# Formation of probability density waves and probability current density waves
by excitation and decay of a doublet of quasistationary states of a three-
barrier heterostructure upon scattering of gaussian wave packets

Yu. G. Peisakhovich  Novosibirsk State Technical University, Novosibirsk,
Russia  A. A. Shtygashev<EMAIL_ADDRESS>Novosibirsk State Technical
University, Novosibirsk, Russia

###### Abstract

Annotation. A numerical-analytical simulation of scattering by a three-barrier
heterostructure of an electronic Gaussian wave packet, the spectral width of
which is on the order of the distance between the levels of the doublet of
quasi-stationary states, is carried out. It is shown that as a result of
scattering, damped waves of electron charge and current densities are formed
outside the double well, their characteristics are determined by the structure
of the initial wave packet and the poles of the scattering amplitudes. The
frequency of these waves is equal to the difference frequency of the doublet,
the wavenumber is the difference between the wave numbers of free motion of
electrons with resonant energies, and the speed of their propagation is the
ratio of these quantities. The system can go into the regime of repetition or
amplification of the emission of electron waves if a periodic resonant pumping
of the doublet population is provided by scattering of a series of coherent
wave packets.

quantum

###### pacs:

84.40.Az, 84.40.Dc, 85.25.Hv, 42.50.Dv, 42.50.Pq

##  I Introduction

The ability of nanoheterostructures to selectively transmit and convert wave
signals of different physical nature makes it possible to create high-speed
and high-frequency devices for optoelectronics, acoustoelectronics,
information transmission systems, laser technology, etc. In recent decades,
laser light sources have been created capable of generating ultrashort pulses
of picosecond, femtosecond, and even attosecond duration Rost2011 -Chek2014 .
This stimulated the intensive development of spectroscopy and high
technologies in the corresponding frequency ranges. The impact of such short-
term signals on microscopic and macroscopic systems and the detection of
responses make it possible to study fast processes, the duration of which is
less than or on the order of the relaxation times in the systems Rost2011
-Ross2002 . In addition to spectroscopic sensing of matter, it is possible to
pose the problem of generating an alternating current in the terahertz range
by converting ultrashort excitation pulses into a system of oscillations and
waves of electron density of charge and current on scales smaller than the
length and time of quantum coherence of electrons. This problem can be solved
using nanoscale heterostructures. It is well known that in thin-film
nanostructures such as a double quantum well with tunnel-transparent walls for
electrons, the energy spectrum of the transverse motion of electrons contains
doublets of resonance levels that are relatively close to each other. In the
forbidden bands of film below the vacuum level, such a spectrum is discrete
and the wave functions of doublet states are localized in the well. In the
allowed bands below and above the vacuum level, the energy spectrum is
continuous and the wave functions of resonance doublets describe delocalized
quasi-stationary states of the transverse scattering problem. The energies of
the doublets and the lifetimes of quasi-stationary states are determined by
the poles of the amplitudes of stationary electron scattering by the
heterostructure, as well as by the shift and smearing of levels due to
inelastic electron scattering. Pulsed excitation and slow decay of a quasi-
resonant nonstationary state formed by the superposition and interference of
quantum states from a narrow band of the electronic spectrum that includes a
doublet can be accompanied by beats of the space-time distributions of the
probability densities and current of electrons whose energies belong to such a
narrow band. This kind of beating often accompany a quantum transient Leo1991
-Cald2016  after a single pulse excitation and last for the lifetime of quasi-
stationary states, which can be much longer than the time period of these
beats if the transparency of the barriers is sufficiently low and the
inelastic processes for electrons are weak. This effect was first observed
indirectly in experiments on differential transmission and four-wave mixing
for femtosecond light pulses in an asymmetric double quantum well Leo1991
-Rosk1992 . In such a well, the quantum beats of the superposition of the wave
functions of the doublet of stationary states of the discrete spectrum of
transverse motion cause the appearance of resonant damped oscillations of the
electron-hole dipole moment and a certain number of registered oscillations of
the dipole electromagnetic radiation at the terahertz difference frequency of
the doublet.

A similar effect should also exist in the case when the doublet of quasi-
stationary states of the transverse scattering problem is located in the
continuous spectrum of the conduction bands above or below the vacuum level
Romo2002 -Peis2008B . In this paper, it will be shown that if the transparency
of the potential barriers of the heterostructure is sufficiently low, then the
coupled oscillations of mixed doublet resonance states should manifest
themselves not only in the periodic flow of the electron density between the
wells through the middle barrier inside the double well Peis2008A -Cald2011 ,
but they should also be accompanied by oscillations of the charge density and
current electrons escaping into outer space through extreme potential
barriers. Outside the double well, these spatiotemporal oscillations of the
envelopes of the charge and current densities can have the character of waves
traveling to the left and right from the heterostructure and decaying in time
and space. The frequency of these waves is equal to the difference frequency
of the doublet, the wavenumber is the difference between the wave numbers of
free motion of electrons with resonant energies, and the speed of their
propagation is the ratio of these quantities. The process of emission of such
electron waves lasts for the lifetime of quasi-stationary states, which can be
much longer than the wave period if the transparency of the barriers is
sufficiently low. With distance from the heterostructure, trains of difference
waves of charge and current densities decay and broaden rather slowly and can
be detected and removed from the system using electric and magnetic fields of
the corresponding structure. The system can switch to the mode of repetition
of the emission of electron waves or even to the mode of self-oscillation if
positive feedback and periodic resonant pumping of the population of the
doublet in the heterostructure are provided.

Population and decay of a doublet of quasistationary states can be provided in
different ways. We theoretically studied and simulated various mechanisms of
this process. The first of them consists in the scattering of a Gaussian
electron wave packet incident on a double-well system from the outside. In the
leading approximation, it can be described by a relatively simple quantum-
mechanical model in the language of only pure one-particle quantum states of
the scattering problem, which makes it possible to rigorously reveal the main
laws of the process and estimate the contributions of the main features. It
turned out that the amplitude of the resonant difference spatio-temporal wave
harmonic can be greater than or of the order of the amplitudes of its smooth
and high-frequency components. These results are presented below in this paper
using the example of a one-dimensional model of scattering of a Gaussian wave
packet by a structure with three identical $\delta$-barriers.

Two other mechanisms of the population and decay of the doublet of quasi-
stationary states that we studied are associated with diffraction by a double-
well heterostructure of photoelectrons arising from the action of an
ultrashort light pulse on the photocathode with the subsequent formation of a
kind of alternating photoemission current. One mechanism is provided by the
incidence of a photoelectron pulse from the outside onto a double-well
heterostructure deposited on a bulk planar photocathode, and the other is
provided by pulsed photoexcitation of electrons directly in thin layers from
the inside of the double-well structure, which itself acts as a very thin
photocathode. To describe these methods of excitation and decay, it is
necessary to consider mixed quantum states taking into account the external
high-frequency electromagnetic pumping field, as well as inelastic scattering
of electrons, using the approximate methods of the nonstationary quantum
theory of many bodies. For this we used the mathematical apparatus of the
density matrix. The results of an approximate description and calculations
will be presented in the following articles, where it will be shown that the
population of quasi-stationary levels can be determined not only by the
explicit pole features of the amplitudes of resonant scattering of electrons
on a double-well heterostructure, but also by the contribution of these
features to the photoexcitation spectrum, and even to the magnitude of the
matrix elements of electronic optical transitions upon photoexcitation from
within the heterostructure.

Here we are interested in the time and space oscillating solutions of the one-
dimensional nonstationary Schrodinger equation with the potential in the form
of wells and barriers, which are located in a finite region. To obtain such
solutions, three methods are most often used: a) direct numerical integration
in finite differences Kons2003 , b) calculation of the dynamic superposition
of solutions of the stationary Schrodinger equation for a boundary value
problem with a continuous and/or discrete spectrum Peis2008A ,Peis2008B ,
Wint1961 ,Cald2013 ,Cald2016 , describing the evolution of a wave packet, c)
representation of the solution in the form of a resonant expansion, the
members of which are the products of the Moshinsky function (associated with
the problem of a quantum gate and diffraction in time) and resonant wave
functions in the internal region of the potential, where they are finite and
normalized by specific conditions. The latter method c) was developed by G.
Garcia-Calderon et al. Camp2009 -Cald2016 . They carried out active research
of transient quantum processes in resonant tunneling and published many
articles containing interesting and important results describing the evolution
and asymptotics of electronic wave functions in different regions of time and
space. Most of these details relate to the internal region of action of the
potential, where the form of the resonant wave functions is known. In
particular, as in our papers Peis2008A -Peis2008B , the impulsive character of
the decay of quasi-stationary states was illustrated Cald2009  if the spectrum
of the initial wave packet covers a small number of quasi-resonant levels; the
flow of the wave function between successive wells was called in Cald2009
,Cald2011  the ”bouncing” and ”breathing” modes. In Cald2013 ,Cald2016 ,
general formulas for decay wave functions outside the region of action of the
potential were also written, the coordinate dependence of these functions was
determined by the Moshinsky functions, and not by the resonance Gamow wave
functions, which exponentially increase with increasing distance from the
system. However, outside the region of action of the potential, the wave
character of the behavior of the probability and current densities during the
decay of a mixture of a doublet of quasistationary states of a two-well
system, considered in our work, was not clearly distinguished and discussed in
Cald2013 ,Cald2016 .

In contrast, in this article, when describing the scattering of one or a
system of Gaussian pulses, we focus on not only the inside, but also on the
outside region of action of the double-well potential. We use method b) to
describe nonstationary probability densities and probability currents at an
arbitrary point in space, and show that in the outer region of a double
quantum well, the envelopes of these quantities demonstrate the properties of
traveling waves. Calculations by the method of continual decomposition b) are
not the calculations of a ””black box” type”, that supposedly ”provide no deep
physical insight” and ”does not provide grasp of the time evolution of the
initial state” Cald2007 -Cald2016 . There is developed by G.F. Drukarev in
1951 Druk1951 ,Baz1969  an elegant version of the saddle-point method, which
allows, within the framework of method b), to identify and estimate the
oscillating contributions of the pole features of the scattering amplitudes to
the wave function of the scattered wave packet in the internal and external
regions of the action of the scattering potential, as was done in Peis2008A
-Peis2008B .

At the end of this introduction, we emphasize that the main thing for us here
is that it is the complete system of wave functions of the stationary
scattering problem of method b) that provides a natural basis for unperturbed
states of the zero approximation for describing and calculating the
interactions of electrons with photons and with other particles in the
subsequent application of the density matrices method to the problem of
photoemission in an open system.

The article is organized as follows. Section II describes a theoretical
quantum mechanical model, provides rigorous formulas, and discusses the
optimal parameters for describing the scattering of a Gaussian wave packet by
a double quantum well. Section III presents the results of rigorous
calculations of space-time oscillations and waves of probability densities and
currents with an explanation of their characteristics. An approximate method
for the analytical identification of these characteristics is described in
Appendix A, and the clear but cumbersome expressions for the probability and
current densities obtained by this method are given in Appendix B.

##  II THEORETICAL MODEL, CALCULATION FORMULAS AND PARAMETERS

To confirm the statements made and to highlight the basic laws of the process,
in accordance with the algorithm described in our articles Peis2008A
-Peis2008B , we will analyze in detail a simple one-dimensional model, which
describes the population and subsequent decay of a doublet of quasi-stationary
states of a three-barrier heterostructure due to scattering of pulsed Gaussian
wave packets arriving from the left and having a spectral width of the order
of the distance between the levels of the doublet.

The double quantum well is assumed to be flat, the axis $x$ is directed
perpendicular to it, and the origin of coordinates is placed on its left
boundary. In order to simplify calculations and interpretation of the results,
we simulate potential barriers for electrons with the mass $m$ by three delta
functions
$U(x)=({{\hbar^{2}}\mathord{\left/{\vphantom{{\hbar^{2}}{2m}}}\right.\kern-1.2pt}{2m}})\sum\nolimits_{n=0}^{2}{\Omega\delta(x-x_{n})}$
of the same power $\Omega$ at a distance $d$ from each other at $x_{0}=0$,
$x_{1}=d$, $x_{2}=2d$. These points on the $x$ axis demarcate the four regions
shown in Fig.1. Delta barrier can be used to model real rather narrow and high
potential barrier, while fair estimate $\Omega\approx 2mU_{b}d_{b}/\hbar^{2}$,
where $U_{b}$ is the height of the barrier and $d_{b}$ is its width. The
electron energy is counted from the vacuum level $U(x)=U=0$, which is the same
to the left ($x<x_{0}=0$) and to the right ($x>2d$) of the heterostructure;
the effective flat bottom of the heterostructure wells ($0<x<2d$) is located
at the potential energy $U(x)=\tilde{U}<0$.

Figure 1: Model three-barrier heterostructure. The vertical lines with arrows
picture the $\delta$-barriers.

The basis wave functions $\psi(E,x)$ of the one-dimensional stationary problem
of scattering of a wave with the energy $E$ incident from the left are
solutions of the Schrodinger equation of the system under consideration and
are given by the expressions

$\psi(E,x)=\left\\{{\begin{array}[]{*{20}c}{A_{0E}\operatorname{e}^{ikx}+B_{0E}\operatorname{e}^{-ikx},\quad\quad\quad\quad
x<0,}\\\
\\!\\!\\!{A_{nE}e^{i\tilde{k}(x-x_{n^{\prime}})}\\!+\\!B_{nE}\operatorname{e}^{-i\tilde{k}(x-x_{n^{\prime}})},\;x_{n^{\prime}}\\!\leqslant\\!x\\!\leqslant\\!x_{n},}\\\
{A_{3E}\operatorname{e}^{ik(x-x_{2})},\quad\quad\quad\quad\quad\quad\quad
x>x_{2}.}\\\ \end{array}}\right.$ (1)

where $n=1,2$; $n^{\prime}=n-1$, $k=\hbar^{-1}\sqrt{2mE}$ is the wave number
outside the heterostructure, $\tilde{k}=\hbar^{-1}\sqrt{2m(E-\tilde{U})}$ is
the wave number inside the potential wells, $A_{jE}$ and $B_{jE}$ are the
partial amplitudes of plane monochromatic waves propagating, respectively, to
the right and left in the regions $j=0,1,2,3$, and $B_{3E}=0$ (the wave
arriving from the right is absent), $A_{0E}=\hbar^{-1}\sqrt{m/2\pi k}$ (which
provides normalization of $\psi(E,x)$ to the energy $\delta$-function).

The transfer matrix method Peis2008A -Peis2008B  allows one to connect seven
partial amplitudes of four regions by linear relations

$\left({\begin{array}[]{*{20}c}{A_{n+1E}}\\\ {B_{n+1E}}\\\
\end{array}}\right)=M_{nef}\left({\begin{array}[]{*{20}c}{A_{0E}}\\\
{B_{0E}}\\\ \end{array}}\right),$ (2)

where $M_{nef}=L^{-1}M_{\Omega}M^{n}L$, $n=0,1,2$, $M=M_{\Omega}M(d)$,

$L=\left({\begin{array}[]{*{20}c}1&1\\\ {ik}&{-ik}\\\
\end{array}}\right),\quad M_{\Omega}=\left({\begin{array}[]{*{20}c}1&0\\\
\Omega&1\\\ \end{array}}\right),$
$M(d)=\left({\begin{array}[]{*{20}c}{\cos\tilde{k}d}&{\tilde{k}^{-1}\sin{\kern
1.0pt}\,\tilde{k}d}\\\ {-\tilde{k}\sin\tilde{k}d}&{\cos\,\tilde{k}d}\\\
\end{array}}\right)$

and express all partial amplitudes in terms of the amplitude of the incident
wave $A_{0E}$. In particular, from (2) at $n=2$ we obtain expressions for the
amplitudes of the reflected $B_{0E}=rA_{0E}$ and transmitted $A_{3E}=tA_{0E}$
waves, where

$r=-\frac{{\tilde{M}_{21}}}{{\tilde{M}_{22}}},\quad
t=\frac{{\det\tilde{M}}}{{\tilde{M}_{22}}}$ (3)

$r$ \- reflection amplitude, $t$ \- transmission amplitude, $\tilde{M}_{il}$
\- matrix elements of a two-dimensional $(i,l=1,2)$ effective transfer matrix
$\tilde{M}\equiv M_{2ef}=L^{-1}M_{\Omega}M^{2}L$.

Hence, it can be seen that all partial amplitudes (except for $A_{0E}$), as
well as the amplitudes of reflection and transmission, can have pole
singularities, which are determined by the zeros of the matrix element
$\tilde{M}_{22}=0$, that is, they can have a resonance character near quasi-
stationary levels. Complex roots of the equation $\tilde{M}_{22}=0$ and quasi-
stationary levels are grouped into doublets
$E_{p}=E^{\prime}_{p}+iE^{\prime\prime}_{p}$ $(p=1,2)$ (Fig.2a). The real
parts of pairs of close roots $E^{\prime}_{1}=\operatorname{Re}E_{1}$ and
$E^{\prime}_{2}=\operatorname{Re}E_{2}$ give the energies of quasi-stationary
levels. The imaginary parts of the roots $E_{1}^{\prime\prime}=ImE_{1}$ and
$E_{2}^{\prime\prime}=ImE_{2}$ determine the spectral widths and lifetimes
$\tau_{1}=\hbar/E_{1}^{\prime\prime}$ and
$\tau_{2}=\hbar/E_{2}^{\prime\prime}$ of these quasi-stationary states. The
dependence $|\tilde{M}_{22}|^{-1}$ on the real energy $E$ in the vicinity of
the doublet has two close peaks, the widths of which are of the order of
$E_{1}^{\prime\prime}$ and $E_{2}^{\prime\prime}$ (Fig.2b).

Figure 2: (Color online) a) the zeros of $\tilde{M}_{22}=0$ lie in the lower
half-plane of the complex energy and for the doublet lowest above the vacuum
level of the heterostructure in Fig. 1 (at $d=125$ Å, $\tilde{U}=-4$ eV,
$\Omega=18.9$ Å${}^{-1}=10$ a.u.) are equal $E_{1}=(0.647-i1.576\cdot
10^{-4})$ eV and $E_{2}=(0.655-i1.576\cdot 10^{-4})$ eV (i.e., the lifetimes
of quasi-stationary states
$\tau_{1}=\hbar/ImE_{1}={\text{4}}{\text{.175}}\cdot 10^{-12}$ s and
$\tau_{2}=\hbar/ImE_{2}={\text{4}}{\text{.175}}\cdot 10^{-12}$ s; b) the red
curve with two peaks and the right scale depict the dependence
$|\tilde{M}_{22}|^{-1}$ on the real energy $E$ for this system, the maxima
$|\tilde{M}_{22}|^{-1}$ are at $E_{1}=0.647$ eV, $E_{2}=0.655$ eV; the black
curve with one maximum and the left scale depict the square of the modulus of
the spectral function $c_{E}$ of the incident wave packet at optimal for the
heterostructure of Fig. 1 and Fig. 2a parameters:
$E_{C}=\hbar^{2}k_{C}^{2}/2m=0.651$ eV, $k_{C}=0.414$ Å${}^{-1}=0.219$ a.u.,
$x_{C}=-5000$Å, $\Delta x=400$Å, $C_{0}=3.755\cdot 10^{3}$
m${}^{-1/2}=2.73\cdot 10^{-2}$ a.u.,
$n(x_{C},0)=|\Psi(x_{C},0)|^{2}=C_{0}^{2}=1.41\cdot 10^{7}$ m-1 $=7.46\cdot
10^{-4}$ a.u. (expressions (4) and (5)). Atomic unit of length 1 a.u.
$(x)=0.529\cdot 10^{-10}$ m, atomic unit of time 1 a.u. $=2.419\cdot 10^{-17}$
s, atomic unit of probability density 1a.u. $(n)=1.89\cdot 10^{10}$ m-1,
atomic unit of probability current density 1 a.u. $(j)=4.1\cdot 10^{16}$ s-1.

Let an electronic Gaussian wave packet fall on the heterostructure from the
left, the wave function of which at the initial time $t=0$ has the form

$\Psi(x,\;0)=C_{0}\exp\left({ik_{C}x-\frac{{(x-x_{C})^{2}}}{{2(\Delta
x)^{2}}}}\right),$ (4)

where $C_{0}=1/\sqrt{\Delta x\sqrt{\pi}}$, $x_{C}<0$ is the initial coordinate
of the center of the packet, $\Delta x$ \- the initial spatial width of the
packet, $C_{0}$ \- the initial amplitude of the packet, $k_{C}$ \- the wave
number of the spectral center of the packet, which corresponds to the energy
$E_{C}=\hbar^{2}k_{C}^{2}/2m$, in the absence of a scattering potential, the
packet moves with the group velocity $v_{C}=\hbar k_{C}/m$.

The spectral function of the wave packet (4) is determined from the stationary
wave functions $\psi(E,x)$ of the scattering problem

$c_{E}=\int_{-\infty}^{\infty}\psi^{*}(E,x)\Psi(x,0)dx.$ (5)

The parameters of the wave packet (4) are chosen so that at $t=0$, the packet
is located far enough to the left of the heterostructure and that its spectral
function $c_{E}$ also has an almost Gaussian form and overlaps mainly only two
considered quasi-stationary levels (Fig.2b). It was shown in Peis2008A  that
this can be easily done by satisfying the conditions

$k_{C}^{-1}\ll\Delta x\ll|x_{C}|\ll k_{C}(\Delta x)^{2},$ (6)

then

$c_{E}\approx\left\\{{\begin{array}[]{*{20}c}{\dfrac{1}{\hbar}\sqrt{\dfrac{{m\Delta
x}}{{\sqrt{\pi}k}}}{e}^{-\frac{{(\Delta
x)^{2}}}{2}(k-k_{C})^{2}}e^{ix_{C}\left({k_{C}-k}\right)},\;E\geqslant 0}\\\
{0,\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad E<0}\\\
\end{array}}\right.$ (7)

and the evolution of the packet is mainly determined by the contribution of
the energy region $E_{\min}<E<E_{\max}$, which includes the selected doublet,
but is far from the neighboring doublets. Therefore, at subsequent times, the
nonstationary wave function is given by the integral

$\Psi(x,t)=\int\limits_{E_{\min}}^{E_{\max}}{c_{E}e^{-iEt/\hbar}\psi(E,x)dE}$
(8)

We are interested in the probability density

$n(x,t)=\left|{\Psi(x,t)}\right|^{2}=\int{\int{\rho_{EE^{\prime}}(t)}n_{EE^{\prime}}(x)dEdE^{\prime}}$
(9)

and the probability current density

$\begin{gathered}j(x,t)=\frac{{i\hbar}}{{2m}}\left({\Psi(x,t)\frac{{d\Psi^{*}(x,t)}}{{dx}}-\Psi^{*}(x,t)\frac{{d\Psi(x,t)}}{{dx}}}\right)=\hfill\\\
\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad=\int{\int{\rho_{EE^{\prime}}(t)j_{EE^{\prime}}(x)}dEdE^{\prime}},\hfill\\\
\end{gathered}$ (10)

where

$n_{EE^{\prime}}(x)=n_{E^{\prime}E}^{*}(x)=\psi(E,x)\psi^{*}(E^{\prime},x)$
(11)
$j_{EE^{\prime}}(x)=\frac{{i\hbar}}{{2m}}\left({\psi(E,x)\frac{{d\psi^{*}(E^{\prime},x)}}{{dx}}-\psi^{*}(E^{\prime},x)\frac{{d\psi(E,x)}}{{dx}}}\right)$
(12)

are ”matrix elements” of density $n_{EE^{\prime}}$ and current density
$j_{EE^{\prime}}(x)$ Land1977 , and

$\rho_{EE^{\prime}}(t)=\rho_{E^{\prime}E}^{*}(t)=c_{E}c_{E^{\prime}}^{*}e^{-i(E-E^{\prime})t/\hbar}$
(13)

is ”density matrix” in a ”pure” quantum-mechanical state $\Psi(x,t)$.

When quasi-stationary states are excited by electromagnetic radiation with
subsequent photoemission, especially from ”inside” the heterostructure, the
states of electrons are ”mixed” and the elements of the density matrix do not
have the form (13), but must be determined from the corresponding kinetic
equations.

The density of the electric charge is $\rho_{e}(x,t)=en(x,t)$ and of the
electric current is $j_{e}(x,t)=ej(x,t)$, where $e$ is the electron charge.
Differentiating (9), (10) and applying the nonstationary Schrodinger equation,
it is easy to make sure that the law of conservation of the probability
density and charge ${{\partial j(x,t)}\mathord{\left/{\vphantom{{\partial
j(x,t)}{\partial x=}}}\right.\kern-1.2pt}{\partial x=}}-{{\partial
n(x,t)}\mathord{\left/{\vphantom{{\partial n(x,t)}{\partial
t}}}\right.\kern-1.2pt}{\partial t}}$ is satisfied at every point in space at
every moment of time.

##  III OSCILLATIONS AND WAVES OF CHARGE AND CURRENT DENSITIES. EVALUATION
FORMULAS AND CALCULATION RESULTS

###  III.1  Pole Contribution Estimation

Our main goal here is to demonstrate and explain the regular space-time
oscillations of the quantities $n(x,t)$ and $j(x,t)$, caused by the population
and decay of quasi-stationary states of a double quantum well after scattering
of a Gaussian wave packet by this well. Substituting (1) and (4) in (5), (8) -
(10) and performing numerical integration in the range of interest, one can
obtain a series of figures (Fig. 3-Fig. 10) illustrating the details of the
phenomenon.

These oscillations can be described analytically with sufficient accuracy by
estimating the integral (8) using the developed by G.F. Drukarev in 1951
Druk1951  a variant of the saddle point (the fastest descent) method, that
allows one to select and evaluate the contributions of the main poles of the
scattering amplitudes in the desired integral value Baz1969 , as was done in
Peis2008A -Peis2008B . A brief explanation of the essence of this method and
the main formulas for calculating the contributions of the saddle points and
poles of the integrands are given in Appendix A.

The result of applying the saddle point method strongly depends on the width
and position of the saddle in the complex plane, its distance from the origin,
as well as on the form of the spectral function of the packet, scattering
amplitudes, and on the location and type of their features. The position on
the complex plane of the mentioned poles, branch points and other features of
the characteristics of stationary scattering and spectral function does not
depend on time and coordinates (see (Fig.16) in Appendix A). However, the
saddle points, and with them the lines of type I, for a fixed $x$ move with
time $t$ to the origin, usually according to the law $k_{S}\sim 1/t$,
capturing the singular points of stationary scattering in sectors II or III.
This determines the appearance of threshold conditions for $x$ and $t$, under
which the singularities make a noticeable contribution to the integrals,
providing the manifestation in the form of an envelope of the wave function
$\Psi(x,t)$ of various moving maxima, fronts, etc.

In the case under consideration, the saddle points of the exponents in the
integrand (8) are responsible for the formation of thresholds and leading
pulses of reflection and transmission of the main body of the scattered wave
packet, in principle, their contribution can be estimated using (A2). The pole
features of $\psi(E,x)$ (i.e., of amplitudes $A_{jE}$ and $B_{jE}$) are
responsible for the formation of the modulation profile of the functions
$\Psi(x,t)$, $n(x,t)$ and $j(x,t)$, which can oscillate and slowly decay in
time and space due to the rather slow oscillatory decay of the superposition
of quasi-stationary states in a double quantum well, which turned out to be
populated after the departure of the main body of the wave packet. Their main
contribution is proportional to the sum of the residues (A3) at the poles of
the integrands (8). In the space-time regions of the steady oscillations, far
enough beyond the thresholds and leading scattering pulses of the main body of
the packet (when the saddle point and straight line I turn out to be to the
left of the spectral center of the initial wave packet and the poles of the
scattering amplitudes), these pole contributions can be large in comparison
with other contributions and the wave function is approximately proportional
to superpositions of damped traveling waves

$\Psi(x,t)\approx\begin{cases}&\\!\\!\\!\Psi_{0}(x,t)+\sum\nolimits_{p=1}^{2}{\tilde{B}_{0E_{p}}e^{-ik_{p}x-iE_{p}t/\hbar},\quad
x<0},\\\
&\\!\\!\\!\Psi_{n}(x,t)+\sum\nolimits_{p=1}^{2}\left(\tilde{A}_{nE_{p}}e^{i\tilde{k}_{p}(x-x_{n^{\prime}})}+\tilde{B}_{nE_{p}}e^{-i\tilde{k}_{p}(x-x_{n^{\prime}})}\right)e^{-iE_{p}t/\hbar},\quad
n=1,2,\;\;n^{\prime}=n-1,\;\;x_{n^{\prime}}\leq x<x_{n},\\\
&\\!\\!\\!\Psi_{3}(x,t)+\sum\nolimits_{p=1}^{2}{\tilde{A}_{3E_{p}}e^{ik_{p}(x-x_{2})-iE_{p}t/\hbar}},\quad
x>x_{2}.\end{cases}$ (14)

The terms $\Psi_{0}(x,t)$, $\Psi_{n}(x,t)$, $\Psi_{3}(x,t)$ come from the
contributions of those parts of the integration contour of the fastest descent
that are far from the poles of the scattering amplitudes; they are smooth
functions of $x$ and $t$ with relatively small magnitude in the regions under
consideration Peis2008A -Peis2008B . The coefficients
$\tilde{A}_{nE_{p}}=|\tilde{A}_{nE_{p}}|e^{i\alpha_{np}}$ ($n=1,2,3$ ) and
$\tilde{B}_{nE_{p}}=|\tilde{B}_{nE_{p}}|e^{i\beta_{np}}$ ($n=0,1,2$) are
proportional to the residues of the integrand (8) at the poles
$E_{p}=E^{\prime}_{p}+iE^{\prime\prime}_{p}$ ($p=1,2$) of the partial
amplitudes $A_{nE}$ and $B_{nE}$ from (1) with taking into account the
explicitly written coordinate-time exponents, and the complex wave numbers are
equal $k_{p}\equiv k(E_{p}{\kern
1.0pt})=\hbar^{-1}\sqrt{2mE_{p}}=k^{\prime}_{p}+ik^{\prime\prime}_{p}$ and
$\tilde{k}_{p}\equiv\tilde{k}(E_{p})=\hbar^{-1}\sqrt{2m(E_{p}-\tilde{U})}=\tilde{k}^{\prime}_{p}+i\tilde{k}^{\prime\prime}_{p}$
(we choose the root branches so as to satisfy the physical conditions of
damping waves in space). We are interested in systems that provide a
sufficiently slow damping, for which
$E^{\prime}_{p}\gg\left|{E^{\prime\prime}_{p}}\right|$ and
$k^{\prime}_{p}\gg|k^{\prime\prime}_{p}|$.

Below we present the results of numerical calculations using exact formulas
(1) and (8) - (12) the quadratic in $\Psi(x,t)$ values of the probability
density $n(x,t)=\left|{\Psi(x,\;t)}\right|^{2}$ and current of the probability
density $j(x,t)$ of electrons inside and outside the considered
heterostructure for the parameters of the wave packet and heterostructure,
which are given in the caption to Fig.2. These calculations show that
approximation (15) provides a reasonable interpretation and estimation of the
considered oscillatory and wave effects.

###  III.2 The region inside the double quantum well

Inside each of the $n=1,2$ wells of heterostructure at $x_{n-1}\leq x\leq
x_{n}$, substituting the expressions of the second line (1) into the exact
formulas (8) - (11) and performing numerical integration, we obtain figures
that demonstrate the probability of finding an electron inside a double
quantum well (Fig.3 and Fig.4) , quasiperiodic flow between the wells of the
wave function and the probability density (Fig.3 and Fig.5), as well as the
corresponding behavior of the probability current density (Fig.6).

The probability of finding an electron inside a double well
$P(t)=\int_{0}^{2d}{|{\Psi(x,\;t)}|^{2}dx}$ with time first increases rather
quickly and then decreases relatively slowly according to a law close to
exponential, while similar probabilities of finding an electron inside each of
the two wells $P_{n}(t)=\int_{(n-1)d}^{nd}{|{\Psi(x,\;t)}|^{2}dx}$ oscillate
with the difference frequency of the doublet
$\omega\equiv\omega_{12}=(E^{\prime}_{2}-E^{\prime}_{1})/\hbar$ and with a
period $T=2\pi/\omega_{12}=2\pi\hbar/(E^{\prime}_{2}-E^{\prime}_{1})\approx
5.27\cdot 10^{-13}$ s$\approx 2.18\cdot 10^{4}$ a.e.$\approx 22000$ a.e.
almost in antiphase with each other (Fig.3):

Figure 3: (Color online) Time dependence of the probabilities of finding an
electron: inside the double well $P(t)$ (black line), in the left well
$P_{1}(t)$ (red line), in the right well $P_{2}(t)$ (blue line). Time is in
atomic units 1 a.u$=2.419\cdot 10^{-17}$ s.

The population of quasi-stationary states occurs approximately during the time
of reflection and transmission of the main body of the wave packet, which is
equal in order of magnitude to $\Delta t\sim d/v_{C}=md/\hbar k_{C}\sim
10^{3}$ a.e.$\ll\tau$, where $v_{C}=\hbar k_{C}/m$. Note that the rate of
increase in the quantity $P(t)$ (the time $\Delta t$ of penetration of an
electron into the well) depends much weaker on the quantity $\Omega$ than the
rate of the subsequent decrease (the time $\tau$ of decay of quasi-stationary
states). Exponential approximation of the decay part of the curve Fig.3 gives
the relaxation time of the population of quasi-stationary states in the
heterostructure $\tau=3.87\cdot 10^{-12}$ s $=1.6\cdot 10^{5}$ a.u. and
effective blur $\hbar/\tau\approx 1.701\cdot 10^{-4}$ eV which is close to
$\operatorname{Im}E_{1}=\operatorname{Im}E_{2}=1.576\cdot 10^{-4}$ eV (see
data Fig.2). The area under the curve $P(t)$ and the maximum value of the
probability of finding an electron inside the double well $P_{\max}$ change
nonmonotonically with increasing value $\Omega$ due to the nonmonotonic
dependence of the transmission coefficients of the $\delta$-barrier: the value
$P_{\max}$ first increases to a certain maximum value at $\Omega\sim d^{-1}$,
and then decreases (Fig.4), but the length of the exponential ”tail” $P(t)$ in
(Fig.3) monotonically increases. The latter is consistent with the statement
proved in our works Peis2008A -Peis2008B  that in a heterostructure formed by
$\delta$-barriers of the same power $\Omega$ located at a distance $d$ from
each other, the lifetime $\tau_{n}$ of the $n$-th quasi-stationary state
increases with increasing $\Omega$ (and $d$), but decreases with increasing
$n$ as $\tau_{n}\propto m\Omega^{2}d^{4}/(n+1)^{3}$. Hence it follows that for
the maximum realization of the studied effects, it is desirable to select the
optimal values of all parameters of the problem (see the caption to (Fig.2)),
so that both $P_{\max}$ and $\tau_{n}$ are as large as possible.

Figure 4: (Color online) Maximum probability of finding a particle inside a
double well $P_{\max}$, depending on the barrier power $\Omega$ with other
fixed parameters (Fig.2).
Figure 5: (Color online) Coordinate-time dependence of the values (color scale
on the right) of the probability density $n(x,t)$ of finding a particle inside
the double well $0\leq x\leq 2d$. Coordinate in angstroms Å, time and
probability density in atomic units.
Figure 6: (Color online) Coordinate-time dependence of the values (color scale
on the right) of the probability current density $j(x,t)$ of a particle inside
the double well $0\leq x\leq 2d$. Coordinate in angstroms Å, time and
probability density in atomic units.

Figures (Fig. 5) and (Fig. 6) are quite well explained by (9), (10) and the
second line (14), at such values of $t$ and $x$, at which it is possible to
neglect $\Psi_{n}(x,t)$. Complete analytical expressions $n(x,t)$ and $j(x,t)$
in this approximation are given by formulas (21) and (22), which are written
out in Appendix B. It can be seen from (21) and (22) that, inside the double
well, the quantities $n(x,t)$ and $j(x,t)$ undergo spatio-temporal
oscillations and weak exponential decay with time. The terms in the first
lines of both expressions (21) and (22) almost do not change with time $t$ and
coordinate $x$, the terms in the second lines weakly decay with time, but
quickly change along the coordinate with spatial periods
$\tilde{\lambda}_{p}=\pi/\tilde{k}^{\prime}_{p}\sim\pi/k_{C}$, which are small
in comparison with the width of the wells $d$. The last four lines in both
expressions (21) and (22) describe plane waves traveling to the right and left
inside the wells, the corresponding wave-like temporal oscillations $n(x,t)$
and $j(x,t)$ occur with the difference frequency of the doublet
$\omega\equiv\omega_{12}=(E^{\prime}_{2}-E^{\prime}_{1})/\hbar$. In this case,
the terms in the third and fourth lines describe traveling waves, the
wavelength of which
$\tilde{\lambda}_{-}=2\pi|{\tilde{k}^{\prime}_{1}-\tilde{k}^{\prime}_{2}}|^{-1}$
is large in comparison with the width of the wells $d$; therefore, such terms
inside the wells are almost independent of $x$ at a fixed $t$, the phase
velocity of these waves is
$\tilde{v}_{-}=\omega\tilde{\lambda}_{-}/2\pi=(E^{\prime}_{2}-E^{\prime}_{1})/\hbar|{\tilde{k}^{\prime}_{1}-\tilde{k}^{\prime}_{2}}|=7.189\cdot
10^{5}$ m/s. However, the fifth and sixth lines describe short-wavelength
waves traveling towards each other, the wavelength of which
$\tilde{\lambda}_{+}=2\pi\left|{\tilde{k}^{\prime}_{1}+\tilde{k}^{\prime}_{2}}\right|^{-1}\sim\pi/k_{C}$
is small compared to the width $d$ of the wells, and the phase velocity of
such waves
$\tilde{v}_{+}=\omega\tilde{\lambda}_{+}/2\pi=(E^{\prime}_{2}-E^{\prime}_{1})/\hbar|\tilde{k}^{\prime}_{1}+\tilde{k}^{\prime}_{2}|\approx
2.202\cdot 10^{3}$ m/s, is small compared to $\tilde{v}_{-}$. Note also that
in expression (21) all terms have almost the same order of magnitude,
therefore, in the figure (Fig.5), the coordinate dependence $n(x,t)$ inside
the wells is dominated by short-wavelength components with a wavelength
$\tilde{\lambda}_{+}\sim\pi/k_{C}$, which rather abruptly change their
amplitude between the wells. On the contrary, in expression (22) such short-
wave components make a relatively small contribution to the coordinate
dependence of $j(x,t)$ in comparison with long-wave components
$\tilde{\lambda}_{-}$: the fifth and sixth lines of expression (22) contain a
small factor $|{\tilde{k}^{\prime}_{1}-\tilde{k}^{\prime}_{2}}|\ll\,\,k_{C}$,
and the third and fourth lines of expression (22) contain a large factor
$|{\tilde{k}^{\prime}_{1}+\tilde{k}^{\prime}_{2}}|\approx 2\,k_{C}$,
therefore, on Figure (Fig. 6) the coordinate dependence of $j(x,t)$ inside the
wells is very smooth with a break at the boundaries of the wells.

###  III.3  The region outside the double quantum well on the left

Similarly, to the left of the double well at $x<0$, substituting the
expression of the first line (1) into the exact formulas (8)-(12) and
performing numerical integration, we obtain figures that demonstrate the
decaying probability density waves (Fig.7) and current density waves traveling
to the left (Fig.8).

Figure 7: (Color online) Wave coordinate-time dependence of the values (color
scale on the right) of the probability density $n([,t)$ of finding a particle
in the corresponding points of the left half-space $x<0$. Coordinate $x$ in
angstroms Å, time and probability density in atomic units.
Figure 8: (Color online) Wave coordinate-time dependence of the values (color
scale on the right) of the current probability density $j(x,t)$ of finding a
particle in the corresponding points of the left half-space $x<0$. Coordinate
$x$ in angstroms Å, time and probability density in atomic units.

These figures (Fig.7) and (Fig.8) are also quite well explained by (9), (10)
and the first line (15), at such values of $t$ and $x$, at which it is
possible to neglect $\Psi_{0}(x,t)$, that gives the main pole contributions to
$n(x,t)$ and $j(x,t)$ in the form of analytical formulas (23) and (24) given
in Appendix B, which describe the probability density and probability current
waves traveling to the left. Figures (Fig.7) and (Fig.8) show that to the left
of the double well, the wave part of $j(x,t)$ changes almost in antiphase to
the wave part of $n(x,t)$. This is explained by the minus sign in (24) and the
fact that we have $k^{\prime}_{1}\approx k^{\prime}_{2}\approx k_{C}=0.219$
a.u.

###  III.4 The region outside the double quantum well on the right

In the same way, to the right of the double well at $x>x_{2}$, after
substituting the expression of the third line (1) into the exact formulas (8)
- (12) and numerical integration, figures are obtained that demonstrate the
decaying waves of the probability density (Fig.8) and the probability current
density traveling to the right.

Figure 9: (Color online) Wave coordinate-time dependence of the values (color
scale on the right) of the probability density $n(x,t)$ of finding a particle
in the corresponding points of the right half-space $x>x_{2}$. Coordinate $x$
in angstroms Å, time and probability density in atomic units.

For the coordinate-time wave dependence of the current density $j(x,t)$ of a
particle at the points of the right half-space $x>x_{2}$, the figure
qualitatively looks like Fig.9, that is, to the right of the double well, the
wave parts of $n(x,t)$ and $j(x,t)$ change almost in phase, but in atomic
units $j(x,t)$ is less than $n(x,t)$ about a decimal order of magnitude.

These dependences are also reasonably well explained by (9), (10) and the
third line (14) at such values of $t$ and $x$ for which it is possible to
neglect $\Psi_{3}(x,t)$, that gives the main pole contributions to $n(x,t)$
and $j(x,t)$ in the form of analytical formulas (25) and (26) given in
Appendix B, which describe the probability density and probability current
waves traveling to the right.

The noted similarity and difference in behavior of $j(x,t)$ and $n(x,t)$ is
explained by the presence in (26) in comparison with (25) of factors
containing $k^{\prime}_{1}\approx k^{\prime}_{2}\approx k_{C}=0.219$ a.u.

###  III.5 The complete picture of generation of probability density and
current waves

Thus, inside a heterostructure in the form of a double quantum well,
oscillations of the electron density and current with the difference frequency
of the doublet $\omega\equiv\omega_{12}=(E^{\prime}_{2}-E^{\prime}_{1})/\hbar$
can occur, which looks like a periodic overflow of the electron wave function
$\Psi(x,t)$ and the probability density $n(x,t)$ between the wells (in time
almost in antiphase to the left and to the right), so that outside the
heterostructure the probability density waves and currents density waves
outgoing to the left and to the right are formed. In this case, outside the
heterostructure, quadratic in magnitude $\Psi(x,t)$ values $n(x,t)$ and
$j(x,t)$ oscillate in time with the difference frequency of the doublet
$\omega\equiv\omega_{12}$, and in space with a wavenumber equal to the
difference $k_{12}=k^{\prime}_{2}-k^{\prime}_{1}$, slowly decaying with
decrements determined by the imaginary parts of $E_{p}$ and $k_{p}$.

Waves of $n(x,t)$ and $j(x,t)$ move to the left and to the right with the same
velocities $\operatorname{v}\approx\lambda/T=4.79\cdot 10^{5}$ m/s, where the
wavelength is $\lambda=2\pi/k_{12}=2\pi/|k^{\prime}_{2}-k^{\prime}_{1}|\approx
2480$Å, and the period of the waves is
$T=2\pi/\omega_{12}=2\pi\hbar/(E^{\prime}_{2}-E^{\prime}_{1})\approx 2.18\cdot
10^{4}$ a.u. $\approx 5.27\cdot 10^{-13}$s. The generation of these waves can
be represented on Fig.10 by level lines on the $t-x$-plane.

Figure 10: (Color online) The calculated relief levels of the quantities a)
probability density $n(x,t)$, b) probability current density $j(x,t)$ in
accordance with the color scales to the right of the figures. On this scale,
the region inside the double well is not allowed, and the two lower stripes on
the left qualitatively represent the main bodies of the incident and reflected
wave packets having $n\sim 10^{-3}$ a.u. and $j\sim 10^{-5}$ a.u. Coordinate
$x$ in angstroms Å, time, probability density and probability current density
in atomic units.

##  IV PROLONGATION AND AMPLIFICATION OF WAVE GENERATION

In the system under consideration, it is possible to organize a mode of
repetition or amplification of the process of radiation of electron
probability density and probability current density waves. If we know the
space-time periods of the waves under study, then in order to prolong the
radiation process and increase the amplitude of the density and current waves,
we can form a quasiperiodic sequence of wave packets similar to the original
packet (4) to the left of the double quantum well and send them in such a way
as to provide an additional resonant pumping of the population of quasi-
stationary states of the heterostructure. To prepare such a coherent chain of
pulses, one can, for example, use two methods:

1) The first of these methods consists in aligning along the axis $x$ of an
equidistant sequence of identical pulses with a spatial period close to a
value that is a multiple of the doubled resonant difference wavelength
$\lambda=2\pi/|k_{12}|$. At the initial moment of time $t=0$, the wave
function should be prepared in the form of a spatial sequence of $N$ identical
wave packets following the head packet (4), in which the initial coordinates
of the centers $x_{Cn}=x_{C}-n\delta x$ are shifted relative to $x_{C}$
($\delta x$ is shift period; $n=1,2,...,N$). If these packets almost do not
overlap and for each of them conditions (6) and (7) are fulfilled with
replacement $x_{C}\to x_{Cn}$, then instead of the spectral function $c_{E}$
in the integrand (8) there appears the spectral function $c_{N}(E)$ of the
entire sequence of packets, which in this case is given by the sum

$c_{N}(E)=c_{E}\sum\limits_{n=0}^{N-1}{{\text{e}}^{-in\delta x\delta
k}}=c_{E}{\text{e}}^{-i(N-1)\delta x\delta k/2}y_{N}\left({\frac{{\delta
x\delta k}}{2}}\right)$ (15)

where $\delta k=k_{C}-k$, and an interference function

$y_{N}(z)\equiv\frac{{\sin(Nz)}}{{\sin z}}$ (16)

is periodic in $z$ with a period $2\pi$ and has the main extrema $y_{N\max}=N$
at the values of the argument $z_{\max}=s\pi$, where $s$ is an integer. In the
theory of diffraction gratings, it describes an increase in the amplitude of
the resultant wave at its main resonance maxima by $N$ times and its intensity
by $N^{2}$ times. In expression (16) we have $z(k)\equiv\delta x\delta k/2$
and it is obvious that at the main extrema of the function $y_{N}(z)$ all
exponentials are equal to one under the sum sign, and the entire sum is equal
$N$. In our case, the integration of (8) with $c_{N}(E)$ instead of $c_{E}$
provides a significant contribution of the poles
$k_{p}=k^{\prime}_{p}+ik^{\prime\prime}_{p}$ of the scattering amplitudes, as
for one wave packet, therefore, due to the superposition of $N$ resonant
diffracted waves, the function $y_{N}(z)$ can also provide up to a close to
$N^{2}$-fold (on conditions $\left|{k^{\prime\prime}_{p}}\right|\ll
k^{\prime}_{p}$) amplification of the wave amplitudes $n(x,t)=|\Psi(x,t)|^{2}$
and $j(x,t)$ in comparison with their values for one ($N=1$) wave packet in
the corresponding intervals of $x$ and $t$. This takes place if the points
$k_{m}$ of the main extrema of the function $y_{N}(z(k))$ are close to the
points $k_{1}\approx k^{\prime}_{1}$ and $k_{2}\approx k^{\prime}_{2}$ of the
resonance maxima of the moduli of the amplitudes of stationary scattering on
the double well, which can be ensured by selecting the value $\delta x$.
Indeed, the period of $y_{N}(z(k))$ by argument $k=k_{C}-\delta k$ is equal to
$4\pi/\delta x$, when $k$ is counted from $k_{C}$, and since our spectral
center $k_{C}$ of the original wave packet is located almost in the middle
between the resonance wave numbers $k_{1}\approx k^{\prime}_{1}$ and
$k_{2}\approx k^{\prime}_{2}$, then at the main extrema there should be
$|\delta k|=|k_{C}-k_{m}|=|k_{12}|/2=\pi/\lambda$, so favorable for maximum
amplification values of the shift periods should be close to $\delta x\approx
4\pi s/k_{12}=2s\lambda$ (Fig.11). Weaker amplification of waves can occur at
such values of $\delta x$ for which
$N>|{y_{N}(z(k_{1}))}|\approx|{y_{N}(z(k_{2}))}|\geq 1$, and the weakening of
the sum wave will occur at $|{y_{N}(z(k_{1}))}|\approx|{y_{N}(z(k_{2}))}|<1$.

Figure 11: (Color online) Spectral functions $c_{N}(E)$ at $s=2$ favorable for
maximizing wave amplification versus resonance peaks $|\tilde{M}_{22}|^{-1}$
(cf. (Fig.2b)): a) for $N=2$ resonance are given by the first main maxima of
$|c_{2}|^{2}$, b) for resonance are given by the second main maxima of
$|c_{3}|^{2}$. The curves are brought to the same unit scale for ease of
comparison
Figure 12:  The spatial profile of the resonant amplification of density and
current probability waves by a sequence of three ($N=3$) identical wave
packets shifted relative to each other by a distance of $\delta x\,\approx
4\pi s/k_{12}=2s\lambda$ = 9828 Å, $s=2$ at the moment of time $t=3\cdot
10^{5}$ a.u. $=7.26\cdot 10^{-12}$ s (the main bodies of the reflected packets
are cut off because they are not of interest to us, they are about an order of
magnitude larger than the vertical size of the panels). Coordinate in
angstroms Å, time, probability density and probability current density in
atomic units.

To find the period $\delta x$, we also used a more general method, which is
also valid in cases where conditions (6) and (7) are violated for all
sequential packets. Namely, the period $\delta x$ was determined numerically
from the points of intersection on the $E-\delta x$-plane of straight lines
$E=E_{1}$, $E=E_{2}$ with the lines of the main extrema of the spectral
function of the entire sequence of wave packets, parametrically depending on
$\delta x$, and calculated not according to (16), but according to the general
formula (5), in which $\Psi(x,0)$ it is taken equal to the initial the wave
function of the entire sequence of wave packets.

Figures (Fig.12) and (Fig.13) demonstrate the resonant coherent amplification
of the probability density and probability current waves by a spatial sequence
of three ($N=3$) identical wave packets shifted in space by $\delta x\,\approx
4\pi s/k_{12}=2s\lambda$ at $s=2$.

Figure 13:  The time profile of the resonant amplification of the probability
density and current waves by a sequence of the same three ($N=3$) identical
wave packets (at $s=2$), as in Fig.12 using the example of points $x=x_{0}=0$
on the left and $x=x_{2}=2d$ on the right boundaries of the double quantum
well. The time, probability density and probability current density in atomic
units.

2) The second method involves the creation of almost identical pulse wave
packets in one place sequentially in time with a period close to a multiple of
the resonant difference time period $T=2\pi/\omega_{12}$ of the wave using the
appropriate time aperture function. In this case, the source of particles
should be arranged in such a way that coherent wave impulses of the form (9),
(10), which follow each other, appear sequentially at the same place with a
time period $\delta t$. If we assume that these packets almost do not overlap
and for each of them conditions of the type (6) and (7) are satisfied, then in
each time interval $(N-1)\delta t\leq t\leq N\delta t$ when the $N$ pulses are
excited (and also for $t>(N_{0}-1)\delta t$ if the pulse with the number
$N=N_{0}$ is the last) in the expressions (9) and (10), instead of the product
of spectral functions $c_{E}c_{E^{\prime}}^{*}$, a function of the entire
sequence of pulses $(c_{E}c_{E^{\prime}}^{*})_{N}$ appears, which is now given
by the sum

$(c_{E}c_{E^{\prime}}^{*})_{N}=c_{E}c_{E^{\prime}}^{*}\sum_{n=0}^{N-1}e^{in\delta
t(E-E^{\prime})/\hbar}=c_{E}c_{E^{\prime}}^{*}e^{i(N-1)z^{\prime}}y(z^{\prime})$
(17)

The interference function $y(z^{\prime})$ has the same form (17), but now its
argument is equal $z^{\prime}\equiv z^{\prime}(E-E^{\prime})=\delta
t(E-E^{\prime})/2\hbar$, as above the function $y(z^{\prime})$ is periodic
with the period $2\pi$ and has main extrema $|y_{\max}|=N$ at the values of
the argument $z^{\prime}_{\max}=s\pi$, where $s$ is an integer. Integration in
(9) and (10) with $(c_{E}c_{E^{\prime}}^{*})_{N}$ instead of
$c_{E}c_{E^{\prime}}^{*}$ provides the determining contribution of the poles
$E_{p}=E^{\prime}_{p}+iE^{\prime\prime}_{p}$ of the scattering amplitudes, so
that in the corresponding intervals $x$ and $t$, in which the $N$ pulses of
$n(x,t)$ and $j(x,t)$ are already excited and undergo diffraction, and due to
their superposition the function $y(z^{\prime})$ can now provide only to near
$N$-fold amplification of the wave amplitudes $n(x,t)=|{\Psi(x,\;t)}|^{2}$ and
$j(x,t)$ (on conditions $|E^{\prime\prime}_{p}|\ll E^{\prime}_{p}$, otherwise
due to attenuation, the amplification is weaker) compared to their values for
one ($N=1$) wave packet. This takes place if equalities $E-E^{\prime}\approx
E^{\prime}_{2}-E^{\prime}_{1}=\hbar\omega_{12}=2\pi\hbar/T=2\pi\hbar s/\delta
t$ are satisfied at the poles $E_{p}=E^{\prime}_{p}+iE^{\prime\prime}_{p}$,
which can be ensured by selecting a value $\delta t$ close to a value that is
a multiple of the period of these waves $\delta t=\delta T$. Weaker
amplification of waves can occur at values of $\delta t$ for which
$N>|y_{N}(z^{\prime})|\geq 1$, and there will be attenuation of waves at
values of $\delta t$ for which $|y_{N}(z^{\prime})|<1$.

The period $\delta t$ favorable for amplification can also be found
numerically by $\delta t$ vertically shifting the patterns of Fig.10a) and/or
Fig.10b) until parallel oblique lines of maxima (and minima) of the shifted
and not shifted patterns are superimposed on each other after the required
number $s$ of periods for the required the number $N$ of wave packets.

In relation to $\Psi(x,t)$ the waves $n(x,t)=|{\Psi(x,\;t)}|^{2}$ and $j(x,t)$
are a kind of ”intensity waves”, in case 2) they experience amplification only
by a factor of $N$, in contrast to the previous case 1) and the situation in
the theory of diffraction gratings, which provide an increase in intensity by
a factor of $N^{2}$.

Figures (Fig.14) and (Fig.15) demonstrate the resonant coherent amplification
of the probability density and current waves by the temporal sequence of two
($N=2$) identical wave packets shifted in time by $\delta t=sT=2\pi
s/|\omega_{12}|$ at $s=3$.

Figure 14:  The spatial profile of the resonant amplification of density and
current probability waves by a sequence of two ($N=2$) identical wave packets
shifted in time relative to each other by $\delta t=sT=65448$ a.e., $s=3$, at
the moment of time $t=3\cdot 10^{5}$ a.u. $=7.26\cdot 10^{-12}$ s (the main
bodies of the reflected packets are cut off because they are not of interest
to us, they are about an order of magnitude larger than the vertical size of
the panels). Coordinate in angstroms Å, time, probability density and
probability current density in atomic units.
Figure 15:  The time profile of the resonant amplification of the probability
density and current waves by a sequence of the same two ($N=2$) identical wave
packets (at $s=3$), as in Fig.14 using the example of points $x=x_{0}=0$ on
the left and $x=x_{2}=2d$ on the right boundaries of the double quantum well.
The time, probability density and probability current density in atomic units.

##  V  CONCLUSION

The generation probability density and probability current density waves of
electrons in the range of terahertz frequencies and micrometer wavelengths is
of interest from the point of view of various applications of micro- and
nanoelectronics. In this paper, we have shown that such generation can be
realized as a result of the excitation by a pulsed electron source of a
doublet of quasi-stationary states of a three-barrier heterostructure in the
form of a symmetric double quantum well. An exciting electron pulse in the
form of a Gaussian wave packet of picosecond duration, in turn, can be
created, for example, by pulsed photoemission when the photocathode is exposed
to a femtosecond light pulse or in some other way. The results of numerical-
analytical modeling of the formation of the probability density waves and the
probability current density waves outside the heterostructure are based on the
solution of the nonstationary Schr?dinger equation describing the scattering
of a Gaussian wave packet on a model structure formed by three tunnel-
transparent dielectric films modeled by $\delta$-barriers of the same power
separated by thin conducting or vacuum nanometer layers thickness. This
simplified model made it possible to implement numerical calculations and
estimate the frequencies, wavelengths, and velocities of such waves, as well
as the amplitudes of oscillations of probability density and current at a
given intensity of the exciting packet and the power of potential barriers.
The characteristics of the generated waves strongly depend on the parameters
of the heterostructure. By varying the parameters of the heterostructure, one
can change the energies, difference frequencies, and lifetimes of quasi-
stationary doublet states. For layer thicknesses of $1-10^{2}$ nm and barrier
heights of 0.5 - 2.5 eV, it is possible to provide the lifetimes of quasi-
stationary states of $10^{-2}-3\cdot 10^{2}$ ps, the generated difference
frequencies for them and the radiated waves of probability and current
densities of $10^{11}-10^{14}$ Hz, and the wavelengths of these waves
$10-10^{3}$ nm. The process of emission of electron waves can be repeated or
even amplified if a periodic resonant pumping of the doublet population in the
heterostructure is provided by a series of Gaussian pulses with a suitable
duty cycle, incident on the heterostructure in phase with oscillations of the
probability and current densities.

The simple quantum mechanical model discussed in this article makes it
possible to rigorously reveal the main regularities and estimate the
contributions of the main characteristics and singularities to the process of
excitation of waves of probability densities and currents during scattering of
wave packets on a double-well heterostructure. This enables us to study in
detail the properties of the complete system of wave functions of the
stationary scattering problem, which forms a natural basis of unperturbed
states of the zero approximation for more realistic models and methods for
describing and calculating the studied generation processes. In particular,
this refers to the models of fast photoemission in an open system, when, in
order to describe the excitation and structure of the scattered wave packet,
it is necessary to take into account the interactions of electrons with
photons and with other particles in the subsequent application of the density
matrix method for mixed quantum states.

*

## Appendix A

Appendix A

The essence of the method proposed by G.F. Drukarev Druk1951 ; Baz1969  for an
analytical estimate of the contributions of singularities of integrands of the
type (8), in short, is that after substituting (1) into (8) and passing to a
variable $k=\hbar^{-1}\sqrt{2mE}$, each of the seven exponential terms (1)
leads to an estimated integral of the form

$I=\int\limits_{0}^{\infty}{F(k)\exp(-i\beta_{t}(k-k_{S})^{2})dk},$ (18)

where $\beta_{t}=\hbar t/2m$, and the quantities $k_{S}$ and $F(k)$ are
different for the seven terms (1), they depend on $x,t$ and the parameters of
the problem, the functions $F(k)$ can have poles $k_{R}$ or other
singularities in the plane of the complex variable $k$, which are determined
by the features of $c_{E}=c(E(k))$ and of amplitudes of the reflected and
transmitted waves. Integral (18) is usually estimated based on the saddle
point method Lavr1967 , Peis2011 .

Figure 16: Deformation of the spectral integral contour in the area of
analyticity to the line I of the largest slope crossing the saddle point
$k_{S}$, the poles $k_{R}=k_{1}$ and $k_{R}=k_{2}$ of the under-integral
expressions are going around by small circles V.

For very large values $\beta_{t}$, the main contribution to it is associated
with the so-called stationary point $k=k_{S}$ on the real axis, which is a
saddle point for a function $\operatorname{Re}(-i\beta_{t}(k-k_{S})^{2})$ with
respect to variables $\operatorname{Re}k$ and $\operatorname{Im}k$, moreover,
the line of the fastest change of this function (the line of the greatest
slope) is a straight line, let us denote it by I, passing on the plane of the
complex variable $k$ through a point $k_{S}$ at an angle $-\pi/4$ to the real
axis (Fig.16). In accordance with the general rule, the contribution from the
$k_{S}$ neighborhood is found by deforming the integration contour in the
region of analyticity of the integrand so that it passes through the saddle
point along the line I of the greatest slope. The contribution of the saddle
point is usually estimated by the Poisson integral along the line I, in our
case it is equal to

$I_{k_{S}}=F(k_{S})\sqrt{\frac{{-i\pi}}{{\beta_{t}}}}$ (19)

the contributions of other distant parts of the deformed contour are usually
small (lines II and III in Fig.16) in comparison with it.

If, when the contour is displaced near the saddle point, a pole $k_{P}$ or a
branch point of the function $F(k)$ is encountered, then they should be
bypassed along a path of type IV, V, as shown in the figure. In the case of
poles, the contribution of sections IV cancels out, and the contribution of
the small circle V around the pole $k_{P}$ is equal to the residue at this
pole

$I_{p}=\pm 2\pi
iRes\\{F(k_{P})\\}\exp\left({-i\beta_{t}(k_{P}-k_{S})^{2}}\right)$ (20)

and may not be small in comparison with the contribution of the saddle point.
We take the plus sign if the pole is located in sector II of the upper half-
plane, minus - in sector III of the lower half-plane, as in the figure Fig.16
(by passing the pole counterclockwise or clockwise). The contributions of type
(20) are the main ones in the ranges of values of $t$ and $x$, which are of
interest to us, describing the oscillatory-wave behavior of the quantities
$n(x,t)$ and $j(x,t)$.

Appendix B

Let us write down analytical formulas that describe the coordinate-time
dependence of quantities $n(x,t)$ and $j(x,t)$ in the region of validity of
expression (14) for such values of $t$ and $x$, at which the oscillatory-wave
mode of beats of quasi-stationary states is established, because the main
contribution is made by the pole features of the scattering amplitudes, and it
is already possible to neglect small terms $\Psi_{0}(x,t)$, $\Psi_{n}(x,t)$,
$\Psi_{3}(x,t)$ in (14).

In the regions inside each of the two wells at $n^{\prime}d\leqslant
x\leqslant nd$, $n^{\prime}\,=n-1$, $n=1,2$, substitution by the second line
(14) in (9) and (10) gives

$\begin{gathered}n(x,t)=\sum\limits_{p=1}^{2}{|{\kern
1.0pt}\tilde{B}_{nE_{p}}|^{2}}e^{2\left|{\tilde{k}^{\prime\prime}_{p}}\right|(x-x_{n-1})-2\left|{E^{\prime\prime}_{p}}\right|t}+\sum\limits_{p=1}^{2}{|\tilde{A}_{nE_{p}}|^{2}}e^{-2\left|{\tilde{k}^{\prime\prime}_{p}}\right|(x-x_{n-1})-2\left|{E^{\prime\prime}_{p}}\right|t}+\hfill\\\
\quad\quad\;\;+2\sum\limits_{p=1}^{2}{|{\kern
1.0pt}\tilde{A}_{nE_{p}}\tilde{B}_{nE_{p}}|}\cos\left({{\kern 1.0pt}{\kern
1.0pt}2\tilde{k}^{\prime}_{p}(x-x_{n-1})+\alpha_{np}-{\kern 1.0pt}{\kern
1.0pt}\beta_{np}}\right)e^{-2\left|{E^{\prime\prime}_{p}}\right|t}+\hfill\\\
\quad+2\left|{\tilde{A}_{nE_{1}}\tilde{A}_{nE_{2}}}\right|\cos\left({\omega{\kern
1.0pt}{\kern 1.0pt}t-{\kern 1.0pt}{\kern
1.0pt}(\tilde{k}^{\prime}_{2}-\tilde{k}^{\prime}_{1})(x-x_{n-1})+\alpha_{n1}-{\kern
1.0pt}{\kern
1.0pt}\alpha_{n2}}\right)e^{-(\left|{\tilde{k}^{\prime\prime}_{1}}\right|+\left|{\tilde{k}^{\prime\prime}_{2}}\right|)(x-x_{n-1})-(\left|{E^{\prime\prime}_{1}}\right|+\left|{E^{\prime\prime}_{2}}\right|)t}+\hfill\\\
\quad+2\left|{\tilde{B}_{nE_{1}}\tilde{B}_{nE_{2}}}\right|\cos\left({\omega{\kern
1.0pt}{\kern 1.0pt}t+{\kern 1.0pt}{\kern
1.0pt}(\tilde{k}^{\prime}_{2}-\tilde{k}^{\prime}_{1})(x-x_{n-1})+\beta_{n1}-{\kern
1.0pt}{\kern
1.0pt}\beta_{n2}}\right)e^{\,\;(\left|{\tilde{k}^{\prime\prime}_{1}}\right|+\left|{\tilde{k}^{\prime\prime}_{2}}\right|)(x-x_{n-1})-(\left|{E^{\prime\prime}_{1}}\right|+\left|{E^{\prime\prime}_{2}}\right|)t}{\kern
1.0pt}{\kern 1.0pt}\,+\hfill\\\
\quad+2\left|{\tilde{A}_{nE_{1}}\tilde{B}_{nE_{2}}}\right|\cos\left({\omega{\kern
1.0pt}{\kern 1.0pt}t+{\kern 1.0pt}{\kern
1.0pt}(\tilde{k}^{\prime}_{2}+\tilde{k}^{\prime}_{1})(x-x_{n-1})+\alpha_{n1}-{\kern
1.0pt}{\kern
1.0pt}\beta_{n2}}\right)e^{-(\left|{\tilde{k}^{\prime\prime}_{1}}\right|-\left|{\tilde{k}^{\prime\prime}_{2}}\right|)(x-x_{n-1})-(\left|{E^{\prime\prime}_{1}}\right|+\left|{E^{\prime\prime}_{2}}\right|)t}+\hfill\\\
\quad+2\left|{\tilde{A}_{nE_{2}}\tilde{B}_{nE_{1}}}\right|\cos\left({\omega{\kern
1.0pt}{\kern 1.0pt}t-{\kern 1.0pt}{\kern
1.0pt}(\tilde{k}^{\prime}_{2}+\tilde{k}^{\prime}_{1})(x-x_{n-1})+\beta_{n1}-{\kern
1.0pt}{\kern
1.0pt}\alpha_{n2}}\right)e^{\;\;(\left|{\tilde{k}^{\prime\prime}_{1}}\right|-\left|{\tilde{k}^{\prime\prime}_{2}}\right|)(x-x_{n-1})-(\left|{E^{\prime\prime}_{1}}\right|+\left|{E^{\prime\prime}_{2}}\right|)t}{\kern
1.0pt}\;+\hfill\\\ \end{gathered}$ (21)

$\begin{gathered}j(x,t)=\frac{\hbar}{m}{\kern
1.0pt}\left[{\sum\limits_{p=1}^{2}{\tilde{k}^{\prime}_{p}\left({|\tilde{A}_{nE_{p}}|^{2}e^{-2\left|{\tilde{k}^{\prime\prime}_{p}}\right|(x-n^{\prime}d)}-\;|{\kern
1.0pt}\tilde{B}_{nE_{p}}|^{2}e^{+2\left|{\tilde{k}^{\prime\prime}_{p}}\right|(x-n^{\prime}d)}}\right)\;}e^{-2\left|{E^{\prime\prime}_{p}}\right|t}-}\right.\hfill\\\
\quad\quad\quad\quad\;\;-2\sum\limits_{p=1}^{2}{k^{\prime\prime}_{p}\,|{\kern
1.0pt}\tilde{A}_{nE_{p}}\tilde{B}_{nE_{p}}|}\;sin\left({{\kern 1.0pt}{\kern
1.0pt}2\tilde{k}^{\prime}_{p}(x-n^{\prime}d)+\;\alpha_{np}-{\kern 1.0pt}{\kern
1.0pt}\beta_{np}}\right)e^{-2\left|{E^{\prime\prime}_{p}}\right|t}+\hfill\\\
+\left({\tilde{k}^{\prime}_{2}+\tilde{k}^{\prime}_{1}}\right)\left|{\tilde{A}_{nE_{1}}\tilde{A}_{nE_{2}}}\right|cos\left({\omega{\kern
1.0pt}{\kern 1.0pt}t-{\kern 1.0pt}{\kern
1.0pt}(\tilde{k}^{\prime}_{2}-\tilde{k}^{\prime}_{1})(x-n^{\prime}d)+\alpha_{n1}-{\kern
1.0pt}{\kern
1.0pt}\alpha_{n2}}\right)e^{-(\left|{\tilde{k}^{\prime\prime}_{1}}\right|+\left|{\tilde{k}^{\prime\prime}_{2}}\right|)(x-n^{\prime}d)-(\left|{E^{\prime\prime}_{1}}\right|+\left|{E^{\prime\prime}_{2}}\right|)t}\\_\hfill\\\
-\left({\tilde{k}^{\prime}_{2}+\tilde{k}^{\prime}_{1}}\right)\left|{\tilde{B}_{nE_{1}}\tilde{B}_{nE_{2}}}\right|cos\left({\omega{\kern
1.0pt}{\kern 1.0pt}t+{\kern 1.0pt}{\kern
1.0pt}(\tilde{k}^{\prime}_{2}-\tilde{k}^{\prime}_{1})(x-n^{\prime}d)+\;\beta_{n1}-{\kern
1.0pt}{\kern
1.0pt}\beta_{n2}}\right)e^{(\left|{\tilde{k}^{\prime\prime}_{1}}\right|+\left|{\tilde{k}^{\prime\prime}_{2}}\right|)(x-n^{\prime}d)-(\left|{E^{\prime\prime}_{1}}\right|+\left|{E^{\prime\prime}_{2}}\right|)t}+\hfill\\\
+\left({\tilde{k}^{\prime}_{1}-\tilde{k}^{\prime}_{2}}\right)\left|{\tilde{A}_{nE_{1}}\tilde{B}_{nE_{2}}}\right|cos\left({\omega{\kern
1.0pt}{\kern 1.0pt}t+{\kern 1.0pt}{\kern
1.0pt}\left({\tilde{k}^{\prime}_{2}+\tilde{k}^{\prime}_{1}}\right)(x-n^{\prime}d)+\;\alpha_{n1}-{\kern
1.0pt}{\kern
1.0pt}\beta_{n2}}\right)e^{-(\left|{\tilde{k}^{\prime\prime}_{1}}\right|-\left|{\tilde{k}^{\prime\prime}_{2}}\right|)(x-n^{\prime}d)-(\left|{E^{\prime\prime}_{1}}\right|+\left|{E^{\prime\prime}_{2}}\right|)t}-\hfill\\\
\left.{-\left({\tilde{k}^{\prime}_{1}-\tilde{k}^{\prime}_{2}}\right)\left|{\tilde{A}_{nE_{2}}\tilde{B}_{nE_{1}}}\right|cos\left({\omega{\kern
1.0pt}{\kern 1.0pt}t-{\kern 1.0pt}{\kern
1.0pt}\left({\tilde{k}^{\prime}_{2}+\tilde{k}^{\prime}_{1}}\right)(x-n^{\prime}d)+\beta_{n1}-{\kern
1.0pt}\alpha_{n2}}\right)e^{(\left|{\tilde{k}^{\prime\prime}_{1}}\right|-\left|{\tilde{k}^{\prime\prime}_{2}}\right|)(x-n^{\prime}d)-(\left|{E^{\prime\prime}_{1}}\right|+\left|{E^{\prime\prime}_{2}}\right|)t}}\right]\hfill\\\
\end{gathered}$ (22)

In (22), we neglected small terms proportional $k^{\prime}_{p}$ everywhere
except for the second line, in which we wrote out a similar negligible sum
just to illustrate the symmetry of the entire expression.

In the region to the left of the double well at $x<0$, substitution of the
first line of (14) in (9) and (10) gives expressions describing damped waves
traveling to the left

$\begin{gathered}n(x,t)=\sum\limits_{p=1}^{2}{|{\kern
1.0pt}\tilde{B}_{0E_{p}}|^{2}}e^{2\left|{k^{\prime\prime}_{p}}\right|x-2\left|{E^{\prime\prime}_{p}}\right|t}+\hfill\\\
\quad\quad\quad+2\left|{\tilde{B}_{0E_{1}}\tilde{B}_{0E_{2}}}\right|cos\left({\omega{\kern
1.0pt}{\kern 1.0pt}t+{\kern 1.0pt}{\kern
1.0pt}(k^{\prime}_{2}-k^{\prime}_{1})x+\beta_{01}-{\kern 1.0pt}{\kern
1.0pt}\beta_{02}}\right)e^{(\left|{k^{\prime\prime}_{1}}\right|+\left|{k^{\prime\prime}_{2}}\right|)x-(\left|{E^{\prime\prime}_{1}}\right|+\left|{E^{\prime\prime}_{2}}\right|)t},\hfill\\\
\end{gathered}$ (23) $\begin{gathered}j(x,t)=-\frac{\hbar}{m}{\kern
1.0pt}\left[{\sum\limits_{p=1}^{2}{k^{\prime}_{p}|\tilde{B}_{0E_{p}}|^{2}}e^{2\left|{k^{\prime\prime}_{p}}\right|x-2\left|{E^{\prime\prime}_{p}}\right|t}}\right.+\hfill\\\
\quad\quad\quad+\left.{(k^{\prime}_{1}+k^{\prime}_{2})|\tilde{B}_{0E_{1}}\tilde{B}_{0E_{2}}|cos\left({\omega{\kern
1.0pt}{\kern 1.0pt}t+{\kern 1.0pt}{\kern
1.0pt}(k^{\prime}_{2}-k^{\prime}_{1})x+\beta_{01}-{\kern 1.0pt}{\kern
1.0pt}\beta_{02}}\right)e^{(\left|{k^{\prime\prime}_{1}}\right|+\left|{k^{\prime\prime}_{2}}\right|)x-(\left|{E^{\prime\prime}_{1}}\right|+\left|{E^{\prime\prime}_{2}}\right|)t}}\right],\quad\hfill\\\
\end{gathered}$ (24)

In the region to the right of the double well at $x>x_{2}$, substitution of
the first line of (14) in (9) and (10) gives expressions describing damped
waves traveling to the right

$\begin{gathered}n(x,t)=\sum\limits_{p=1}^{2}{|\tilde{A}_{3E_{p}}||^{2}}e^{-2\left|{k^{\prime\prime}_{p}}\right|(x-x_{2})-2\left|{E^{\prime\prime}_{p}}\right|t}+\hfill\\\
\quad\quad+2\left|{\tilde{A}_{3E_{1}}\tilde{A}_{3E_{2}}}\right|cos\left({\omega{\kern
1.0pt}{\kern 1.0pt}t-{\kern 1.0pt}{\kern
1.0pt}(k^{\prime}_{2}-k^{\prime}_{1})(x-x_{2})+\;\alpha_{31}-{\kern
1.0pt}{\kern
1.0pt}\alpha_{32}}\right)e^{-(\left|{k^{\prime\prime}_{1}}\right|+\left|{k^{\prime\prime}_{2}}\right|)(x-x_{2}d)-(\left|{E^{\prime\prime}_{1}}\right|+\left|{E^{\prime\prime}_{2}}\right|)t},\hfill\\\
\end{gathered}$ (25) $\begin{gathered}j(x,t)=\frac{\hbar}{m}{\kern
1.0pt}\left[{\sum\limits_{p=1}^{2}{k^{\prime}_{p}|\tilde{A}_{3E_{p}}|^{2}}e^{-2\left|{k^{\prime\prime}_{p}}\right|(x-x_{2})-2\left|{E^{\prime\prime}_{p}}\right|t}}\right.+\hfill\\\
\quad\quad\quad+\left.{(k^{\prime}_{1}+k^{\prime}_{2})\,\left|{\tilde{A}_{3E_{1}}\tilde{A}_{3E_{2}}}\right|cos\left({\omega{\kern
1.0pt}{\kern 1.0pt}t-{\kern 1.0pt}{\kern
1.0pt}(k^{\prime}_{2}-k^{\prime}_{1})(x-x_{2})+\;\alpha_{31}-{\kern
1.0pt}{\kern
1.0pt}\alpha_{32}}\right)e^{-(\left|{k^{\prime\prime}_{1}}\right|+\left|{k^{\prime\prime}_{2}}\right|)(x-x_{2})-(\left|{E^{\prime\prime}_{1}}\right|+\left|{E^{\prime\prime}_{2}}\right|)t}}\right].\hfill\\\
\end{gathered}$ (26)

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