Quantum Concepts and Techniques in Classical Domains Demonstrated in Bulk Phonons and Plasmons

Abstract

The turning point that sparked the initiation of quantum theory was the Planck–Einstein postulate that the energy of a monochromatic radiation field is quantized in terms of photons, and this was followed by the development of the principles of quantum mechanics. Although some conceptual issues remain to be resolved, quantum mechanics is regarded as a well-established discipline which may lead to the unraveling of the nature of matter in general. Today, the influence of quantum mechanics is evident in its applications, with remarkable technological advances involving diverse aspects of the physical world. What appears to need particular attention, however, (after a hundred years have elapsed since the birth of quantum mechanics) is the impact that the concept of the ‘quantum’ has had beyond traditional quantum mechanics. The paper describes how the ‘quantum’ concept has influenced and continues to influence developments in physical systems, which are essentially classical, in that they are basically governed, entirely, or in part, by non-quantum laws, but in which, the physics is distinguished by its own special quantum—the photon analogue. The paper illustrates this by considering, as prototype examples, bulk plasmons and phonons. The study outlines the systematic quantization of plasmons and phonons, both of the polariton (transverse) forms and their longitudinal forms, and discusseshow these interact with quantum systems such as electrons, atoms, and condensed matter. It is demonstarted using one case, namely, involving longitudinal plasmons, how utilizing quantum concepts and techniques facilitate their interaction with matter, as in electron energy loss spectroscopy.

1. Introduction

Condensed matter physics is one of the major areas where quantum concepts and methodology are used, not just, necessarily, from the point of view of determining the microscopic quantum properties of the condensed matter itself [1,2] but also in order to deal with situations that focus on implementing the instructive concept of the collective excitations [3] or the quanta—the photon analogues involved in classical contexts [4,5]. The quantum (or elementary excitation) concept [3], can fortunately be generalized to encompass a large number of cases and has proved to be particularly helpful for the study of a number of essential features of condensed matter physics. The main point here is essentially to elucidate this point and to highlight the role that quantum concepts and techniques have played in classical domains.

In most cases, the basic theory relies on assumptions involving the existence of well-defined quantizable fields. Like non-relativistic perturbative quantum electrodynamics and quantum optics [6,7], problems involving condensed-matter quanta are concerned with their exchange via processes of absorption and emission that contribute non-trivially to the macroscopic and quantum properties of condensed matter [1]. They also feature whenever external probes in real experiments involving beams of various species (for example electrons, photons, neutrons, and ions) are used to study the elementary excitations of the system [8]. Such problems and many others are especially well suited for the rigorous application of perturbation techniques in conjunction with quantization methods of the fields in condensed matter. This framework is known to have exhibited many distinct successes, but this appears to be unfamiliar to scholars concerned primarily with traditional quantum mechanics.

In the context of condensed matter physics, there are numerous examples of elementary excitations [4]. The best known are the phonons, magnons, plasmons, helicons, excitons, polaritons, and other hybrid modes of various kinds. One cannot elaborate on the full set and consider it sufficient to focus on plasmons and phonons as illustrative examples to discuss in some detail.

A collective excitation in a condensed matter system can be classified as pure, involving oscillations of one kind, or a hybrid between two or more strongly coupled oscillations [9]. The phonon polariton quanta [10], for example, are excitations associated with a coupled phonon–photon field described as an electromagnetic wave with lattice vibrations propagating in a medium, forming a well-defined kind of elementary excitation. Another example of hybrid modes are the phonon–plasmon polaritons which involve phonons and plasmons coupled to light [11].

A typical quantizable field is characterized by a field equation, most commonly second order in space and time. The field equation could have arisen directly, as in the case of the electromagnetic field in free space, or has emerged as a result of a linearization step for some coupled equations imposed, for example, by the continuum or long-wavelength approximation. A comparably large number of problems fall in this category, but as stated above, we only consider in some detail phonons and plasmons. The field equation normally emerges from an appropriate Lagrangian by application of the known Euler–Lagrange equation. The corresponding Hamiltonian can then be defined, and its quantization can be carried out using standard quantization methods. The appearance of a Hamiltonian of the system is indicative of the presence of an energy–momentum tensor and associated conservation laws involving continuity of energy–momentum flow.

The general plan of this article is as follows. Section 2 discusses the basic concepts leading to the emergence of the quantum of the elementary excitation of a general field in unbounded space. Then, as an illustration, the simplest one-dimensional case (namely the elastic line) is considered, which is characterized by a classical scalar field. Section 3 focuses on classical fluids as a more complex system in three dimensions, where quantum concepts of such a classical field are not normally explored. Section 4 is concerned with plasmons as a real system, where mechanical motion of charge carriers in three dimension are governed by Newton’s laws when coupled to electromagnetic fields. This system supports elementary excitations in the form of plasmons, both transverse and longitudinal, and the discussion proceeds to their quantization. Section 5 discusses the general case of field materials with well-defined dielectric functions. The specific materials in this case involve the coupling of lattice vibrations to the electromagnetic fields leading to phonons both (transverse) phonon polaritons and (longitudinal) acoustic phonons, and the calculation proceeds to quantize the corresponding fields. Section 6 deals in general terms with the interactions of the quantized fields with quantum systems, and as an application, the theory of the scattering of electrons from quantized bulk plasma modes is outlined, leading to a description of electron energy loss spectroscopy of electronically-dense metals. Section 7 summarizes the essential points highlighted in this paper with a general discussion about the contexts where the quanta of plasmons and phonons have featured in practical applications involving real materials.

Note that the choice of plasmons and phonons to discuss elementary excitations makes no claim regarding thorough coverage of these two separate sub-fields in condensed matter. They just happen to constitute obvious representatives where quantzsation has proved to be practical, not just from a procedural point of view, but for real evaluations, as in the context of longitudinal plasmons in electron energy loss spectroscopy (EELS).

2. Free Continuous Fields

The concept of a field is indeed a classical one. Defining a field in a given region of space amounts to associating with every point of the given region a property, which can be a scalar (in which case, it is said one has a scalar field) or a vector (vector field), and, more generally, one may be concerned with a tensor field. The fields one is concerned with here are functions of position r $\equiv (x,y,z)$, as well as time t. The version of the action principle for this case is obtained in terms of a total Lagrangian L and a Lagrangian density $\mathcal{L}$ such that

$$L=\int d^{3}r\mathcal{L}(\left\{{\varphi }_i\right\},\left\{{\dot{\varphi }}_i\right\},\{\mathsf{\nabla }{\varphi }_i\},t).$$

(1)

One should, therefore, talk about functionals, i.e., functions of the fields ${\varphi }_i$, their spatial derivatives $\mathsf{\nabla }{\varphi }_i,$ and velocities, i.e., time derivatives ${\dot{\varphi }}_i$. The symbol $\left\{{\varphi }_i\right\}$ designates a set of scalar functions either individually or in components forming, for example, vector fields. Here, and henceforth, the dot (for example, ${\dot{\varphi }}_i$) indicates a single time derivative.

The action integral, denoted I, and its variation $\delta I$ are as follows

$$I={\int }_{t_1}^{t_2}Ldt,\delta I=0,$$

(2)

but the variational calculus with $\mathcal{L}$ as a functional leads to the Euler–Lagrange equations, with $i,j=(x,y,z)=(1,2,3)$:

$${\displaystyle \frac{\partial }{\partial t}}\left({\displaystyle \frac{\partial \mathcal{L}}{\partial {\dot{\varphi }}_i}}\right)+\sum _{j=1}^3{\nabla }_j{\displaystyle \frac{\partial \mathcal{L}}{\partial \left({\nabla }_j{\varphi }_i\right)}}-{\displaystyle \frac{\partial \mathcal{L}}{\partial {\varphi }_i}}=0.$$

(3)

A dynamical system characterized by a field ${\varphi }_i$ minimizes the action integral, provided it satisfies the Euler–Lagrange Equation (3), and conversely, a field system with a given Lagrangian density satisfying the Euler–Lagrange equation automatically conforms with the action principle.

The formalism needed to define a given field ${\varphi }_i$ then involves a definite Lagrangian density, which enables the canonical momentum ${\Pi }_i$ conjugate to the field ${\varphi }_i$ to be derived. One has

$${\Pi }_i={\displaystyle \frac{\partial \mathcal{L}}{\partial {\dot{\varphi }}_i}}.$$

(4)

The Hamiltonian density follows by evaluating the expression

$$\mathcal{H}=\sum _i{\Pi }_i{\dot{\varphi }}_i-\mathcal{L}.$$

(5)

The final stage in the field quantization procedure involves the imposition of the commutation relation between ${\varphi }_i$ and its canonically conjugate momentum ${\Pi }_i$ at equal times t. The role of ${\varphi }_i$ in field theory can be seen by analogy with that played by the coordinate $q_i$ in particle dynamics. However here, ${\varphi }_i$ (in view of its continuous dependence on both space and time) pertains to a system that possesses an infinite number of degrees of freedom. One needs a commutation relation that is the analogue of $[p_i,q_j]=-i\hslash {\delta }_{ij}$, with ℏ the reduced Planck constant and ${\delta }_{ij}$ the Kronecker delta, and to derive this let us begin by dividing space into discrete elements each of volume ${\left(\delta \tau \right)}_n$, and assuming that ${\varphi }_n$ is the value of $\varphi$ at element n positioned at point $\mathbf{r}$ at time t. Let the corresponding Lagrangian density at the element be ${\mathcal{L}}_n$. Thus, one can write for the corresponding momentum canonically conjugate to ${\varphi }_n$

$$p_n={\displaystyle \frac{\partial L}{\partial {\dot{\varphi }}_n}}={\left(\delta \tau \right)}_n{\displaystyle \frac{\partial {\mathcal{L}}_n}{\partial {\dot{\varphi }}_n}}={\left(\delta \tau \right)}_n{\Pi }_n\left(\mathbf{r}\right).$$

(6)

The commutation relations are then

$$[p_{n^{\prime }},{\varphi }_n]=-i\hslash {\delta }_{n{n}^{\prime }},$$

(7)

and on substituting for $p_{n^{\prime }}$ in terms of ${\Pi }_{n^{\prime }}$, one gets

$$[{\Pi }_{n^{\prime }}\left(\mathbf{r}\right),{\varphi }_n\left({\mathbf{r}}^{\prime }\right)]=-i\hslash {\displaystyle \frac{1}{{\left(\delta \tau \right)}_{n^{\prime }}}}{\delta }_{n{n}^{\prime }}.$$

(8)

In the limit ${\left(\delta \tau \right)}_n\to 0$, one has

$$[\Pi \left(\mathbf{r}\right),\varphi \left({\mathbf{r}}^{\prime }\right)]=-i\hslash \delta \left(\mathbf{r}-{\mathbf{r}}^{\prime }\right),$$

(9)

with $\delta \left(\mathbf{r}-{\mathbf{r}}^{\prime }\right)$ the Dirac delta function. Therefore, one has in general,

$$[{\Pi }_i(\mathbf{r},t),{\varphi }_j({\mathbf{r}}^{\prime },t)]=-i\hslash {\delta }_{ij}\left(\mathbf{r}-{\mathbf{r}}^{\prime }\right).$$

(10)

There is no need, of course, to derive the commutation rules using the method just described. It is sufficient to impose the commutation relations in the continuum form as a requirement of the quantum field theory. The canonical commutation relations are needed in order to ensure the consistency embodied in the requirement that the field equations follow as Heisenberg equations involving the total Hamiltonian

$${\displaystyle \frac{d\Pi }{dt}}={\displaystyle \frac{i}{\hslash }}[H,\Pi ].$$

(11)

It is shown later here that, at the final stage of the second quantization, the requirement of satisfying the commutation relations between $\Pi$ and $\Phi$ is equivalent to the imposition of commutation conditions between annihilation and creation operators of the corresponding field quanta. The above canonical procedure is an essential prerequisite for constructing a quantum field theory. However, the most practical format of the theory as regards utility for actual applications is that obtained by carrying out the next and final step using the expansion postulate leading to the concept of quanta.

Quantization of Free Fields

The canonical procedure, when applied to the cases of free fields, leads to the identification of field equations, the construction of a suitable Lagrangian enabling the derivation of the canonical momenta conjugate to the field variables, followed by the imposition of the appropriate commutation relations between them. Finally, the Hamiltonian is constructed and expressed in terms of the canonical variables and conjugate momenta.

Quantization, the ultimate stage in the procedure, involves making use of the superposition principle, according to which, a physical system can be described entirely in terms of its normal modes. This is quite an old concept that can be traced back to the work of Daniel Bernoulli and others [12]. The concept was later extended and adapted to the case of electrodynamics, which laid the foundations of quantum electrodynamics, regarded to be the gateway to the general subject of quantum field theory. Here, it is worth to emphasize that this theoretical framework is not a preserve of quantum field theory but has proved to be effective in many non-quantum contexts.

The stage leading to the appearance of quanta begins with the field equation that should be solved subject to a set of physically motivated boundary conditions. The boundary conditions can, in general, be satisfied by well defined frequencies that determine the spectrum of the excitations that correspond to the given field equation. Each solution is a normal mode with a characteristic energy, and the requirement that normal modes form a linearly independent set means that such modes must also be orthogonal.

Once a description in terms of the normal modes is realized, it is said one has, in general, an exactly solvable problem. The reason is that any state of the system can be described in terms of a function satisfying the given boundary conditions, and it must be expressible as a sum in terms of the normal mode functions. Thus, the goal is to use the amplitudes of the normal modes as dynamical coordinates to characterize the Lagrangian and Hamiltonian in a specially simple form, namely, the form in which there is no coupling between the coordinates in the equations of motion. The system then becomes equivalent to a set of isolated (i.e., non-interacting) harmonic oscillators.

To illustrate the procedure, let us consider the simplest one-dimensional case, namely the elastic line which is characterized by a scalar field $\varphi$ that satisfies a given field equation and is subject to a specific set of boundary conditions in the region $(x_1,x_2)$. Let the set $\left\{{\tilde{\varphi }}_n(x,t)\right\}$ be the set of normal modes, defined as the solutions of the field equation conforming with the given boundary conditions. Let ${\omega }_n$ be the corresponding frequency characterizing the normal mode, and let ${\tilde{\varphi }}_n$ satisfy the condition

$${\int }_{x_1}^{x_2}{\tilde{\varphi }}_n\left(x\right){\tilde{\varphi }}_m\left(x\right)dx={\delta }_{nm}.$$

(12)

The superposition principle means that it is allowed to express a general state $\Phi (x,t)$ of the field as a sum,

$$\Phi (x,t)=\sum _n{q}_n\left(t\right){\tilde{\varphi }}_n(x,t).$$

(13)

In actually all of the physically interesting cases, the Lagrangian L can be cast in a simpler form using the orthonormality relation Equation (12). One has

$$L={\displaystyle \frac{1}{2}}\sum _n\left({\dot{q}}_n^2-{\omega }_n^2{q}_n^2\right).$$

(14)

One may now regard the set of coefficients $q_n$ as the generalized dynamical variables and the Lagrangian in Equation (14) as the starting point. One then has canonical momenta $p_n$ corresponding to $q_n$ defined by

$$\begin{array}{c}\hfill p_n={\displaystyle \frac{\partial L}{\partial {\dot{q}}_n}}={\dot{q}}_n.\end{array}$$

(15)

The Lagrange equation with L given by Equation (14) and $q_n$ as a variable gives

$${\ddot{q}}_n+{\omega }_n^2{q}_n=0.$$

(16)

The Hamiltonian is obtainable in the common fashion as follows:

$$H=\sum _n\left(p_n^2+{\omega }_n^2{q}_n^2\right).$$

(17)

Thus, the problem has been exceptionally simplified. One now has a basis for the standard treatment of dealing with the mechanics of continuous systems. The dynamics of such a system is mathematically equivalent to that of a system of non-interacting harmonic oscillators.

The transition from the description of the normal modes in terms of the variables $q_n$ and $p_n$ to those appropriate for the corresponding harmonic oscillator is now straightforward. Let us define the lowering and raising operator of the nth mode by

$$\begin{array}{c}\hfill a_n={\left({\displaystyle \frac{1}{2\hslash {\omega }_n}}\right)}^{{\displaystyle \frac{1}{2}}}({\omega }_n{q}_n+i{p}_n),a_n^{†}={\left({\displaystyle \frac{1}{2\hslash {\omega }_n}}\right)}^{{\displaystyle \frac{1}{2}}}({\omega }_n{q}_n-i{p}_n).\end{array}$$

(18)

The relations (18) are formally equivalent to those encountered in the case of the single harmonic oscillator. In particular, one can write

$$\begin{array}{c}\hfill [a_n,a_m^{†}]={\delta }_{nm},p_n^2+{\omega }_n^2{q}_n^2={\displaystyle \frac{1}{\hslash {\omega }_n}}\left(a_n{a}_n^{†}+a_n^{†}{a}_n\right).\end{array}$$

(19)

Therefore, one has

$$\begin{array}{c}\hfill H={\displaystyle \frac{1}{2}}\sum _n\hslash {\omega }_n\left(a_n{a}_n^{†}+a_n^{†}{a}_n\right)=\sum _n\hslash {\omega }_n\left(a_n^{†}{a}_n+{\displaystyle \frac{1}{2}}\right).\end{array}$$

(20)

From the definitions of $a_n$ and $a_n^{†}$, we can write

$$q_n={\left({\displaystyle \frac{\hslash }{2{\omega }_n}}\right)}^{{\displaystyle \frac{1}{2}}}(a_n+a_n^{†}).$$

(21)

The required quantized field $\Phi (x,t)$ emerges at once by substitution in Equation (13). The result can be written as follows

$$\Phi (x,t)=\sum _n{\left({\displaystyle \frac{\hslash }{2{\omega }_n}}\right)}^{{\displaystyle \frac{1}{2}}}\{a_n{\tilde{\varphi }}_n+a_n^{†}{\tilde{\varphi }}_n^{*}\}.$$

(22)

The field $\Phi$ given in Equation (22) is now an operator.

The formal step of the ‘quantization’ of any given theory begins after the Hamiltonian is derived. It involves, once a commutation relation between the field and its canonically conjugate momentum is written, the immediate recognition of the field as a quantum rather than a classical system. The continuum case emerges from the above discrete form by converting the summation into an integral. One has

$$\Phi =\int d^3\mathbf{k}{C}_k\left\{{\tilde{\varphi }}_{k}a\left(\mathbf{k}\right)+{\tilde{\varphi }}_k^{*}{a}^{†}\left(\mathbf{k}\right)\right\}.$$

(23)

Let us now apply the procedure described in this Section to the case of a fluid, which is an essentially classical system, but which is unfamiliar in contexts of quantization.

3. Quantized Classical Fluids

By a fluid, it is meant a physical system consisting of a continuous distribution of a prevalent state of matter in the form of a gas or a liquid. The interest here is primarily in the macroscopic attributes of fluids [13,14], and not their microscopic structure or their truly quantum properties are considered, as in the context of quantum fluids [15]. It is expected the fluid system under question to possess a continuous distribution of mass and that the mass be conserved. It may be electrically neutral or it may possess a continuous distribution of monopole charge, as in the case of an electron gas, or dipole charge, as in the case of an overall neutral but polarizable medium. For simplicity and in order to highlight the general procedure, only the simplest case of an incompressible fluid is considered here.

The treatment here is restricted to to the case of ideal fluids, defined as those that are non-viscous, since viscosity forces originate in part in friction forces involving microscopic attributes. The main interest here is their field theoretical description and the quantization of some of the relevant attributes of fluids. Let us begin, however by outlining some of the basic theory of hydrodynamics, which sets the scene for the more involved aspects of the corresponding canonical theory prior to quantization. First, let us focus on an electrically neutral fluid.

The dynamics of a neutral fluid are characterized by a mass density, which is a scalar field $\rho (\mathbf{r},t)$, a fluid displacement vector field $\mathbf{u}$ (or, equivalently, a velocity field $\mathbf{v}=\dot{\mathbf{u}}$), and a pressure scalar field p. In a linearized treatment, these fields are connected by three basic relations. The first is the equation of continuity, which relates the mass density field to the velocity field, and is basically a statement of conservation of fluid mass. The second is the equation of motion, which is Newton’s law, equating the mass acceleration of an element of the fluid to the pressure forces due to the rest of the fluid. The third relates the mass density field to the pressure field.

The mass density $\rho$ of the fluid is defined as typical, such that the mass $dm$ contained in a fluid element of volume $dV$ situated at point $\mathbf{r}$ at time t is

$$dm=\rho (\mathbf{r},t)dV,$$

(24)

and for a finite volume, the amount of mass contained at a specific time t is

$$m={\int }_V\rho (\mathbf{r},t)dV.$$

(25)

Associated with fluid motion is a fluid displacement field $\mathbf{u}(\mathbf{r},t)$ and a current density vector $\mathbf{J}(\mathbf{r},t)$. The current density vector field is defined by

$$\mathbf{J}(\mathbf{r},t)=\dot{\mathbf{u}}(\mathbf{r},t)\rho (\mathbf{r},t).$$

(26)

The mass conservation law demands that, in a region of fluid of volume V enclosed by a surface S, the amount of inward fluid flow crossing the surface per unit time must be balanced by the rate of increase in the mass within the volume

$${\displaystyle \frac{dm}{dt}}=-{\int }_S\mathbf{J}·\widehat{\mathbf{n}}dS={\int }_V\mathsf{\nabla }·\mathbf{J}dV,$$

(27)

where $\widehat{n}$ is the unit vector normal to the surface S and use of Gauss’s theorem has been made. On making use of Equations (24)–(26) one has

$${\displaystyle \frac{\partial \rho }{\partial t}}+\mathsf{\nabla }·\left(\rho \dot{\mathbf{u}}\right)=0,$$

(28)

which is the equation of continuity relating the mass density of the fluid to its velocity fields.

The equation of motion in the fluid is an equality between the mass acceleration and the gradient of the pressure. The dynamics can be described either in terms of the fluid displacement vector field $\mathbf{u}$ or in terms of the scalar potential field $\varphi$, such that the velocity field $\mathbf{v}=-\mathsf{\nabla }\varphi$.

The Lagrangian density $\mathcal{L}$ can thus be expressed either in terms of $\mathbf{u}$ or in terms of $\varphi$. However, from the point of view of physical interpretation, the most direct form is that in terms of $\mathbf{u}$. The kinetic energy density to leading order is $\mathcal{T}={\displaystyle \frac{1}{2}}{\rho }_0{\dot{\mathbf{u}}}^2$, while the potential energy is $\mathcal{V}={\displaystyle \frac{1}{2}}{\beta }^2{(\mathsf{\nabla }·\mathbf{u})}^2$. Then, one has for the Lagrangian density

$$\mathcal{L}=\mathcal{T}-\mathcal{V}={\displaystyle \frac{1}{2}}{\rho }_0\left\{{\dot{\mathbf{u}}}^2-{\beta }^2{(\mathsf{\nabla }·\mathbf{u})}^2\right\},$$

(29)

where ${\rho }_0$ and $\beta$ are the equilibrium mass density and the oscillations velocity parameter entering the fluid pressure p. The theory is then in terms of a vector field (i.e., three scalar fields) rather than just one if one were to proceed in terms of $\varphi$. Let us consider the $\mathbf{u}$ form of this field theory first and deal with the $\varphi$ form next in order to compare the procedures and physical contents of each. With $\mathcal{L}$ given by Equation (29) and $\mathbf{u}$ as the field variable, the conjugate momentum $\mathit{\pi }$ is defined now by

$$\mathit{\pi }={\displaystyle \frac{\partial \mathcal{L}}{\partial \dot{\mathbf{u}}}}={\rho }_0\dot{\mathbf{u}}.$$

(30)

Thus the Euler–Lagrange equations with $\mathcal{L}$ as the Lagrangian density yields

$$\ddot{\mathbf{u}}-{\beta }^2{\nabla }^2\mathbf{u}=0.$$

(31)

The corresponding Hamiltonian density is

$$\mathcal{H}=\mathit{\pi }·\dot{\mathbf{u}}-\mathcal{L}={\displaystyle \frac{1}{2}}{\rho }_0\left({\dot{\mathbf{u}}}^2+{\beta }^2{(\mathsf{\nabla }.\mathbf{u})}^2\right).$$

(32)

The interpretation is straightforward: the Hamiltonian is just the sum of the kinetic and potential energies.

Quantization may now proceed as follows, anticipating that $\mathbf{u}$ is an irrotational vector, i.e., $\mathsf{\nabla }\times \mathbf{u}=0$. Let us expand $\mathbf{u}$ in plane waves in a cubic cavity of volume $d^3$ subject to periodic boundary conditions. Thus,

$$\mathbf{u}(\mathbf{r},t)=\sum _n{\widehat{\mathbf{k}}}_n{C}_n\left(a_n{e}^{i({\mathbf{k}}_n.\mathbf{r}-{\omega }_{n}t)}+a_n^{†}{e}^{-i({\mathbf{k}}_n.\mathbf{r}-{\omega }_{n}t)}\right),$$

(33)

where ${\widehat{\mathbf{k}}}_n$ is a unit vector along ${\mathbf{k}}_n$, $C_n$ is the mode amplitude to be determined, and ${\omega }_n$ is the mode frequency for a plane wave satisfying the wave equation Equation (31):

$${\omega }_n=\beta k_n,k_n=\left|{\mathbf{k}}_n\right|.$$

(34)

Application of the periodic boundary conditions gives

$${\mathbf{k}}_n={\displaystyle \frac{2\pi }{d}}(n_1,n_2,n_3),$$

(35)

where $n_1,n_2$, and $n_3$ are integers ranging from $-\infty$ to ∞. Thus, each mode is specified by three indices. It is convenient however to continue to adopt the single index n to stand for the three.

From Equation (33):

$$\dot{\mathbf{u}}(\mathbf{r},t)=-i\sum _n{\widehat{\mathbf{k}}}_n{\omega }_n{C}_n\left(a_n{e}^{i({\mathbf{k}}_n.\mathbf{r}-{\omega }_{n}t)}-a_n^{†}{e}^{-i({\mathbf{k}}_n.\mathbf{r}-{\omega }_{n}t)}\right),$$

(36)

and

$$\mathsf{\nabla }.\mathbf{u}=i\sum _n{\mathbf{k}}_n{C}_n\left(a_n{e}^{i({\mathbf{k}}_n.\mathbf{r}-{\omega }_{n}t)}-a_n^{†}{e}^{-i({\mathbf{k}}_n.\mathbf{r}-{\omega }_{n}t)}\right).$$

(37)

Substituting in the field Hamiltonian, one has

$$\begin{array}{c}\hfill H={\displaystyle \frac{1}{2}}{\rho }_0\int dV\left\{\dot{\mathbf{u}}·{\dot{\mathbf{u}}}^{†}+{\beta }^2(\mathsf{\nabla }·\mathbf{u}){(\mathsf{\nabla }·\mathbf{u})}^{†}\right\}.\end{array}$$

(38)

Substituting and making use of the integral

$${\int }_{V}dV{e}^{i({\mathbf{k}}_n-{\mathbf{k}}_{n^{\prime }}).\mathbf{r}}=V{\delta }_{n{n}^{\prime }},$$

(39)

one finds that the Hamiltonian reduces to the canonical form

$$H={\displaystyle \frac{1}{2}}\sum _n\hslash {\omega }_n\left(a_n{a}_n^{†}+a_n^{†}{a}_n\right),$$

(40)

provided that the mode coefficient $C_n$ is given by

$$C_n=\sqrt{{\displaystyle \frac{\hslash }{2V{\rho }_0{\omega }_n}}}.$$

(41)

In the infinite domain, $d\to \infty$ is taken and one obtains

$$\mathbf{u}(\mathbf{r},t)=\int d^3\mathbf{k}{\left({\displaystyle \frac{\hslash }{2{\left(2\pi \right)}^3{\rho }_0\omega }}\right)}^{{\displaystyle \frac{1}{2}}}\widehat{\mathbf{k}}\left[a\left(\mathbf{k}\right)e^{i(\mathbf{k}.\mathbf{r}-\omega t)}+a^{†}\left(\mathbf{k}\right)e^{-i(\mathbf{k}.\mathbf{r}-\omega t)}\right].$$

(42)

The commutation relation that should accompany the formalism with $\mathbf{u}$ as the field variable is between the components of $\mathit{\pi }$ and $\mathbf{u}$:

$$[{\pi }_i(\mathbf{r},t),u_j({\mathbf{r}}^{\prime },t)].$$

(43)

One obtains, by direct substitution of $\mathit{\pi }={\rho }_0\dot{\mathbf{u}}$,

$$[{\pi }_i(\mathbf{r},t),u_j({\mathbf{r}}^{\prime },t)]={\delta }_{ij}^{\left|\right|}\left(\mathbf{r}-{\mathbf{r}}^{\prime }\right),$$

(44)

where ${\delta }_{ij}^{\left|\right|}\left(\mathbf{r}-{\mathbf{r}}^{\prime }\right)$ is the longitudinal delta function

$${\delta }_{ij}^{\left|\right|}(\mathbf{r}-{\mathbf{r}}^{\prime })=\int d^3\mathbf{k}{\widehat{k}}_i{\widehat{k}}_j{e}^{i\mathbf{k}.\left(\mathbf{r}-{\mathbf{r}}^{\prime }\right)}.$$

(45)

The emergence of the result of the commutator in this manner is a direct consequence of the feature that $\mathbf{u}$ is irrotational.

4. Plasmons

One of the best examples of a quantizable classical system that has been and continues to be extensively studied is distinguished by well-defined elementary excitations with quanta called plasmons [16]. These excitations arise in the context of a model quasi-continuum material called jellium. The jellium model is exemplified by the case of valence electrons in a metal (where there are typically approximately ${10}^{28}$ electrons ${\mathrm{m}}^{-3}$) or the conduction electrons in a doped semiconductor (approximately 10²⁵ ${\mathrm{m}}^{-3}$). The relevant fields are those pertaining to a charged fluid representing the electrons interacting with the electromagnetic fields, which are, themselves, modified by the polarization provided by the charged fluid. Electrons in a metal can thus be viewed as a charged fluid in the presence of an overall neutralizing background charge due to the fixed uniformly distributed nuclei. Since the periodicity of the lattice ions is ignored, the situation resembles a jellium whereby the electron fluid oscillates relative to the positive background provided by the ions. In a linearized approximation, the properties of the charged fluid are represented by the polarization amplitude field $\mathit{\xi }(\mathbf{r},t)$, via which, deviations from equilibrium manifest themselves. If n is the equilibrium volume density of electrons (of effective mass $m^{*}$ and charge e), the deviation $\Delta n$ of the charge density from equilibrium can be expressed in terms of $\mathit{\xi }(\mathbf{r},t)$ as follows:

$$\Delta n=-n\mathsf{\nabla }·\mathit{\xi }.$$

(46)

$\mathit{\xi }$ may be referred to as the collective amplitude field and is related to the conserved fluid charge density $\rho$ and current density $\mathbf{J}$ by

$$\rho =-ne\mathsf{\nabla }.\mathit{\xi },\mathbf{J}=ne\dot{\mathit{\xi }}.$$

(47)

The basic equations for this so-called dispersive hydrodynamic model in the absence of any free charges are as follows. First, the fields are coupled to the polarization charge density and current density via Maxwell’s equations

$$\begin{array}{ccc}\hfill \mathsf{\nabla }·\mathbf{E}& =& -{\displaystyle \frac{1}{{ϵ}_0}}ne\mathsf{\nabla }·\mathit{\xi },\hfill \\ \hfill \mathsf{\nabla }\times \mathbf{E}& =& -\dot{\mathbf{B}},\hfill \\ \hfill \mathsf{\nabla }\times \mathbf{B}& =& {\mu }_0{ϵ}_0\dot{\mathbf{E}}+{\mu }_{0}ne\dot{\mathit{\xi }},\hfill \\ \hfill \mathsf{\nabla }·\mathbf{B}& =& 0.\hfill \end{array}$$

(48)

for electric E and magnetic B filelds, with ${ϵ}_0$ and ${\mu }_0$ denoting the permittivity and permeativity in the vacuum. Second, the fields modify the mechanical motion of the fluid via Newton’s law. For an element of the fluid, the equation of motion can be written as follows

$$m^{*}\ddot{\mathit{\xi }}\left(\mathbf{r},\mathbf{t}\right)=e\mathbf{E}(\mathbf{r},t)+m^{*}{\beta }_p^2\mathsf{\nabla }(\mathsf{\nabla }·\mathit{\xi }),$$

(49)

with ${\beta }_p$ the plasma velocity parameter. The spatial dispersion is manifest on inclusion of the last term, identified as the pressure term in Equation (49). The hydrodynamic pressure is

$$\Delta p=-n{m}^{*}{\beta }_p^2\mathsf{\nabla }·\mathit{\xi }.$$

(50)

The force corresponding to the last term in Equation (49) is equal to $-\mathsf{\nabla }(\Delta p)$.

The procedure demands first finding the Lagrangian as the basis for the description of this system. The point is to express fields in terms of electromagnetic potentials. In an arbitrary gauge, one writes

$$\mathbf{E}=-\dot{\mathbf{A}}-\mathsf{\nabla }\Phi ,\mathbf{B}=\mathsf{\nabla }\times \mathbf{A},$$

(51)

with A the vector potential. The equations of motion are now as follows. The first equation in the Maxwell’s equations (48) becomes

$$\mathsf{\nabla }·\dot{\mathbf{A}}+{\nabla }^2\Phi ={\displaystyle \frac{1}{{ϵ}_0}}ne\mathsf{\nabla }·\mathit{\xi },$$

(52)

while the third equation in the Maxwell’s equations (48) now reads

$$\mathsf{\nabla }\times \mathsf{\nabla }\times \mathbf{A}=-{\mu }_0{ϵ}_0\ddot{\mathbf{A}}-{\mu }_0{ϵ}_0\mathsf{\nabla }\dot{\Phi }+{\mu }_0{ϵ}_0\dot{\mathit{\xi }}.$$

(53)

Finally Newton’s law (49) becomes

$$m^{*}\ddot{\mathit{\xi }}=-e(\dot{\mathbf{A}}+\mathsf{\nabla }\Phi )+m^{*}{\beta }_p^2\mathsf{\nabla }(\mathsf{\nabla }·\mathit{\xi }).$$

(54)

The Lagrangian must reproduce the above equations and must account for both the electromagnetic and the mechanical contributions together with their coupling interaction. Thus, the total Lagrangian density is the sum of three contributions

$$\mathcal{L}={\mathcal{L}}_{\mathrm{em}}+{\mathcal{L}}_{\xi }+{\mathcal{L}}_{\mathrm{int}},$$

(55)

where ${\mathcal{L}}_{\mathrm{em}}$ pertains to the electromagnetic fields

$${\mathcal{L}}_{\mathrm{em}}={\displaystyle \frac{1}{2}}{ϵ}_0\left({[\dot{\mathbf{A}}+\mathsf{\nabla }\Phi ]}^2-c^2{[\mathsf{\nabla }\times \mathbf{A}]}^2\right).$$

(56)

${\mathcal{L}}_{\xi }$ pertains to the mechanical motion of the electron fluid

$${\mathcal{L}}_{\xi }={\displaystyle \frac{1}{2}}{m}^{*}n\left({\dot{\mathit{\xi }}}^2-{\beta }_p^2{(\mathsf{\nabla }·\mathit{\xi })}^2\right).$$

(57)

Finally, ${\mathcal{L}}_{\mathrm{int}}$ represents the coupling between the fields and the fluid’s charge and current densities

$${\mathcal{L}}_{\mathrm{int}}=ne\dot{\mathit{\xi }}·\mathbf{A}+ne(\mathsf{\nabla }·\mathit{\xi })\Phi .$$

(58)

Consider now the Euler–Lagrange equation with $\mathit{\xi }$ as a variable:

$${\displaystyle \frac{\partial \mathcal{L}}{\partial \dot{\mathit{\xi }}}}=m^{*}n\dot{\mathit{\xi }}+ne\mathbf{A}=\mathcal{P},$$

(59)

$$\sum _i{\partial }_i{\displaystyle \frac{\partial \mathcal{L}}{\partial \left({\partial }_i\dot{\mathit{\xi }}\right)}}=ne\mathsf{\nabla }\Phi -n{m}^{*}{\beta }_p^2(\mathsf{\nabla }·\mathit{\xi }),$$

(60)

$${\displaystyle \frac{\partial \mathcal{L}}{\partial \mathit{\xi }}}=0,$$

(61)

where ${\partial }_i\equiv \partial /\partial x_i$. On substituting in the Euler–Lagrange equations, Equation (54) is recovered. Consider next the field equation with $\mathbf{A}$ as a variable:

$${\displaystyle \frac{\partial \mathcal{L}}{\partial \dot{\mathbf{A}}}}={ϵ}_0(\dot{\mathbf{A}}+\mathsf{\nabla }\Phi )=\mathsf{\Pi },$$

(62)

$$\sum _i{\partial }_i{\displaystyle \frac{\partial \mathcal{L}}{\partial \left({\partial }_i\mathbf{A}\right)}}=-{ϵ}_0{c}^2\mathsf{\nabla }\times \mathsf{\nabla }\times \mathbf{A},$$

(63)

$${\displaystyle \frac{\partial \mathcal{L}}{\partial \mathbf{A}}}=ne\dot{\mathit{\xi }}.$$

(64)

It can be checked that the Euler–Lagrange equation with $\mathbf{A}$ as a variable yields Equation (53). Finally, consider the case of $\Phi$ as a variable. One finds

$${\displaystyle \frac{\partial \mathcal{L}}{\partial \dot{\Phi }}}=0,$$

(65)

$$\sum _i{\partial }_i{\displaystyle \frac{\partial \mathcal{L}}{\partial ({\partial }_i\Phi )}}={ϵ}_0(\mathsf{\nabla }·\dot{\mathbf{A}}+{\nabla }^2\Phi ),$$

(66)

$${\displaystyle \frac{\partial \mathcal{L}}{\partial \Phi }}=ne\mathsf{\nabla }·\mathit{\xi }.$$

(67)

Once again, one can see that the Euler–Lagrange equation with $\Phi$ as a variable leads to the first Maxwell equation in Equation (48).

The Hamiltonian density now follows straightforwardly using the familiar prescription

$$\mathcal{H}=\mathcal{P}·\dot{\mathit{\xi }}+\mathsf{\Pi }·\dot{\mathbf{A}}-\mathcal{L}.$$

(68)

Substituting for the canonical momenta from Equations (59) and (62) and after cancellations and other simplifications, one finds

$$\begin{array}{l}\mathcal{H}={\displaystyle \frac{1}{2}}{m}^{*}n({\dot{\mathit{\xi }}}^2+{\beta }_p^2{[\mathsf{\nabla }·\mathit{\xi }]}^2)+{\displaystyle \frac{1}{2}}{ϵ}_0\left({\dot{\mathbf{A}}}^2+c^2{[\mathsf{\nabla }\times \mathbf{A}]}^2\right)\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} +{ϵ}_0\mathsf{\nabla }\Phi ·\dot{\mathbf{A}}+{\displaystyle \frac{1}{2}}{ϵ}_0\mathsf{\nabla }\Phi ·\mathsf{\nabla }\Phi -ne(\mathsf{\nabla }·\mathit{\xi })\Phi .\end{array}$$

(69)

This is the Hamiltonian for the electromagnetic fields coupled to the jellium, and it must be emphasized that, since no gauge condition has so far been imposed on fields $\mathbf{A}$ and $\Phi$, the fields conform with an arbitrary gauge.

One may now make the division of each of the vector fields (represented by V) into its transverse (divergence free) and longitudinal (curl free) components such that

$$\begin{array}{c}\hfill \mathbf{V}={\mathbf{V}}^T+{\mathbf{V}}^L,\mathsf{\nabla }·{\mathbf{V}}^T=0,\mathsf{\nabla }\times {\mathbf{V}}^L=0,{\mathbf{V}}^T·{\mathbf{V}}^L=0.\end{array}$$

(70)

One therefore ends up with two sets of field equations, which may be proceeded to deal with separately. The transverse set of equations to be referred to as the plasmon polariton set and the longitudinal as the longitudinal plasmon set. The division applies also to the canonical momenta, the Lagrangian, and the Hamiltonian.

4.1. TransverseFields: Plasmon Polaritons

The transverse fields conform with the following set of equations

$$\begin{array}{c}\hfill \mathsf{\nabla }·{\mathbf{E}}^T=0=\mathsf{\nabla }·{\mathit{\xi }}^T,\end{array}$$

(71)

$$\mathsf{\nabla }\times {\mathbf{E}}^T=-\dot{\mathbf{B}},$$

(72)

$$\mathsf{\nabla }\times \mathbf{B}={\mu }_0{ϵ}_0{\dot{\mathbf{E}}}^T+{\mu }_{0}ne{\dot{\mathit{\xi }}}^{\mathbf{T}},$$

(73)

$$\mathsf{\nabla }·\mathbf{B}=0,$$

(74)

$$m^{*}{\ddot{\mathit{\xi }}}^T=e{\mathbf{E}}^T.$$

(75)

Let us now associate the transverse electromagnetic fields with the transverse vector potential ${\mathbf{A}}^T$ and the longitudinal fields with $\Phi$. The vector potential is required to be transverse everywhere satisfying $\mathsf{\nabla }{·\mathbf{A}}^T=0$. Then, all terms in the Hamiltonian containing both $\mathbf{A}$ with $\mathsf{\nabla }\Phi$ may be dropped, since ${\mathbf{A}}^T·\mathsf{\nabla }\Phi =0$. Equating the transverse components in Equation (53), we have

$${\nabla }^2{\mathbf{A}}^T-{\mu }_0{ϵ}_0{\ddot{\mathbf{A}}}^T=-{\mu }_{0}ne{\dot{\mathit{\xi }}}^T.$$

(76)

The relation (76) also follows from Equation (73) with the substitution

$${\mathbf{E}}^T=-{\dot{\mathbf{A}}}^T,\mathbf{B}=\mathsf{\nabla }\times {\mathbf{A}}^T.$$

(77)

The ${\mathit{\xi }}^T$ equation now involves ${\mathbf{A}}^T$ as follows

$$m^{*}{\ddot{\mathit{\xi }}}^T=-e{\dot{\mathbf{A}}}^T.$$

(78)

The Hamiltonian density can be written immediately as the transverse part of the full Hamiltonian density. It is

$${\mathcal{H}}^T={\displaystyle \frac{1}{2}}{m}^{*}n{[{\dot{\mathit{\xi }}}^T]}^2-ne\dot{\mathit{\xi }}·\mathbf{A}+{\displaystyle \frac{1}{2}}{ϵ}_0\left({[{\dot{\mathbf{A}}}^T]}^2+c^2{[\mathsf{\nabla }\times {\mathbf{A}}^T]}^2\right).$$

(79)

It can also be checked that the Hamiltonian density (79) follows from the transverse Lagrangian density with transverse vector potential and transverse canonical momentum.

The Hamiltonian density (79) of the transverse fields can be cast in an alternative form, characteristic of polaritons as follows. From Equation (78), $\mathit{\xi }=e\dot{\mathbf{A}}/m^{*}{\omega }^2$. The first mechanical energy term and the second (interaction) term in the Hamiltonian (79) can now be rewritten as

$${\displaystyle \frac{1}{2}}{m}^{*}n{[{\dot{\mathit{\xi }}}^T]}^2-ne\dot{\mathit{\xi }}·\mathbf{A}=-{\displaystyle \frac{1}{2}}{ϵ}_0{\displaystyle \frac{{\omega }_p^2}{{\omega }^2}}{\dot{\mathit{A}}}^2+{ϵ}_0{\displaystyle \frac{{\omega }_p^2}{{\omega }^2}}{\dot{\mathit{A}}}^2={\displaystyle \frac{1}{2}}{ϵ}_0{\displaystyle \frac{{\omega }_p^2}{{\omega }^2}}{\dot{\mathit{A}}}^2,$$

(80)

where ${\omega }_p$ is the plasma frequency

$${\omega }_p^2={\displaystyle \frac{n{e}^2}{{ϵ}_0{m}^{*}}}.$$

(81)

The Hamiltonian density can now be written as

$${\mathcal{H}}^T={\displaystyle \frac{1}{2}}{ϵ}_0\left(\left\{1+{\displaystyle \frac{{\omega }_p^2}{{\omega }^2}}\right\}{[{\dot{\mathbf{A}}}^T]}^2+c^2{[\mathsf{\nabla }\times {\mathbf{A}}^T]}^2\right).$$

(82)

The Hamiltonian density can further be written in a form that is applicable to all transverse polaritons [17]:

$${\mathcal{H}}^T={\displaystyle \frac{1}{2}}\left(\left\{{\displaystyle \frac{\partial [\omega ϵ]}{\partial \omega }}\right\}{[{\dot{\mathbf{A}}}^T]}^2+{\displaystyle \frac{1}{{\mu }_0}}{[\mathsf{\nabla }\times {\mathbf{A}}^T]}^2\right),$$

(83)

where, in this plasmon polariton case, the corresponding dielectric function $ϵ$,

$$ϵ\left(\omega \right)={ϵ}_0\left(1-{\displaystyle \frac{{\omega }_p^2}{{\omega }^2}}\right),$$

(84)

have been introduced. One can see that the wave equation Equation (76) can now be rewritten as

$${\nabla }^2{\mathbf{A}}^T+{\displaystyle \frac{{\omega }^2ϵ\left(\omega \right)}{c^2}}{\mathbf{A}}^T=0.$$

(85)

For plane waves, the plasmon frequency $\omega$ versus the wavevector k is then expressed as follows

$${\omega }^2ϵ\left(\omega \right)=c^2{k}^2,$$

(86)

which yields

$${\omega }^2={\omega }_p^2+k^2{c}^2,$$

(87)

and $\hslash \omega$ is the quantum energy of the plasmon polariton.

4.2. Quantized Plasmon Polaritons

The quantization of the plasmon polaritons follows the common procedure so that the total field is written as a superposition of plane wave polariton modes. The transversality of the field indicates the existence of two polarization states $\widehat{\mathit{e}}(\mathbf{k},j)$, labeled $j=1,2$. One thus has

$$\mathbf{A}(\mathbf{r},t)=\sum _j\int d^3\mathbf{k}C\left(\mathbf{k}\right)\left\{\widehat{\mathit{e}}(\mathbf{k},j)b(\mathbf{k},j)e^{i(\mathbf{k}·\mathbf{r}-\omega t)}+\mathrm{H}.\mathrm{c}.\right\},$$

(88)

where “H.c.” stands for Hermitian conjugate, and the components of the polarization vectors satisfy the rule

$$\sum _{j=1}^2{\widehat{\mathit{e}}}_{\alpha }(\mathbf{k},j){\widehat{\mathit{e}}}_{\beta }^{*}(\mathbf{k},j)=\left({\delta }_{\alpha \beta }-{\displaystyle \frac{k_{\alpha }{k}_{\beta }}{k^2}}\right).$$

(89)

The boson operators $b(\mathbf{k},j)$ satisfy the commutation rules

$$[b(\mathbf{k},j),b^{†}({\mathbf{k}}^{\prime },j^{\prime })]={\delta }_{j{j}^{\prime }}\delta \left(\mathbf{k}-{\mathbf{k}}^{\prime }\right).$$

(90)

Finally, $C\left(\mathbf{k}\right)$ is the normalization factor arising from ensuring that the Hamiltonian of the transverse fields reduces to the canonical form

$$H^T={\displaystyle \frac{1}{2}}\sum _{j=1}^2\int d^3\mathbf{k}\hslash \omega (\mathbf{k},j)\left[b^{†}(\mathbf{k},j)b(\mathbf{k},j)+b(\mathbf{k},j)b^{†}(\mathbf{k},j)\right].$$

(91)

4.3. Longitudinal Plasmons

For the longitudinal fields, one has the following set of equations:

$$\mathsf{\nabla }·{\mathbf{E}}^L=-{\displaystyle \frac{1}{{ϵ}_0}}ne\mathsf{\nabla }·{\mathit{\xi }}^L,$$

(92)

$${\dot{\mathbf{E}}}^L+{\displaystyle \frac{1}{{ϵ}_0}}ne\dot{{\mathit{\xi }}^L}=0,$$

(93)

$$m^{*}{\ddot{\mathit{\xi }}}^L\left(\mathbf{r},\mathbf{t}\right)=e{\mathbf{E}}^L(\mathbf{r},t)+m^{*}{\beta }_p^2\mathsf{\nabla }(\mathsf{\nabla }·{\mathit{\xi }}^L).$$

(94)

The following set of equations follow from the set of Equations (92)–(94):

$${\dot{\mathit{\xi }}}^L=-{\displaystyle \frac{1}{ne}}{ϵ}_0{\dot{\mathbf{E}}}^L,$$

(95)

$$-m^{*}{\omega }^2{\mathit{\xi }}^L=-{\displaystyle \frac{n{e}^2}{{ϵ}_0}}{\mathit{\xi }}^L+m^{*}{\beta }_p^2\mathsf{\nabla }(\mathsf{\nabla }·{\mathit{\xi }}^L),$$

(96)

and since $\mathsf{\nabla }(\mathsf{\nabla }·{\mathit{\xi }}^L)={\nabla }^2{\mathit{\xi }}^L+\mathsf{\nabla }\times \mathsf{\nabla }\times {\mathit{\xi }}^L$, but the last term is zero, this equation can be rearranged to read

$$\left\{{\omega }^2-{\omega }_p^2+{\beta }_p^2{\nabla }^2\right\}{\mathit{\xi }}^L=0,$$

(97)

where ${\omega }_p$ is the plasma frequency

$${\omega }_p^2={\displaystyle \frac{n{e}^2}{m^{*}{ϵ}_0}}.$$

(98)

For plane waves, the dispersion relation is

$${\omega }^2={\omega }_P^2+{\beta }_p^2{k}^2.$$

(99)

The same Equation (97) is satisfied by ${\mathbf{E}}^L$ and by the corresponding scalar potential:

$${\mathbf{E}}^L=-{\displaystyle \frac{ne}{{ϵ}_0}}{\mathit{\xi }}^L=-\mathsf{\nabla }\Phi ,$$

(100)

$$\left\{{\omega }^2-{\omega }_p^2+{\beta }_p^2{\nabla }^2\right\}\Phi =0.$$

(101)

The Hamiltonian density is the longitudinal part of the total density. Then,

$$\mathcal{H}={\displaystyle \frac{1}{2}}{m}^{*}n({[{\dot{\mathit{\xi }}}^L]}^2+{\beta }^2{[\mathsf{\nabla }.{\mathit{\xi }}^L]}^2)-{\displaystyle \frac{1}{2}}{ϵ}_0\mathsf{\nabla }\Phi .\mathsf{\nabla }\Phi -ne(\mathsf{\nabla }.{\mathit{\xi }}^L)\Phi .$$

(102)

Expressing $(\mathsf{\nabla }·{\mathit{\xi }}^L)$ in terms of $\Phi$ in the last term, one can write

$$\mathcal{H}={\displaystyle \frac{1}{2}}{m}^{*}n({[{\dot{\mathit{\xi }}}^L]}^2+{\beta }_p^2{[\mathsf{\nabla }·{\mathit{\xi }}^L]}^2)+{\displaystyle \frac{1}{2}}{ϵ}_0\mathsf{\nabla }\Phi ·\mathsf{\nabla }\Phi .$$

(103)

The division is apparent between the mechanical and electrical contributions. Finally, the last term can be eliminated in favor of another involving ${\mathit{\xi }}^L$ using Equation (100):

$$\mathcal{H}={\displaystyle \frac{1}{2}}{m}^{*}n\left({[{\dot{\mathit{\xi }}}^L]}^2+{\omega }_p^2{[{\mathit{\xi }}^L]}^2+{\beta }_p^2{[\mathsf{\nabla }·{\mathit{\xi }}^L]}^2\right).$$

(104)

Alternatively, one may write the Hamiltonian density (104) entirely in terms of $\Phi$:

$$\mathcal{H}={\displaystyle \frac{1}{2}}{ϵ}_0\left\{\mathsf{\nabla }\dot{\Phi }·\mathsf{\nabla }\dot{\Phi }+{\omega }_p^2\mathsf{\nabla }\Phi ·\mathsf{\nabla }\Phi +{\beta }_p^2{({\nabla }^2\Phi )}^2\right\}.$$

(105)

4.4. Quantizing the Longitudinal Plasmons

In Section 4.1, it was shown how the longitudinal plasmons arise from the hydrodynamic jellium model and may be quantized as a Coulomb field in the manner just described. The quantized field $\Phi$ in the three dimensional bulk is written as

$$\Phi =\int d^3\mathbf{k}C\left(\mathbf{k}\right)\left\{a\left(\mathbf{k}\right)e^{i\mathbf{k}.\mathbf{r}}-a^{†}\left(\mathbf{k}\right)e^{-i\mathbf{k}.\mathbf{r}}\right\},$$

(106)

where $C\left(\mathbf{k}\right)$ is the energy normalization (as that in Equation (88)) factor such that the Hamiltonian (105) reduces to the canonical form

$$H=\int d^3\mathbf{k}\omega \left(k\right)\left\{a^{†}\left(\mathbf{k}\right)a\left(\mathbf{k}\right)+{\displaystyle \frac{1}{2}}\right\},$$

(107)

where ${\omega }^2={\omega }_p^2+{\beta }_p^2{k}^2$, with ${\omega }_p$ as the characteristic plasma frequency, and ${\beta }_p$ is the velocity parameter responsible for dispersion. In the literature, a distinction is made between the low- and high-frequency properties of the electron system [18]. These limits give rise to two alternative choices of ${\beta }_p$:

$${\beta }_p=\sqrt{{\displaystyle \frac{1}{3}}}{v}_F\mathrm{and}{\beta }_p=\sqrt{{\displaystyle \frac{3}{5}}}{v}_F,$$

(108)

where $v_F$ is the Fermi velocity. The operators $a\left(\mathbf{k}\right)$ and $a^{†}\left(\mathbf{k}\right)$ are the boson annihilation and creation operators of the bulk plasmon field such that

$$[a\left(\mathbf{k}\right),a^{†}\left({\mathbf{k}}^{\prime }\right)]=\delta \left(\mathbf{k}-{\mathbf{k}}^{\prime }\right).$$

(109)

The most significant role played by quantized longitudinal plasmons is in electron energy loss spectroscopy (EELS), which is discussed in Section 6 below.

5. Phonons

5.1. Phonon Polaritons

Another quantizable field is in the context of dielectrics and semiconductors and involves dipole active excitations representing matter waves coupled to the electromagnetic fields. The hybrid modes are transverse elementary excitations called phonon polaritons [10,19,20]. The most direct route to dealing with phonon polaritons is by making use of the known dielectric function of these excitations. We shall not pursue the details of the derivation of the dielectric function and the general formalism in a dielectric [21]. With the dielectric function assumed known we refer to the Hamiltonian density form we derived above for plasmon polaritons, namely Equation (83). This form applies in general, and once the dielectric function is known, one can apply the formalism to phonon polaritons as well. The wave equation for phonon polaritons has the form given in Equation (85), given by

$${\nabla }^2{\mathbf{A}}^T+{\displaystyle \frac{{\omega }^2ϵ\left(\omega \right)}{c^2}}{\mathbf{A}}^T=0,$$

(110)

where for the phonon polariton, the dielectric function $ϵ\left(\omega \right)$ can be written as [1]

$$ϵ\left(\omega \right)={ϵ}_{\infty }{\displaystyle \frac{({\omega }^2-{\omega }_L^2+{\beta }_L^2{k}^2)}{({\omega }_T^2-{\omega }^2)+{\beta }_L^{\prime 2}{k}^2}},$$

(111)

where ${\omega }_T$ is the transverse phonon frequency, ${\omega }_L$ is the longitudinal phonon frequency, and these are connected by the relation

$${\omega }_L={\left({\displaystyle \frac{{ϵ}_s}{{ϵ}_{\infty }}}\right)}^{{\displaystyle \frac{1}{2}}}{\omega }_T,$$

(112)

where ${ϵ}_s$ and ${ϵ}_{\infty }$ are, respectively, the low frequency and the high frequency dielectric functions. Finally, the parameter ${\beta }_L\ll c$ is a velocity characterizing the dispersion of the longitudinal oscillations, and ${\beta }_L^{\prime }={\beta }_L/{ϵ}_{\infty }$.

Since ${\beta }_L$ is significant only in the context of longitudinal phonons, one may ignore it when focusing on the phonon polaritons. The dielectric function Equation (111) becomes

$$ϵ\left(\omega \right)={ϵ}_{\infty }{\displaystyle \frac{{\omega }_L^2-{\omega }^2}{{\omega }_T^2-{\omega }^2}}.$$

(113)

The dispersion relation giving $\omega$ against k is

$${\omega }^2ϵ\left(\omega \right)=c^2{k}^2.$$

(114)

Substituting for $ϵ\left(\omega \right)$ from Equation (113) and solving the resulting equation for $\omega$ against k, one finds there are two branches ${\omega }_{+}\left(k\right)$ and ${\omega }_{-}\left(k\right)$ of bulk phonon polaritons:

$${\omega }_{\pm }={\displaystyle \frac{1}{2}}\left\{{\omega }_L^2+\left({\displaystyle \frac{c^2{k}^2}{{ϵ}_{\infty }}}\right){\left[{\left\{{\omega }_L^2-{\displaystyle \frac{c^2{k}^2}{{ϵ}_{\infty }}}\right\}}^2+{\displaystyle \frac{4c^2{k}^2({\omega }_L^2-{\omega }_T^2)}{{ϵ}_{\infty }}}\right]}^{{\displaystyle \frac{1}{2}}}\right\}.$$

(115)

Let us therefore write for the Hamiltonian

$$H={\displaystyle \frac{1}{2}}{ϵ}_0\sum _i\int d^3\mathbf{r}\left\{{\displaystyle \frac{\partial [\omega ϵ(\omega )]}{\partial \omega }}{\mathbf{E}}_i^{T2}+c^2{\mathbf{B}}_i^2\right\},$$

(116)

where ${\mathbf{E}}^T=-{\dot{\mathbf{A}}}^T$, and $\mathbf{B}=\mathsf{\nabla }\times {\mathbf{A}}^T$.

The quantization of the ith branch now follows by writing for the electric field associated with branch i

$${\mathbf{E}}_i\left(\mathbf{r}\right)=\int d^3\mathbf{k}\left({\mathcal{E}}_{0i}{e}^{i\mathbf{k}.\mathbf{r}}{b}_j\left(\mathbf{k}\right)+\mathrm{H}.\mathrm{c}.\right),$$

(117)

where $b_i$ and $b_i^{†}$ are mode annihilation and creation operators satisfying boson commutation relations

$$[b_i\left(\mathbf{k}\right),b_{i^{\prime }}^{†}\left({\mathbf{k}}^{\prime }\right)]={\delta }_{i{i}^{\prime }}\delta \left(\mathbf{k}-{\mathbf{k}}^{\prime }\right).$$

(118)

The mode vector amplitudes ${\mathcal{E}}_{0i}$ are to be determined using the canonical procedure. Since the consideration here is dealing with transverse modes, there are two unit vectors involved in each phonon polariton branch, as was the case for plasmon polaritons. Phonon polaritons interact with electrons in semiconductor structures. For details, see [19]).

5.2. Longitudinal Phonons

Assuming that the material has no free charges, one has $\mathsf{\nabla }·\mathbf{D}=0$, which for the longitudinal phonons is satisfied by $\mathbf{D}=0$. This amounts to the condition

$$ϵ(\omega ,k)=0.$$

(119)

Thus, from Equation (111), one has the dispersion relation

$${\omega }^2={\omega }_L^2-{\beta }_L^2{k}^2.$$

(120)

For an isotropic medium, the parameter ${\beta }_L$ is generally recognized as the sound velocity in the medium.

The quantization of the longitudinal phonon modes is quite similar to the longitudinal plasmons, except for the sign of the second term in the dispersion relation.

The Lagrangian density can be written in terms of the displacement field ${\dot{\mathbf{u}}}^L$

$$\mathcal{L}={\displaystyle \frac{1}{2}}\left\{{\omega }_L^2{{\mathbf{u}}^L}^2-{{\dot{\mathbf{u}}}^L}^2-{\beta }_L^2(\mathsf{\nabla }·{{\mathbf{u}}^L)}^2\right\}.$$

(121)

The inclusion of the spatial dispersion allows the standard quantization program to be carried out in terms of plane waves, as performed for longitudinal plasmons; so, no further details are provided here.

6. Interactions

In Section 4, it has been shown how the (transverse) bulk plasmon polaritons and the corresponding longitudinal plasmons can be quantized. In Section 5, it has been also discussed the quantization of bulk (transverse) phonon polaritons together with the corresponding longitudinal phonon fields. We are thus in a position to consider interaction processes involving these fields. Typical processes consist of the emission and absorption of field quanta, leading to measurable changes in the properties of the systems with which the quantized fields interact. As it was pointed out in Section 1, the contexts in which the above quantized fields arise include electronically-dense metals and doped semiconductors of the category III-V compounds such as GaAs and GaAlAs. Such semiconductors are typically doped with Si, which introduces additional conduction electrons and leads to the creation of plasmons with which the host electrons interact This leads to changes to the relaxation rates and to the electric transport properties of the semiconductor itself. The plasmons in electronically-dense metals manifest themselves in processes involving EELS. Semiconductors also support phonons, both transverse (phonon polaritons) and longitudinal (acoustic), and these influence the electric transport properties in which excited electrons relax their energy by the emission of phonons and phonon polaritons.

One can proceed to consider examples where exchanges of quanta with other physical systems occur and lead to real effects that are calculable, and the results can be compared with experiments. In the case considered here, one could consider the exchange of plasmon polaritons, longitudinal plasmons, phonon polaritons, and longitudinal phonons. However, due to space limitations, let us now consider a concrete example of how quantization enables the evaluation of physical properties by focusing on EELS of electronically-dense metals, for example, aluminium. This would illustrate how quantum techniques have proved to be highly powerful and physically transparent. EELS involves the emission of longitudinal plasmons by an electron beam traversing a metallic or a semiconductor slab. It is an analysis that uses focused beams of electrons to determine a material’s properties. As the electrons pass through the sample, they lose energy through inelastic scattering. In the case of an electronically dense material here, the inelastic scattering is dominated by the excitation of longitudinal plasmons. The Hamiltonian of the system is [22]

$$H={\displaystyle \frac{{\left|\mathbf{P}\right|}^2}{2M}}+V\left(\mathbf{r}\right)+\int d^3\mathbf{k}\omega \left(k\right)\left\{a^{†}\left(\mathbf{k}\right)a\left(\mathbf{k}\right)+{\displaystyle \frac{1}{2}}\right\},$$

(122)

where M is the electron mass, and $\mathbf{P}$ is its linear momentum. The first term represents the kinetic energy of the electrons in the beam, and the last term represents the Hamiltonian of the quantized longitudinal bulk plasmons, while $V\left(\mathbf{r}\right)$ is the interaction Hamiltonian with $a^{†}\left(\mathbf{k}\right)$ and $a\left(\mathbf{k}\right)$, the creation and annihilation operators of longitudinal plasmons. It is distinctly tempting to write $V\left(\mathbf{r}\right)=Q\Phi \left(\mathbf{r}\right)$, where Q is the electric charge and $\Phi \left(\mathbf{r}\right)$ is the quantized Coulomb potential of the longitudinal plasmons. However, it has been shown that agreement of the theory with the experimental EELS results emerges when a canonical transformation [22] is applied leading to the following form of $V\left(\mathbf{r}\right)$:

$$V\left(\mathbf{r}\right)={\displaystyle \frac{Q}{2M\omega }}\left(\mathbf{P}·\mathsf{\nabla }\Phi +\mathsf{\nabla }\Phi ·\mathbf{P}\right).$$

(123)

Interestingly, when one writes $\mathbf{P}/\mathbf{M}=\mathbf{u}$, the velocity, and sets $\mathsf{\nabla }\Phi =-\mathcal{E}$, the electric field, one can write

$${\displaystyle \frac{Q}{2M}}\left(\mathbf{P}·\mathsf{\nabla }\Phi +\mathsf{\nabla }\Phi ·\mathbf{P}\right)=-Q\mathbf{u}·\mathcal{E},$$

(124)

which is the classical expression of the energy loss rate. Thus, the above canonically transformed potential has the advantage of clear classical correspondence.

The standard formalism can now be used to evaluate the cross section for scattering from an initial state $\left|i,0\right.〉$ to a final state $\left|f,\omega \right.〉$. The initial state $\left|i,0\right.〉$ is a plane wave electron state of energy–momentum $(E_i,{\mathbf{P}}_i)$ and no plasmon excited, and the final state $\left|f,\omega \right.〉$ is of energy–momentum $(E_f,{\mathbf{P}}_f)$, where a plasmon of frequency $\omega$ is excited. This process thus involves and energy loss E and momentum transfer q such that

$$E=E_i-E_f,q=|{\mathbf{P}}_i-{\mathbf{P}}_f|.$$

(125)

The evaluation of the cross section $\sigma (E,q)$ follows a standard procedure, and it is convenient to set $\hslash =1$. Then, [23]

$$\sigma (E,q)=\left({\displaystyle \frac{{\beta }_L^2{q}^2}{{[\omega \left(q\right)]}^3}}\right){\displaystyle \frac{\gamma (q,\omega )}{[{\left\{E-\omega \left(q\right)\right\}}^2+{\gamma }^2(q,\omega )]}}.$$

(126)

The function $\gamma$, calculated using the Golden rule, is found to be

$$\gamma (q,\omega )={\gamma }_0+\left({\displaystyle \frac{Q^2{\omega }_p^2}{4M^2{\beta }^2}}\right){\displaystyle \frac{q^3}{{[\omega \left(q\right)]}^2}},$$

(127)

where ${\gamma }_0$ is an adjustable parameter representing the residual spectrum when $q=0$. It is the variation of the cross section with the energy loss E for a fixed momentum transfer q that is experimentally relevant here. It is seen that, for a fixed q, the spectrum $\sigma$ against E given by Equation (126) is a Lorentzian with a half width at half maximum given by Equation (127), which changes with q but is constant for a given q. The theoretical predictions outlined above are generally consistent with the measurement of EELS of electronically dense metals [24].

7. Summary and Conclusions

The primary aim of this paper has been to highlight how the quantum concept has influenced developments in areas of physics that are unfamiliar or seldom addressed by scholars whose main concerns are the fundamentals and applications of traditional quantum theory. The interest has been in the concepts and techniques appropriate for the vast subject of long range processes involving collective excitations. However, the focus here was necessarily restricted to a narrow section in condensed matter contexts, namely plasmons and phonons. These sub-fields of condensed matter have a long history, but they continue to develop, encompassing modern issues of two-dimensional systems [25,26,27,28,29,30,31,32,33,34,35]. However, the study has focused on bulk plasmons and phonons, as the aim was to highlight the concepts and techniques in these contexts.

The paper began with the classical ideal fluid as a simple system governed by Newton’s laws, which serves to illustrate the methodology that leads to the emergence of the classical fluid quanta.

The study then tackled the significant topic of plasma oscillations in bulk metals and doped semiconductors. These oscillations are modeled in terms of a charged fluid of electrons with a neutralizing positive background charge, and the physics is governed by Newton’s laws with coupling to electromagnetic fields. It was shown that there are two types of plasma oscillations, namely the transverse (divergence free) also called plasmon polaritons and the longitudinal (curl-free) plasmons. It was shown that both types are amenable to quantization, and the quanta can be exchanged in processes of emission and absorption by a quantum system with which the plasmons interact. It was shown how discrete quanta of longitudinal plasmons are emitted by fast electron beams traversing thin metals. The quantization scheme used to explore the physics in the context of EELS, illustrates the power of the ‘quantum’ concept, which is distinctly confirmed by experiment.

It was also considered, as a second significant case, the lattice vibrations in polar semiconductors, which couple to electromagnetic fields, constituting phonon polaritons and longitudinal phonons, for example, as in the case of the III-V compounds. Like plasmons, these are a mixture of mechanical and electromagnetic oscillations. They too are characterized by a frequency-dependent dielectric function, which incorporates the mechanical as well as the electromagnetic components responsible for both the transverse phonon polaritons and the longitudinal phonons. The study could have proceeded to discuss the coupling of the phonon polaritons and longitudinal phonons to electrons in semiconductors. Longitudinal optic phonons, in particular, are of importance in that they contribute to the energy relaxation of hot electrons [36] and hot phonons [37], which determine the electric transport in semiconductor devices.

Even within this relatively narrow perspective of plasmons and phonons, there are further essential related issues, which, due to need for brevity, have not been covered in this paper. These issues are concerned with the presence of surfaces and boundaries between different materials. In the context here, the presence of surfaces gives rise to new quanta, namely surface plasmons and surface polar optical phonons.

Here too, the concept of quantized plasmons and phonons as well as localized plasmons (as in the case of plasmons on metal nano-particles) has proved to be exceptionally practical [38,39,40] with direct techniques to quantize localized plasmons [39,40]. Other and parallel aspects also exist in the context of phonons. For example, the two dimensional quantum, well formed of a thin layer of GaAs sandwiched between two thicker layers of AlAs, involves the trapping of an electron gas inside the GaAs layer, but the phonons, both transverse and longitudinal, are also modified due to the presence of interfaces between the two materials. Multiple quantum wells such as this may form semiconductor superlattices [41,42], in which the periodicity gives rise to artificially formed matter with special features, such as bands and band-gaps for both the electrons and the plasmons and phonons [43]. The modified excitations are normally probed using Raman scattering [8,44,45]. Finally, it must be emphasized that the subject of collective excitations is vast and covers quite a large number of contexts. It is thus beyond the narrow coverage here. Only plasmons and phonons are considered in order to illustrate how quantum concepts are techniques that have proved to be instructive when dealing with the physics in these two sub-fields of condensed matter.