# Can Wigner distribution functions with collisions satisfy complete
positivity and energy conservation?

Matteo Villani1, Xavier Oriols1

(1: Department of Electronic Engineering, Universitat Autònoma de Barcelona,
Campus de la UAB, 08193 Bellaterra, Barcelona, Spain)

###### Abstract

To avoid the computational burden of many-body quantum simulation, the
interaction of an electron with a photon (phonon) is typically accounted for
by disregarding the explicit simulation of the photon (phonon) degree of
freedom and just modelling its effect on the electron dynamics. For quantum
models developed from the (reduced) density matrix or its Wigner-Weyl
transformation, the modelling of collisions may violate complete positivity
(precluding the typical probabilistic interpretation). In this paper, we show
that such quantum transport models can also strongly violate the energy
conservation in the electron-photon (electron-phonon) interactions. After
comparing collisions models to exact results for an electron interacting with
a photon, we conclude that there is no fundamental restriction that prevents a
collision model developed within the (reduced) density matrix or Wigner
formalisms to satisfy simultaneously complete positivity and energy
conservation. However, at the practical level, the development of such
satisfactory collision model seems very complicated. Collision models with an
explicit knowledge of the microscopic state ascribed to each electron seems
recommendable, since they allow to model collisions of each electron
individually in a controlled way satisfying both conditions.

#### keywords:

Matter-light interaction; Wigner Function; Complete positivity; Energy
conservation; Bohmian conditional wavefunctio

#### Corresponding author:

<EMAIL_ADDRESS>
Electron devices are quantum systems outside of thermodynamic equilibrium with
many interacting particles (electrons, atoms, photons, etc). In addition, one
is not only interested on the time-independent or steady state (DC) simulation
of such devices, but also on their time-dependent (AC, transients)
performance, and even on their noise properties. As a result, from a
computational point of view, an electron devices is one of the most difficult
quantum systems to be simulated. Any attempt to directly get the device
performance from the simulation of all particles, in a type of solution of the
many-body Schrödinger equation, is directly impossible. This difficulty has
the origin in the so-called many body problem [1]. The typical strategy to
reduce such inaccessible computational burden is introducing an artificial
division between the simulated particles, usually named the open system, and
the rest of non-simulated particles, which are usually referred as the
environment [1, 2]. Such division, in turn, requires introducing the effect of
the non-simulated (environment) particles onto the simulated ones through some
new term in the equations of motion of simulated particles. This new element
in the equation of motion is what is called the collision term. Typical
examples found in the literature following this strategy are Green Function
[3, 4, 5], the density matrix [6, 7], Wigner distribution function [8, 9, 10,
11, 12, 13, 14], the master equation [15, 16], Kubo formalism [17],
conditional wave functions [18, 19, 20, 21, 22], etc.

There are two main difficulties in these quantum transport approaches with
dissipation. The first difficulty is that open quantum systems cannot be
described by an (orthodox) pure state, but by a density matrix (or some
transformation of it). Thus, we cannot assign a pure state to an electron and
we have difficulties to identify which are the properties of each electron
that are modified by collisions. The second difficulty appears because the
collisions are, at best, a reasonable approximation of the real interaction
between the simulated and non-simulated degrees of freedom (but never an exact
result). Both difficulties make the evaluation of the physical soundness of a
collision model for quantum transport a difficult task. A useful criteria for
evaluating collisions models is checking about its complete positivity. The
complete positivity means that the (reduced) density matrix will not provide
negative values of the probability presence along the device. The presence of
such negative values is unphysical because the typical probabilistic
interpretation is precluded then [1, 2]. The Wigner distribution function, as
a Wigner-Weyl transformation of the density matrix [8, 23], does also suffer
from such lack of complete positivity. We clarify that we are not referring
here to the fact that the Wigner function is a quasi-probability [24], but to
the fact of obtaining negative probability of finding electrons at some
positions of the device. For example, it is well-known that the typical use of
a Boltzmann-like superoperator in the description of the collision term in the
Wigner formalism can produce such regions of negative probability presence [6,
7, 25]. Other issues like the lack of energy conservation have been underlined
in the use of a Boltzmann superoperator, [26, 27].

In this work, we will focus on an additional criteria to verify the physical
soundness of collision models. It is very reasonable to assume that most
collisions between particles induce a well-defined change of the energy of the
electron. The physical justification of such assumption is that an
hypothetical exact solution of the interaction of the particles involved in a
collision, through a many-body Schrödinger equation, would show that the total
energy (the ensemble value of the Hamiltonian) is a constant of motion.
However, in the density matrix (or some transformation of it), it is not
always possible to access to the energy of each particle. Without such
information, it is not clear if a collision model satisfies or not the
mentioned energy conservation requirements. A clear example of such
difficulties appears in the formulation of quantum transport through the
Wigner distribution function. In the Wigner function, the information of the
quantum system is given by one position and one momentum degree of freedom. No
information of the energy of each individual particle is given by the Wigner
formalism. One can assume that the kinetic energy of particles can be obtained
from the value of their momentum. However, in many practical scenarios, there
is no one-to-one relation between energy and momentum (the energy eigenstates
are not momentum eigenstates, and viceversa). Then, a reasonable change in the
momentum in the Wigner function after a collision even can be translated into
an unexpected/uncontrolled change in the total energy, making the final result
of the collision process unphysical.

After this introduction, in Sec. 1, we will define the two conditions,
complete positivity and energy conservation, for modeling collisions in a
general density matrix formulation of a quantum system. In Sec. 2, we will
show an exact solution for the electron-photon interaction emphasizing the
importance of energy conservation during the electron-photon interaction.
Then, we will perform a comparison between approximate collision models that
modify the energy of the electron and approximate collision models that modify
the momentum of the electron, respectively. In Sec. 3, we will show the
Wigner-Weyl representation of such energy and momentum collisions models for
several cases showing the non-physical results originated by the collisions
models that exchange momentum in scenarios where energy and momentum do not
commute. In Sec. 4, we define the two practical requirements that are
mandatory to be satisfied by a collision model implemented in the phase-space
(Wigner) description to satisfy complete positivity and energy conservation.
In Sec. 4 we also add a subsection about time reversibility and how this
influenced by collisions. We conclude in Sec. 5 indicating that alternative
collision models with an explicit knowledge of the state ascribed to each
electron seems strongly recommendable to avoid the previous two problems.

##  1 Problems in modelling collisions

The density matrix deals with mixed states which arise in quantum mechanics
when the preparation of the system is not fully known, or when one wants to
describe an open system which is entangled to an environment [1]. Since both
conditions are typically observed in electron devices, the density operator
seems an adequate tool to study quantum transport. The typical equation of
motion for the density operator $\hat{\rho}(t)$ is given by the Liouville-Von
Neumann equation

$\frac{d\hat{\rho}(t)}{dt}=\frac{1}{i\hbar}[\hat{H},\hat{\rho}(t)]=\frac{1}{i\hbar}[\hat{H}_{0},\hat{\rho}(t)]+\hat{C}[\hat{\rho}(t)].$
(1)

The first term in the right hand side of (1) provide the unitary evolution of
the systems with $H_{0}$ a single-particle Hamiltonian, while the collisions
are introduced by the new term given by the superoperator $\hat{C}$ acting on
the density matrix operator.

Due to a collision at time $t_{s}$, the new density matrix
$\hat{\rho}^{s}(t_{s}+\Delta t)$ from (1) will be equal to the unitary density
matrix with a free evolution until time $t_{s}$ given by
$\hat{\rho}_{0}(t_{s})=\hat{\rho}(t_{s})+\frac{\Delta
t}{i\hbar}[\hat{H}_{0},\hat{\rho}(t_{s})]$, plus some perturbation due to the
collisions given by $\Delta\hat{\rho}(t_{s})=\Delta
t\hat{C}[\hat{\rho}(t_{s})]$:

$\displaystyle\hat{\rho}^{s}(t_{s}+\Delta t)$ $\displaystyle=$
$\displaystyle\hat{\rho}(t_{s})+\frac{\Delta
t}{i\hbar}[\hat{H}_{0},\hat{\rho}(t_{s})]+\Delta t\hat{C}[\hat{\rho}(t_{s})]$
(2) $\displaystyle=$
$\displaystyle\hat{\rho}_{0}(t_{s})+\Delta\hat{\rho}(t_{s}).$

Hereafter, when needed, we will use the superindex $s$ to indicate elements
that refer to the system after the scattering event has taken place. Under the
assumption that changes in the state of the system need some time to occur (an
example of such time delay will be provided in the exact solution of the
electron-photon in Sec. 2.1), a reasonable condition for the application of
the model in (2) is a finite value of the scattering time $\Delta t$ [26, 27,
28].

In most computational algorithms, it is not possible to know which are the
individual (pure) states that build the density matrix. However, such
(microscopic) knowledge can be obtained within the computational technique
known as stochastic Schrödinger equation (for Markovian systems) or within the
Bohmian theory (for either Markovian or non-Markovian systems) [29, 30, 31].
Within such techniques, we can assume that the exact single-particle wave
function $\psi_{j}(x,t)=\langle x|\psi_{j}(t)\rangle$ is known for each $j$-th
electron to build the density operator $\hat{\rho}(t)$ as:

$\hat{\rho}(t)=\frac{1}{M}\sum_{j=1}^{N}M_{j}|\psi_{j}(t)\rangle\langle\psi_{j}(t)|,$
(3)

where $M_{j}$ is the number of states $|\psi_{j}(t)\rangle$ that are present
in the system with $M=\sum_{j=1}^{N}M_{j}$ and $N$ the maximum number of
possible types of states (to be able to deal with annihilation and creation of
electronic states, $M_{j}$ can be incremented or decremented by one at the
scattering time, $t_{s}$, but we do not write its time dependence explicitly
to simplify the notation).

To better understand the problems of complete positivity and energy
conservation, in the next two subsections, we will discuss the change of the
density matrix due to a collision in (2) by distinguishing between the
algorithms that have access to the additional knowledge given by (3) and the
ones that do not have access. The computation of collisions without such
knowledge corresponds to most of the algorithms presented in the literature
dealing with quantum transport. We will see that without the additional
knowledge of (3) the collisions process implemented in (2) can lead to
violations of the complete positivity and energy conservation. Such violations
can easily be avoided using computational algorithms that have access to the
additional information provided by (3).

###  1.1 The problem of complete positivity

For algorithms [29, 30, 31] that have access to the additional information in
(3), the interaction of one electron with a photon at time $t_{s}$, can be
understood as an electron changing from its initial state
$|\psi_{2}(t_{s})\rangle$ to its final state $|\psi^{s}_{2}(t_{s})\rangle$.
Thus, the new scattered density matrix after the scattering is:

$\displaystyle\hat{\rho}^{s}(t)=\hat{\rho}_{0}(t)-\frac{1}{M}|\psi_{2}(t)\rangle\langle\psi_{2}(t)|+\frac{1}{M}|\psi^{s}_{2}(t)\rangle\langle\psi^{s}_{2}(t)|.$
(4)

The last two terms in the above expression correspond to annihilating the old
(description of the) electron $|\psi_{2}(t)\rangle$ and creating a new
(description of the) electron $|\psi^{s}_{2}(t)\rangle$. Such collision
corresponds to the new collision term in (2) as:

$\Delta\hat{\rho}(t)=-\frac{1}{M}|\psi_{2}(t)\rangle\langle\psi_{2}(t)|+\frac{1}{M}|\psi^{s}_{2}(t)\rangle\langle\psi^{s}_{2}(t)|.$
(5)

The presence density (also known as the charge density) of the density matrix
in (4) at any time $t>t_{s}$ can be computed as:

$\displaystyle\langle Q(x,t)\rangle$ $\displaystyle=$ $\displaystyle\langle
x|\hat{\rho}(t)|x\rangle=\sum_{j=1}^{N}\frac{M_{j}}{M}|\psi_{j}(x,t)|^{2}$ (6)
$\displaystyle-$
$\displaystyle\frac{1}{M}|\psi_{2}(x,t)|^{2}+\frac{1}{M}|\psi^{s}_{2}(x,t)|^{2}.$

Notice that $|\psi_{2}\rangle$ and $|\psi^{s}_{2}\rangle$ are totally
different states because the collision can change momentum, energy, spatial
distribution of the probability, etc. The problem with collisions for those
algorithms that have no access to the information in (3) is that the
implementation of $\Delta\hat{\rho}(t)$ can correspond to subtracting a state
that is not present in the initial density matrix $\hat{\rho}(t_{s})$. Then,
the term -$\frac{1}{M}|\psi_{2}(x,t)|^{2}$ can provide negative presence
density $\langle Q(x,t)\rangle<0$ in some spatial regions or times, which is
clearly unphysical. Certainly, the computational algorithms with the
additional knowledge of (3) could avoid such unphysical result by just not
performing the scattering process from $|\psi_{2}(t)\rangle$ to
$|\psi^{s}_{2}(t)\rangle$ in (2) because such algorithms knows that there is
no such initial state $|\psi_{2}(t)\rangle$. The problem is that most of the
computational algorithms dealing with quantum transport do not have access to
such microscopic information from (3) and they have to blindly apply a change
on the density matrix given by $\Delta\hat{\rho}(t)$ without knowing which are
the implications of such change in terms of states. In other words, the study
of quantum transport in terms of the (macroscopic) matrix
$\rho(x,x^{\prime})=\langle x^{\prime}|\hat{\rho}(t)|x\rangle$ has the great
computational advantage of not needing the (microscopic) knowledge of the
states building such matrix, but it has the drawback of losing control on the
physical meaning of a change of density matrix by an amount equal to
$\Delta\rho(x,x^{\prime})=\langle x^{\prime}|\Delta\hat{\rho}(t)|x\rangle$
after a collision.

A more formal discussion of this problem, without the previous classification
in terms of knowledge of information given by (3), shows that non-Lindblad
type of collisions can violate complete positivity [1, 2]. It is also well-
known that the Boltzmann-like collision operator applied to the Wigner
distribution function can violate such complete positivity [6, 7, 25].

###  1.2 The problem of energy conservation

We discuss now an additional problem, not related with the probability
presence, but with the conservation of energy. For those algorithms [29, 30,
31] that have access to the information in (3), the electron-photon collision
means that an electron with (ensemble) energy $\langle
E_{2}(t_{s})\rangle=\langle\psi_{2}(t_{s})|\hat{H}_{0}|\psi_{2}(t_{s})\rangle$
at time $t_{s}$ change its energy, due to interaction with a photon of energy
$E_{\gamma}$, to the new value $\langle
E^{s}_{2}(t_{s})\rangle=\langle\psi_{2}^{s}(t_{s})|\hat{H}_{0}|\psi_{2}^{s}(t_{s})\rangle$.
The requirement of conservation of energy with such algorithms that have
access to the microscopic states, $\langle E^{s}_{2}(t_{s})\rangle=\langle
E_{2}(t_{s})\rangle+E_{\gamma}$, is trivially satisfied.

The energy of the electron system at any time $t>t_{s}$ can be computed as:

$\displaystyle\langle E^{s}(t)\rangle\rangle$ $\displaystyle=$
$\displaystyle\text{Tr}\left(\hat{\rho}^{s}(t)H_{0}\right)=\sum_{j=1}^{N}\frac{M_{j}}{M}\langle
E_{j}(t)\rangle$ (7) $\displaystyle-$ $\displaystyle\frac{1}{M}\langle
E_{2}(t)\rangle+\frac{1}{M}\langle E^{s}_{2}(t)\rangle,$

where $\langle
E_{j}(t)\rangle=\langle\psi_{j}(t)|\hat{H}_{0}|\psi_{j}(t)\rangle$. Again, for
algorithms that are not allowed to deal with the microscopic information of
the state of each electron, before an after the collision, the change of
energy due to the collision done in (2) has to satisfy:

$\displaystyle E_{\gamma}$ $\displaystyle=$ $\displaystyle\langle
E^{s}_{2}(t)\rangle-\langle
E_{2}(t)\rangle=\text{Tr}\left(\Delta\hat{\rho}(t)H_{0}\right),$ (8)

with $\Delta\hat{\rho}(t)$ defined in (5). The problem is that it is not
trivial to ensure that expression (8) is satisfied, only using the macroscopic
information that we have $\rho(x,x^{\prime})=\langle
x^{\prime}|\hat{\rho}(t)|x\rangle$, and without knowing the information of the
states involved in such collision.

In summary, as we will see along the paper, the computational advantage of the
density operator of encapsulating all the physical information into the
density matrix $\rho(x,x^{\prime})=\langle x^{\prime}|\hat{\rho}(t)|x\rangle$,
avoiding to treat the heavy microscopic description of the state of each
electron, also presents some drawbacks when dealing with collisions. The
representation of the density matrix in coordinate space can be changed into
the so-called Wigner distribution function $f_{W}(x,k,t)$ through the Wigner-
Weyl transformation [8, 23]:

$f_{W}(x,k,t)=\frac{1}{2\pi}\int dx^{\prime}\,\,\,\langle
x+\frac{x^{\prime}}{2}|\hat{\rho}(t)|x-\frac{x^{\prime}}{2})\rangle\,\,e^{-ikx^{\prime}}.$
(9)

Again, in general, the Wigner distribution function has no information at all
about the microscopic states. Then, the collision operator can provoke
unphysical violations of the conservation of energy, as we will see in
following sections.

##  2 Exact and approximate models for matter-light interaction

We develop here an example of the role of the energy conservation in the
collision of an electron with a photon. First, we explain and exact electron-
photon Schrödinger equation and, then, we present two approximate collision
models. The first approximate model for collisions will be based on changing
the electron energy during the scattering process, while the second one is
based on change of the value of momentum.

###  2.1 Exact electron-photon interaction

The electromagnetic field in our quantum description is assumed to be a
monochromatic field inside a optical cavity of length $L_{\gamma}$ with the
shape $E(x,t)\propto q\;cos(k_{\gamma}x-\omega t)$ with $\omega$ the angular
frequency of the field and $k_{\gamma}$ the wave vector giving a speed of
light $c=\omega/k_{\gamma}$. The dependence of $E(x,t)$ on $x$ can be removed
by assuming $L_{x}<<L_{\gamma}=\frac{2\pi}{\omega}$ where $L_{x}$ is the
length of the active region of the device, where electrons are simulated. The
quantization of the electromagnetic field appears because the amplitude $q$ is
not a fixed value, but a variable degree of freedom. Then, the Hamiltonian of
the electromagnetic field in the $q$-representation $H_{\gamma}$ can be
written as [32]:

$\displaystyle\hat{H}_{\gamma}=-\frac{\hbar^{2}}{2}\frac{\partial^{2}}{\partial
q^{2}}+\frac{\omega^{2}}{2}q^{2}.$ (10)

Inside the optical cavity, the vacuum state related to the absence of photons
is $|0\rangle$, corresponding to the ground state of an harmonic oscillator
$\psi_{0}(q)=\langle q|0\rangle$ of (10). Identically, the first state of the
harmonic oscillator in (10) corresponds to the presence of one mode, or one
photon, inside the optical cavity described by $\psi_{1}(q)=\langle
q|1\rangle$. For more details, see [22].

The electron part of the system follows the well known electron Hamiltonian,
in the $x$-representation:

$H_{0}=-\frac{\hbar^{2}}{2m}\frac{\partial^{2}}{\partial x^{2}}+V(x),$ (11)

where $V(x)$ includes both internal and external scalar potential and $m$ is
the electron effective mass. The Hamiltonian $H_{0}$ is written assuming a 1D
electron system.

Finally, the electron-photon wavefunction $\Psi(x,q,t)$ will be guided by the
following Schrödinger-like equation of motion:

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

where the last term in (12) is the interaction term between the optical and
electron parts in the typical dipole approximation [32]. The parameter
$\alpha^{\prime}$ is the interaction strength.

Since we are only interested in a dynamic process involving only two possible
photons states, $\psi_{0}(q)$ for zero photons and $\psi_{1}(q)$ for one
photon, the whole wave function $\Psi(x,q,t)$ can be decomposed as:

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

with

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

The wave functions $\psi_{A}(x,t)$ and $\psi_{B}(x,t)$ describe how the total
wave function is projected in $\psi_{0}(q)$ and $\psi_{1}(q)$ respectively.

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

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

and

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

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

In this way the closed electron-photon system is now described with an exact
model acting only on two electron wavefunctions. The present model can be
extended to an open system through the use of the so-called Bohmian
conditional wavefunction for electron wavefunctions $\psi_{A}(x,t)$ and
$\psi_{B}(x,t)$. For explanation about Bohmian mechanics, we recall [18, 19],
and for a definition and properties of the Bohmian conditional wavefunction
see [20, 21, 22].

###  2.2 Modelling electron-photon collisions as energy exchange

We consider now an electron defined by a single-particle state $\psi(x,t)$. As
indicated above, for an open system like an electron device, such state can be
understood as a Bohmian conditional wavefunction [20, 21, 22] where only the
degree of freedom of the electron is considered
$\psi(x,t)\equiv\psi(x,q(t),t)$ while the degree of freedom of the photon $q$
is fixed to some particular (Bohmian) value $q(t)$. See [22] for more details.

At time $t_{s}$ the electron undergoes a collision with a photon. It is worth
mentioning that the use of a finite time for the implementation of the
scattering event is key to maintain continuity of the (conditional) wave
function in space and time, but to simplify the discussion we will assume an
instantaneous scattering process here. Later, in the numerical results, we
will explicitly consider the finite duration of the scattering process.

The process of collision $\psi(x,t_{s})\to\psi^{s}(x,t_{s})$ can be understood
as a transition between the initial and final states $\psi(x,t_{s})$ and
$\psi^{s}(x,t_{s})$, respectively, which satisfy $\langle
E^{s}(t_{s})\rangle=\langle E(t_{s})\rangle+E_{\gamma}$, 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 with hamitonian $\hat{H}_{0}$ in (11) as:

$\psi(x,t_{s})=\int dE\;a(E,t_{s})\;\phi_{E}(x),$ (17)

with $a(E,t_{s})=\int dx\;\psi(x,t_{s})\;\phi_{E}^{*}(x)$. The central energy
$\langle E^{s}(t_{s})\rangle$ after the scattering has to increase (or
decreases) an amount $E_{\gamma}$, so the new state after the collision can be
written as:

$\displaystyle\psi^{s}(x,t_{s})$ $\displaystyle=$ $\displaystyle\int
dE\;a(E-E_{\gamma},t_{s})\;\phi_{E}(x)$ (18) $\displaystyle=$
$\displaystyle\int dE\;a^{s}(E,t_{s})\;\phi_{E}(x),$

where we have defined $a^{s}(E,t_{s})=a(E-E_{\gamma},t_{s})$. This transition
corresponds to absorption of energy by the electron. Emission can be
identically modelled by using $\langle E^{s}(t_{s})\rangle=\langle
E(t_{s})\rangle-E_{\gamma}$. In few words, the collision model performs a
change in the wave packet central energy of an amount $\pm E_{\gamma}$.

###  2.3 Modeling electron-photon collisions as momentum exchange

It is straightforward to see that the process
$\psi(x,t_{s})\to\psi^{s}(x,t_{s})$ satisfying a controlled change of the
momentum can be done using
$\psi^{s}(x,t_{s})=e^{ip_{\gamma}x/\hbar}\psi(x,t_{s})$, which ensures that
the state $\psi^{s}(x,t_{s})$ has a increase of momentum with respect to
$\psi(x,t_{s})$ of $p_{\gamma}$. In fact, it can be easily demonstrated that
such change on the momentum of $\psi^{s}(x,t)$ can be produced from the
Schrödinger equation of the form:

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

where $\Theta_{t_{s}}$ is a function equal to zero for $t<t_{s}$ and $1$ for
$t>t_{s}$. Notice that the probability presence of the scattered wave packet
satisfies $|\psi^{s}(x,t_{s})|^{2}=|\psi(x,t_{s})|^{2}$ because only a global
phase $e^{ip_{\gamma}x/\hbar}$ is added. For a deeper explanation of the
derivation of (19) we recall [33].

In an analogous way as the previous section, before the scattering, the state
can be written as a superposition of momentum eigenstates $\phi_{p}(x)$ (which
is also a basis of the electron in the x space) as:

$\psi(x,t)=\int dp\;b(p,t)\;\phi_{p}(x),$ (20)

with $b(p,t)=\int dx\;\psi(x,t)\;\phi_{p}^{*}(x)$. The central energy $\langle
p(t)\rangle$ is increase of an amount $p_{\gamma}$, so in the momentum
representation the state after the collision is:

$\displaystyle\psi^{s}(x,t_{s})$ $\displaystyle=$ $\displaystyle\int
dp\;b(p-p_{\gamma},t_{s})\;\phi_{p}(x)$ (21) $\displaystyle=$
$\displaystyle\int dp\;b^{s}(p,t_{s})\;\phi_{p}(x)$ $\displaystyle=$
$\displaystyle e^{ip_{\gamma}x/\hbar}\psi(x,t_{s}),$

where we have defined $b^{s}(p,t)=b(p-p_{\gamma},t)$. Notice that:

$\displaystyle b(p-p_{\gamma},t)$ $\displaystyle=$ $\displaystyle\int
dx\;\psi(x,t)\;\phi_{p-p_{\gamma}}^{*}(x)$ (22) $\displaystyle=$
$\displaystyle e^{ip_{\gamma}x/\hbar}\int dx\;\psi(x,t)\;\phi_{p}^{*}(x).$

Unfortunately, as it will be demonstrated in the following numerical results,
a global mechanism of scattering valid for scenarios with potential barriers
requires dealing with the energy conservation, and not with momentum
conservation.

For the sake of clarity, the relationship of the approximated models in Sec.
2.2 and 2.3 with the exact model of Sec. 2.1 is here described. In the exact
model, for each electron, both wavefunctions $\psi_{A}(x,t)$ and
$\psi_{B}(x,t)$ are computed through (2.1) and (2.1). This ensures the
possibility of a continuous transition from $\psi_{A}(x,t)$ (zero photon) to
$\psi_{B}(x,t)$ (one photon), and viceversa. In the approximate models, we
just simulate one wave function $\psi(x,t)$ for each electron that takes into
account the perturbation of the state due to scattering. In particular, in the
case of a emission of a photon, the initial state is $\psi(x,0)=\psi_{B}(x,0)$
(one photon), and we wanted the final state at time $t_{s}$ provided by
approximate model to be as similar as possible to
$\psi^{s}(x,t_{s})\approx\psi_{A}(x,t_{s})$ (zero photon). For photon
absorption, we have as the initial state $\psi(x,0)=\psi_{A}(x,0)$ (zero
photon), and the final state $\psi^{s}(x,t_{s})\approx\psi_{B}(x,t_{s})$ (one
photon).

Figure 1: Wigner function of the total electron wavefunction interacting with
a double barrier structure (green lines), at different times of its evolution:
(a) at the beginning of the evolution, (b) when first entering inside the
double barrier structure with $\langle E(t)\rangle=0.096\,eV$, (c) after
emitting a photon, so that the electron energy is $\langle
E(t)\rangle=0.023\,eV$. In (d), (e), (f) are shown projections along the
energy (top), momentum (middle) and position (bottom) axis of the Wigner
transform respectively of (a), (b), and (c).

##  3 Numerical results

In this section, we will compare the exact collision approach developed in the
previous section, with the two collisions models mentioned there. Apart from
plotting the evolution of the involved states before and after the collisions,
we will also plot the Wigner distribution through the Wigner-Weyl transform
defined from Eq. (9). In particular, we will consider scenarios where the
electrons are suffering emission and absorption of photons, while impinging on
a resonant tunneling diode. Such structure is composed by a double barrier
with thickness 2nm, height 0.3 eV and a distance between the two barriers of
16 nm. The electron has effective mass $m=0.041\,m_{0}$, where $m_{0}$ is the
mass of the electron at rest. The resonant energies of the double barrier
structure are $0.023$ eV for first resonant state and $0.096$ eV for the
second resonant state.

###  3.1 Exact evolution of the Wigner Function

In this subsection we will show electron wave functions as defined in (13). In
particular, we start by considering $\psi_{A}(x,0)$ as a Gaussian wave packet
in the left contact outside the barrier with an energy equal to the second
resonant energy, while $\psi_{B}(x,0)=0$, meaning that initially there is no
photon in the structure. Since both electron wavefunctions have a coupled
evolution in (2.1) and (2.1), step by step, the wave function $\psi_{B}(x,t)$
grows and $\psi_{A}(x,t)$ decreases indicating that a photon is being created.
The Wigner function of the whole process is computed as the sum of the Wigner
function linked to $\psi_{A}(x,t)$ (with (9)) plus the Wigner function linked
to $\psi_{B}(x,t)$ (again with (9)). Notice that, from (13), we get $\int
dq|\Psi(x,q,t)|^{2}=|\psi_{A}(x,t)|^{2}+|\psi_{B}(x,t)|^{2}$ because
$\psi_{0}(q)$ and $\psi_{1}(q)$ are orthogonal. The Wigner function is plotted
in Fig. 1, from (a) to (c) in three different times corresponding to the
electron impinging upon a double barrier structure with an (initial) energy
equal to the second resonant level of the double barrier. Inside the well the
electron is interacting with the electromagnetic field and Rabi oscillations
of the electron between first and second resonant levels are observed. This
can be seen from the wavefunction that, showing one maximum in the probability
distribution inside the quantum well, is occupying the first level in 1(b) and
(e). On the other hand, in Fig 1(c) and (f) two maxima are observable inside
the well (see (f)), so that the second level of the double barrier is
occupied.

###  3.2 Approximate evolution of the Wigner Function in free space

In this subsection we show the evolution of the Wigner function defined from a
single electron evolving in free space and undergoing scattering through the
(approximate) collision models explained in the Sec. 2.2 and 2.3. The goal of
this section is to prove the equivalence of two previous collisions models
(energy exchange and momentum exchange) in the case of an electron system in
free space. In free space, the momentum of a photon is much smaller than the
momentum of an electron and the conservation of momentum (not only of energy)
has to be satisfied. Thus, in this Sec. 3.2, from a physical point of view, we
consider the interaction of an electron with a phonon (instead of a photon).

In (2), is has been shown [26, 27, 28] that the time of scattering $\Delta t$
has to be finite. In the present work, the transition shown in (18) for a
scattering of energy $E_{s}$, is divided into $40$ steps of energy change
$\Delta E_{s}=E_{s}/40$. Between two steps, the system is evolved for a time
of around $6$ fs. As a result, the total scattering time for the approximate
model will be $40\cdot 6=240$ fs. An analogous implementation is done for
(21). In Fig. 2, the electron is injected with an central energy equal to the
first resonant level, $\langle E(t)\rangle=0.023$ eV, meaning a momentum of
$\langle p(t)\rangle=1.573\cdot 10^{8}$ m-1, This electron undergoes
scattering for a finite time of approximately 0.1 ps to a final wave packet
given by the collision models in Sec. 2.2 or Sec. 2.3 with an expectation
value of the final energy $\langle E^{s}(t)\rangle=0.096$ eV, corresponding to
a momentum of $\langle p^{s}(t)\rangle=3.214\cdot 10^{8}$ m-1. This process
$\psi(x,t_{s})\to\psi^{s}(x,t_{s})$ mimics, for example, an electron absorbing
a phonon. The Wigner function is computed from (9) for the states
$\psi(x,t_{s})\to\psi^{s}(x,t_{s})$.

In Fig. 3, the electron is injected with initial central energy $\langle
E(t)\rangle=0.096$ eV, or momentum of $\langle p(t)\rangle=3.214\cdot 10^{8}$
m-1. This electron also undergoes scattering in a finite time of approximately
0.1 ps and emits a phonon reaching a central energy of the wave packet equal
to $\langle E^{s}(t)\rangle=0.023$ eV, corresponding to a momentum of $\langle
p^{s}(t)\rangle=1.573\cdot 10^{8}$ m-1. This process corresponds to an
emission of phonon and the transition $\psi(x,t_{s})\to\psi^{s}(x,t_{s})$ is
computed from the collision models in Sec. 2.2 or Sec. 2.3 and the Wigner
function linked to such states from (9).

We conclude that the evolution of the Wigner function of an electrons
scattered in free space behaves equivalently with both models, the one of Sec.
2.2 with energy conservation and the one of Sec. 2.3 with momentum
conservation. The physical reason of such agreement is because, in free space,
there is a one-to-one correspondence between the eigenvalues of momentum and
energy operators (since these commute with each other).

Figure 2: Wigner function for photon absorption in free space: (a) before
scattering (b) scattered with the energy exchange and the momentum exchange
model, which is equivalent to the former in free space conditions. In (c) and
(d) are shown projections along the energy (top), momentum (middle) and
position (bottom) axis of the Wigner functions respectively of (a) and (b).

Figure 3: Wigner function for photon emission in free space: (a) before
scattering (b) scattered with the energy exchange and the momentum exchange
models, which is equivalent to the former in free space conditions. In (c) and
(d) are shown projections along the energy (top), momentum (middle) and
position (bottom) axis of the Wigner functions respectively of (a) and (b).

###  3.3 Approximate evolution of the Wigner Function with potential barriers

In this subsection, we show a comparison between the energy exchange and
momentum exchange scattering models, respectively explained in Sec. 2.2 and
2.3, for an electron wavefunction interacting with the double barrier
structure described at the beginning of Sec. 3. The momentum of an electron
inside a quantum well tends to be very small (the wave function tends to
become real) and the conservation of momentum is not a requirement because of
the translation symmetry is broken by the barriers. Thus, in this Sec. 3.3, we
do consider the interaction of an electron with a photon.

In Fig. 4, the electron is first injected at the first resonant level of the
double barrier structure and undergoes a photon absorption, thus occupying,
after the scattering, the second level of the double barrier structure.
Energetically the initial an final energy values are the same as in Fig. 2.
This transition is clear in Fig. 4(b) and (e), where the energy exchange model
is used, in fact, while in 4(d) there is a single maximum of the probability
density in the bottom image, there are two maxima in 4(d). This is a proof of
a transition from the first to the second resonant state of the well. However
this is not observed in 4(c) and (f), where a momentum exchange model is used.
Now, inside the well, the maximum of probability after the scattering belongs
still to the first resonant level. This is an unphysical result whose physical
reason is that there is no one-to-one correspondence between energy and
momentum. Thus, despite having control on the change of the momentum, we do
not have control on the change of the energy. In the case of photon emission,
shown in Fig. 5, the same physical transition is observed from (a) to (b) (and
on their respective projections in (d) and (e), where the two maxima of
probability inside the quantum well are transformed into just one maxima
later), while the transition between levels is not observed from (a) to (c),
where the number of maxima remain equal to two. This is another proof of the
difficulties to use a momentum exchange model in case of an arbitrary
potential. In other words, in typical device scenarios for nanoscale devices,
the energy and momentum operators do not commute so that the eigenstates of
the momentum are not eigenstates of the energy and vice-versa. Then, the
change in momentum can have an arbitrary translation into change of energy.
The dramatic consequence is that the conservation of energy during the
electron-photon collision is not guaranteed.

Figure 4: Comparison of Wigner functions undergoing photon absorption in a
double barrier structure (green lines), at different times: (a) before
scattering (b) scattered with the energy exchange model and (c) with the
momentum exchange model. In (d), (e), (f) are shown projections along the
energy (top image), momentum (middle) and position (bottom) axis of the Wigner
transform respectively of (a), (b), and (c). Notice the change for the
transition from the first resonant level to the second from (d) to (e), while
such transition is not present from (d) to (f).

Figure 5: Comparison of Wigner functions undergoing photon emission in a
double barrier structure (green lines), at different times: (a) before
scattering (b) scattered with the energy exchange model and (c) with the
momentum exchange model. In (d), (e), (f) are shown projections along the
energy (top), momentum (middle) and position (bottom) axis of the Wigner
transform respectively of (a), (b), and (c). Notice the change for the
transition from the second resonant level to the first from (d) to (e), while
such transition is not present from (d) to (f).

##  4 Does Wigner Function satisfy complete positivity and energy
conservation?

The typical way of reducing the computational burden by including the role of
photons and phonons in quantum transport is through a collision model. Such
collision model introduces some new term in the equation of motion of the
simulated degrees of freedom. For example, in the Wigner distribution
formalism, we can visualize the effect of the collision as a change from the
initial (before collision) Wigner function $f_{W}(x,k,t)$ to the final (after
collision) Wigner function $f_{W}^{s}(x,k,t)$. But, is there any mandatory
requirement that such change of the Wigner function has to satisfy? Or is any
possible change physically acceptable?

By construction, the Wigner function without external collisions do satisfy
the complete positivity in the sense that
$\int^{+\infty}_{-\infty}dkf_{W}(x,k,t_{s})>0$ for all positions $x$. The
Wigner function, after being modified by the collision operator, does also
have to satisfy the same complete positivity condition:

$\displaystyle Tr(\hat{\rho}(t_{s})|x\rangle\langle
x|)=\int^{+\infty}_{-\infty}dkf^{s}_{W}(x,k,t_{s})>0\;\;\;\forall\;\;x.$ (23)

Notice that for a discrete Wigner function (used in the computational
algorithms), for a grid with $N_{x}$ position points and $N_{k}$ wave vector
points, the condition (23) imposes $N_{x}$ equations, each equation involving
$N_{k}$ elements of the estimated discrete $f^{s}_{W}(x,k,t_{s})$.

On the other hand, during the collision we can assume a controlled change of
the energy of the system. To verify such energy conservation, the expectation
value of the electron energy within the Wigner approach is defined as:

$\langle E(t)\rangle=Tr(\hat{\rho}\hat{H_{0}})=\int
dk\,dx\,f_{W}(x,k,t)\,h_{0}(x,k),$ (24)

where $h_{0}(x,k)=\int dx^{\prime}\langle
x+\frac{x^{\prime}}{2}|\hat{H_{0}}|x-\frac{x^{\prime}}{2}\rangle
e^{-ikx^{\prime}}$ with $\hat{H_{0}}$ the Hamiltonian without collisions
defined in (11).

The initial energy of the Wigner function is $\langle
E(t_{s})\rangle=\int^{+\infty}_{-\infty}dx\int^{+\infty}_{-\infty}dkf_{W}(x,k,t_{s})h_{0}(x,k)$,
then after the absorption (or emission) of a photon with energy $E_{\gamma}$,
the final Wigner function has to satisfy:

$\displaystyle\langle E^{s}(t_{s})\rangle$ $\displaystyle=$
$\displaystyle\langle E(t_{s})\rangle\pm E_{\gamma}=Tr(\hat{\rho}^{s}\hat{H})$
(25) $\displaystyle=$
$\displaystyle\int^{+\infty}_{-\infty}dx\int^{+\infty}_{-\infty}dk\,f_{W}^{s}(x,k,t_{s})h_{0}(x,k).$

The condition (25) implies one additional equation involving $N_{x}\times
N_{k}$ elements of the new Wigner function. Together, conditions (23) and (25)
do only define $N_{x}+1$ elements of the total $N_{x}\times N_{k}$ elements of
the Wigner function after the scattering $f^{s}_{W}(x,k,t_{s})$. Notice that
the Wigner distribution function has $N_{x}\times N_{k}$ elements so that
these two conditions do not totally determine which is the new Wigner function
after scattering. In other words, there are many possible Wigner functions
that can satisfy (23) and (25), but there are also many Wigner function that
are invalid because they do not satisfy (23) and (25).

We want to emphasize again that there is no fundamental limitation for
developing a successful collisions model from the Wigner function formalism.
In other words, it is possible to find $f_{W}(x,k,t_{s})$ and
$f^{s}_{W}(x,k,t_{s})$, before and after the scattering respectively, that
satisfy complete positivity and energy conservation. To underline this point,
we explain an algorithm that use the collision model developed in Sec. 2.2 for
wavefunctions. The algorithm explained below will not be easily implementable
in a general Wigner transport formalism, but it will clearly confirm that it
is possible to develop a successful transition $f_{W}(x,k,t_{s})\to
f^{s}_{W}(x,k,t_{s})$ due to the collision of an electron with a photon
(phonon), while satisfying conditions (23) and (25). The algorithm deals with
a single electron (as we have done in all this paper), and it has the
following three steps:

  * •

First step: The Wigner function before the scattering $f_{W}(x,k,t)$ is
translated into a wave function $\psi(x,t_{s})$ in the following way:

$\displaystyle\psi(x,t_{s})=\rho(x,0,t_{s})\frac{1}{\psi^{*}(0,t_{s})}$ (26)
$\displaystyle=$
$\displaystyle\frac{1}{2\pi\psi^{*}(0,t_{s})}\int\\!\\!\\!\int
dx^{\prime}dk\;\rho\left(y,y^{\prime},t_{s}\right)e^{ik(x-x^{\prime})}$
$\displaystyle=$ $\displaystyle\frac{1}{\psi^{*}(0,t_{s})}\int
f_{W}\left(\frac{x}{2},k,t_{s}\right)e^{ikx}dk,$

where we have used that $\rho(x,0)=\psi^{*}(0,t)\psi(x,t)$ for a pure state,
and $y=(x+x^{\prime})/2$ and $y^{\prime}=(x-x^{\prime})/2$. The complex number
$\psi^{*}(0,t_{s})$ is irrelevant here since it can be understood as a
normalization constant [34].

  * •

Second step: Once we have $\Psi(x,t_{s})$ we can apply the collision algorithm
explained in Sec. 2.2 to get the transition
$\Psi(x,t_{s})\to\Psi^{s}(x,t_{s})$ that we know satisfy complete positivity
and energy conservation.

  * •

Third step: Once we get the pure state after the collision
$\Psi^{s}(x,t_{s})$, we use expression (9) to compute the Wigner distribution
function after the collision: $f^{s}_{W}(x,k,t_{s})$.

Indeed, an example of such Wigner functions, from
$f_{W}(x,k,t_{s})$ before collision to $f^{s}_{W}(x,k,t_{s})$ after collision,
is already plotted from Fig. 4(a) to (b), respectively, for photon absorption.
Identically, another example is provided from Fig. 5(a) to (b) for photon
emission. In Fig. 6 it is shown how such algorithm (based on the electron-
photon collision model of Sec. 2.2) reproduce the exact electron-phonon
interaction computed from Sec. 2.1. The evolution of the ensemble value of the
energy for simulations in Fig. 1 (black dashed line), 4(a) and 4(b) (blue
line), 5(a) and 5(b)(red line). The ensemble value $\langle E(t)\rangle$ is
computed using expression (24) at different times. To better approximate the
exact result the change of energy $E_{\gamma}$ is not done in a single time
step of the simulation, but in a time interval identical to the time it takes
the exact solution to produce such energy transition (which is related to the
frequency of the Rabi oscillations [32]).

Figure 6: Comparison between the expectation value of energy of the system
undergoing collision modelled from the exact model (black dashed line) and
with the energy exchange algorithm, for photon emission (red line) or photon
absorption (blue line). These models are implemented from the Wigner
distribution function of a pure state.

The excellent agreement in Fig. 6 was possible because the evolution of such
system is treated on the level of the Bohmian conditional wavefunction.
Without such knowledge, the task of satisfying conditions (23) and (25)
becomes very complicated. In a real electron device, more than one electron
have to be simultaneously simulated and then expression (26) cannot be applied
because the density matrix, and its Wigner transform, will hide the knowledge
about every single-particle state. We conclude that without the knowledge of
the state it seems quite complicated to model a reasonable collision model
that satisfies energy conservation. For example, there is no guarantee at all
that the direct implementation of the Boltzmann collision operator in the
Wigner distribution can satisfy the above conditions.

###  4.1 Are collisions a source of time irreversibility ?

At this point, although not directly related with the goal of this paper, it
is interesting to discuss if collisions in electron devices are a source of
time-irreversibility or not. While typical microscopic laws are time-
reversible, an arrow of time appears in macroscopic phenomena. For example, we
know that electrons lose energy, in average, when traversing the device (Joule
effect). Therefore, we can know if we are looking _forward_ or _backward_ in
time by looking at the macroscopic heating or cooling of the device. But, why
microscopic time-reversible laws become time-irreversible at the macroscopic
level? What is the role of collisions?

We have shown in Sec. 3.1 that electron-photon interaction can be studied in
an exact way as a solution of the many-body Schrödinger equation in a closed
system including the degrees of freedom of one electron and one photon. Thus,
we can conclude that interaction of a single electron with a single photon are
a time-reversible phenomena, either for emission or absorption, when described
by (12) because we know exactly how to go back in time in this exact model,
and the process of emission and absorption level are time-reversible at the
microscopic.

In the approximations done in Sec. 2.2 and Sec. 2.3 we eliminate the explicit
simulation of the photon degree of freedom and introduce the effect of the
photon into the electron as a collision. The electron alone, without the
photon, is an open (non-Markovian) system. But, if we know exactly when the
collision with the electron takes place, then we can still consider that
approximations done in Sec. 2.2 and Sec. 2.3 as time-reversible (as it was Eq.
(12) itself) because we know how to go back in time, and the process of
emission and absorption level are time-reversible at the microscopic.

The conclusions above for time-reversibility of the equation of motion of one
electron interacting with one photon can be straightforwardly generalized to
the equation of motion of thousands of electrons interacting with many
thousands of photons, as far as we only look at the microscopic information.
However, for such many-body system, if instead of looking at the microscopic
degrees of freedom, we are interested in a macroscopic equation of motion for
the average energy translated from the electrons to the photons and viceversa,
the conclusion can be different. If we were able to simulate a many-particle
Schrödinger equation of such system, including the electron device and the
surroundings in an scenario outside of thermodynamic equilibrium, we will
realize that a photon emitted from the electron device will hardly be able to
return to the electron device again because the surroundings are larger
extension, in space, than the electron device. Thus, inside the electron
device there will be more emissions than absorption, in average. In fact, the
expected net dissipation of electron energy (Joule effect) will take place. We
conclude that the equation of motion for the macroscopic average energy in the
electrons is time-irreversible. Notice that the time-irreversibility of the
(macroscopic) equation of motion of the average energy of electrons is fully
compatible with the time-reversibility of the (microscopic) equation of motion
of electrons and photons.

In conclusion, _are collisions a source of time irreversibility?_ Not in a
microscopic description, but yes in a macroscopic one. Microscopically the
collisions provide an equation of motion for the dynamics of electrons and
photons that is time-reversible. Macroscopically the collisions ensure that
the dissipation of the average energy of the electrons is time-irreversible
and satisfy the Joule effect.

We have learn a practical lesson to implement a collision model here. Since
the exact solution of thousands of electrons interacting with thousands of
photons is not possible, the practical implementation of the scattering model
with the approximations done in Sec. 3.2 has to satisfy the microscopic time-
reversibility of the equation of motion and, simultaneously, the time-
irreversibility of the macroscopic average energy. This is typically achieved
by randomly selecting the scattering rates with a distributions that satisfies
the macroscopic requirement: more emission that absorptions.

##  5 Conclusions

In this paper we have shown that a physically reasonable collision model,
within the Wigner description, has to satisfy the following two mandatory
requirements: Complete positivity and energy conservation. After presenting
the two conditions and studying its implementation for an electron inside a
double barrier interacting with a photon, we compare exact results to a
collision model based on energy exchange in Sec. 2.2 and to a collision model
based on momentum exchange in Sec. 2.3. We conclude that, unfortunately, it is
very complicated to develop a practical collision model in the Wigner
formalism that satisfies both conditions.

The good conclusion is that there is no fundamental reasons that disqualifies,
a priori, the possibility of implementing collisions in the Wigner
distribution framework. The practical requirement in (23) and in (25) provides
less restrictions than the degrees of freedom (number of elements) of a
(discretized) Wigner distribution function. In fact, for a pure state, we have
shown an algorithm that models electron-phonon collisions in the Wigner
function formalism with an excellent agreement. The problem for its
generalization to realistic quantum transport is that the mentioned algorithm
needs the wave function information of each electron inside the device, but
such additional information is not available in the typical Wigner function
algorithms.

Finally, we want to mention that most of the difficulties of the Wigner
distribution function to satisfy energy conservation in collisions, are
inherited form the difficulties of its father description, the density matrix,
to tackle the properties of individual particles. At the end of the day, these
problems are just a manifestation of the orthodox statements that negates any
microscopic properties for particles, unless such microscopic properties are
being measured explicitly [20]. But, a collision is not a measurement, so that
the orthodox theory forbids to access to the microscopic information on what
has happened to each electron during the collision. For these reasons, we
argue that collisions models based on the (Bohmian) conditional wave function
are very promising because they have the ability to describe the microscopic
properties of individual particles and satisfy conditions (23) and (25) in a
very _natural_ way [20, 21, 22, 35, 36, 37, 38].

#### Contributions

Conceptualization: Matteo Villani, Xavier Oriols; Methodology: Matteo Villani,
Xavier Oriols; Formal analysis and investigation: Matteo Villani, Xavier
Oriols; Writing - original draft preparation: Matteo Villani, Xavier Oriols;
Writing - review and editing: Matteo Villani, Xavier Oriols; Funding
acquisition: Xavier Oriols; Resources: Xavier Oriols; Supervision: Xavier
Oriols. All authors read and approved the final manuscript.

#### Funding

This project has received funding from the European Union’s Horizon 2020
research and innovation programme under the Maria Skłodowska-Curie grant
agreement No. 765426 (TeraApps). This research was also 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).

#### Conflicts of interest

The authors have no conflicts of interest to declare that are relevant to the
content of this article. The funders had no role in the design of the study;
in the collection, analysis, or interpretation of data; in the writing of the
manuscript, or in the decision to publish the results.

#### Compliance with Ethical Standards

Not applicable.

#### Availability of data and material

The datasets generated during and/or analysed during the current study are
available from the corresponding author on reasonable request.

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