#  A survey on the intrinsic decoherence in the Milburn formalism

S. V. Mousavi<EMAIL_ADDRESS>Department of Physics, University of Qom,
Ghadir Blvd., Qom 371614-6611, Iran  S. Miret-Artés<EMAIL_ADDRESS>Instituto de Física Fundamental, Consejo Superior de Investigaciones
Científicas, Serrano 123, 28006 Madrid, Spain

###### Abstract

In this work, we consider different theoretical aspects and simple
applications of the Milburn equation which is governed by a parameter
controlling what is known as intrinsic decoherence. The main goal is to show
some similarities observed also when external decoherence is considered.
Purity, Ehrenfest relations, probability density current, the Wigner
representation as well as the relation to a Linbladian master equation are
analyzed in terms of this intrinsic decoherence, leading to new insights on
the Milburn dynamics. Interference of two wave packets, tunneling and the
bouncing ball problem are also analyzed under this perspective.

Keywords: Intrinsic decoherence; Purity; Ehrenfest relations; Arrival time;
Wigner function; Tunneling; Bouncing ball.

##  I Introduction

Quantum mechanics is a linear theory, and thus it admits coherent
superposition of distinct physical states. This property is the basis of the
measurement problem in this theory. However, this superposition principle does
not act at the macroscopic scale, although nothing shows this in the
formulation of the quantum mechanics. The problem of the standard quantum
mechanics is to explain non-observability of macroscopic distinct states. Two
routes have been tackled in the literature to overcome this problem. First,
the role played by the environment. In this way, the whole system, the system
of interest and the environment, is considered a closed system evolving on the
standard Schrödinger equation. Then, by integrating out the environment
degrees of freedom, an equation known as the master equation is derived for
the reduced density matrix of the system of interest. This formalism is known
as environment-induced decoherence started by the work of Zeh Ze-FP-1970  and
developed by Zurek Zu-RMP-2003 . Decoherence is important, not only from the
foundational point of view but also in the realm of quantum information
theory, since quantum information processing and quantum technologies depend
on our ability in creation, maintenance and manipulation of non-classical
superposed states Sc-PR-2019 . In the second approach, the Schrödinger
equation is itself modified by adding some additional terms which induce
decoherence within the quantum mechanics itself. Among the various models have
been proposed in this regard, the famous one is maybe the model of Ghirardi,
Rimini and Weber, the so-called GRW model GRW-PRD-1986 . In another approach,
decoherence has been related to the gravity effects ElMoNa-PLB-1989 .

Over the years, several different paths have been taken to break up with the
standard quantum mechanics; some of the well-known attempts include the
postulation of extra nonlinear terms or stochastic terms in the Schrödinger
equation; and gravitational effects invalidates this equation St-PTRSA-2012 .
Several years ago, Milburn Mi-PRA-1991  modified the von Neumann equation of
the density operator in a way to include the decoherence, what is known as
intrinsic decoherence. He postulated that, at sufficiently short time steps,
the system does not evolve continuously under unitary evolution but by a
stochastic sequence of identical unitary transformations. This means that a
minimum time step in the universe is introduced. The inverse of this time step
is the mean frequency of the unitary steps. The system is then considered
isolated and described by a state whose evolution is governed by a generalized
von Neumann equation having this mean frequency as a parameter. For free
particles, it can be seen that the Milburn equation is a Lindbladian master
equation, where the only Lindblad operator is proportional to the square of
linear momentum MoMi-JPA-2022 . It is notable that in studying a Jaynes-
Cummings model with phase damping, Kuang et. al Kuetal-PRA-1997  have
introduced a master equation describing phase damping under the Markovian
approximation which has the same form as the Milburn equation. If the
frequency of time steps is sufficiently high, evolution in the experimental
time-scales seems approximately continuous. In the zeroth order of the inverse
of this frequency, the usual von Neumann equation is recovered. The first
order correction yields decoherence in the energy eigenbasis. One aspect of
this model is that constants of motion in the standard quantum mechanics
approach remain constants of motion of the model and thus, stationary states
remain stationary. Dynamics of open quantum systems has been considered within
the Milburn framework BuKo-PRA-1998 . Furthermore, a Lorentz invariant model
of intrinsic decoherence has been presented Mi-NJP-2006 .

The Milburn equation has attracted many attentions within the quantum
information community in recent years. Effect of the intrinsic decoherence on
quantum communications and various quantum correlations has been studied
LiShZo-PLA-2010 ; JaNa-IJQI-2016 ; ZhZh-EPJD-2017 ; NaJa-IJTP-2019 ; AbMu-
PS-2021 ; EsKhMaDa-OQE-2022 ; NaMuCh-PA-2022 ; OuChMa-IJTP-2023 ; KhMu-
APB-2023 ; ChMa-APB-2023 . Decaying dynamics of a displaced harmonic
oscillator UrMC-Pramana-2022  and in the mirror-field interaction UrMC-
arxiv-2023  have been studied within this framework. In spite of these
studies, only two works, to the best of our knowledge, have applied the
Milburn equation in the coordinate representation: (i) Milburn himself
investigated, in his original paper, the effect of the intrinsic decoherence
on the interference fringes of the position probability density for a harmonic
oscillator prepared in an initial superposition state of two coherent states
Mi-PRA-1991  and (ii) this equation has been used by the same authors in the
context of the quantum backflow MoMi-JPA-2022 . Thus, we think the Milburn
equation deserves further attention by analyzing different theoretical
aspects. In this paper, we are going to examine this equation in the
coordinate representation.

This work is organized as follows. Section II discusses some new aspects of
the Milburn equation and the role played by the intrinsic decoherence by
analyzing purity, Ehrenfest relations, the probability current density,
arrival times, the Wigner representation, the relation to a Linbladian master
equation and transmission through a localized barrier. In section III, three
simple problems are analyzed by comparing the von Neumann and Milburn
dynamics: interference of two Gaussian wave packet, tunneling through a
rectangular barrier and the so-called bouncing ball problem. Finally, in the
last section, some conclusions are reported.

##  II  Intrinsic decoherence in the Milburn formalism

The intrinsic decoherence within the Milburn formalism is based on the
equation of motion of the density operator Mi-PRA-1991  given by

$\displaystyle\frac{\partial\hat{\rho}}{\partial t}$ $\displaystyle=$
$\displaystyle\gamma\left(e^{-i\hat{H}/\hbar\gamma}\hat{\rho}~{}e^{i\hat{H}/\hbar\gamma}-\hat{\rho}\right).$
(1)

In this equation of motion, $\hat{H}$ is the Hamiltonian of the system and
$\gamma$ a parameter which is the inverse of the minimum time step in the
universe responsible for a stochastic sequence of identical unitary
transformations at sufficiently short time steps. Expansion of Eq. (1) up to
first order in $\gamma^{-1}$ reads as

$\displaystyle\frac{\partial\hat{\rho}}{\partial t}$ $\displaystyle=$
$\displaystyle-\frac{i}{\hbar}[\hat{H},\hat{\rho}]-\frac{1}{2\hbar^{2}\gamma}[\hat{H},[\hat{H},\hat{\rho}]],$
(2)

which reduces to the well-known von Neumann equation in the limit
$\gamma\rightarrow\infty$. In the energy representation, this equation can be
written

$\displaystyle\frac{\partial}{\partial t}\rho_{EE^{\prime}}(t)$
$\displaystyle=$
$\displaystyle-\frac{i}{\hbar}(E-E)\rho_{EE^{\prime}}(t)-\frac{1}{2\hbar^{2}\gamma}(E-E)^{2}\rho_{EE^{\prime}}(t),$
(3)

$\\{|E\rangle\\}$ being the energy eigenfunctions of the Hamiltonian $\hat{H}$
expressed as

$\displaystyle\hat{H}|E\rangle$ $\displaystyle=$ $\displaystyle E|E\rangle,$
(4)

where $\rho_{EE^{\prime}}(t)=\langle E|\hat{\rho}(t)|E^{\prime}\rangle$ are
the density matrix elements in the energy eigenbasis. Eq. (3) is easily solved
leading to

$\displaystyle\rho_{EE^{\prime}}(t)$ $\displaystyle=$
$\displaystyle\exp\left[-\frac{i}{\hbar}(E-E^{\prime})t-\frac{1}{2\hbar^{2}\gamma}(E-E^{\prime})^{2}t\right]\rho_{EE^{\prime}}(0).$
(5)

Thus, for the density operator one has that

$\displaystyle\hat{\rho}(t)$ $\displaystyle=$
$\displaystyle\sum_{E}\sum_{E^{\prime}}\rho_{EE^{\prime}}(t)|E\rangle\langle
E^{\prime}|$ (6) $\displaystyle=$
$\displaystyle\sum_{E}\sum_{E^{\prime}}e^{-i(E-E^{\prime})t/\hbar-(E-E^{\prime})^{2}t/(2\hbar^{2}\gamma)}\rho_{EE^{\prime}}(0)|E\rangle\langle
E^{\prime}|,$ (7)

which shows that if the initial state is an energy eigenstate then it does not
evolve in time i.e., stationary states of the von Neumann equation remains so
in the Milburn framework. Note that sums over $E$ and $E^{\prime}$ turn into
integrals when the energy spectrum is continuous. From Eq. (6), in the
position representation one has that

$\displaystyle\rho(x,x^{\prime},t)$ $\displaystyle=$
$\displaystyle\sum_{E}\sum_{E^{\prime}}e^{-i(E-E^{\prime})t/\hbar-(E-E^{\prime})^{2}t/(2\hbar^{2}\gamma)}\rho_{EE^{\prime}}(0)u_{E}(x)u_{E^{\prime}}^{*}(x^{\prime}),$
(8)

where the corresponding diagonal elements of the density operator give us the
probability density

$\displaystyle P(x,t)$ $\displaystyle=$
$\displaystyle\sum_{E}\sum_{E^{\prime}}e^{-i(E-E^{\prime})t/\hbar-(E-E^{\prime})^{2}t/(2\hbar^{2}\gamma)}\rho_{EE^{\prime}}(0)u_{E}(x)u_{E^{\prime}}^{*}(x),$
(9)

where $u_{E}(x)=\langle x|E\rangle$. Now, if the initial state is considered
to be a pure state

$\displaystyle\hat{\rho}(0)$ $\displaystyle=$
$\displaystyle|\psi_{0}\rangle\langle\psi_{0}|,$ (10)

then

$\displaystyle P(x,t)$ $\displaystyle=$
$\displaystyle\sum_{E}\sum_{E^{\prime}}e^{-i(E-E^{\prime})t/\hbar-(E-E^{\prime})^{2}t/(2\hbar^{2}\gamma)}C_{E}C^{*}_{E^{\prime}}u_{E}(x)u_{E^{\prime}}^{*}(x),$
(11)

where the expansion coefficients are given by

$\displaystyle C_{E}$ $\displaystyle=$ $\displaystyle\langle
E|\psi_{0}\rangle=\int dx~{}u^{*}_{E}(x)\psi_{0}(x).$ (12)

Due to the time-dependent damping exponential factor, the long time behavior
of the probability density is given by

$\displaystyle\lim_{t\to\infty}P(x,t)$ $\displaystyle\approx$
$\displaystyle\sum_{E}|C_{E}|^{2}|u_{E}(x)|^{2}$ (13)

which is independent on time. In this limit, non-diagonal elements of density
matrix decay exponentially. This is the way the intrinsic decoherence works in
the Milburn framework. It is a gradual and continuous process depending on the
$\gamma^{-1}$ value. This behavior is the same observed for the external
decoherence.

In the context of the standard quantum mechanics, when the Hamiltonian of a
many body system has the additive form

$\displaystyle\hat{H}$ $\displaystyle=$ $\displaystyle\sum_{i}\hat{H}_{i},$
(14)

a possible solution of the Schrödinger equation is written as a product of
single particle states,

$\displaystyle\Psi(x_{1},x_{2},\cdots x_{N},t)$ $\displaystyle=$
$\displaystyle\Pi_{i}\psi_{i}(x_{i},t),$ (15)

i.e., provided that the state is in the product form in an instant of time,
this property is maintained over the time evolution. However, this is not
valid in the context of the Milburn framework. In fact, one can show that if
the one-particle state $\hat{\rho}_{i}$ evolves under the Milburn equation
with the Hamiltonian $\hat{H}_{i}$, then the two-particle separable state
$\hat{\rho}=\hat{\rho}_{1}\otimes\hat{\rho}_{2}$ is not evolved under the
Milburn equation with the two-particle Hamiltonian $\hat{H}_{1}+\hat{H}_{2}$
since

$\displaystyle\frac{\partial}{\partial t}\hat{\rho}_{1}\otimes\hat{\rho}_{2}$
$\displaystyle=$ $\displaystyle\frac{\partial\hat{\rho}_{1}}{\partial
t}\otimes\hat{\rho}_{2}+\hat{\rho}_{1}\otimes\frac{\partial\hat{\rho}_{2}}{\partial
t}$ (16) $\displaystyle=$
$\displaystyle\left(-\frac{i}{\hbar}[\hat{H}_{1},\hat{\rho}_{1}]-\frac{1}{2\hbar^{2}\gamma}[\hat{H}_{1},[\hat{H}_{1},\hat{\rho}_{1}]]\right)\otimes\hat{\rho}_{2}+\hat{\rho}_{1}\otimes\left(-\frac{i}{\hbar}[\hat{H}_{2},\hat{\rho}_{2}]-\frac{1}{2\hbar^{2}\gamma}[\hat{H}_{2},[\hat{H}_{2},\hat{\rho}_{2}]]\right)$
$\displaystyle=$
$\displaystyle-\frac{i}{\hbar}[\hat{H}_{1}+\hat{H}_{2},\hat{\rho}_{1}\otimes\hat{\rho}_{2}]-\frac{1}{2\hbar^{2}\gamma}[\hat{H}_{1}+\hat{H}_{2},[\hat{H}_{1}+\hat{H}_{2},\hat{\rho}_{1}\otimes\hat{\rho}_{2}]]+\frac{1}{\gamma\hbar^{2}}[\hat{H}_{1},\hat{\rho}_{1}]\otimes[\hat{H}_{2},\hat{\rho}_{2}].$

The last term in the third line is responsible for this different behavior.
Note that for convenience we have written $\hat{H}_{1}+\hat{H}_{2}$ instead of
$\hat{H}_{1}\otimes\hat{\mathds{1}}_{2}+\hat{\mathds{1}}_{1}\otimes\hat{H}_{2}$;
$\hat{\mathds{1}}_{i}$ being the identity operator in the vector space of
particle $i$.

###  II.1 Purity

Once it is well established this gradual intrinsic decoherene process, a
standard quantity used in this context is the so-called purity. Purity is
defined as the subtraction of the trace of the square of the density operator
from unity

$\displaystyle\xi(t)$ $\displaystyle=$ $\displaystyle
1-\text{tr}(\hat{\rho}(t)^{2}).$ (17)

Since the trace operation is a basis independent quantity, from Eq. (1), and
writing the density matrix elements in the energy representation as
$\rho(E,E^{\prime},t)=\langle E|\hat{\rho}(t)|E^{\prime}\rangle$, one has that

$\displaystyle\rho_{EE^{\prime}}(t)$ $\displaystyle=$
$\displaystyle\rho_{EE^{\prime}}(0)~{}\exp\left[\gamma
t\left(e^{-i(E-E^{\prime})/\hbar\gamma}-1\right)\right],$ (18)

and the purity is expressed as

$\displaystyle\xi(t)$ $\displaystyle=$ $\displaystyle 1-\sum_{E}~{}\langle
E|\hat{\rho}(t)^{2}|E\rangle=1-\sum_{E}\sum_{E^{\prime}}~{}\langle
E|\hat{\rho}(t)|E^{\prime}\rangle\langle E^{\prime}|\hat{\rho}(t)|E\rangle$
(19) $\displaystyle=$ $\displaystyle 1-\sum_{E}\sum_{E^{\prime}}~{}\langle
E|\hat{\rho}(0)|E^{\prime}\rangle\langle
E^{\prime}|\hat{\rho}(0)|E\rangle\exp\left[-4\gamma
t\sin^{2}\left(\frac{E-E^{\prime}}{2\hbar\gamma}\right)\right],$

where in the second line of this equation we have used Eq. (18). Thus, purity
decays also exponentially in time. Its short linear time behavior is then
given by

$\displaystyle\xi(t)$ $\displaystyle\approx$ $\displaystyle\xi(0)-4\gamma
t\sum_{E}\sum_{E^{\prime}}~{}|\langle
E|\hat{\rho}(0)|E^{\prime}\rangle|^{2}\sin^{2}\left(\frac{E-E^{\prime}}{2\hbar\gamma}\right).$
(20)

Here, we have used the Hermiticity property of the density operator. The other
interesting limit of purity is its behavior when $\gamma\rightarrow\infty$,
i.e., the limit of the standard quantum mechanics. In this limit, to first
order in $\gamma^{-1}$, one obtains from Eq. (19)

$\displaystyle\xi(t)$ $\displaystyle\approx$
$\displaystyle\xi(0)+\frac{t}{\gamma\hbar^{2}}\text{tr}([\hat{H},\hat{\rho}(0)]^{2}).$
(21)

Note that the operator $[\hat{H},\hat{\rho}(0)]$ is anti-Hermitian and thus
its eigenvalues are imaginary. Thus, since the second term in Eq. (21) is
negative, the intrinsic decoherence reduces purity following a linear behavior
in time.

###  II.2 Ehrenfest relations

In the standard quantum mechanics, the Ehrenfest relations are

$\displaystyle\frac{d}{dt}\langle\hat{x}\rangle=\frac{\langle\hat{p}\rangle}{m}$
(22)
$\displaystyle\frac{d}{dt}\langle\hat{p}\rangle=-\left\langle\frac{\partial
V}{\partial\hat{x}}\right\rangle.$ (23)

It is instructive to see the effect of the intrinsic decoherence on these
relations. From Eq. (1), one obtains

$\displaystyle\frac{d}{dt}\langle\hat{A}\rangle_{t}$ $\displaystyle=$
$\displaystyle\gamma\left\langle
e^{-i\hat{H}/\hbar\gamma}\hat{A}e^{i\hat{H}/\hbar\gamma}-\hat{A}\right\rangle$
(24)

for the rate of change of the expectation value of an observable $\hat{A}$. To
first order in $\gamma^{-1}$, one has that

$\displaystyle\frac{d}{dt}\langle\hat{A}\rangle_{t}$ $\displaystyle\approx$
$\displaystyle\frac{i}{\hbar}\langle[\hat{H},\hat{A}]\rangle_{t}-\frac{1}{2\hbar^{2}\gamma}\langle[\hat{H},[\hat{H},\hat{A}]]\rangle_{t}.$
(25)

The rates of change of the expectation values of position and momentum
operators are then

$\displaystyle\frac{d}{dt}\langle\hat{x}\rangle_{t}$ $\displaystyle\approx$
$\displaystyle\frac{\langle\hat{p}\rangle_{t}}{m}-\frac{1}{2m\gamma}\left\langle\frac{\partial
V}{\partial\hat{x}}\right\rangle_{t}$ (26)
$\displaystyle\frac{d}{dt}\langle\hat{p}\rangle_{t}$ $\displaystyle\approx$
$\displaystyle-\left\langle\frac{\partial
V}{\partial\hat{x}}\right\rangle_{t}-\frac{1}{4m\gamma}\left\langle\left\\{\hat{p},\frac{\partial^{2}V}{\partial\hat{x}^{2}}\right\\}\right\rangle_{t},$
(27)

where $\\{\hat{A},\hat{B}\\}=\hat{A}\hat{B}+\hat{B}\hat{A}$ is the anti-
commutator of observables $\hat{A}$ and $\hat{B}$. It is assumed here that the
interaction potential is only a function of the position operator i.e.,
$V=V(\hat{x})$. These relations show that only for non-interacting (free)
particles the first Ehrenfest relation (26) has the same formal expression as
in the standard QM while for potentials, at most linear in the position
operator, the second Ehrenfest relation (27) also does. Then, one concludes
that, only for free particles, $\langle\hat{x}\rangle_{t}$ follows a classical
trajectory. In fact, the intrinsic decoherence does not affect the first
moments $\langle\hat{x}\rangle_{t}$ and $\langle\hat{p}\rangle_{t}$ (Mi-
PRA-1991, ).

As an illustration, let us consider the motion in a uniform gravity field for
which $V(\hat{x})=mg\hat{x}$. In this case, solutions of Eqs. (26) and (27)
yield

$\displaystyle\langle\hat{x}\rangle_{t}\approx\langle\hat{x}\rangle_{0}+\left(\frac{\langle\hat{p}\rangle_{0}}{m}-\frac{g}{2\gamma}\right)t-\frac{1}{2}gt^{2},$
(28)
$\displaystyle\langle\hat{p}\rangle_{t}\approx\langle\hat{p}\rangle_{0}-mgt.$
(29)

Note that while the expectation value of momentum is not affected by the
intrinsic decoherence, the rate of change of the expectation value of position
is reduced by the amount of $g/2\gamma$. In a certain sense, one could speak
about an effective friction. In the upward motion of a projectile, the peak
height is
$\dfrac{\langle\hat{p}\rangle_{0}^{2}}{2m^{2}g}\left(1-\frac{mg}{\gamma\langle\hat{p}\rangle_{0}}\right)$.
Note that in order to reach this height, one needs
$\langle\hat{p}\rangle_{0}>mg/\gamma$. If a harmonic potential
$V(\hat{x})=m\omega^{2}\hat{x}^{2}/2$ is now considered, solutions of Eqs.
(26) and (27) are given by

$\displaystyle\langle\hat{x}\rangle_{t}\approx
e^{-\omega^{2}t/2\gamma}\left(\langle\hat{x}\rangle_{0}\cos(\omega
t)+\frac{\langle\hat{p}\rangle_{0}}{m\omega}\sin(\omega t)\right),$ (30)
$\displaystyle\langle\hat{p}\rangle_{t}\approx
e^{-\omega^{2}t/2\gamma}(-m\omega\langle\hat{x}\rangle_{0}\sin(\omega
t)+\langle\hat{p}\rangle_{0}\cos(\omega t)).$ (31)

It is clearly seen that the parameter $\gamma^{-1}$ acts as a true friction
coefficient; when it is zero, one recovers the corresponding Ehrenfest
relations.

###  II.3 The continuity equation, the probability current density and arrival
times

The Milburn equation (2) in the position representation reads as

$\displaystyle\frac{\partial}{\partial t}\rho(x,x^{\prime},t)$
$\displaystyle=$
$\displaystyle-\frac{i}{\hbar}\left[\frac{-\hbar^{2}}{2m}(\partial_{x}^{2}-\partial_{x^{\prime}}^{2})+V(x)-V(x^{\prime})\right]\rho(x,x^{\prime},t)$
(32) $\displaystyle-$
$\displaystyle\frac{1}{2\hbar^{2}\gamma}\left[\frac{-\hbar^{2}}{2m}(\partial_{x}^{2}-\partial_{x^{\prime}}^{2})+V(x)-V(x^{\prime})\right]^{2}\rho(x,x^{\prime},t).$

This equation in the center of mass and relative coordinates, defined by
$R=(x+x^{\prime})/2$ and $r=x-x^{\prime}$, is written as

$\displaystyle\frac{\partial}{\partial t}\rho(R,r,t)$ $\displaystyle=$
$\displaystyle-\frac{i}{\hbar}\left[\frac{-\hbar^{2}}{m}\frac{\partial^{2}}{\partial
R\partial r}+V(R+r/2)-V(R-r/2)\right]\rho(R,r,t)$ (33) $\displaystyle-$
$\displaystyle\frac{1}{2\hbar^{2}\gamma}\left[\frac{-\hbar^{2}}{m}\frac{\partial^{2}}{\partial
R\partial r}+V(R+r/2)-V(R-r/2)\right]^{2}\rho(R,r,t),$

which in the absence of interactions i.e., for free particles, it can be
written as a continuity equation

$\displaystyle\frac{\partial}{\partial t}\rho(R,r,t)+\frac{\partial}{\partial
R}j(R,r,t)$ $\displaystyle=$ $\displaystyle 0,$ (34)

where we have introduced the probability current density matrix $j(R,r,t)$ as

$\displaystyle j(R,r,t)$ $\displaystyle=$
$\displaystyle\left(-\frac{i\hbar}{m}+\frac{\hbar^{2}}{2\gamma
m^{2}}\frac{\partial^{2}}{\partial r\partial R}\right)\frac{\partial}{\partial
r}\rho(R,r,t).$ (35)

On the other hand, the density matrix in the position representation can be
seen as the inverse Fourier transform of its representation in the momentum
space,

$\displaystyle\rho(R,r,t)$ $\displaystyle=$
$\displaystyle\frac{1}{2\pi\hbar}\int_{-\infty}^{\infty}dp\int_{-\infty}^{\infty}dp^{\prime}e^{i(p-p^{\prime})R/\hbar}e^{i(p+p^{\prime})r/2\hbar}\langle
p|\hat{\rho}(t)|p^{\prime}\rangle,$ (36)

where, from the solution of Eq. (2) for free particles, we have that the
matrix elements in this representation are written as

$\displaystyle\langle p|\hat{\rho}(t)|p^{\prime}\rangle$ $\displaystyle=$
$\displaystyle\exp\left[-\frac{i}{\hbar}\frac{p^{2}-p^{\prime
2}}{2m}t-\frac{1}{2\hbar^{2}\gamma}\left(\frac{p^{2}-p^{\prime
2}}{2m}\right)^{2}t\right]\langle p|\hat{\rho}(0)|p^{\prime}\rangle,$ (37)

where the standard result is again reached when $\gamma^{-1}\rightarrow 0$.

By substituting now Eq. (37) into Eq. (35), the diagonal elements of the
probability current density are given by

$\displaystyle J(x,t)$ $\displaystyle=$ $\displaystyle\frac{1}{4\pi
m\hbar}\int_{-\infty}^{\infty}dp\int_{-\infty}^{\infty}dp^{\prime}e^{i(p-p^{\prime})x/\hbar}(p+p^{\prime})\left(1-i\frac{p^{2}-p^{\prime
2}}{4\hbar m\gamma}\right)\rho(p,p^{\prime},t).$ (38)

Note that, since the density operator is Hermitian, one has that $\langle
p|\hat{\rho}(t)|p^{\prime}\rangle^{*}=\langle
p^{\prime}|\hat{\rho}(t)|p\rangle$ which, as expected, $J(x,t)$ is real.

As a particular case, let us see if the product of states in the context of
the standard quantum mechanics is kept in this formalism. If the wave function
of a system is initially factorized as follows

$\displaystyle\psi(\mathbf{r})$ $\displaystyle=$
$\displaystyle\psi_{1}(x)\psi_{2}(y)\psi_{3}(z),$ (39)

one sees that it will be preserved along time when the potential is additive
such as

$\displaystyle V(\mathbf{r})$ $\displaystyle=$ $\displaystyle
V_{1}(x)+V_{2}(y)+V_{3}(z),$ (40)

(including $V=0$). However, this result is not longer valid in the context of
the Milburn equation. If one considers free propagation, Eq. (37) explicitly
shows, even if the initial state is factorized as

$\displaystyle\rho(\mathbf{p},\mathbf{p^{\prime}},0)$ $\displaystyle=$
$\displaystyle\rho_{1}(p_{x},p^{\prime}_{x},0)\rho_{2}(p_{y},p^{\prime}_{y},0)\rho_{3}(p_{z},p^{\prime}_{z},0),$
(41)

that the exponential term
$e^{-[(\mathbf{p}^{2}-\mathbf{p^{\prime}}^{2})/2m]^{2}t/2\hbar^{2}\gamma}$
appears in Eq. (37) and this factorization is not maintained at later times.

Within the framework of the Bohmian mechanics, it is proved that the ideal
(measurement-independent) arrival time distribution at the detector location
$x_{d}$ is uniquely given by the modulus of the probability current density
when starting from the continuity equation Leavens

$\displaystyle\Pi(x_{d},t)$ $\displaystyle=$
$\displaystyle\frac{|J(x_{d},t)|}{\int_{0}^{\infty}dt^{\prime}~{}|J(x_{d},t^{\prime})|}.$
(42)

Here, we use this formalism to see how arrival times are affected by the
intrinsic decoherence. For a given state, the probability current density is
positive for all times at the location of the detector which, by assumption,
is taken to be at the origin. Then, the normalized arrival time distribution
is

$\displaystyle\Pi(0,t)$ $\displaystyle=$
$\displaystyle\frac{J(0,t)}{\int_{0}^{\infty}dt^{\prime}~{}J(0,t^{\prime})}.$
(43)

The mean arrival time at the detector location $x_{d}=0$ and its variance are
then given by

$\displaystyle\langle t\rangle$ $\displaystyle=$
$\displaystyle\int_{0}^{\infty}dt^{\prime}~{}t^{\prime}~{}\Pi(0,t^{\prime}),$
(44) $\displaystyle(\Delta t)^{2}$ $\displaystyle=$ $\displaystyle\langle
t^{2}\rangle-\langle t\rangle^{2},$ (45)

respectively. The standard deviation and the square root of the variance can
be used as a measure of the width of the arrival time distribution, i.e., the
rms width. Then, from Eqs. (37) and (38), we have that

$\displaystyle\int_{0}^{\infty}dt^{\prime}~{}J(0,t^{\prime})$ $\displaystyle=$
$\displaystyle\frac{1}{2\pi
i}\int_{-\infty}^{\infty}du\int_{-\infty}^{\infty}dv\frac{\rho(u,v,0)}{v},$
(46)
$\displaystyle\int_{0}^{\infty}dt^{\prime}~{}t^{\prime}~{}J(0,t^{\prime})$
$\displaystyle=$
$\displaystyle-\frac{m\hbar}{2\pi}\int_{-\infty}^{\infty}du\int_{-\infty}^{\infty}dv\frac{\rho(u,v,0)}{uv^{2}\left(1-\frac{iuv}{2m\hbar\gamma}\right)},$
(47) $\displaystyle\int_{0}^{\infty}dt^{\prime}~{}t^{\prime
2}~{}J(0,t^{\prime})$ $\displaystyle=$
$\displaystyle-\frac{m^{2}\hbar^{2}}{i\pi}\int_{-\infty}^{\infty}du\int_{-\infty}^{\infty}dv\frac{\rho(u,v,0)}{u^{2}v^{3}\left(1-\frac{iuv}{2m\hbar\gamma}\right)^{3}},$
(48)

where we have used the new variables

$\displaystyle u=\frac{p+p^{\prime}}{2}$ (49) $\displaystyle v=p-p^{\prime}.$
(50)

Eq. (46) shows that the normalization factor (or time integral of this
current) is independent on $\gamma^{-1}$ i.e., the intrinsic decoherence does
not affect this quantity. The first and second moment have the same functional
dependence of the intrinsic decoherence. From Eqs. (46), (47) and (48), one
writes to first order in $\gamma^{-1}$

$\displaystyle\langle t\rangle$ $\displaystyle\approx$ $\displaystyle\langle
t\rangle_{\text{q}}+\frac{1}{2\gamma}$ (51) $\displaystyle(\Delta t)^{2}$
$\displaystyle\approx$ $\displaystyle(\Delta
t)^{2}_{\text{q}}+\frac{2}{\gamma}\langle t\rangle_{\text{q}},$ (52)

where the sub-index “q” stands for the corresponding quantity within the
standard quantum mechanics. Eqs. (51) and (52) confirm that both the mean
arrival time and the rms width of the arrival time distribution are increased
by the intrinsic decoherence and the amount of increment is linear in
$\gamma^{-1}$. We emphasize that the above results are only valid for states
where the current density is positive.

###  II.4 The Wigner representation

The Wigner function is defined as the partial Fourier transform of the density
matrix in the $(R,r)$ representation with respect to the relative coordinate
$r$ MoMi-Sym-2023

$\displaystyle W(R,u,t)$ $\displaystyle=$
$\displaystyle\frac{1}{2\pi\hbar}\int dr~{}e^{-iur/\hbar}\rho(R,r,t).$ (53)

From Eq. (33) one obtains an equation of motion for the Wigner function
expressed as

$\displaystyle\frac{\partial}{\partial t}W(R,u,t)$ $\displaystyle=$
$\displaystyle-\frac{u}{m}\frac{\partial}{\partial R}W(R,u,t)+\int
du^{\prime}~{}K(R,u^{\prime}-u,t)~{}W(R,u^{\prime},t)$ $\displaystyle+$
$\displaystyle\frac{1}{2\pi\hbar}\frac{-1}{2\hbar^{2}\gamma}\int
du^{\prime}~{}\int
dr~{}e^{-iur/\hbar}\left(\frac{-\hbar^{2}}{m}\frac{\partial^{2}}{\partial
R\partial
r}+V(R+r/2)-V(R-r/2)\right)^{2}e^{iu^{\prime}r/\hbar}W(R,u^{\prime},t),$

where the kernel of the first integral is defined as

$\displaystyle K(R,q,t)$ $\displaystyle=$
$\displaystyle\frac{1}{2\pi\hbar}\int
dr~{}e^{iqr/\hbar}\frac{V(R/2+r)-V(R/2-r)}{\hbar}.$ (55)

Note that the first line of Eq. (LABEL:eq:_W-motion) is the equation of motion
for the Wigner distribution function within the von Neumann formalism; the
second line is the contribution due to the intrinsic decoherence. As an
illustration, let us consider now two simple examples, the motion for a free
particle and under a linear potential or constant force.

For free particles, Eq. (LABEL:eq:_W-motion) is reduced to

$\displaystyle\frac{\partial}{\partial
t}W(x,p,t)+\frac{p}{m}\frac{\partial}{\partial
x}W(x,p,t)-\frac{p^{2}}{2m^{2}\gamma}\frac{\partial^{2}}{\partial
x^{2}}W(x,p,t)$ $\displaystyle=$ $\displaystyle 0.$ (56)

Apart from the last term which is responsible for the intrinsic decoherence,
this equation has the same formal expression as that of the classical
Liouville equation

$\displaystyle\frac{\partial}{\partial
t}f(x,p,t)+\frac{p}{m}\frac{\partial}{\partial
x}f(x,p,t)+\dot{p}\frac{\partial}{\partial p}f(x,p,t)$ $\displaystyle=$
$\displaystyle 0,$ (57)

where $f(x,p,t)$ is the phase space distribution function. Note that in this
case $\dot{p}=0$ and thus the last term of Eq. (57) is zero. In the second
example, the motion under the presence of a linear potential

$\displaystyle V(x)=c_{1}~{}x,$ (58)

one obtains from Eq. (LABEL:eq:_W-motion)

$\displaystyle\frac{\partial}{\partial t}W(x,p,t)$ $\displaystyle+$
$\displaystyle\frac{p}{m}\frac{\partial}{\partial
x}W(x,p,t)+c_{1}\frac{\partial}{\partial p}W(x,p,t)$ (59) $\displaystyle-$
$\displaystyle\frac{1}{\gamma}\left(\frac{p^{2}}{2m^{2}}\frac{\partial^{2}}{\partial
x^{2}}-\frac{c_{1}}{2m}\frac{\partial}{\partial
x}-c_{1}\frac{p}{m}\frac{\partial}{\partial x}\frac{\partial}{\partial
p}+\frac{c_{1}^{2}}{2}\frac{\partial^{2}}{\partial p^{2}}\right)W(x,p,t)=0.$

which again, apart from the last term, one recovers the classical Liouville
equation (57).

###  II.5 The Milburn and Lindblad equations

In the context of open quantum systems by taking into account the role played
by the environment one obtains the Lindbladian master equation Sch-book-2007

$\displaystyle\frac{\partial}{\partial t}\hat{\rho}(t)$ $\displaystyle=$
$\displaystyle-\frac{i}{\hbar}[\hat{H}^{\prime},\hat{\rho}(t)]-\frac{1}{2}\sum_{i}\kappa_{i}[\hat{L}_{i},[\hat{L}_{i},\hat{\rho}(t)]],$
(60)

for the evolution of the reduced density operator describing the system of
interest where we have assumed Hermitian Lindblad operators $\hat{L}_{i}$ and
parameters $\kappa_{i}>0$. The so-called Lamb-shifted Hamiltonian
$\hat{H}^{\prime}$ is not generally identical to the self Hamiltonian
$\hat{H}$ of the system containing additional terms caused by the interaction
with the environment. For free particles with
$\hat{H}^{\prime}=\hat{p}^{2}/2m$, this master equation with only one Lindblad
operator reads

$\displaystyle\frac{\partial}{\partial t}\hat{\rho}(t)$ $\displaystyle=$
$\displaystyle-\frac{i}{\hbar}[\frac{\hat{p}^{2}}{2m},\hat{\rho}(t)]-\frac{1}{2}\kappa[\hat{L},[\hat{L},\hat{\rho}(t)]].$
(61)

Comparison of Eq. (61) with the Milburn equation

$\displaystyle\frac{\partial}{\partial t}\hat{\rho}(t)$ $\displaystyle=$
$\displaystyle-\frac{i}{\hbar}[\frac{\hat{p}^{2}}{2m},\hat{\rho}(t)]-\frac{1}{2\hbar^{2}\gamma}[\frac{\hat{p}^{2}}{2m},[\frac{\hat{p}^{2}}{2m},\hat{\rho}(t)]],$
(62)

for free particles shows that the Milburn equation (62) has the same
mathematical structure than the Lindbladian equation (61) with a Lindblad
operator proportional to $\hat{p}^{2}$. Let us consider two simple cases. If
we consider $\hat{L}=f(\hat{p})$, $f$ being an arbitrary well-defined
function, from Eq. (61) in the momentum representation we obtain

$\displaystyle\frac{\partial}{\partial t}\rho(p,p^{\prime},t)$
$\displaystyle=$ $\displaystyle\left\\{-\frac{i}{\hbar}\frac{p^{2}-p^{\prime
2}}{2m}-\frac{1}{2}\kappa(f(p)-f(p^{\prime}))^{2}\right\\}\rho(p,p^{\prime},t),$
(63)

whose solution reads as

$\displaystyle\rho(p,p^{\prime},t)$ $\displaystyle=$
$\displaystyle\exp\left[\left(-\frac{i}{\hbar}\frac{p^{2}-p^{\prime
2}}{2m}-\frac{1}{2}\kappa(f(p)-f(p^{\prime}))^{2}\right)t\right]\rho(p,p^{\prime},0),$
(64)

revealing the probability density $\rho(p,p,t)$ is time independent. Now, with
the choice $\hat{L}=\hat{x}$, Eq. (61) can be recast as

$\displaystyle\frac{\partial}{\partial t}\rho(x,x^{\prime},t)$
$\displaystyle=$
$\displaystyle\left[\frac{i\hbar}{2m}\left(\frac{\partial^{2}}{\partial
x^{2}}-\frac{\partial^{2}}{\partial x^{\prime
2}}\right)-\frac{1}{2}\kappa(x-x^{\prime})^{2}\right]\rho(x,x^{\prime},t),$
(65)

in the coordinate representation being the environmental scattering Sch-
book-2007  equation of motion. Again, comparison of the high temperature limit
of the well-known Caldeira-Leggett master equation

$\displaystyle\frac{\partial}{\partial t}\rho(x,x^{\prime},t)$
$\displaystyle=$
$\displaystyle\left[\frac{i\hbar}{2m}\left(\frac{\partial^{2}}{\partial
x^{2}}-\frac{\partial^{2}}{\partial x^{\prime
2}}\right)-\lambda(x-x^{\prime})\left(\frac{\partial}{\partial
x}-\frac{\partial}{\partial
x^{\prime}}\right)-\frac{D}{\hbar^{2}}(x-x^{\prime})^{2}\right]\rho(x,x^{\prime},t)$
(66)

with Eq. (65) reveals that the former is Lindbladian only in the negligible
dissipation limit i.e, when the second term can be neglected. Note that
$\lambda$ is the relaxation rate or damping constant and $D=2m\lambda
k_{B}T_{e}$ plays the role of the diffusion coefficient ($k_{B}$ is the
Boltzmann constant and $T_{e}$ is the temperature of the environment).

###  II.6 Transmission through a localized barrier

Consider now the solution of the time independent Schrödinger equation for a
localized barrier $V(x)$,

$\displaystyle u_{k}(x)$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2\pi}}\begin{cases}e^{ikx}+R(k)e^{-ikx}&x<0,\\\
A(k)f(x)+B(k)g(x)&0\leq x<L,\\\ T(k)e^{ikx}&L\leq x,\end{cases}$ (67)

where $k=\sqrt{2mE/\hbar^{2}}$ is the wave number, $[0,L]$ is the support of
the barrier and $R(k)$ and $T(k)$ are the reflection and transmission
coefficients, respectively. When $E<V_{\max}$, tunneling is present. Finally,
$f(x)$ and $g(x)$ are two independent solutions of the Scrödinger equation in
the barrier region. Note that here we have preferred to use $u_{k}(x)$ instead
of $u_{E}(x)$.

Let us now consider transmission of a localized wave packet through the
barrier. If the system is initially prepared in the state $\psi_{0}(x)$ in the
far left side of the barrier and the wavepacket includes only positive
momenta, the solution of the Schrödinger equation is given by

$\displaystyle\psi(x,t)$ $\displaystyle=$
$\displaystyle\int_{0}^{\infty}dk~{}\phi(k)e^{-iE_{k}t/\hbar}u_{k}(x),\qquad
E_{k}=\frac{\hbar^{2}k^{2}}{2m},$ (68)

where $\phi(k)=\langle k|\psi_{0}\rangle$ denotes the momentum amplitude (the
Fourier transform of $\psi_{0}(x)$) of initial wave packet in the absence of
the barrier.

From Eq. (8), the solution of the corresponding problem in the Milburn
framework is given by

$\displaystyle\rho(x,x^{\prime},t)$ $\displaystyle=$
$\displaystyle\int_{0}^{\infty}dk~{}\int_{0}^{\infty}dk^{\prime}~{}e^{-i(E_{k}-E_{k^{\prime}})t/\hbar-(E_{k}-E_{k^{\prime}})^{2}t/(2\hbar^{2}\gamma)}\phi(k)\phi^{*}(k^{\prime})u_{k}(x)u_{k^{\prime}}^{*}(x^{\prime}).$
(69)

The time-dependent transmission probability is defined as

$\displaystyle P_{\text{tr}}(t)$ $\displaystyle=$
$\displaystyle\int_{L}^{\infty}dx~{}\rho(x,x,t)$ (70)

and the stationary value of this quantity is independent on the lower limit of
the corresponding integral. The transmission probability for a monochromatic
beam with momentum $p=\hbar k$ is $|T(k)|^{2}$. Due to the fact that the
initial probability density in momentum is $|\phi(k)|^{2}$, then the
stationary transmission probability for a wave packet is

$\displaystyle P_{\text{tr}}(\infty)$ $\displaystyle=$
$\displaystyle\int_{0}^{\infty}dk~{}|\phi(k)|^{2}|T(k)|^{2}.$ (71)

This is applicable even in the framework of the Milburn equation. In this way,
one sees that $P_{\text{tr}}(\infty)$ is independent on the intrinsic
decoherence parameter $\gamma$.

##  III Results and discussions

In this section, several problems are analyzed within the Milburn context in
order to see how the intrinsic decoherence process is settled down in the
dynamics. We are going to work in a system of units where $\hbar=m=1$.

###  III.1 Interference of two wave packets

As a first application, we analyze the effect of the intrinsic decoherence on
the interference of two wave packets in free space i.e., in the absence of
external interactions. Consider the superposition

$\displaystyle\psi(x,0)$ $\displaystyle=$
$\displaystyle\mathcal{N}(\psi_{a}(x,0)+\psi_{b}(x,0))$ (72)

of two well-separated Gaussian wave packets, each one described by Eq. (82)
with the same width $\sigma_{0}$, well-localized at $x_{0a}$ and $x_{0b}$ and
kick momenta $p_{0a}$ and $p_{0b}$. With these initial conditions,

$\displaystyle x_{0a}=-x_{0b}=x_{0},$ (73) $\displaystyle
p_{0a}=-p_{0b}=p_{0},$ (74)

the normalization constant $\mathcal{N}$ is given by

$\displaystyle\mathcal{N}$ $\displaystyle=$
$\displaystyle\left\\{2\left(1+\exp\left[-\frac{x_{0}^{2}}{2\sigma_{0}^{2}}-2\frac{p_{0}^{2}\sigma_{0}^{2}}{\hbar^{2}}\right]\right)\right\\}^{-1/2}.$
(75)

The time evolution is given by Eq. (36) and the corresponding probability
density reads as

$\displaystyle P(x,t)$ $\displaystyle=$
$\displaystyle\rho(R=x,r=0,t)=\mathcal{N}^{2}[P_{aa}(x,t)+P_{bb}(x,t)+2\text{Re}\\{P_{ab}(x,t)\\}],$
(76)

where $P_{ij}(x,t)=\varrho_{ij}(R=x,r=0,t)$ give the diagonal elements of the
time evolution of $\psi_{i}(x,0)\psi_{j}^{*}(x^{\prime},0)$

$\displaystyle P_{ij}(x,t)$ $\displaystyle=$
$\displaystyle\frac{1}{2\pi\hbar}\int_{-\infty}^{\infty}dp\int_{-\infty}^{\infty}dp^{\prime}~{}e^{i(p-p^{\prime})x/\hbar}\exp\left[-\frac{i}{\hbar}\frac{p^{2}-p^{\prime
2}}{2m}t-\frac{1}{2\hbar^{2}\gamma}\left(\frac{p^{2}-p^{\prime
2}}{2m}\right)^{2}t\right]$ (77) $\displaystyle\times$
$\displaystyle\sqrt{\frac{2}{\pi}}\frac{\sigma_{0}}{\hbar}\exp\left[-\frac{(p-p_{0i})^{2}+(p^{\prime}-p_{0j})^{2}}{\hbar^{2}}-i\frac{x_{0i}~{}p-x_{0j}~{}p^{\prime}}{\hbar}\right].$

Figure 1:  The probability density for the cat state (72) in free propagation
under the Milburn equation versus position at different times $t=0$ (left top
panel) $t=1$ (middle top panel), $t=3$ (right top panel), $t=5$ (left bottom
panel), $t=10$ (middle bottom panel), $t=15$ (right bottom panel) for
$\gamma^{-1}=0$ (black curves), $\gamma^{-1}=0.2$ (red curves) and
$\gamma^{-1}=0.5$ (green curves). For numerical calculations, we have used
$\sigma_{0}=1$, $x_{0}=5$, $p_{0}=1$, $x_{0a}=-x_{0b}=x_{0}$ and
$p_{0a}=-p_{0b}=-p_{0}$.  Figure 2:  The visibility function
$\mathcal{V}(x,t_{0})$ versus $x$ at time $t_{0}=mx_{0}/p_{0}=5$ for different
values of the intrinsic decoherence parameter: for $\gamma^{-1}=0$ (black,
dashed curve),$\gamma^{-1}=0.2$ (red curve), $\gamma^{-1}=0.5$ (green curve)
and $\gamma^{-1}=0.8$ (blue curve). The same parameters are used as Figure 1.

In Figure 1, the probability density (76) at different times for different
values of the intrinsic decoherence parameter is plotted. As this figure
shows, interference fringes are apparent for the von Neumann dynamics while
disappearing with increasing $\gamma^{-1}$. When particles propagate under the
usual von Neumann equation i.e., for $\gamma^{-1}=0$, the Gaussian wave
packets $\psi_{a}$ and $\psi_{b}$ with the conditions given by Eqs. (73) and
(74) overlap at time $t_{0}=mx_{0}/p_{0}$. For these initial conditions, from
Eq. (77) and for $\gamma^{-1}=0$ one can easily show that

$\displaystyle P_{aa}(x,t_{0})$ $\displaystyle=$ $\displaystyle
P_{bb}(x,t_{0})=|P_{ab}(x,t_{0})|=\frac{1}{\sqrt{2\pi}\sigma_{t_{0}}}\exp\left[-\frac{2p_{0}^{2}\sigma_{0}^{2}}{\hbar^{2}x_{0}^{2}+4p_{0}^{2}\sigma_{0}^{4}}x^{2}\right],$
(78) $\displaystyle\sigma_{t}$ $\displaystyle=$
$\displaystyle\sigma_{0}\sqrt{1+\frac{\hbar^{2}t^{2}}{4m^{2}\sigma_{0}^{4}}}.$
(79)

Following Mi-PRA-1991 , one can define the fringe visibility as

$\displaystyle\mathcal{V}$ $\displaystyle=$
$\displaystyle\frac{|P_{ab}(x,t_{0})|}{\sqrt{P_{aa}(x,t_{0})P_{bb}(x,t_{0})}},$
(80)

which is a way to quantify the degree of interference. From Eqs. (78) and
(80), one sees that the visibility equals unity, independent on $x$. Figure 2
shows the visibility for different values of $\gamma^{-1}$. According to this
figure, and as one expects, around the central peak, the visibility in the
Milburn framework is a decaying function of $|x|$ and is smaller for higher
values of $\gamma^{-1}$.

###  III.2 Tunneling through a rectangular barrier

We now investigate tunneling through a rectangular barrier

$\displaystyle V(x)$ $\displaystyle=$ $\displaystyle
V_{0}\theta(x)\theta(L-x).$ (81)

Let us consider the Gaussian wave packet

$\displaystyle\psi(x)$ $\displaystyle=$
$\displaystyle\frac{1}{(2\pi\sigma_{0}^{2})^{1/4}}\exp\left[-\frac{x-x_{0}}{4\sigma_{0}^{2}}+i\frac{p_{0}}{\hbar}x\right],$
(82)

which is incident from left. Parameters of the wave packet and the barrier
have been chosen in a way that: (i) contribution of negative momenta to the
wave packet is negligible and (ii) the mean energy of the wave packet is
smaller than the height of the barrier. Thus, transmission is present due to
tunneling. In particular, we use: $\sigma_{0}=1$, $p_{0}=2$, $x_{0}=-10$,
$V_{0}=3$ and $L=1$.

Figure 3:  Probability density for incident of a Gaussian wave packet on a
rectangular barrier for the von Neumann dynamics (left column) and the Milburn
dynamics with $\gamma^{-1}=0.8$ (right column) at different times: $t=1$
(first row), $t=5$ (second row), $t=10$ (third row) and $t=40$ (fourth row).
For numerical calculations we have used: $\hbar=m=1$, $\sigma_{0}=1$,
$p_{0}=2$, $x_{0}=-10$, $V_{0}=3$ and $L=1$.

In Figure 3, the probability density is plotted at different times for the von
Neumann (left column) and Milburn (right column) dynamics. It is clearly seen
that the reflection and transmission parts of the wave packet are quite
reduced for $\gamma^{-1}=0.8$. The oscillations of the reflection part of the
wave packet are also nearly suppressed. This behavior is better understood
from Figure 4 where both the transmission probability and the probability
inside the barrier region are depicted for both dynamics. As calculations
confirm, the stationary value of the transmission probability and the
probability inside the barrier region given by

$\displaystyle P_{\text{D}}(t)$ $\displaystyle=$
$\displaystyle\int_{0}^{L}dx~{}\rho(x,x,t)$ (83)

and the dwell time by

$\displaystyle\tau_{\text{D}}$ $\displaystyle=$
$\displaystyle\int_{0}^{\infty}dt~{}P_{\text{D}}(t)$ (84)

are the same for both dynamics. However, note that in the Milburn dynamics,
particles arrive at the barrier sooner while leave later. For these chosen
parameters, the stationary transmission probability is $0.24536$; independent
on the value of the intrinsic decoherence parameter $\gamma$ and the dwell
time is $0.41845$, again the same for both dynamics.

Figure 4:  Time dependent transmission probability (left panel) and
probability inside the rectangular barrier region (right panel) for an
incident Gaussian wave packet for $\gamma^{-1}=0$ (black
curves),$\gamma^{-1}=0.2$ (red curve), $\gamma^{-1}=0.5$ (green curve) and
$\gamma^{-1}=0.8$ (blue curves). Orange dashed line on the left panel depicts
the stationary value of the transmission probability which is the same for
both, the von Neumann and Milburn dynamics. The same parameters as Figure 3
have been used.

###  III.3 The bouncing ball

As the last application of the Milburn formalism, we analyze a bound system
with discrete energy spectrum, the bouncing ball. This ball has mass $m$ and
is bouncing vertically over a table in the presence of the gravity field $g$,

$\displaystyle V(z)$ $\displaystyle=$ $\displaystyle\begin{cases}mgz&z>0,\\\
+\infty&z\leq 0.\end{cases}$ (85)

Here, we have denoted the space coordinate by $z$ instead of $x$ in order to
emphasize that motion is in the vertical direction. The solution of the time-
independent Schrödinger equation

$\displaystyle\left(\frac{\hbar^{2}}{2m}\frac{d^{2}}{dz^{2}}+mgz\right)u_{n}(z)$
$\displaystyle=$ $\displaystyle E_{n}u_{n}(z),\qquad z>0$ (86)

yields for the eigenvalues and eigenfunctions VaSo-book-2004

$\displaystyle E_{n}=-R_{n}\left(\frac{1}{2}mg^{2}\hbar^{2}\right)^{1/3}$ (87)
$\displaystyle
u_{n}(z)=\sqrt{\alpha}\frac{1}{\text{Ai}^{\prime}(R_{n})}\text{Ai}(\alpha
z+R_{n}),\qquad\alpha=\left(\frac{2m^{2}g}{\hbar^{2}}\right)^{1/3},$ (88)

respectively. Here $\text{Ai}(z)$ and $\text{Ai}^{\prime}(z)$ are the Airy
function and its derivative, $R_{n}$ is the n${}^{\text{th}}$ root of the Airy
function i.e., $\text{Ai}(R_{n})=0$. Let the initial wave function be a
Gaussian wave packet (82) with $p_{0}=0$. In this case, the expansion
coefficients (12) are given by

$\displaystyle C_{n}$ $\displaystyle=$
$\displaystyle\frac{\sqrt{\alpha}}{\text{Ai}^{\prime}(R_{n})}\frac{1}{(2\pi\sigma_{0}^{2})^{1/4}}\int_{0}^{\infty}dz~{}\text{Ai}(\alpha
z+R_{n})\exp\left[-\frac{(z-z_{0})^{2}}{4\sigma_{0}^{2}}\right].$ (89)

Figure 5:  Expansion weights $|C_{n}|^{2}$ versus $n$ (left panel) computed by
Eq. (89) and purity versus time for different values of $\gamma^{-1}$, in
units of $(2\hbar/mg^{2})^{1/3}$. For numerical calculations, we have used
$\sigma_{0}=1$ and $z_{0}=5$ in units of $(\hbar^{2}/2m^{2}g)^{1/3}$.

From Eq. (13), one obtains

$\displaystyle\lim_{t\to\infty}\langle z\rangle(t)$ $\displaystyle\approx$
$\displaystyle\sum_{n}|C_{n}|^{2}\int
dz~{}z|u_{n}(z)|^{2}=\sum_{n}|C_{n}|^{2}\langle u_{n}|z|u_{n}\rangle$ (90)

for the asymptotic value of the position expectation value. By substituting
Eq. (88) into (90), one reaches Va-AJP-2000

$\displaystyle\lim_{t\to\infty}\langle z\rangle(t)$ $\displaystyle\approx$
$\displaystyle-\frac{2}{3\alpha}\sum_{n}|C_{n}|^{2}R_{n}.$ (91)

On left panel of Figure 5, we have plotted the weights $|C_{n}|^{2}$ versus
$n$ for $\sigma_{0}=1$ and $z_{0}=5$ in units of $(\hbar^{2}/2m^{2}g)^{1/3}$
as the natural unit of length. Furthermore, natural units for energy and time
are $(mg^{2}\hbar^{2}/2)^{1/3}$ and $(2\hbar/mg^{2})^{1/3}$, respectively. As
this figure shows, the first few terms are enough for calculations. The right
panel of this figure depicts the evolution of purity for different values of
$\gamma^{-1}$. Concerning the bouncing motion, Figure 6 shows that
oscillations are killed by increasing the Milburn parameter $\gamma^{-1}$.
Taking only the first ten expansion coefficients for the probability density,
one has that $\lim_{t\to\infty}\langle z\rangle(t)\approx 3.5$ from Eq. (91)
which coincides with Figure 6.

Figure 6:  Position expectation value $\langle z\rangle(t)$ for the Milburn
dynamics. For numerical calculations we have used $\sigma_{0}=1$ and
$z_{0}=5$. Space coordinates in units of $(\hbar^{2}/2m^{2}g)^{1/3}$ and time
in units of $(2\hbar/mg^{2})^{1/3}$.

##  IV Conclusions

In this work, we have analyzed several theoretical and practical aspects of
the Milburn equation which comes from a first order expansion on the parameter
$\gamma^{-1}$ governing the decoherence, the so-called intrinsic decoherence.
In the limit of $\gamma^{-1}\to 0$, one recovers the von Neumann equation for
the density operator. Purity is reduced following a linear behavior in time.
Ehrenfest relations and the probability current density and arrival times are
also modified by the presence of the intrinsic decoherence. In particular, the
extension of the Milburn formalism to include the Wigner representation of the
corresponding equation and the relation to a Linbladian master equation are
important steps to better understand the importance of the corresponding
quantum dynamics. Moreover, the interference displayed by two Gaussian wave
packets leads to a decreasing function of the visibility as the intrinsic
decoherence increases. The same is observed for the tunneling transmission
probability. For the bouncing ball problem, the oscillations displayed by the
average value of the height are clearly attenuated. In a certain sense, this
behavior is the one should expect when the external decoherence is present; a
gradual and continuous process. This work stresses thus new insights in the
Milburn dynamics compared to the von Neumann dynamics of the density operator.

###### Acknowledgements.

SVM acknowledges support from the University of Qom. SMA would like to thank
support from the Fundación Humanismo y Ciencia.

Data availability: This manuscript has no associated data.

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