Guiding of waves between absorbing walls

Guiding of waves between absorbing walls is phenomenon based on the reflection of waves coming at the grazing angle to a plane separating the media with different value of the index of refraction, whenever the step of index of refraction is real or complex. The imaginary part of the index of refraction represents the absorption. The guiding can be interpreted as the multiple reflection.

Warning! this article is under construction.

Absorbing Schroedinger

In the simple case, the propagation of scalar waves between absorbing waves can be described with the Absorbing Schroedinger equation:

\( \displaystyle \!\!\!\!\!\!\!\!\!\! (1) ~ ~ ~ \mathrm i \hbar \dot \Psi = \hat H ~ \Psi -\mathrm i ~ \hat V ~ \Psi \)

where \(\Psi\) is wave function, the dot differentiats it with respect to time, \(\hat H\) is the conventional, Hermitian part of Hamiltonian, and \(\hat V\) is the Hermitian operators that represents the entanglement of the state with some other degrees of freedom, that are taken into account only phenomenologocally. As usually, \(\hbar\) is the Planck constant; with the appropriate set of physical unite, this constant can be set to unity.

The Absorbing Schroedinger describes the evolution of the quantum state of a particle slightly coupled to many other particles, environment, giving evolution of the component of the wave function of the particle that "did not yet interact" with the environment.

Assume the empty space between parallel walls, and uniform absorption with rate \(\gamma\) within walls. In the coordinate representation, for a particle of mass \(m\), this corresponds to

\( \displaystyle \!\!\!\!\!\!\!\!\!\! (2) ~ ~ ~ \hat H = \frac{-\hbar^2}{2m} \nabla^2\)

and

\( \displaystyle \!\!\!\!\!\!\!\!\!\! (3) ~ ~ ~ \hat V = \hbar \gamma~ \mathrm{UnitStep} \left( |x|-d \right)\)

where \(2d>0\) is distance betwen walls, and \(x\) is transversal coordinate.

In the quasimonochromatic approximation, the exponential dependence of wave function on time \(t\) is assumed:

\( \displaystyle \!\!\!\!\!\!\!\!\!\! (4) ~ ~ ~ \Psi = \psi ~ \exp(\mathrm i \omega t)\)

where \(\omega\) (or \(\hbar \omega\), if the Planxk constant is drawn through the deduction) has sense of energy of the particle, and \(\psi\) is spacial part of the wave function, that is supposed to depend on time very slowly if at all; in the simple approximation \(\psi\) is function of only few spacial coordinates; for example, \(x\) and \(z\).

The substitution of (2),(3),(4) into (1) gives the "monochromatic" equation

\( \displaystyle \!\!\!\!\!\!\!\!\!\! (5) ~ ~ ~ \hbar\, \omega\, \psi = \frac{-\hbar^2}{2m} \nabla^2 \psi - \mathrm i ~\hbar ~\gamma~ \mathrm{UnitStep} \left( |x|-d \right) ~ \psi\)

where \(\psi= \psi(x,z)\) and \(\nabla\) differentiates with respect to \(x\) and \(z\).

At propagation of wave in the uniform absorbing medium with absorption rate \(\gamma\) (case \(d\rightarrow 0\)) the solution of equation (5) can be expressed as follows:

\( \displaystyle \!\!\!\!\!\!\!\!\!\! (6) ~ ~ ~ \psi = \exp( \mathrm i k z )\)

The substitution to (5) gives

\( \displaystyle \!\!\!\!\!\!\!\!\!\! (7) ~ ~ ~ \hbar \omega = \frac{(\hbar k)^2}{2m} - \mathrm i \hbar \gamma\)

which means leads to

\( \displaystyle \!\!\!\!\!\!\!\!\!\! (8) ~ ~ ~ k = \sqrt{ \frac{2m \omega}{\hbar} + \frac{2 \mathrm i \gamma m }{ \hbar} } = \sqrt{ b^2 + \frac{2 \mathrm i \gamma m }{ \hbar} } \)

where

\( \displaystyle \!\!\!\!\!\!\!\!\!\! (9) ~ ~ ~ b= \sqrt{ \frac{2m \omega}{\hbar}}\)

is real quantity, that can be interpreted as wavenumber of the particle in the empty space.

Relation to the Zeno effect

The absorbing walls can be considered as detectors, that check, wether the particle has left from the region between walls or not yet. In Wuantum mechanics, the detection can be interpreted as absorption annihilation of a particle localized between walls (perhaps, wiht creation of a particle(s) in some other states. The detection of the transition suppress it. The absorbing Schroedinger above describes this process, giving the shape of th eguided mode and its effective absorption. Such an estimate can be applied also to the case of discrete periodic absorbers. Fore the reflection of cold atoms from ridged mirrors, the estimate gives good agreement with experiments . The good guiding can be interpreted as multiple reflection, giving the same estimate for the absorption of the giuded mode .

Keywords

Zeno effect, Quantum reflection,