# Scattering in terms of Bohmian conditional wave functions for scenarios with
non-commuting energy and momentum operators

Matteo Villani1, Guillermo Albareda2,3, Carlos Destefani1, Xavier Cartoixà1,
Xavier Oriols1

(1: Department of Electronic Engineering, Universitat Autònoma de Barcelona,
Campus de la UAB, 08193 Bellaterra, Barcelona, Spain
2: Max Planck Institute for the Structure and Dynamics of Matter, Luruper
Chaussee 149, 22761 Hamburg, Germany
3: Institute of Theoretical and Computational Chemistry, Universitat de
Barcelona, Gran Via de les Corts Catalanes 585, 08007 Barcelona, Spain)

###### Abstract

Without access to the full quantum state, modeling quantum transport in
mesoscopic systems requires dealing with a limited number of degrees of
freedom. In this work, we analyze the possibility of modeling the perturbation
induced by the non-simulated degrees of freedom on the simulated ones as a
transition between single-particle pure states. First, we show that Bohmian
conditional wave functions (BCWF) allow a rigorous discussion of the dynamics
of electrons inside open quantum systems in terms of such single-particle pure
states, either under Markovian or non-Markovian conditions. Second, we discuss
the practical application of the method for modeling light-matter interaction
phenomena in a resonant tunneling device (RTD), where a single photon is
interacting with a single electron. Third, we emphasize the importance of
interpreting such scattering mechanism as a transition between initial and
final single-particle BCWF with well-defined energies (rather than with well-
defined momenta).

#### keywords:

quantum dissipation; Bohmian mechanics; collision; conditional wave function;
decoherence; open systems; many-body problem

#### Corresponding author:

<EMAIL_ADDRESS>
##  1 Introduction

Due to the well-known many-body problem, electron transport in nanoscale
devices has to be modeled as an open quantum system [1]. The contacts, cables,
atoms, electromagnetic radiation, etc. are commonly considered to be part of
the environment. The effect of this environment on the dynamics of the
simulated degrees of freedom, i.e., the electrons in the active region, can be
recovered using some kind of perturbative approximation. There are different
formalisms in the literature to deal with such _environmental_ perturbation
(Green’s functions [4, 2, 3], density matrix [5, 6], Wigner distribution
function [7, 9, 8, 10, 11], Kubo formalism [12], Pauli quantum Master equation
[13, 14], pure states [15, 16], etc). In this work we analyze the possibility
of modeling the quantum nature of such simulated degrees of freedom with
single-particle pure states and the environmental perturbations on the
simulated degrees of freedom by the non-simulated ones as a transition between
initial and final single-particle pure states.

In particular, we are interested in modeling the collision of an electron with
a phonon or/and photon in an active region with tunneling barriers, i.e. in a
scenario where the energy and momentum operators do not commute. The path for
achieving this goal requires first addressing the answer of the following
question: Is it possible to model an open system in terms of single-particle
pure states?. Once this conceptual question is answered, the next practical
question that needs to be addressed is: How do we select the single-particle
pure states before and after the collision? In this paper we answer both
questions. It will be shown that the alternative Bohmian formulation of
quantum transport [17] provides a rigorous and versatile tool to describe
collisions in open quantum systems in terms of single-particle pure wave
functions. This work is part of a long-term research project for the
development of a general-purpose nanoelectronic device simulator, the so-
called BITLLES simulator [18], using Bohmian trajectories.

The structure of the paper is the following. In Sec. 2 the answer to the first
question about using single-particle pure states for open systems is provided
from the Bohmian description of quantum phenomena. In Sec. 3 we provide an
exact model for matter-light interaction in a closed system, and some
simulation results are reported for different conditions of the total energy
of the closed system, with a final discussion on the interaction between
active region and environment, to extend this description to an open system.
In Sec. 4, the practical implementation of the transition between a pres-
selected $|i\rangle$ and a post-selected $|f\rangle$ states:
$|i\rangle\to|f\rangle$ is discussed. This done for two different models:
model A deals with the energy conservation and model B deals with the momentum
conservation. In Sec. 5 these two models are compared, for a flat potential
and for an arbitrary potential, to each other and to the exact model of Sec.
3. Our conclusions are summarized in Sec. 6.

##  2 Is it possible to model open system in terms of single-particle pure
states?

As we have stated the active region of an electron device is, strictly
speaking, an open quantum system interacting with the contacts, atoms in
thermal motion, radiation, etc. As a consequence, in principle one is not
allowed to describe the electron in the active region in terms of pure states,
but one has to rely on the use of the reduced density matrix.

Most approaches to open systems revolve around the reduced density matrix
built by tracing out the degrees of freedom of the environment [1]. The
ability to describe open systems with pure states can be partially justified
when dealing with Markovian systems. In a pragmatical definition of
Markovianity [19], the correlations between system and environment decay in a
time scale that is much smaller than the observation (or relevant) time
interval of the system. Thus, it can be assumed that every time we observe the
system it is defined by a pure state. For Markovian evolutions, the Lindblad
master equation [20] for the reduced density matrix is a standard simulation
tool. In addition, in Markovian scenarios where the off-diagonal terms of the
reduced density matrix become irrelevant, a quantum master equation can be
implemented dealing with transitions between pure states [13, 14].

In fact, it is possible to develop stochastic Schrödinger equations to unravel
the reduced density matrix in terms of a pure-state solution for either
Markovian or non-Markovian systems. The pure-state solution of stochastic
Schrödinger equations can be interpreted as the state of the Markovian system
while the environment is under (continuous) observation. However, such
physical interpretation cannot be given to the solutions of the stochastic
Schrödinger equations for non-Markovian systems [22, 23, 24, 25, 26, 27, 28,
29, 30, 21], where pure states can provide the correct one-time ensemble
value, but cannot be used to compute time correlations.

Therefore, for general non-Markovian quantum processes, when we are interested
in a time-resolved description of the electron device performance, it is not
possible to define the open system in terms of orthodox pure states. As
described in [31] and explained below, a proper solution for treating
electrons in non-Markovian open systems as single-particle pure states comes
from the Bohmian formalism.

To explain how the Bohmian theory allows a general description of a many-body
quantum system in terms of wave functions, we consider a simplified scenario
with only two degrees of freedom: one degree of freedom $x$ belonging to the
system, plus one degree of freedom $y$ belonging to the environment. Thus, the
pure state in the position representation solution of the unitary Schrodinger
equation is $\Psi(x,y,t)$. For each experiment, labelled by $j$, a Bohmian
quantum state is defined by this $\Psi(x,y,t)$ plus two well-defined
trajectories, $X^{j}[t]$ in the $x$-physical space and $Y^{j}[t]$ in the
$y$-physical space. The role of the many-body wavefunction $\Psi(x,y,t)$ is
guiding each trajectory $X^{j}[t]$ with a velocity that reads [17, 32, 33]

$\displaystyle
v_{x}^{j}[t]=\frac{dX^{j}[t]}{dt}=\frac{J_{x}(X^{j}[t],Y^{j}[t],t)}{|\Psi(X^{j}[t],Y^{j}[t],t)|^{2}}=\frac{1}{m^{*}}\left.\frac{\partial
S(x,y,t)}{\partial x}\right|_{x=X^{j}[t],y=Y^{j}[t]},$ (1)

where $J_{x}(x,y,t)=\hbar\mathop{\rm
Im}\left(\Psi^{*}(x,y,t)\frac{\partial}{\partial_{x}}\Psi(x,y,t))\right)/m^{*}$
is the current density with $m^{*}$ the mass of the $x$-particle, and
$S(x,y,t)$ is the phase of the wave function written in polar form
$\Psi(x,y,t)=|\Psi(x,y,t)|e^{iS(x,y,t)/\hbar}$. Analogous definitions are
possible for the $Y^{j}[t]$ trajectory. The two positions
$\\{X^{j}[t],Y^{j}[t]\\}$ in different $j=1,...,W$ experiments are distributed
(obeying quantum equilibrium [32, 33]) as

$\displaystyle|\Psi(x,y,t)|^{2}=\frac{1}{W}\sum_{j=1}^{W}\delta(x-X^{j}[t])\delta(y-Y^{j}[t]).$
(2)

The identity in (2) requires $W\to\infty$. Numerically, we will just require a
large enough $W$ to reproduce ensemble values given by the Born law in
agreement with the orthodox theory.

The Bohmian theory opens the possibility to deal with a wave function of a
subsystem, through the concept of Bohmian conditional wave function (BCWF)
[33, 34]. The BCWF is defined for the $x$-degree of freedom during the $j$-th
experiment as

$\psi^{j}(x,t)\equiv\Psi(x,Y^{j}[t],t).$ (3)

We emphasize that $\psi^{j}(x,t)$ provides a rigorous (Bohmian) definition of
a single-particle wave function for an open system [32] that still includes
the correlations with the other degree of freedom $y$. Notice that the reason
why the BCWF has a relevant role in the Bohmian theory is because the
trajectory $X^{j}[t]$ is equivalently guided by $\Psi(x,y,t)$ or by
$\psi^{j}(x,t)$. In other words, the velocity $v^{j}_{x}[t]$ in (1) can be
equivalently computed from the wave function as

$\displaystyle
v^{j}_{x}[t]=\frac{dX^{j}[t]}{dt}=\frac{J_{x}^{j}(X^{j}[t],t)}{|\psi^{j}(X^{j}[t],t)|^{2}}=\frac{1}{m^{*}}\left.\frac{\partial
s(x,t)}{\partial x}\right|_{x=X^{j}[t]},$ (4)

where $|\psi^{j}(X^{j}[t],t)|^{2}=|\Psi(X^{j}[t],Y^{j}[t],t)|^{2}$,
$J_{x}^{j}(x,t)=\hbar\mathop{\rm
Im}\left(\psi^{j,*}(x,t)\frac{\partial}{\partial_{x}}\psi^{j}(x,t))\right)/m^{*}$,
and $s(x,t)$ is the angle of the BCWF in polar form
$\psi^{j}(x,t)=|\psi^{j}(x,t)|e^{is(x,t)/\hbar}$. Notice that we have not done
any approximation about the Markovianity of the quantum system in the
definition of the BCWF. Thus, at the conceptual level, we conclude that any
quantum open system can be analyzed in terms of single-particle pure states
(i.e., BCWF) using the Bohmian formalism. This is a well-known result [31] and
provides a definitive positive answer to the initial question: Is it possible
to model open system in terms of single-particle pure states?

Let us discuss now a more realistic scenario with $N$ electrons inside the
active region with degrees of freedom $\\{x_{1},x_{2},...,x_{N}\\}$ that we
want to simulate explicitly (for simplicity each electron is assumed to be
defined in a 1D space). There are, however, $M$ environmental degrees of
freedom $\\{y_{1},y_{2},...,y_{M}\\}$that we do not want to simulate
explicitly. The new many-body wave function of such scenario is
$\Psi(x_{1},x_{2},...,x_{N},y_{1},y_{2},...,y_{M})$, which is numerically
inaccessible. We define
$\bar{X}^{j}_{i}[t]=\\{x^{j}_{1}[t],..,x_{i-1}[t],x^{j}_{i+1}[t],...,x^{j}_{N}\\}$
as the set of all Bohmian trajectories of the system except $x^{j}_{i}(t)$ for
the $j$-experiment. Notice that we are dealing now with a superindex $j$
indicating the experiment and subindex $i$ indicating each particle in a given
experiment. We also define $Y^{j}[t]=\\{y^{j}_{1}[t],.....,y^{j}_{M}[t]\\}$ as
the set of all trajectories of the environment at the $j$-experiment. Then,
the set of equations of motion of the resulting $N$ single-electron
conditional wave functions
$\psi^{j}(x_{1},t)\equiv\Psi(x_{1},\bar{X}^{j}_{1}[t],Y^{j}[t],t),...,\psi^{j}(x_{N},t)\equiv\Psi(x_{N},\bar{X}^{j}_{N}[t],Y^{j}[t],t)$
inside the active region can be written as:

$\displaystyle i\hbar\frac{d\psi^{j}(x_{1},t)}{dt}$ $\displaystyle=$
$\displaystyle\left[-\frac{\hbar^{2}}{2m}\nabla^{2}_{x_{1}}+U_{eff}^{j}(x_{1},t)\right]\psi^{j}(x_{1},t)$
(5) $\displaystyle\vdots$ $\displaystyle i\hbar\frac{d\psi^{j}(x_{N},t)}{dt}$
$\displaystyle=$
$\displaystyle\left[-\frac{\hbar^{2}}{2m}\nabla^{2}_{x_{N}}+U_{eff}^{j}(x_{N},t)\right]\psi^{j}(x_{N},t).$

The effective single-particle potential $U_{eff}^{j}(x_{i},t)\equiv
U_{eff}^{j}(x_{i},\bar{X}^{j}_{i}[t],Y^{j}[t],t)$ is

$\displaystyle U_{eff}^{j}(x_{i},t)$ $\displaystyle=$
$\displaystyle{U}^{j}(x_{i},t)+V^{j}(x_{i},t)$ (6) $\displaystyle+$
$\displaystyle\mathcal{A}^{j}(x_{i},t)+i\mathcal{B}^{j}(x_{i},t),$

where $U^{j}(x_{i},t)$ is an external potential acting only on the system
degrees of freedom $x_{i}$, $V^{j}(x_{i},t)$ is the Coulomb potential between
$x_{i}$ and the rest of particles at fixed positions $\bar{X}^{j}_{1}[t]$ and
$Y^{j}[t]$, and $\mathcal{A}^{j}(x_{i},t)$ and $\mathcal{B}^{j}(x_{i},t)$ are
potentials responsible for the remaining of quantum correlations between the
degrees of freedom of the system and the environment [31]. A mandatory
clarification is needed here. Are the set of the BCWF in (5) solving the many-
body problem? No. If you want to use the coupled system of equations of motion
of the $N$ BCWF in (5) to describe a given experiment, first, you have to
solve the Poisson (Gauss) equation to find $U^{j}(x_{i},t)$ and
$V^{j}(x_{i},t)$ explicitly and, second, you have to know the exact solution
of the many-body wave function
$\Psi(x_{1},x_{2},...,x_{N},y_{1},y_{2},...,y_{M})$ to find
$\mathcal{A}^{j}(x_{i},t)$ and $\mathcal{B}^{j}(x_{i},t)$ for all
electrons[31]. The last step is numerically inaccessible. The merit of the
system of equations in (5) is showing that such type of solutions of the many-
body function exists and that we can look for educated guesses on the shape of
$\mathcal{A}^{j}(x,t)$ and $\mathcal{B}^{j}(x,t)$ to provide reasonable
approximations. Notice that a similar procedure is followed in Density
Functional Theory: it shows a way of rewriting the many-body wave function in
terms of single-particle wave functions, but the procedure requires the
knowledge of the exchange-correlation functional, which is only known once the
many-body wave function is known. See further details and an explanation on
$\mathcal{A}^{j}(x,t)$ and $\mathcal{B}^{j}(x,t)$ in [33, 31, 32, 18, 35, 36].

To better appreciate the details of this simulation technique for electron
devices, we notice that the total current $I^{j}(t)$ at time $t$ for the
$j$-experiment, after solving the set of BCWF from (5) with the appropriate
approximations for $\mathcal{A}^{j}(x,t)$ and $\mathcal{B}^{j}(x,t)$ can be
defined from the Bohmian trajectories with the help of a quantum version of
the Ramo–Schokley–Pellegrini theorem [37] as:

$I^{j}(t)=\frac{e}{L}\sum_{i=1}^{n(t)}v_{x_{i}}^{j}(x_{i}^{j}[t],\bar{X}^{j}_{i}[t],Y^{j}[t]),$
(7)

where $L$ is the distance between the two (metallic) contacts, $e$ is the
electron charge, and $v_{x_{i}}(x^{j}_{i}[t],\bar{X}^{j}_{i}[t],Y^{j}[t]t)$ is
the Bohmian velocity of the $i$-th electron inside the active region in the
$j$-experiment. Notice that the observables are computed from the trajectories
(not from the BCWF) and they are linked to a particular experiment $j$ (which
can be understood as a single configuration of the environment). The different
possible values of $x^{j}_{i}[t]$,$\bar{X}^{j}_{i}[t]$ and $Y^{j}[t]$ for the
same (preparation of the) many-body wave function
$\Psi(x_{1},x_{2},...,x_{N},y_{1},y_{2},...,y_{M})$ introduce the inherent
quantum randomness in any experiment. As such, if one is interested in
ensemble average values, one can repeat the calculation for all environment
configuration $Y^{j}[t]$ and particle distributions $x^{j}_{i}[t]$ and
$\bar{X}^{j}_{i}[t]$. Typically, in electronics, this ensemble average of the
current $I^{j}(t)$ over many experiments $j=1,...,\infty$ are interesting for
evaluating DC values of the electrical current under ergodic assumptions. In
the laboratory, a large time-average of the current $I^{j}(t)$ in a single
$j$-experiment is performed. However, if one is interested in noise or time-
correlations of the current at different times, $I^{j}(t_{1})$ and
$I^{j}(t_{2})$, then the access to the individual experiment offered by the
BCWF is very relevant.

Finally, we mention which are the computational advantages of this simulation
framework. It is a microscopic description of the transport in the sense that
it provides an individual description for each electron inside the active
region. It provides a rigorous estimation (a part from the approximations for
$\mathcal{A}^{j}(x_{i},t)$ and $\mathcal{B}^{j}(x_{i},t)$) to the quantum
dynamics of electrons in the active region (open quantum system) for Markovian
and non-Markovian systems. It is a versatile approach in the sense that it can
simulate many different scenarios, from steady-state DC to transient and AC,
including the fluctuations of the current (noise). Notice that $I^{j}(t)$ in
(7) includes the particle and displacement current, even at THz frequencies,
when multi-time measurements are implicit. In this sense, we argue that the
amount of information that this simulator framework can provide in the quantum
regime is comparable to the predicting capabilities of the traditional Monte
Carlo solution of the Boltzmann transport equation [38] in the semi-classical
regime.

##  3 How do we select the single-particle pure states before and after the
collision?

To discuss how electron-photon scattering can be included in this simulation
framework, we provide, first, an exact computation of the interaction between
a single electron and a single photon in a closed system in terms of BCWF and
Bohmian trajectories and, second, we provide some indications on how such
interaction can be modeled (without the explicit consideration of the photon)
in an open system, giving attention to how the initial and final states have
to be modeled.

###  3.1 Exact solution in a closed system

The full quantum Hamiltonian
$\hat{H}=\hat{H}_{e}+\hat{H}_{\gamma}+\hat{H}_{I}$ that describes light-matter
interaction is given by the sum of the electron Hamiltonian $\hat{H}_{e}$, the
electromagnetic field Hamiltonian $\hat{H}_{\gamma}$, and the electron-photon
interaction Hamiltonian $\hat{H}_{I}$. In particular, for a single electron in
a semiconductor, the position representation for $\hat{H}_{e}$ (assuming a 1D
system for the electron with degree of freedom $x$) is given by

$H_{e}=-\frac{\hbar^{2}}{2m^{*}}\frac{\partial^{2}}{\partial x^{2}}+V(x),$ (8)

where $V(x)$ includes both internal and external electrostatic potential. See
the blue electron wave packet and the scalar potential $V(x)$ for a double
barrier region of length $2L_{x}$ in the horizontal $x$-axis of Fig. 1(a).

We consider that the electromagnetic field is described by a single mode with
angular frequency $\omega$ inside a closed cavity of length $2L_{M}$. See the
cyan mirrors in the horizontal $x$-axis of Fig. 1(a) and (b). The typical
description of the electric field will be $E(x,t)\propto q\;cos(kx-\omega t)$
with the wave vector $k=2\pi/\lambda$ related to the angular frequency as
$c=\omega/k$ with $c$ the speed of light. The variable $q$ represents the
instantaneous amplitude of the electromagnetic field along the polarization
vector. Under the assumption $L_{M}\gg L_{x}$ meaning that the wave-length for
the electromagnetic wave ($\approx 500$ nm) is much larger than the active
region ($\approx 20$ nm), we can neglect the spatial dependence $x$ of the
electromagnetic field. Then, the Hamiltonian of the electromagnetic field in
second quantization can be written as
$\hat{H}_{\gamma}=\hbar\omega\left(1/2+\hat{a}^{\dagger}\hat{a}\right)$. The
relationship between the now quantized amplitude of the electric field $q$ and
the creation $\hat{a}^{\dagger}$ and annihilation operators $\hat{a}$ is given
by

$\hat{a}=\sqrt{\frac{\omega}{2\hbar}}\left(q+\frac{\hbar}{\omega}\frac{\partial}{\partial
q}\right)\;\;\;\;,\;\;\;\hat{a}^{\dagger}=\sqrt{\frac{\omega}{2\hbar}}\left(q-\frac{\hbar}{\omega}\frac{\partial}{\partial
q}\right).$ (9)

Then, the q-representation of $\hat{H}_{\gamma}$ is

$\displaystyle H_{\gamma}=-\frac{\hbar^{2}}{2}\frac{\partial^{2}}{\partial
q^{2}}+\frac{\omega^{2}}{2}q^{2},$ (10)

where the electromagnetic vacuum state with zero photons $|0\rangle$ solution
of $\hat{H}_{\gamma}$ corresponds to the ground state of a harmonic oscillator
$\psi_{0}(q)=\langle q|0\rangle$, while the state solution of
$\hat{H}_{\gamma}$ with one photon corresponds to the first excited state of
an harmonic oscillator $\psi_{1}(q)=\langle q|\hat{a}^{\dagger}|0\rangle$.

Figure 1: Schematic representation of the time evolution of the wave function
for an electron impinging upon a double barrier region with electromagnetic
radiation. In (a) and (b) we consider a cavity small enough so that the
electromagnetic light does not radiate and no interaction with an environment
degree of freedom outside the active region is included. Only the information
on the electron degree of freedom $x$ and the internal degree of freedom of
the light $q$ (not plotted) are relevant. The Bohmian position of the electron
$X[t]$ is indicated as a solid black circle. The $Q[t]$ trajectory of the
electromagnetic field is not indicated. Notice that in (a) the initial
electron wave function is $\psi_{A}(x,t=0)\neq 0$ (blue curve for the
electron) and $\psi_{B}(x,t=0)=0$, while in (b) we get $\psi_{B}(x,t=0)\neq 0$
due to spontaneous emission.

The interaction Hamiltonian in the dipole approximation can be written as
$\hat{H}_{I}=-e\hat{x}\hat{E}$, where $e$ is the (unsigned) electron charge
and the electrical field operator is given by
$\hat{E}=\epsilon\left(\hat{a}+\hat{a}^{\dagger}\right)$, with $\epsilon$ the
strength of the electric field, or explicitly as

$\displaystyle H_{I}=\alpha^{\prime}xq$ (11)

where $\alpha^{\prime}$, which depends on $\epsilon$ and other parameters of
the cavity, controls the strength of the light-matter interaction. Finally,
the wave function $\Psi(x,q,t)$ that describes the quantum nature of electrons
and the electromagnetic field simultaneously in the $q$-representation is
solution of the following two-dimensional Schrödinger equation,

$\displaystyle i\hbar\frac{\partial\Psi(x,q,t)}{\partial t}=$ $\displaystyle-$
$\displaystyle\frac{\hbar^{2}}{2m}\frac{\partial^{2}\Psi(x,q,t)}{\partial
x^{2}}+V(x)\Psi(x,q,t)$ (12) $\displaystyle-$
$\displaystyle\frac{\hbar^{2}}{2}\frac{\partial^{2}\Psi(x,q,t)}{\partial
q^{2}}+\frac{\omega^{2}}{2}q^{2}\Psi(x,q,t)$ $\displaystyle+$
$\displaystyle\alpha^{\prime}xq\Psi(x,q,t).$

To simplify our discussion of emission and absorption of a photon by an
electron, let us assume that only the zero photon state, $\psi_{0}(q)=\langle
q|0\rangle=\langle q|\psi_{0}\rangle$, and the one photon state,
$\psi_{1}(q)=\langle q|\hat{a}^{\dagger}|0\rangle=\langle q|\psi_{1}\rangle$,
are relevant in our active region. Notice that we are discussing the
interaction of a single electron with a single photon. Then, we can rewrite
the wave function $\Psi(x,q,t)$ solution of (12) as

$\displaystyle\Psi(x,q,t)=\psi_{A}(x,t)\psi_{0}(q)+\psi_{B}(x,t)\psi_{1}(q),$
(13)

with

$\displaystyle\psi_{A}(x,t)=\int\;\psi_{0}^{*}(q)\Psi(x,q,t)dq$ (14)
$\displaystyle\psi_{B}(x,t)=\int\;\psi_{1}^{*}(q)\Psi(x,q,t)dq.$ (15)

The equation of motion of $\psi_{A}(x,t)$ and $\psi_{B}(x,t)$ can be obtained
by introducing the definition (13) into (12) and using the orthogonality of
$\psi_{0}(q)$ and $\psi_{1}(q)$ as follows

$\displaystyle i\hbar\frac{\partial\psi_{A}(x,t)}{\partial t}=$
$\displaystyle-$
$\displaystyle\frac{\hbar^{2}}{2m}\frac{\partial^{2}\psi_{A}(x,t)}{\partial
x^{2}}$ (16) $\displaystyle+$
$\displaystyle\left(V(x)+\frac{1}{2}\hbar\omega\right)\psi_{A}(x,t)$
$\displaystyle+$ $\displaystyle\alpha x\psi_{B}(x,t),$ $\displaystyle
i\hbar\frac{\partial\psi_{B}(x,t)}{\partial t}=$ $\displaystyle-$
$\displaystyle\frac{\hbar^{2}}{2m}\frac{\partial^{2}\psi_{B}(x,t)}{\partial
x^{2}}$ (18) $\displaystyle+$
$\displaystyle\left(V(x)+\frac{3}{2}\hbar\omega\right)\psi_{B}(x,t)$
$\displaystyle+$ $\displaystyle\alpha x\psi_{A}(x,t),$

where we have defined $\alpha=\alpha^{\prime}\int\psi_{0}(q)q\psi_{1}(q)dq$
and we have used
$\int\psi_{0}(q)q\psi_{0}(q)\;dq=\int\psi_{1}(q)q\psi_{1}(q)\;dq=0$.

We simulate now an initial electron impinging on a double barrier with a
potential energy $V(x)$ as shown in Fig. 2 (a). It corresponds to the
conduction band of a typical resonant tunnelling diode (RTD) with a 10 nm-well
width, barrier thickness of 2 nm, and barrier height of 0.5 eV. In Fig. 2 (b),
the transmission coefficient of the double barrier is plotted showing two
resonant energies inside the well at $E_{1}=0.058$ eV and $E_{2}=0.23$ eV. The
positive energies correspond to energy eigenstates impinging from the left and
negative energies from the right.

Figure 2: (a) Band structure and (b) Transmission coefficient of a GaAs/AlGaAs
RTD device with 10 nm well width. Positive energies means electrons injected
from the left and negative energies electrons injected from the right.

At the initial time, we assume that there are no photons inside the active
region. In other words, the (vacuum) electromagnetic field is given by an
amplitude $q$ with probability $|\psi_{0}(q)|^{2}$. Thus, we define
$\psi_{A}(x,0)$ as a Gaussian wave packet outside of the barrier region with a
central energy equal to the second resonant level of the double barrier
$E_{2}$ and a spatial dispersion of $30$ nm, as seen in the blue wave packet
in the $x$-axis of Fig. 1(a), and $\psi_{B}(x,0)=0$. Thus, the initial
electron-photon wave function in expression (13) is given only by
$\Psi(x,q,t)=\psi_{A}(x,t)\psi_{0}(q)$. When solving (16) and (18), together,
with $\alpha=2.5\cdot 10^{7}eV/m$ and $\omega=(E_{2}-E_{1})/\hbar$ we obtain
that $\psi_{B}(x,t)\neq 0$ so that the global wave function in (13) becomes
$\Psi(x,q,t)=\psi_{A}(x,t)\psi_{0}(q)+\psi_{B}(x,t)\psi_{1}(q)$. This process
of spontaneous emission cannot be understood without the quantization of the
electromagnetic field done in (12).

Next, to compute how much probability inside the well can be assigned to
$\psi_{A}(x,t)$ and $\psi_{B}(x,t)$, at each resonant level, we define

$\displaystyle
P_{A,1}(t)=\frac{1}{N}\int_{0}^{\frac{E_{1}+E_{2}}{2}}|c_{A}(E,t)|^{2}dE,$
(19) $\displaystyle
P_{A,2}(t)=\frac{1}{N}\int_{\frac{E_{1}+E_{2}}{2}}^{\infty}|c_{A}(E,t)|^{2}dE,$

with

$c(E,t)=\int_{-L_{x}}^{L_{x}}\psi(x,t)\phi_{E}^{*}(x)dx,$ (20)

The subindex $A$ in $c(E,t)$ and $\psi(x,t)$ is assumed in (20). We omit it to
use later, in the numerical results, the same expression for a general BCWF.
The functions $\phi_{E}(x)$ are the energy eigenstates of the electron
Hamiltonian $H_{e}$ in (8). Notice that we are only interested in the
probability inside the barrier region with limits given by $x=\pm L_{x}$.
Identical definitions can be provided for $P_{B,1}(t)$ and $P_{B,2}(t)$ with
the normalization constant $N$ ensuring that
$P_{A,1}(t)+P_{A,2}(t)+P_{B,1}(t)+P_{B,2}(t)=1$.

In Fig. 3 we plot $P_{A,1}(t)$, $P_{A,2}(t)$, $P_{B,1}(t)$ and $P_{B,2}(t)$,
show the typical Rabi oscillation. The initial value $P_{A,2}(0)\equiv 1$ in
Fig. 3 indicates an electron injected with a central energy equal to the
second eigenvalue of the well without photons. The vertical dashed lines in
Fig 3 indicates two times where the system passes from one electron in the
first level and one photon $P_{A,2}\approx 0$ and $P_{B,1}\approx 1$ (blue
dashed line), to one electron in the second level and zero photons
$P_{A,2}\approx 1$ and $P_{B,1}\approx 0$ (red dashed line).

Figure 3: Evolution of the $P_{A,1}$, $P_{A,2}$, $P_{B,1}$, $P_{B,2}$ for the
first (a) and second (b) eigenstates of the quantum well described in Fig.2,
when the BCWF is injected in the second eigenstate of the quantum well.

From the whole wave function
$\Psi(x,q,t)=\psi_{A}(x,t)\psi_{0}(q)+\psi_{B}(x,t)\psi_{1}(q)$ we can compute
the probability presence in the $x$-space as:

$P_{e}(x,t)=\int dq\Psi(x,q,t)=|\psi_{A}(x,t)|^{2}+|\psi_{B}(x,t)|^{2}$ (21)

In Fig. 4(a) we have represented the evolution of $P_{e}(x,t)$ computed from
(21) as a function of time together with some selected trajectories
$X^{j}[t]$. Such trajectories $X^{j}[t]$ are computed from the guiding total
wave function $\Psi(x,q,t)=\psi_{A}(x,t)\psi_{0}(q)+\psi_{B}(x,t)\psi_{1}(q)$
together with the trajectories $Q^{j}[t]$ belonging to the electromagnetic
degree of freedom $q$ following the velocities defined in (1) for the same
simulation presented before. The evolution shows qualitatively the alternate
transition from the first to the second eigenstate of the quantum well. The
Bohmian trajectories follow this evolution, since they alternatively move from
the side to the center of the quantum well. This evolution suggests a type of
Rabi oscillations where the electron emits a photon into a single-mode
electromagnetic cavity and then reabsorbs it. The evolution of $P_{e}(x,t)$
inside the well shows qualitatively the alternate transition from one maximum
(first eigenstate) to two maxima (second eigenstate). The Bohmian trajectories
show a velocity close to zero when each eigenstate is well-defined, and a
large velocity during the transitions between the two eigenstates. All this
dynamical information is in agreement with the physics of the Rabi
oscillations depicted in Fig. 2 where the electron emits a photon into a
single-mode electromagnetic cavity and then reabsorbs it. As a technical
detail, we mention that as expected Bohmian trajectories do not cross in the
$x-q$ space (not plotted), but they cross in the subspace $x$ of Fig. 4(a). In
addition, one can expect some chaotic behavior in 2D systems [39, 40] that is
not present in the 1D system that is shown in the Fig. 4(a).

Figure 4: (a) Evolution of $P_{e}(x,t)$ for the electron interacting with the
RTD device described in Fig. 2, while emitting and absorbing electromagnetic
radiation. The solid black lines show Bohmian trajectories $X^{j}[t]$ for a
selected set of experiments. The green vertical lines indicate the position of
the potential barriers. (b) Probability distribution of the Hamiltonian
eigenstates for the BCWF given by
$\psi(x,t)=\Psi(x,Q^{j}[t],t)=\psi_{A}(x,t)\psi_{0}(Q^{j}[t])+\psi_{B}(x,t)\psi_{1}(Q^{j}[t])$
for a selected trajectory $Q^{j}[t]$ at two different times indicated by the
horizontal dashed lines in (a). We define the scattering time $t_{s}$ as the
time of the blue horizontal dashed line.

In Fig. 4(b) we plot the probability of the energy states $|c(E,t)|^{2}$ given
by Eq. (20) at the two times indicated by horizontal read and blue dashed
lines in Fig. 4(a) that corresponds to the vertical dashed lines in Fig 3. The
BCWF in Eq. (20) has been defined as
$\psi(x,t)=\Psi(x,Q^{j}[t],t)=\psi_{A}(x,t)\psi_{0}(Q^{j}[t])+\psi_{B}(x,t)\psi_{1}(Q^{j}[t])$
for a selected trajectory $Q^{j}[t]$ of the $j$-experiment. Notice that such
definition of the BCWF corresponds in this case to
$\psi(x,t)\approx\psi_{B}(x,t)$ for the blue wave packet, while the red wave
packet corresponds to $\psi(x,t)\approx\psi_{A}(x,t)$, because of the values
of $P_{A,2}$ and $P_{B,1}$ when Fig. 4(b) is estimated.

As expected, the fact that the conservation of the total energy has to be
satisfied from (12) has important consequences on the type of electron-photon
interaction allowed. We now repeat the simulation when the electron (with no
photon) is injected with a central energy corresponding to the first resonant
level. No electron transition (or spontaneous emission) takes place, giving
$\psi_{B}(x,t)\approx 0$ because the initial energy $E_{1}+\hbar\omega/2$
cannot be converted into a much higher final energy $E_{2}+3\hbar\omega/2$.
The result is shown in Fig. 5.

Figure 5: Evolution of the $P_{A,1}$, $P_{A,2}$, $P_{B,1}$, $P_{B,2}$ for the
first (a) and second (b) eigenstates of the quantum well when the initial
electron is injected in the first resonant level of the quantum well. Because
of the conservation of energy, no matter-light interaction is possible.

We now repeat the same simulation done in Fig 3, where the initial electron
had mean energy equal to the second eigenvalue of the well, $E_{2}$, but
considering a new photon energy $\hbar\omega=0.26$ eV much larger than
$E_{2}-E_{1}=0.172$ eV. In this case no light-matter interaction takes place
since it would imply a violation of the conservation of whole energy. The
initial energy $E_{2}+\hbar\omega/2$ do not coincide with a possible final
energy $E_{1}+3\hbar\omega/2$. This simulation is shown in Fig. 6.

Figure 6: Evolution of the $P_{A,1}$, $P_{A,2}$, $P_{B,1}$, $P_{B,2}$ for the
first (a) and second (b) eigenstates of the quantum well when the BCWF is
injected in the second eigenstate of the quantum well and $\hbar\omega=0.26$
eV. Because of the conservation of energy, no matter-light interaction is
possible.

###  3.2 Approximate solution with BCWF for an open system

In the previous subsection we discussed the interaction of a single electron
with a single photon in a closed system. Here, we discuss how such interaction
can be generalized to include the possibility to detect a photon at a position
$y$, far from the active region.

The proper simulation of such scenario as a closed system is far from the
scope of the present paper. Apart from considering the detector outside of the
active region as a new electron with degree of freedom $y$, the transition of
the electromagnetic energy from the active region to the environment will
require an electromagnetic field with an arbitrary shape different from the
one considered in the previous section. A Fourier transform of such arbitrary
electromagnetic field will imply dealing with several components
$E(x,t)\propto q\;cos(kx-\omega t)$ at different frequencies. In any case,
without an explicit solution of such problem, only from the conservation of
energy, we can anticipate what will be the expected behavior of the whole
system.

The process of spontaneous emission of a photon inside the active region, and
its posterior detection far from the active region, can be anticipated as:

  * •

At the initial time, $t=0$, we consider an electron in the active region, with
degree of freedom $x$ with a central energy $E_{2}$, linked to zero photons
wave function $\psi_{0}(q)$, plus another electron far from the active region,
with degree of freedom $y$ and energy $E_{ext}$, linked to zero photons
$\psi_{0}(q)$. At this initial time, thus, the total energy involved in such
scenario is $E_{2}+\hbar\omega/2$, in the active region, plus the energy
$E_{ext}+\hbar\omega/2$ outside.

  * •

At the intermediate time, we consider that a spontaneous emission of a photon
happens inside the active region. As seen in Fig. 3, such internal process
ensure energy conservation. So that the new photon inside the active region
implies a change of energy there, $E_{2}+\hbar\omega/2\to
E_{1}+3\hbar\omega/2$, while the energy outside of the active region remains
the same as before, $E_{ext}+\hbar\omega/2$. The total energy is the same as
the initial one.

  * •

At the final time $t$, we detect a photon at the position $y$ far from the
active region. Thus, the electron at $y$ is now linked to one photon wave
function $\psi_{1}(q)$ which implies an increment of the energy of
$\hbar\omega$ far from the active region, $E_{ext}+\hbar\omega/2\to
E_{ext}+3\hbar\omega/2$. The conservation of the total energy implies that the
same amount of energy is eliminated in the active region when the photon is
leaving it, $E_{1}+3\hbar\omega/2\to E_{1}+\hbar\omega/2$. The electron in the
active region will have a new energy $E_{1}$ linked to the zero photons wave
function $\psi_{0}(q)$. As we have seen in Fig. 5, under such new energy
conditions in the active region, such electron will not be able any more to
generate spontaneous emission inside the RTD. Thus, the Rabi oscillations seen
in Fig. 3 for a closed system, will not be present when we assume that the
photon is leaving the cavity.

In summary, using the BCWF to focus only on the description of the electron
inside the active region as a single-particle pure state $\psi(x,t)$, we
conclude that the spontaneous emission in the active region can be modeled by
an initial BCWF $\psi(x,0)$ with central energy $E_{2}$ that changes to a
final BCWF $\psi(x,t)$ with energy $E_{1}$. Such process will be allowed as
far as the photon energy coincides with $E_{2}-E_{1}$. Identically, absorption
in the active region can be modeled by an initial BCWF $\psi(x,0)$ with
central energy $E_{1}$ that changes to a final BCWF $\psi(x,t)$ with energy
$E_{2}$ with the photon energy given by $E_{2}-E_{1}$. The conservation of
energy enables the photon absorption to be accompanied by a subsequent process
of spontaneous emission, that returns the photon energy to the environment
outside of the active region.

##  4 Implementation of the transition from pre- to post-selected BCWF

In this section we describe practical issues on how such type of transitions
between initial and final states can be implemented in a transport simulator
for real electron devices based on Bohmian mechanics. Some additional
information about these types of simulators for quantum transport can be found
in the BITLLES simulator[18].

To implement the transition from pre- to post-selected BCWF, a definition of
the initial $|i\rangle$ and the final $|f\rangle$ states is needed. Although
the contacts do not allow us to perfectly prepare the wave description of
electron, we can have some reasonable arguments to anticipate some of its
properties. One option could be to deal with Hamiltonian eigenstates, which
extend to infinite in both sides (left and right) of the device. Although
these infinitely-extended states are useful tools to model (steady-state) DC
transport properties of quantum devices, they are less useful to describe
other device performances as, for example, the fluctuations of the electrical
current due to the partition noise in a tunneling barrier. The initial
electron, after impinging with the barrier, is either located at the left
(reflection) or at the right (transmission) of a barrier, but not at both
sides of it. Such randomness (transmission of reflection) translates into
current fluctuations. To model such fluctuations, a localized wave function
seems appropriate to model electrons. However, the wave function cannot have a
very narrow localization in position since the Heisenberg uncertainty
principle would lead to extremely large momentum and energy uncertainties
(larger than thermal energies). Thus, a definition of an electron, deep inside
the contact, as a Gaussian wave packet with well-defined central position and
central energy seems reasonable. We add that such limited extension of the
electron wave function can be related to the coherence length of the sample.

In classical mechanics, an electron having a well-defined energy is compatible
with an electron having a well-defined momentum. However, this is not the case
for quantum electrons. As a general rule, two properties can be simultaneously
well-defined if their operators commute. In our case, the energy (linked to
the Hamiltonian operator $\hat{H}$) and the momentum (linked to the momentum
operator $\hat{p}$) can be simultaneously defined when $[\hat{H},\hat{p}]=0$.
In the position representation, knowing that the Hamiltonian operator is the
sum of the kinetic energy operator $(\hat{p})^{2}/2m$, which obviously
commutes with $\hat{p}$, plus the potential energy operator $\hat{V}$,
momentum and energy are well-defined properties when

$\big{[}H,-i\hbar\frac{\partial}{\partial_{x}}\big{]}=\big{[}V(x),-i\hbar\frac{\partial}{\partial_{x}}\big{]}=i\hbar\frac{\partial
V(x)}{\partial_{x}}=0.$ (22)

Thus, only when dealing with flat potentials we can assume that a wave packet
with a reasonable well-defined energy has also a reasonable well-defined
momentum. This discussion seems relevant to transport models developed in
phase-space (the Wigner distribution function), where information on only
momenta and positions are available.

In the next two subsections, we discuss the implementation of the transition
from a pre- to a post-selected BCWF when using well-defined energies (model A)
or momenta (model B). In Sec. 5 we compare the numerical results of these two
different implementations.

###  4.1 Model A: change of the central energy

We consider an electron defined by a single-particle BCWF that at time $t_{s}$
undergoes a scattering event. We define $t_{s}^{-}=t_{s}-\Delta t_{s}$ as the
time just before and $t_{s}^{+}=t_{s}+\Delta t_{s}$ as the time just after the
scattering event. For simplicity, we consider $\Delta t_{s}\to 0$, but we have
seen in Sec. 3 that such transition between initial and final BCWF takes a
finite time because, from a conceptual point of view, it has to guarantee the
continuity of the BCWF in space and time. The initial and final BCWFs are
$\psi(x,t_{s}^{-})$ and $\psi(x,t_{s}^{+})$, which satisfy $\langle
E(t_{s}^{+})\rangle=\langle E(t_{s}^{-})\rangle+E_{\gamma}$, with $E_{\gamma}$
the energy of a photon. Within the energy representation, the wave packet can
be decomposed into a superposition of Hamiltonian eigenstates $\phi_{E}(x)$ of
the electron $\hat{H}_{e}$ in (8) as

$\psi(x,t_{s^{-}})=\int dE\;a(E,t_{s}^{-})\;\phi_{E}(x),$ (23)

with $a(E,t)=\int dx\;\psi(x,t)\;\phi_{E}^{*}(x)$. The central energy $\langle
E(t_{s}^{-})\rangle$ is

$\langle E(t_{s}^{-})\rangle=\int dE\;E\;|a(E,t_{s}^{-})|^{2},$ (24)

which can be increased to get the new central energy at $t_{s}^{+}$ as

$\displaystyle\langle E(t_{s}^{+})\rangle$ $\displaystyle=$
$\displaystyle\langle E(t_{s}^{-})\rangle+E_{\gamma}$ (25) $\displaystyle=$
$\displaystyle\int dE\;(E+E_{\gamma})\;|a(E,t_{s}^{-})|^{2}$ $\displaystyle=$
$\displaystyle\int
dE^{\prime}\;E^{\prime}\;|a({E^{\prime}-E_{\gamma}},t_{s}^{-})|^{2}$
$\displaystyle=$ $\displaystyle\int
dE^{\prime}\;E^{\prime}\;|a^{\prime}(E^{\prime},t_{s}^{+})|^{2},$

where we have defined $a^{\prime}(E,t_{s}^{+})=a(E-E_{\gamma},t_{s}^{-})$.
Thus, the new wavepacket after the collision is

$\displaystyle\psi(x,t_{s^{+}})$ $\displaystyle=$ $\displaystyle\int
dE\;a^{\prime}(E,t_{s}^{+})\;\phi_{E}(x)$ (26) $\displaystyle=$
$\displaystyle\int dE\;a(E^{\prime}-E_{\gamma},t_{s}^{-})\;\phi_{E}(x).$

This transition corresponds to absorption of energy by the electron. Emission
can be identically modeled by using $\langle E(t_{s}^{+})\rangle=\langle
E(t_{s}^{-})\rangle-E_{\gamma}$. If required, the _technical_ discontinuity
between $\psi(x,t_{s^{-}})$ and $\psi(x,t_{s^{+}})$ can be solved by just
assuming that the change of energy is produced in a finite time interval
$\Delta t_{s}=N_{t_{s}}\Delta t$, with $\Delta t$ the time step of the
simulation. Then, at each time step of the simulation, the change in the wave
packet central energy is $E_{\gamma}/N_{t_{s}}$. A continuous change of both
energy and wave packet will be obtained as far as $\Delta t\to 0$.

This continuous evolution of the BCWF can be represented as a Schrödinger-like
equation as explained in [31]. In any case, the shape of the Hamiltonian of
this new Schrödinger equation describing the collision is subjected to the
post-selection of the state.

###  4.2 Model B: change of central momentum

In Ref. [34] we explain how a change of momentum $p_{\gamma}$ in a wave packet
in free space can be performed with a unitary Schrödinger equation. That
algorithm can be understood as a pre- and a post-selection of the initial
BCWF, $\psi(x,t_{s}^{-})$, and of the final BCWF, $\psi(x,t_{s}^{+})$,
respectively. At time $t_{s}^{-}$, the BCWF can be written as a superposition
of momentum eigenstates $\phi_{p}(x)$ (which are a basis of the electron in
the x space) as

$\psi(x,t_{s^{-}})=\int dp\;b(p,t_{s^{-}})\;\phi_{p}(x),$ (27)

with $b(p,t_{s^{-}})=\int dx\psi(x,t_{s^{-}})\;\phi_{p}^{*}(x)$. The central
momentum $\langle p(t_{s}^{-})\rangle$ is

$\langle p(t_{s}^{-})\rangle=\int dp\;p\;|b(p,t_{s^{-}})|^{2},$ (28)

which can be increased to get the new central momentum $\langle
p(t_{s}^{+})\rangle=\langle p(t_{s}^{-})\rangle+p_{\gamma}$ at $t_{s}^{+}$ as

$\displaystyle\langle p(t_{s}^{+})\rangle$ $\displaystyle=$
$\displaystyle\langle p(t_{s}^{-})\rangle+p_{\gamma}$ (29) $\displaystyle=$
$\displaystyle\int dp\;(p+p_{\gamma})\;|b(p,t_{s^{-}})|^{2}$ $\displaystyle=$
$\displaystyle\int
dp^{\prime}\;p^{\prime}\;|b({p^{\prime}-p_{\gamma}},t_{s^{-}})|^{2}$
$\displaystyle=$ $\displaystyle\int
dp^{\prime}\;p^{\prime}\;|b(p^{\prime},t_{s^{+}})|^{2},$

where we have defined $b(p,t_{s^{+}})=b(p-p_{\gamma},t_{s^{-}})$. In this
particular scenario, we know the explicit shape of the momentum eigenstates,
$\phi_{p}(x)=1/\sqrt{2\pi}\exp(ipx/\hbar)$, so that

$\displaystyle\psi(x,t_{s}^{+})$ $\displaystyle=$ $\displaystyle\int
dp\;b(p,t_{s^{+}})\;\phi_{p}(x)$ $\displaystyle=$ $\displaystyle\int
dp\;b(p-p_{\gamma},t_{s^{-}})\;\phi_{p}(x)$ $\displaystyle=$
$\displaystyle\int dp\int
dx^{\prime}\psi(x,t_{s^{-}})\;\phi_{p-p_{\gamma}}^{*}(x^{\prime})\phi_{p}(x)$
$\displaystyle=$ $\displaystyle\int dp\int
dx\;\psi(x,t_{s^{-}})\;\frac{1}{2\pi}e^{ip(x^{\prime}-x)/\hbar}e^{ip_{\gamma}x^{\prime}/\hbar}$
$\displaystyle=$ $\displaystyle e^{ip_{\gamma}x/\hbar}\psi(x,t_{s^{-}}).$

With the condition
$\psi(x,t_{s}^{+})=e^{ip_{\gamma}x/\hbar}\psi(x,t_{s^{-}})$, it can be easily
demonstrated which one is the unitary equation satisfied by the BCWF. If we
define $\psi^{\prime}(x,t)$ as the wave function solution of the following
Schrödinger equation, $i\hbar\frac{\partial\psi^{\prime}(x,t)}{\partial
t}=\frac{1}{2m^{*}}\left(-i\hbar\frac{\partial}{\partial_{x}}\right)^{2}\psi^{\prime}(x,t)+V(x)\psi^{\prime}(x,t)$,
with initial condition at $t=t_{s}$ given by
$\psi^{\prime}(x,t_{s})=\psi(x,t_{s}^{+})$, then, the solution
$\psi^{\prime}(x,t)$ for $t>t_{s}$ is identical to the following Schrödinger
equation, $i\hbar\frac{\partial\psi(x,t)}{\partial
t}=\frac{1}{2m^{*}}\left(-i\hbar\frac{\partial}{\partial_{x}}+p_{\gamma}\right)^{2}\psi(x,t)+V(x)\psi(x,t)$,
for the original $\psi(x,t)$ and with its original initial condition for
$t>t_{s}$. Finally, a single equation for $\psi(x,t)$ valid for all times is
just

$\displaystyle i\hbar\frac{\partial\psi(x,t)}{\partial t}$ $\displaystyle=$
$\displaystyle\frac{1}{2m^{*}}\left(-i\hbar\frac{\partial}{\partial_{x}}+p_{\gamma}\Theta_{t_{s}}\right)^{2}\psi(x,t)$
(31) $\displaystyle+$ $\displaystyle V(x)\psi(x,t),$

where $\Theta_{t_{s}}$ is a Heaviside function equal to $1$ for $t>t_{s}$ and
zero otherwise. Thus, a description of the evolution of the wave function
$\psi(x,t)$ during the collision process can be done from a unitary
Schrödringer equation, where the momentum operator
$-i\hbar\frac{\partial}{\partial_{x}}$ is changed for the new momentum
operator $-i\hbar\frac{\partial}{\partial_{x}}+p_{\gamma}\Theta_{t_{s}}$, as
indicated in [34]. Notice that the probability presence of the scattered wave
packet satisfies $|\psi(x,t_{s^{+}})|^{2}=|\psi(x,t_{s^{-}})|^{2}$ because
only a global phase $e^{ip_{\gamma}x/\hbar}$ is added.

It is quite easy to see from (4) that the Bohmian velocity of the electron
after the collision computed from $\psi(x,t_{s}^{+})$ is just the old velocity
computed from $\psi(x,t_{s}^{-})$ plus $p_{\gamma}/m^{*}$,

$\displaystyle v_{x}^{j}[t_{s}^{+}]$ $\displaystyle=$
$\displaystyle\frac{1}{m^{*}}\frac{\partial s(x,t_{s}^{+})}{\partial
x}|_{x=X^{j}[t]}$ (32) $\displaystyle=$
$\displaystyle\frac{1}{m^{*}}\frac{\partial s(x,t_{s}^{-})}{\partial
x}|_{x=X^{j}[t]}+p_{\gamma}/m^{*}.$

The collision increases the velocity of the electron by the same amount that
we add in (31). Unfortunately, as discussed at the beginning of the section, a
global mechanism of scattering valid for scenarios with potential barriers
requires dealing with change of the energy as presented in Model A (not with
change of the momentum as presented in Model B).

##  5 Numerical results

We present now the numerical results of our two models for the transition
between initial and final single-particle BCWF, as explained in the previous
section. We first study electron-photon collisions in free space, when energy
and momentum operators commute, and then electron-photon collisions in a
scenario with a double barrier potential profile, when energy and momentum
operators do not commute. This last case will be compared with numerical
results of the exact model presented in Sec. 3, and used to verify the
physical soundness of the two models.

###  5.1 Collisions in flat potentials

In this section, we study the interaction of an electron and a photon in free
space. The electron evolves in a flat potential. We consider the absorption of
a photon by an electron. In flat potential, the momentum and energy
conservation is ensured during the collision. Thus, since the momentum of the
photon is negligible, in this Sec. 5.1, we assume that the electron is
interacting with a phonon and a photon. The phonon will not be needed in Sec.
5.2. We consider that the final BCWF will be modeled by a final electron
(post-selected state) with an energy increase of $\hbar\omega$
($E_{\gamma}>0$) plus the corresponding increase of momentum (provided by the
phonon) with respect to the initial electron energy (pre-selected state).

Figure 7: The evolution of the BCWF $\psi^{j}(x,t)$, undergoing photon
absorption with $E_{\gamma}=0.1eV$, shown as function of position and time.
The wavefunctions are simulated (a) without collision, (b) with collision
using model A, (c) without scattering, (d) scattered using model B. The
trajectories $X^{j}[t]$ guided by the BCWF $\psi^{j}(x,t)$, where
$j=1,\ldots,10$, are some representative experiments and are shown in black.
In a flat potential, the results of models A and B are identical.

In Fig. 7 we show the simulation of the electron-photon collision in a flat
potential. The collision is modeled by exchanging the energy
$E_{\gamma}=0.1eV$ in Fig. 7(a) $\to$ (b), and by exchanging the momentum
$p_{\gamma}=\sqrt{2E_{\gamma}/m^{*}}$ in Fig. 7(c) $\to$ (d). As expected, in
this scenario, both models give identical results. After the scattering event,
the Gaussian wave function evolves with a higher velocity, as indicated in
(32). We notice that the wave function suffers a continuous evolution during
the collision because it is solution of the Schrödinger-like equation (31).
Analogous results (not shown) are obtained for emission. The main conclusion
of this subsection is that model A and model B are, as expected, numerically
equivalent in the case of a flat potential.

###  5.2 Collisions in arbitrary potentials

Figure 8: Gaussian wavefunctions interacting with a double barrier potential
profile with and without scattering with a photon: (a) a wavepacket and some
selected trajectories with unitary evolution (without scattering). (b) The
same wave packet and the same selected trajectories when scattering with
energy $E_{\gamma}=0.186\,$eV using model A occurs. (c) and (d) are identical
to (a) and (b) when model B is used. In all figures, the Gaussian wave packet
is injected from the left at energy $E=E_{1}=0.058\,$eV. The trajectories
$X^{j}[t]$ guided by the BCWF $\psi^{j}(x,t)$ are plotted in black. The set of
trajectories in plot (a) is different from the one in plot (c), with the goal
of selecting those trajectories that most interact in the quantum well in each
case. The trajectories in plot (b) are the same as in plot (a) and the
trajectories in plot (d) are the same as in plot (c). The energy of the photon
is equal to the distance between the two first energy levels,
$E_{\gamma}=E_{2}-E_{1}$.

As in Sec. 5.1, we study the absorption of a photon by an electron modeled by
a final electron (post-selected state) with an energy increase of
$\hbar\omega$ ($E_{\gamma}>0$) with respect to the initial electron energy
(pre-selected state). But now we use a double barrier potential $V(x)$
identical to the one mentioned in Sec. 3.1, with the same two resonant
energies $E_{1}=0.058\,$eV and $E_{2}=0.23\,$eV.

In Fig. 8 the evolution of $\psi^{j}(x,t)$ and the trajectories $X^{j}[t]$ are
shown when the electron absorbs a photon while impinging on the potential
barrier of the RTD. The position of the barriers is shown by the green
vertical lines. The energy of the photon is equal to the difference of the
resonant energies in the quantum well, $E_{\gamma}=E_{2}-E_{1}$, and the BCWF
is injected with a central energy equal to the first resonant energy
$E=E_{1}$. A transition from $E_{1}$ to $E_{2}$ is expected during the
collision $\psi_{A}(x,t_{s}^{-})\to\psi_{B}(x,t_{s}^{+})$.

In Fig. 8 (a), we plot the time evolution of the electron interacting with the
barrier, but without photon collision. In Fig. 8 (b), an electron-photon
collision is produced at $t_{s}=150fs$ using model A. The wavepacket undergoes
a shift of the energy probability distribution of the Hamiltonian eigenstates
$\phi_{E}(x)$ towards higher values. As expected, the evolution of $\psi(x,t)$
is a transition from the first eigenstate of the well (with one peak of
probability in the middle of the well) to the second one (with two probability
peaks). The same trajectories $X^{j}[t]$ that were first reflected by the
barrier in Fig. 8 (a) are now transmitted through the well in Fig. 8 (b)
because the second resonant level has a wider transmission probability, as
shown in Fig. 2 (b). The results in Fig. 8 (b) have a reasonable agreement
with the results in Fig. 4 (a) at times equivalent to the blue and red
horizontal lines of Fig. 4 (a). Clearly, we also notice that the simulated
result in Fig. 4 (a) belongs to a simulation with the active region as a
closed system, where the photon energy does not disappear, an the electron is
continuously emitting and absorbing such photon energy, as explained in Sec.
3.1. On the contrary, Fig. 8 (b) corresponds to a simulation of the active
region as an open system, where the photon energy appear/disappear at/from the
active region only once, as explained in Sec. 3.2.

Figure 9: (a) Probability distribution of the Hamiltonian eigenstates for
Model A (spatial evolution shown in Fig.8(b)). (b) Probability distribution of
the Hamiltonian eigenstates for Model B (spatial evolution shown in Fig.8(d)).
Blue lines represent the probability distribution of the Hamiltonian
eigenstates before the scattering at $t<t_{s}$, while the red lines show it at
$t>t_{s}$.

The same plots are reproduced in Fig. 8 (c) and (d) when using model B. Now,
an oscillatory behaviour on the BCWF and on the trajectories $X^{j}[t]$ is
shown after time $t_{s}=250$ fs. Such results can be understood by noticing
that model B produces an increase of the velocity in Bohmian trajectories, but
such faster Bohmian trajectories are not the _natural_ behavior of the
trajectories in the well when associated to just one eigenstate (they are
expected to remain inside the well for a large time with a velocity close to
zero). But since the eigenstates of the quantum well form a complete basis
inside the well, the mentioned oscillatory BCWF can be a solution of the
Schrödinger equation there at the price of using many more eigenstates (with
higher energies) to describe the new accelerated wave packet. Thus, the
combination of several eigenstates in the well produces the oscillatory
behaviour that we see in Fig. 8(d).

To better understand that model A provides a _natural_ transition, while model
B provides an _unnatural_ one, we show in Fig. 9 the probability of the energy
states $|c(E,t)|^{2}$ given by Eq. (20) at $t=0$ and $t=t_{s}^{+}$. The
positive and negative energies just indicate scattering states injected from
the left (positive) and injected from the right (negative). The blue line is
the probability distribution of the energy eigenstates at the initial time
$c(E,0)$, while the red line is the same distribution but after the scattering
$c(E,t_{s}^{+})$. In Fig. 9 (a) for model A, we observe a _natural_ shift in
the central energy given by $\langle E(t_{s}^{+})\rangle=\langle
E(t_{s}^{-})\rangle+E_{\gamma}$, as expected. A definite argument in favor of
model A (and against model B) is that the results in Fig. 9 (a) have an almost
perfect agreement with the result in Fig. 3(b) that where computed without
approximation: the same transition happens from the first to the second energy
eigenvalues of the quantum well. On the contrary, in Fig. 9 (b) for model B, a
large amount of Hamiltonian eigenstates with negative energies are _created_
after the scattering process. As explained, these additional energy components
are the reason why we observe an oscillatory behaviour inside the well in Fig.
8 (d). Model B is nonphysical because it does not satisfy the requirement of
conservation of energy in the electron and photon collision. Since we are
dealing with a wave packet (with some uncertainty on its energy), some
deviation on the requirement of conservation of energy in each experiment is
reasonable, but not the deviations plotted in Fig. 9 (b) where energies as
high as 1eV are involved.

In conclusion, model B can only describe electron collisions when an
approximation of flat potential is reasonable to describe the dynamic of the
unperturbed electron. We get exactly the same conclusions when evaluating the
emission process (not plotted) instead of the absorption process. In the
authors’ opinion, this conclusion about model B has dramatic consequences for
quantum transport formalisms that introduce scattering in the position-
momentum space, like the Wigner distribution function.

##  6 Conclusions

Quantum transport formalisms require the modeling of the perturbation induced
by the non-simulated degrees of freedom (like photons or phonons) on degrees
of freedom of the simulated active region (the electrons). Among a number of
different algorithms that allow to include scattering events, here we explore
the possibility of implementing such scattering events as transitions between
single-particle pure-states. We have shown that the Bohmian theory, through
the use of BCWF, allows a rigorous implementation of transitions between pre-
and post-selected single-particle pure states in the active device that is
valid for both Markovian and non-Markovian conditions. Furthermore, we have
shown that the practical implementation of such transitions requires one to
model scattering events as a shift of central energies of BCWFs instead of a
shift of central momenta. This last result seems to indicate dramatic
consequences for quantum transport formalisms that introduce collisions
through changes in momentum, e.g., the Wigner function approach, when dealing
with non-flat potential profiles where energy and momentum are non-commuting
operators. The paper is part of a global and long-term research project
developing the so-called BITLLES simulator[18]. We argue that the amount of
information that this simulator framework can provide (from steady-state DC
till transient and AC including the fluctuations of the current) in the
quantum regime is comparable to the predicting capabilities of the traditional
Monte Carlo solution of the Boltzmann transport equation in the semi-classical
regime.

#### Contributions

“Conceptualization, M.V., X.O., X.C., C.D. and G.A.; methodology, M.V., X.O.,
X.C., C.D. and G.A.; software, M.V., X.O.; validation, M.V., X.O. X.C., C.D.
and G.A.; investigation, M.V., X.O., X.C., C.D. and G.A.; writing—original
draft preparation, M.V., X.O., X.C., C.D. and G.A.; writing–review and
editing, M.V., X.O., X.C., C.D. and G.A.; visualization, M.V. and X.O.;
supervision, X.O., X.C., G.A. and C.D.; project administration, X.O.; funding
acquisition, X.O., G.A. and X.C. All authors have read and agreed to the
published version of the manuscript.”

#### Funding

This research was funded by Spain’s Ministerio de Ciencia, Innovación y
Universidades under Grant No. RTI2018-097876-B-C21 (MCIU/AEI/FEDER, UE), the
”Generalitat de Catalunya” and FEDER for the project 001-P-001644
(QUANTUMCAT), the European Union’s Horizon 2020 research and innovation
programme under Grant No. 881603 GrapheneCore3 and under the Marie Skłodowska-
Curie Grant No. 765426 TeraApps.

#### Conflicts of interest

The authors declare no conflict of interest. The funders had no role in the
design of the study; in the collection, analyses, or interpretation of data;
in the writing of the manuscript, or in the decision to publish the results.

#### Abbreviations

The following abbreviations are used in this manuscript:

BCWF | Bohmian Conditional Wave Function
---|---
RTD | Resonant Tunnelling Diode

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