Quantum Light and Coherent States in Conducting Media

We present a simple description of classical and quantum light propagating through homogeneous conducting linear media. With the choice of Coulomb gauge, we demonstrate that this description can be performed in terms of a damped harmonic oscillator which is governed by the Caldirola-Kanai Hamiltonian. By using the dynamical invariant method and the Fock states representation we solve the time-dependent Schrödinger equation associated with this Hamiltonian and write its solutions in terms of a special solution of the Milne-Pinney equation. We also construct coherent states for the quantized light and show that they are equivalent to the well-known squeezed states. Finally, we evaluate some important properties of the quantized light such as expectation values of the amplitude and momentum of each mode, their variances and the respective uncertainty principle.


Introduction
For a long time, the old and fascinating problem (from classical and quantum viewpoint) of the interaction of light with matter has received considerable attention of physicists. The story of the solution of this problem is a familiar one. Further, the solution of this problem has been of crucial importance for the development of our understanding of nature.
In order to obtain the basic concepts to study the classical and quantum behavior of light we must take into account Maxwell's equations. In the quantum case, the quantization of these equations is traditionally performed in free space or in empty cavities by associating a time-independent mechanical oscillator Now, in order to obtain a solution of this equation we consider light waves in a certain volume of space. So, by the familiar procedure of separation of variables, we write the vector potential in terms of the mode  is the velocity of light in the medium.
In the follows let us discuss the solutions of Equations (8) and (9). The solution of Equation (9) can be written in the form where l A and l δ are constants to be determined by the initial conditions and l Ω is given by  [33]. Hence, the total Hamiltonian of the electromagnetic field is a sum of individual Hamiltonians corresponding to each mode, that is, In the following discussion we focus our attention on the solution of Equation (8). Considering that the electromagnetic field is contained in a certain cubic volume V of side L of nonrefracting media, the mode functions are required to satisfy the transversality condition and to form a complete orthonormal set. Furthermore, assuming periodic boundary conditions on the surface, the mode function

,t A r
(see Equation (7)) by using Equations (10) and (13). Hence, using Equations (7) and (13) we can write, for each mode l, the electric and magnetic fields (see Equation (5) where we have used that  . Therefore, the above results give us a complete classical description of the propagation of light in conducting linear media since the electric E and magnetic B fields are completely specified. Here it is worth noticing that in the previous description we have associated a damped harmonic oscillator to the each mode of the electromagnetic field. Let us also observe that in the absence of the dissipation, that is, 0 σ = the Hamiltonian (12) reduces to that of the standard harmonic oscillator with the permittivity playing the role of the mass of the mechanical oscillator. As a consequence, all of our previous results coincide with those of the propagation of light in empty cavities.

Quantum Light Propagation in Conducting Media
In order to obtain a quantum description of light propagating in a conducting linear media we need to quantize the electromagnetic field. Now as the spatial mode functions ( ) l u r are completely determined, the amplitude of each nor-Journal of Applied Mathematics and Physics mal mode in Equation (7) needed to specify a particular field configuration is ( ) l q t [1]. Thus, for each canonical operator l q the electric E and magnetic B fields operators may be derived from the potential vector A by using Equation (5). So, let us move our attention to the canonical operator ( ) l q t in order to obtain the vector potential. For this purpose, let us solve the Schrödinger equation associated with the Hamiltonian (12) , , , where the coordinate ( ) l q t and the momentum l p are now canonically con- We can obtain the solutions of this equation with the aid of the dynamical invariant method developed by Lewis and Riesenfeld [23]. According to this method, we must look for a nontrivial Hermitian operator ( ) l I t which satisfies the equa- Then, the solutions of the Schrödinger Equation (16)  In what follows, let us consider a quadratic invariant that satisfies Equation (17). Here, we assume an invariant in the form We must now find the eigenstates of the invariant ( ) l I t . To this end, we will use the Fock representation since, as is well-known, the quantum behavior of some quantum systems, in particular quantum harmonic oscillator-type systems, is most obvious in Fock ststes, which are states with specific numbers of energy quanta. Then, let us introduce annihilation and creation-type operators ( ) l a t and ( ) † l a t defined by [16] [23] ( ) In terms of these operators, the invariant (21) can be factored as From Equations (26) and (27) (24) and (25). From the expressions of these operators, we obtain that Thus, using Equations (7), (13), (23), (33) and (36) In the above expression we have written the annihilation and creation operators The above field operators describe the quantum propagation of light in conducting linear media. We also see that both electric and magnetic fields decrease exponentially in time due the conductivity of the medium proportionally to ( ) Further, in the absence of the dissipation, that is, 0 σ = these fields reduce to that in empty cavities [1].
In what follows, we use the Fock states to calculate the expectation values of the amplitude l q , momentum l p , their variances and the respective uncertainty principle. Hence, making use of Equations (30) and (31) and after a little of algebra, we find that The quantum variances are given by ( ) By using the above expressions we obtain the uncertainty principle as media. We have used the Coulomb gauge and considered light waves confined in a cubical volume of side L filled with a conductive medium as well as light propagating under periodic boundary conditions. We have demonstrated that this propagation can be performed by associating a damped mechanical oscillator with each mode of the electromagnetic field. As a consequence, we have established a unification of the procedure to obtain the classical and quantum propagation of light in empty cavities (or free space) and cavities filled with a material medium. In the former case, it is usually performed by associating an ordinary harmonic oscillator with each mode of the electromagnetic field, and in the latter one it can be performed by the association of a damped harmonic oscillator. Further, using the invariant method, appropriated annihilation and creation-type operators and the Fock states we have easily solved the time-dependent Schrödinger for our problem and write its solutions in terms of a special solution of the Milne-Pinney equation. We have also constructed coherent states for the quantized light and have calculated the quantum variances of the amplitude ( ) l q t and momentum ( ) l p t as well as the uncertainty principle for each mode of the electromagnetic field in both states, namely, Fock and coherent states. We have seen that the uncertainty product in the coherent states is equal to the minimum value of that of the number states. In addition, we have seen that the uncertainty principle in the coherent states, in general, does not attain its minimum value. By employing a direct procedure we have shown that this latter result occurs because the coherent states constructed previously correspond to the squeezed states. Finally, we expected that the simple procedure developed in this work can be helpful to investigate subjects related to the interaction of light with material media.