of the oscillating displacements of electrons, the fluctuation of the electron density n t r ( , ) generates the total kinetic energy of the electron gas. Denote δ t r r ( , ) the displacement of the electron having the coordinate r at the time moment t, and m the electron mass. Since the electron has the velocity the whole electron gas has following total kinetic energy ∫ δ= T t m d n t t r r r r ( ) 1 2 ( , ) ( , ) .
2 Thus we have derived following expression of the Lagrangian of the electron gas r r r r r r r r r r r r r r r r r r r r r r r The dynamical variables δ t r r ( , ) and ρ t r ( , ) cannot be completely independent, and we must establish the relationship between these physical quantities. We note that since oscillating displacements of electrons cause the fluctuations of the electron number n t r ( , ), there must exist some direct relationship between n t r ( , ) and δ t r r ( , ). For establishing this relationship we consider any finite volume Ω bounded by a closed surface Σ in the spatial region of the electron gas. In comparison with the average electron number density n 0 needed for compensating the positive charge of the ions in the lattice of the metal, the number of excess electrons in the volume Ω is 0 Due to the fluctuation of n t r ( , ), this number changes with the time and its increment during a very short time On the other side, the fluctuation of n t r ( , ) is caused by the oscillating displacements of electrons. Denote δ Σ N t ( ) the number of electrons displacing across the boundary Σ of the volume Ω and leaving this volume, i.e. moving from the inside of the closed surface Σ to its outside during the same time interval (t, t + δt). We have By means of the Ostrogradski-Gauss theorem we transform the surface integral over Σ in the r.h.s. of formula (13) into a volume integral over Ω and obtain Because the number δ Σ N t ( ) of electrons leaving the volume Ω across its surface Σ must be equal to the decrement δ − Ω N t ( ) of the number of electrons contained in the volume Ω 0 Thus the dynamical variables δ t r r ( , ) and ρ t r ( , ) in the Lagrangian (9) are not independent. They must satisfy the subsidiary condition (16).
The appearance of the small effective charge density ρ t r ( , ) in comparison with the average charge density (having the absolute value en 0 ) of the electron gas is the consequence of the very small oscillating displacements δ t r r ( , ) of the electrons. In the Lagrangian (9) and the subsidiary condition (16) they are two very small quantities of the same order. Consider Lagrangian (9) and the subsidiary condition (16) in the lowest order with respect to these very small quantities. Then the Lagrangian has the following approximate expression

Canonical formalism in the harmonic approximation
Now we establish the canonical formalism of classical mechanics and then apply the quantization procedure to the study of the electron gas with the charge density ρ t r ( , ) and the electron displacement vectors δ t r r ( , ) being the solution of the system of differential equations consisting of Lagrange equations with the approximate Lagrangian (17) and the subsidiary condition (18). For this purpose we decompose functions ρ t r ( , ) and δ t r r ( , ) into the Fourier series of plane waves normalized in a cube with the volume V and orthogonalized by means of the periodic boundary conditions: Since ρ t r ( , ) and δ t r r ( , ) are the real quantities, their Fourier transforms must satisfy conditions Substituting the Fourier series (19) and (20) into the r.h.s. of formula (17), we obtain in the r.h.s. of formula (27) can be discarded. The absence of these constants in the Lagrangian has the following clear physical meaning: only the longitudinal displacements of electrons can cause the wave of propagating fluctuations of electron number density n t r ( , ), i.e. of charge density, in the electron gas. Therefore instead of the expansion formula (20) we shall use following expression i k k kr In this case the Lagrangian (26) becomes   The quantization procedure consists of the replacement of dynamical canonical variables q t ( ) ( )  Hamiltonian of the quantized system is the operator Instead of two hermitian operatorsq ( )