# Focused Ion Beam fabrication of Janus bimetallic cylinders acting as drift
tube Zernike phase plates

###### Abstract

Modern nanotechnology techniques offer new opportunities for fabricating
structures and devices at the micron and sub-micron level. Here, we use
focused ion beam techniques to realize drift tube Zernike phase plates for
electrons, whose operation is based on the presence of contact potentials in
Janus bimetallic cylinders, in a similar manner to the electrostatic Aharonov-
Bohm effect in bimetallic wires. We use electron Fraunhofer interference to
demonstrate that such bimetallic pillar structures introduce phase shifts that
can be tuned to desired values by varying their dimensions, in particular
their heights.

Paolo Rosi1,Gian Carlo Gazzadi2,Stefano Frabboni1,2, Vincenzo Grillo2,Amir H.
Tavabi3, Rafal E. Dunin-Borkowski3, Giulio Pozzi3,4

1 Department FIM, University of Modena and Reggio Emilia, via G. Campi 213/a,
Modena, 41125, Italy
2 Department of Physics, University of Ottawa, 25 Templeton Street, Ottawa,
Ontario K1N 6N5, Canada
3 Ernst Ruska-Centre for Microscopy and Spectroscopy with Electrons and Peter
Grünberg Institute, Forschungzentrum Jülich, 52425 Jülich, Germany
4 Department of Physics and Astronomy, University of Bologna, viale B. Pichat
6/2, 40127 Bologna, Italy

Developments in state-of-the-art electron microscopes (including aberration
correctors, field emission guns and single particle detectors), in the
fabrication of sub-micron devices (using focused ion beam instruments,
electron beam and optical lithography, _etc._) and in biological specimen
preparation (_e.g._ , the use of frozen-hydrated specimens in cryo-electron
microscopy) have stimulated renewed interest in using phase plates as devices
that can be used to apply phase shifts to scattered electrons with respect to
unscattered beams, in order to improve contrast in images of weak phase
objects (including unstained biological molecules) more efficiently than using
standard methods, such as defocusing the objective lens.

A survey of the history of phase plates has been presented by Nagayama, [1]
while the state of the art in the field has been reported by Glaeser,[2] who
summarised the strengths of each device, highlighted remaining problems and
presented future perspectives. In his analysis, he listed electrostatic drift
tubes, _i.e._ , electrically biased drift tubes surrounded by grounded guard
electrodes,[3] as a promising approach. One of the problems with fabricating
such devices is the need to connect and control the electrical bias to each
nanoscale electrode, with limited space available for cabling. Here, we show
how similar devices can be realized by making use of contact potential
differences between metals to generate potential differences. This concept
loses the advantage of tuneability, but is simpler in realization and offers
prospects for further reductions in device dimensions. A desired constant
phase shift can be chosen by selecting the dimensions of such a device, in
particular its height. A similar use of contact potentials has been reported
by two groups[4, 5] for a two-metal Einzel lens configuration (Boersch phase
plate),[2] which has a different configuration from the present “drift tube”
concept.

The working principle of the proposed device can be understood within the
framework of the Aharonov-Bohm effect [6, 7, 8] in its electrostatic
version,[9, 10] as demonstrated previously for a bimetallic wire.[11, 12, 13,
14] It is straightforward to demonstrate the equivalence of the electron
optical phase shifts produced by magnetic and electrostatic dipoles that are
rotated with respect to each other by $90^{\circ}$. [15] For a toroidal
instead of a linear geometry, a circular distribution of elementary magnetic
dipoles, which is shown in discrete form in Fig. 1(a), is equivalent to a
circular distribution of electrostatic dipoles, as shown in Fig. 1(b). Each
arrangement produces a constant phase shift between an electron beam passing
inside and outside the toroid. On a macroscopic level, this equivalence is
retained between a toroidal magnet, as shown in Fig. 1(c), and a cylindrical
distribution of charge that has a radial electric dipole moment, such as the
drift tube shown in Fig. 1(d).

In order to test this proposal, we fabricated Janus bimetallic cylinders and
measured their phase shifts with respect to nearby holes of nearly equal
diameter. We recorded Fraunhofer images of each pair of holes, from which the
phase shift of the drift tube with respect to the nearby hole could be
measured in the form of a lateral displacement of the interference fringes
with respect to the diffraction envelope.[16] This effect was enhanced by
optimizing the radii of the holes and their separations. [17, 15]

A rough estimate of the phase difference can be obtained by assuming that the
potential inside a cylinder of height $h$ is constant and equal to the contact
potential difference $V_{C}$ between the two dissimilar metals, in our case Pt
and Al, which is $\sim 1V$.[18, 19] The corresponding phase difference is
given by the expression [20, 21, 15]

$\phi=C_{E}V_{C}h~{},$ (1)

where $C_{E}$ is an interaction constant that takes a value of $7.3{\rm
rad}/({\rm V\mu m})$ for $\rm 200~{}keV$ electrons. The expected phase shift
is $2\pi$ for a tube length $h\sim 0.86~{}\mu$m.

Figure 1: Analogies between arrangements of magnetic (left) and electric
(right) dipoles. (a) Toroidal magnetic flux tube and (b) toroidal ring of
electrostatic dipoles, represented by discrete sets of dipoles. (c) Toroidal
magnet and (d) drift tube formed from coaxial cylinders of opposite charge.

We used focused ion beam (FIB) techniques to realize a toroidal version of a
bimetallic wire. The starting point was a commercial silicon nitride membrane
of thickness $200~{}\rm nm$, with nine $100\times 100~{}\rm\mu m$ windows,
onto both sides of which $\sim 50~{}\rm nm$ of Al was evaporated. Pairs of
circular holes, with a pillar on one hole in each pair, were fabricated in
four steps to realize the cross-sectional structures shown in Fig. 2. First,
Pt-C cylinders were deposited by FIB-induced deposition (FIBID) using a Pt-
organic precursor gas. For each cylinder, the ion beam was scanned during gas
injection over a ring-shaped area with an inner radius of $1.15~{}\rm\mu m$
and an outer radius of $1.5~{}\rm\mu m$. Five pillars of varying height were
deposited. The total scan time was $\sim 1$ minute for the shortest pillar and
$\sim 9$ minutes for the tallest pillar (Fig. 3.d). A distance of at least 35
$\rm\mu m$ was kept between pillars and no more than three pillars were
deposited on each window. In the second step, a bimetallic structure was
created by depositing $\sim 100~{}\rm nm$ of Al over the entire sample using
thermal evaporation. In the third step, a further $\sim 150~{}\rm nm$ of Au
was deposited onto the “lower” side of the membrane using sputter coating, in
order to make the support opaque to electrons. In the final step, a double
aperture was formed by boring pairs of holes with radii of $1.2~{}\rm\mu m$
using FIB milling, one through each tube and one adjacent to it. The distances
between the centers of adjacent holes was approximately twice the diameter of
each hole.

Figure 2: Schematic diagram of the cross-sectional structures of the cylinders
and adjacent holes.

Figure 3 shows scanning electron microscopy (SEM) images recorded in tilted
and top view configurations. A reference structure of two holes without a
fabricated pillar structure is shown in Fig. 3(a). The tilted views shown in
Figs 3(b-d) indicate that the pillar heights are 0.7, 2.2 and 3.3 $\mu$m,
respectively.

Figure 3:  SEM tilted and top views of the double circular aperture
structures. One aperture in each pair takes the form of a FIBID-fabricated Al-
covered Pt-C tube. Several representative values of tube height are shown. The
scale bar in each image is $2~{}\rm\mu m$. The measured pillar heights are (a)
0 (_i.e._ , no pillar structure), (b) 0.7, (c) 2.2 and (d) 3.3 $\mu$m.

Experiments were carried out at 200 keV on a Talos X TEM in low magnification,
large angle diffraction mode (camera length 1.4 km). Results obtained from the
structures shown in Fig. 3 are reported in Fig. 4.

Figure 4: TEM images and diffraction patterns of the double apertures shown in
Fig. 3. Line scans of the diffraction patterns are shown, with the markers
representing experimental data points and solid black lines fits. The
direction of the line scan is marked by an arrow in each diffraction pattern.
The phase differences (modulo $2\pi$) obtained from fits to Eq. 2 are
indicated in the top right corner of each line scan.

Fraunhofer images corresponding to the zero height pillar (Fig. 4 (a)) and the
0.7 $\mu$m pillar (Fig. 4 (b)) are found to be symmetrical, with a central
maximum with respect to the diffraction envelope, indicating that their phase
difference is a multiple of 2$\pi$. In contrast, the results shown in Fig. 4
(c) exhibit a marked asymmetry, suggesting a detectable phase difference,
while the results shown in Fig. 4 (d) have a central minimum instead of a
central maximum, indicating that the phase difference is approximately $\pi$.
Values of the phase difference were measured by performing fits to the data
based on the formula

$I(x)=b\frac{J_{1}^{2}(xD)}{(xD)^{2}}\left[1+\mu
cos\left(xd+\Delta\varphi\right)\right]~{},$ (2)

which was derived by considering the Fraunhofer images of two circular
apertures. The multiplicative factor is the Fourier transform of a circular
aperture (_i.e._ , an Airy disk), while the interference term in square
brackets includes a damping factor $\mu$, which accounts for partial coherence
effects resulting from the finite dimensions of the electron source (_i.e._ ,
spatial coherence).[17] $J_{1}$ is a Bessel function of the first kind, $b$ is
a fitting parameter that allows a non-normalized intensity profile to be
fitted, $\Delta\varphi$ is the phase difference between the apertures and $D$
and $d$ are coefficients that are proportional to the aperture diameter and
separation, respectively. As a result of the high coherence of the field
emission gun, the influence of spatial coherence was found to be less than
$10\%$ of the overall intensity.

The fitted values of the phase difference, which are indicated in Fig. 4, are
consistent with the generation of constant phase shifts by Janus bimetallic
cylindrical structures that behave as drift tubes. More systematic experiments
will be required in the future to assess the influence on the phase shift of
experimental parameters associated with the fabrication process, as well as
possible effects of electron beam induced charging of both the structures and
their supports. When better control of the phase shift is achieved, it may be
possible to use such structures as Zernike phase plates in transmission
electron microscopy, as they are more compact than tuneable drift tube devices
and therefore promise to allow a greater number of spatial frequencies to
contribute to phase contrast images.

Beyond their use as Zernike phase plates, we foresee the application of such
bimetallic devices for different forms of electron beam shaping. For example,
they could be used to replace electrostatic elements in applications such as
spiral phase plates,[22, 23, 24, 25, 26] Hilbert phase plates and
electrostatic orbital angular momentum sorters.[27, 28] They could also be
used to replace electrostatic phase plates for conformal mapping
operations.[29] A two-dimensional array of such cylinders could be used to
produce an arbitrary phase landscape without the problems that come with using
material-based holograms. A similar concept was recently proposed for
pixelated programmable phase plates.[30, 31] When using bimetallic cylinders,
the loss of _in situ_ programmability would be compensated by the ability to
fabricate a greater number of pixels per unit of area.

## Acknowledgments

PR and SF aknowledge the staff at the CIGS facility in Modena for helping in
setting up the experiments. The research leading to these results has received
funding from the European Union’s Horizon 2020 Research and Innovation
Programme under Grant Agreements: 1. Grant No 766970 Q-SORT
(H2020-FETOPEN-1-2016-2017); 2. Grant No. 823717 (project ESTEEM3).

## Data Availability Statement

The data that support the findings of this study are available from the
corresponding author upon reasonable request.

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eqn

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# Quantum gravitational decoherence from Generalized Uncertainty Principle
with stochastic deformation parameter

Eissa Al-Nasrallah<EMAIL_ADDRESS>Department of Physics,
Kuwait University, P.O. Box 5969, Safat 13060, Kuwait  Energy and Building
Research Center, KISR, P.O. Box 24885, Safat 13109, Kuwait  Saurya Das
<EMAIL_ADDRESS>Theoretical Physics Group and Quantum Alberta, Department
of Physics and Astronomy, University of Lethbridge, 4401 University Drive,
Lethbridge, Alberta, T1K 3M4, Canada  Fabrizio Illuminati
<EMAIL_ADDRESS>Dipartimento di Ingegneria Industriale, Universita degli
Studi di Salerno, Via Giovanni Paolo II, 132 - 84084 Fisciano, Salerno, Italy
INFN, Sezione di Napoli, Gruppo collegato di Salerno, Italy  Luciano
Petruzziello<EMAIL_ADDRESS>Dipartimento di Ingegneria Industriale,
Universita degli Studi di Salerno, Via Giovanni Paolo II, 132 - 84084
Fisciano, Salerno, Italy
INFN, Sezione di Napoli, Gruppo collegato di Salerno, Italy  Elias C. Vagenas
<EMAIL_ADDRESS>Theoretical Physics Group, Department of Physics,
Kuwait University, P.O. Box 5969, Safat 13060, Kuwait

###### Abstract

Several theories of quantum gravity propose the existence of a minimal
measurable length and maximum measurable momentum near the Planck scale. When
integrated into the framework of quantum mechanics, such restrictions lead to
the generalization of the Heisenberg uncertainty principle, which is commonly
referred to as the generalized uncertainty principle (GUP). The GUP has been
applied to a plethora of physical models in order to provide further insights
on our understanding of the Universe, both in the macroscopic and the
microscopic regime. Nevertheless, the corrections related to GUP are expected
to be extremely small, which thus renders experimental verification a
difficult task. Therefore, phenomenological approaches on a qualitative level
are typically addressed with the aim of enhancing our possibility of detecting
these tiny effects. Motivated by a recent work on a universal decoherence
mechanism stemming from the standard formulation of GUP, we extend the ideas
contained therein by considering a more general form of GUP which includes
linear and quadratic momentum terms to examine its implications. To achieve
such a picture, we equip the deformation parameter appearing in front of all
momentum terms with a stochastic nature, so that the effective decoherence
mechanism emerges by taking the average over fluctuations. We find that,
despite the apparently small differences among the two generalizations of the
Heisenberg uncertainty principle, the consequences at the level of the
universal decoherence mechanism they entail is significant, as they predict
decoherence times that are completely uncorrelated and distinct.

##  I Introduction

Quantum decoherence has been an active area of research for more than 50
years. It was originally proposed by Zeh as an explanation for the quantum-to-
classical transition Zeh:1970zz . The conundrum of quantum systems behaving
classically at the macroscopic scale has been a source of various
interpretations and intense discussions. The phenomenon of decoherence
attempts to solve this issue by taking into account the unavoidable
entanglement shared between any quantum system and its environment. A similar
entanglement affects the set of possible observable states of the system in a
measurement and in turn gives rise to a net loss of quantum information. In
other words, the system tends to lose all the characteristic quantum features
(such as for instance the superposition of states which are predicted quantum
mechanically and yet unobserved on a macroscopic scale, and in some
circumstances even prohibited Zurek:2003zz ; Schlosshauer:2019ewh ;
Schlosshauer:2003zy ; Hornberger ; Anastopoulos:2000hg ).
In the quest for seeking a faithful and rigorous representation of natural
phenomena, it has been observed that our most successful theories, namely
quantum field theory (QFT) and general relativity, encounter non-trivial
adversities that prevent them from providing an accurate description of
reality when approaching the Planck scale. This stimulated the search for a
unifying _theory of everything_ which has introduced many valuable candidates;
among these, it is worth mentioning String Theory, Loop Quantum Gravity,
Causal Dynamical Triangulations, Modified Dispersion Relations (MDR), and
Doubly Special Relativity (DSR). All these models predict a minimum measurable
length which is typically proportional to the Planck length
$\mathcal{O}(\ell_{p})$, where $\ell_{p}\sim 10^{-35}$ m. Such a constraint on
the minimum measurable length necessarily requires a modification of the
Heisenberg Uncertainty Principle (HUP), which goes by the name of Generalized
Uncertainty Principle (GUP) (see Refs. Kempf:1994su ; noncom2 ; noncom3 ; bis
; quar ; ses ; ong ; set ; ot ; dieci ; qft ; qft4 ; plenio2 ; Das:2009hs ;
Das:2010sj  and therein for a comprehensive overview of the subject).
Understanding decoherence in the framework of quantum gravity is thought to
bring an insight on the nature of quantum gravitational interactions and
provide possibilities for their experimental tests. Decoherence stemming from
general relativity and different models of quantum gravity has been thoroughly
treated in the recent literature (see for instance Refs. Asprea:2019dok ;
Allali:2021puy ; Anastopoulos:2013zya ; Bonder:2015hja ; Miki:2020hvg ;
Podolskiy:2015wna ; Pang:2016foq ; ellis ; milburn ; bargueno  and therein).
Among these proposals, two of the present authors have investigated the
consequences of deformed canonical commutation relation (DCCR) inspired by
quantum gravitational theories to derive a Lindblad-type master equation valid
for a generic quantum system and obtain the ensuing decoherence time
associated with the DCCR Petruzziello:2020wkd ; Kempf:1994su ; Kempf:1996fz ;
Das:2017qwp

$[x_{i},p_{j}]=i\hbar\left[\delta_{ij}+\beta\delta_{ij}p^{2}+2\beta
p_{i}p_{j}\right],\ \ \beta=\beta_{0}\frac{\ell_{p}^{2}}{\hbar^{2}},~{}$ (2)

which leads to an uncertainty relation of the form

$\Delta x\Delta p\geq\frac{\hbar}{2}\left(1+\beta\Delta p^{2}\right)~{}.$ (3)

In this paper, we extend the above considerations by addressing a more general
form of GUP that includes both linear and quadratic momentum terms known as
LQGUP. Such a generalization has been analyzed in different contexts (see
Refs. Vagenas:2019wzd ; Das:2008kaa ; Das:2020ujn ; Ali:2010yn ; Ali:2009zq ;
Das:2020ujn ; Ali:2010yn ) and it is compatible with DSR. Specifically, it
takes the form

$\displaystyle\Delta x\Delta p$
$\displaystyle\geq\frac{\hbar}{2}\left[1-2\alpha\langle
p\rangle+4\alpha^{2}\langle p^{2}\rangle\right]$ (4)
$\displaystyle\geq\frac{\hbar}{2}\left[1+\left(\frac{\alpha}{\sqrt{\langle
p^{2}\rangle}}+4\alpha^{2}\right)\Delta p^{2}+4\alpha^{2}\langle
p\rangle^{2}-2\alpha\sqrt{\langle p^{2}\rangle}\right]~{}.$

In this case, the DCCR becomes

$[x_{i},p_{j}]=i\hbar\left[\delta_{ij}-\alpha\left(p\delta_{ij}+\frac{p_{i}p_{j}}{p}\right)+\alpha^{2}(p^{2}\delta_{ij}+3p_{i}p_{j})\right]~{},$
(5)

where $x_{i}$ and $p_{i}$ are defined as

$x_{i}=x_{0i},\ p_{i}=p_{0i}(1-\alpha p_{0}+2\alpha^{2}p_{0}^{2})~{},\\\ $ (6)

with

$[x_{0i},p_{0j}]=i\hbar\delta_{ij},$ (7)

and $[x_{i},x_{j}]=0,[p_{i},p_{j}]=0$,
$p^{2}=\sum_{j=1}^{3}p_{j}p_{j},p_{0}^{2}=\sum_{j=1}^{3}p_{0j}p_{0j}$. The
parameter $\alpha$ is defined as
$\alpha=\frac{\alpha_{0}}{M_{P}c}=\frac{\alpha_{0}\ell_{p}}{\hbar}$ where
$\alpha_{0}$ is often assumed to be of order unity Ali:2010yn .
The paper is structured as follows: in Section II we derive the Lindblad-type
master equation in the context of LQGUP, whereas in Section III we obtain the
entropy variation and the decoherence time due to the LQGUP-modified Lindblad
master equation. Finally, in Section IV we summarize our results and provide
final remarks.

##  II LQGUP-induced master equation

Following the analysis of Ref. Petruzziello:2020wkd , we derive the Lindblad-
type master equation in the context of LQGUP. First, we note that we can start
without loss of generality from the canonical Schrödinger equation

$i\hbar\partial_{t}|\psi\rangle=H|\psi\rangle=(\frac{p^{2}}{2m}+V)|\psi\rangle$
(8)

and use the definition of the GUP-modified momentum as given by Eq. (6) so as
to get

$i\hbar\partial_{t}|\psi\rangle=\left(p_{0}^{2}\frac{\left(1-\alpha
p_{0}+2\alpha^{2}p_{0}^{2}\right)^{2}}{2m}+V\right)|\psi\rangle=\left(p_{0}^{2}\frac{\left(1-2\alpha
p_{0}+5\alpha^{2}p_{0}^{2}-4\alpha^{3}p_{0}^{3}+4\alpha^{4}p_{0}^{4}\right)}{2m}+V\right)|\psi\rangle$
(9)

and thus

$i\hbar\partial_{t}|\psi\rangle=\left(\frac{p_{0}^{2}}{2m}+V-\alpha\frac{p_{0}^{3}}{m}+5\alpha^{2}\frac{p_{0}^{4}}{2m}-2\alpha^{3}\frac{p_{0}^{5}}{m}+2\alpha^{4}\frac{p_{0}^{6}}{m}\right)|\psi\rangle~{}.$
(10)

We observe that several orders of the deformation parameter appear in the
perturbation of the standard Hamiltonian. For our purposes, we will consider a
first-order treatment of the factor $\alpha$. However, for the sake of
completeness, we show in Appendix A that considerations involving second-order
corrections produce the same outcome.
Hence, if we account for a first-order treatment only of the GUP parameter
$\alpha$ and ignore factors that go like $\mathcal{O}(\alpha^{2})$, then the
Schrödinger equation is given by

$i\hbar\partial_{t}|\psi\rangle=H|\psi\rangle=\left(H_{0}+H_{1}+\mathcal{O}(\alpha^{2})\right)|\psi\rangle~{},$
(11)

where

$H_{0}=\frac{p_{0}^{2}}{2m}+V,\ \ H_{1}=-2\alpha\frac{p_{0}^{3}}{2m}~{}.$ (12)

Now, by introducing the density matrix

$\varrho=|\psi\rangle\langle\psi|~{},$ (13)

we can cast the Liouville-von Neumann equation as follows

$\partial_{t}\varrho=-\frac{i}{\hbar}[H_{0}+H_{1},\varrho]~{}.$ (14)

Next, to solve the equation for the density matrix we shift our analysis to
the interaction picture, where the state takes the form

$|\psi\rangle=e^{-\frac{iH_{0}t}{\hbar}}|\tilde{\psi}\rangle~{}.$ (15)

According to this representation, we can rewrite Schrödinger equation as

$i\hbar\partial_{t}|\tilde{\psi}\rangle=\tilde{H_{1}}|\tilde{\psi}\rangle\
\text{where}\ \
\tilde{H_{1}}=e^{\frac{iH_{0}t}{\hbar}}H_{1}e^{-\frac{iH_{0}t}{\hbar}}|\tilde{\psi}\rangle~{},$
(16)

and substituting back in the Liouville-von Neumann equation, i.e., Eq. (14),
we get

$\partial_{t}\tilde{\varrho}(t)=-\frac{i}{\hbar}[\tilde{H_{1}}(t),\tilde{\varrho}(t)]~{}.$
(17)

A formal solution of the above equation can be achieved by integrating both
sides over the interval $[0,t]$, which thus yields

$\int_{0}^{t}\partial_{t}\tilde{\varrho}(t^{\prime})dt^{\prime}=\int_{0}^{t}-\frac{i}{\hbar}[\tilde{H_{1}}(t^{\prime}),\tilde{\varrho}(t^{\prime})]dt^{\prime}$
(18)

and results in

$\tilde{\varrho}(t)=\tilde{\varrho}(0)-\frac{i}{\hbar}\int_{0}^{t}[\tilde{H_{1}}(t^{\prime}),\tilde{\varrho}(t^{\prime})]dt^{\prime}.$
(19)

Subsequently, Eq. (19) is inserted into the r.h.s. of Eq. (17), thereby giving

$\partial_{t}\tilde{\varrho}(t)=-\frac{i}{\hbar}[\tilde{H_{1}}(t),\tilde{\varrho}(0)]-\frac{1}{\hbar^{2}}\int_{0}^{t}[\tilde{H_{1}}(t),[\tilde{H_{1}}(t^{\prime}),\tilde{\varrho}(t^{\prime})]]dt^{\prime}~{}.$
(20)

The next step consists in relying on the Born-Markov approximation to let the
density matrix of the r.h.s. of the above equation depend on $t$ rather than
$t^{\prime}$, i.e.,

$\partial_{t}\tilde{\varrho}(t)=-\frac{i}{\hbar}[\tilde{H_{1}}(t),\tilde{\varrho}(0)]-\frac{1}{\hbar^{2}}\int_{0}^{t}[\tilde{H_{1}}(t),[\tilde{H_{1}}(t^{\prime}),\tilde{\varrho}(t)]]dt^{\prime}.$
(21)

This is a common procedure in the derivation of Lindblad-type master equations
for weak interactions between the system and reservoir Petruzziello:2020wkd ;
Xu:2020pzr ; Kolovsky_2020 . However, we note that it has been argued that
this approximation may not be valid for certain regimes, such as composite
systems in strong coupling regime Nakatani2010  and low temperature systems
Vadimov2021 .
At this point, we observe that the GUP based on theories such as Loop Quantum
Gravity and DSR should contain the information of space-time fluctuations near
the Planck scale, which is a typical by-product of the aforementioned quantum
gravity models. To comply with the existence of a fluctuating minimum length
scale, one can think of encoding a similar feature in the present analysis by
demanding a fluctuating deformation parameter, which requires a stochastic
treatment of the problem. By averaging over the fluctuations of the GUP
parameter $\alpha$ we can allow for the development of the master equation
from the mean stochastic Liouville-von Neumann equation. Therefore, we have to
regard $\alpha$ as a random variable; specifically, for the sake of simplicity
we can interpret it as being a Gaussian white noise with fixed mean and sharp
auto-correlation. Inspired by the considerations contained in Ref.
Petruzziello:2020wkd , we can then impose that the dimensionless GUP parameter
$\alpha_{0}$ is given by

$\alpha_{0}=\sqrt{t_{p}}\chi(t),\ \ \langle\chi(t)\rangle=\bar{\alpha}_{0},\ \
\langle\chi(t)\chi(t^{\prime})\rangle=\delta(t-t^{\prime})~{},$ (22)

with $t_{p}$ being the Planck time. In order to be in accordance with the
standard literature, we can assume $\alpha_{0}$ to be of order unity on
average. For this purpose, if we average over its fluctuations we can take
$\langle\alpha_{0}\rangle=\sqrt{t_{p}}\langle\chi(t)\rangle=1$ and
consequently define the fixed mean as $\bar{\alpha}_{0}={1}/{\sqrt{t_{p}}}$.
Alternatively, we can follow a more conservative approach and impose the mean
value of $\alpha_{0}$ to not exceed the experimental bound available for the
deformation parameter, that is $\langle\alpha_{0}\rangle\simeq 10^{15}$. In
any case, we will show that the arbitrary choice made to fix
$\bar{\alpha}_{0}$ will not play a relevant role in the upcoming
investigation, as the important quantity is represented by the auto-
correlation and not the mean value.
Now, by averaging over the fluctuations and introducing the shorthand notation
$\langle\varrho\rangle=\rho$, the derivative will take the form

$\partial_{t}\langle\tilde{\varrho}\rangle=\partial_{t}\tilde{\rho}(t),$ (23)

and Eq. (21) can be rewritten as

$\partial_{t}\tilde{\rho}(t)=\left\langle-\frac{i}{\hbar}\left[\tilde{H}_{1}(t),\tilde{\varrho}(0)\right]\right\rangle-\frac{1}{\hbar^{2}}\int_{0}^{t}\left\langle\left[\tilde{H}_{1}(t),\left[\tilde{H}_{1}(t^{\prime}),\tilde{\varrho}(t)\right]\right]\right\rangle
dt^{\prime}.$ (24)

The first term on the r.h.s. of Eq. (24) vanishes since $\tilde{\varrho}(0)$
is constant. For the second term, in order to further manipulate the factor
inside the integral, we recall the definition of $H_{1}$ in Eq. (12) to
substitute it and obtain

$\partial_{t}\tilde{\rho}(t)=-\frac{4}{\hbar^{2}}\int_{0}^{t}\left\langle\alpha(t)\alpha(t^{\prime})\right\rangle\left[\frac{\tilde{p}^{3}_{0}(t)}{2m},\left[\frac{\tilde{p}^{3}_{0}(t^{\prime})}{2m},\tilde{\rho}(t)\right]\right]dt^{\prime}.$
(25)

In terms of $\alpha={\alpha_{0}\ell_{p}}/{\hbar}$, we rewrite the above
equation as

$\partial_{t}\tilde{\rho}(t)=-\frac{4\ell_{p}^{2}}{\hbar^{4}}\int_{0}^{t}\left\langle\alpha_{0}(t)\alpha_{0}(t^{\prime})\right\rangle\left[\frac{\tilde{p}^{3}_{0}(t)}{2m},\left[\frac{\tilde{p}^{3}_{0}(t^{\prime})}{2m},\tilde{\rho}(t)\right]\right]dt^{\prime}~{}.$
(26)

Now since $\alpha_{0}$ is a Gaussian white noise as explained in Eq. (22), the
integral can be easily evaluated (See Appendix A), thereby yielding

$\partial_{t}\tilde{\rho}(t)=-\sigma\left[\frac{\tilde{p}^{3}_{0}(t)}{2m},\left[\frac{\tilde{p}^{3}_{0}(t)}{2m},\tilde{\rho}(t)\right]\right],$
(27)

where $\sigma={4t_{p}\ell_{p}^{2}}/{\hbar^{4}}$.

At this point, we can recover the density matrix in the Schrödinger
representation by means of the following transformation:

$\rho(t)=e^{-\frac{iH_{0}t}{\hbar}}\tilde{\rho}(t)e^{\frac{iH_{0}t}{\hbar}}~{},$
(28)

so that we can go back to the Liouville-Von Neumann equation given by Eq. (14)
with the dissipator term that depends on the characteristic features of the
stochastic LQGUP model treated so far. In particular, by requiring the system
to not be subject to any external potential, i.e., $V=0$, we are left with

$\partial_{t}\rho(t)=-\frac{i}{\hbar}\left[\frac{p_{0}^{2}}{2m},\rho(t)\right]-\sigma\left[\frac{p^{3}_{0}}{2m},\left[\frac{p^{3}_{0}}{2m},\rho(t)\right]\right]~{}.$
(29)

By employing the definitions in Eq. (12), the above equation can be rewritten
in terms of Hamiltonians as

$\partial_{t}\rho(t)=-\frac{i}{\hbar}\left[H_{0},\rho(t)\right]-2m\sigma\left[H^{3/2}_{0},\left[H^{3/2}_{0},\rho(t)\right]\right]~{}.$
(30)

Equations (29) and (30) represent the LQGUP-modified Lindblad-type master
equation, where the dissipator is identified with the second term of the
r.h.s.. Clearly, in the limit $\sigma\approx 0$ we recover the standard
unitary dynamics predicted by the unmodified Liouville-von Neumann equation.
Furthermore, it is worth remarking that the power of $H_{0}$ appearing in Eq.
(30) is different from the usual one that appears in other gravitational
decoherence mechanisms, according to which the dependence on the free
Hamiltonian is typically linear ellis ; milburn . This unconventional behavior
is shared also with the other work dealing with a GUP-induced decoherence
process Petruzziello:2020wkd , but interestingly the power of $H_{0}$ is
different in the two scenarios, thus suggesting that the intrinsic
characteristics of the emergent decoherence phenomenon strictly depend on the
GUP model used for computations.

##  III Application and Physical Results

In this Section, we look at the temporal evolution of an entropy quantifier,
i.e., linear entropy (for which calculations can be carried out analytically),
to analyze the irreversibility of the decoherence mechanism and provide a
solution for the differential equation Eq. (30). By means of this study, we
are able to derive the decoherence time associated with the LQGUP, which
foresees energy localization in momentum space as it occurs also in Ref.
Petruzziello:2020wkd .

###  III.1 Entropy Variation

Linear entropy corresponds to the mixedness of the quantum state and is
therefore associated with the quantum state purity. It is defined as

$S(t)=1-tr\left(\rho^{2}(t)\right),$ (31)

and we recall that a given state is pure if and only if the correlated density
matrix is idempotent, namely $\rho^{2}=\rho$ . Now, the time evolution of the
linear entropy is obtained by differentiating Eq. (31), that is

$\partial_{t}S(t)=\partial_{t}(1-tr\left(\rho^{2}(t)\right))=-2tr\left(\rho(t)\partial_{t}\rho(t)\right).$
(32)

At this point, by resorting to the LQGUP-modified Lindblad-type master
equation from the previous Section, namely Eq. (29), we can substitute the
factor $\partial_{t}\rho(t)$ so as to give

$\partial_{t}S=\frac{2i}{\hbar}tr\left(\rho(t)\left[\frac{p_{0}^{2}}{2m},\rho(t)\right]\right)+2\
\sigma\
tr\left(\rho(t)\left[\frac{p^{3}_{0}}{2m},\left[\frac{p^{3}_{0}}{2m},\rho(t)\right]\right]\right).$
(33)

By manipulating the r.h.s of the above equation and using the cyclic property
of the trace, it is possible to obtain that

$\partial_{t}S=2\ \sigma\ \left[2\
tr\left(\rho\frac{p^{6}_{0}}{4m^{2}}\rho\right)-2\
tr\left(\rho\frac{p^{3}_{0}}{2m}\right)^{2}\right].$ (34)

If we now introduce operator $O$ defined as

$O=\left[\frac{p^{3}_{0}}{2m},\rho\right],$ (35)

then it can be shown that

$\left[2\ tr\left(\rho\frac{p^{6}_{0}}{4m^{2}}\rho\right)-2\
tr\left(\left(\rho\frac{p^{3}_{0}}{2m}\right)^{2}\right)\right]=tr\left(O^{{\dagger}}O\right).$
(36)

Next, by going back to Eq. (34), we are left with

$\partial_{t}S=2\ \sigma\ tr\left(O^{{\dagger}}O\right)\geq 0~{},$ (37)

where $\sigma$ and $tr\left(O^{{\dagger}}O\right)$ are positive quantities,
which entails that the linear entropy is monotonically increasing with time
and the initial quantum state tends to increase mixedness. Furthermore, this
achievement clearly conveys the intrinsic irreversible nature of the studied
process, as the entropy can never decrease with time.

###  III.2 Decoherence Time

We are now ready to calculate the decoherence time associated with the LQGUP-
modified Lindblad-type master equation, i.e., Eq. (30). To this aim, we point
out that the computation is easier if performed in the momentum
representation, where the elements of the density matrix are

$\rho_{p,p^{\prime}}(t)=\langle\bm{{\rm p}}|\rho(t)|\bm{{\rm
p}}^{\prime}\rangle~{},$ (38)

and consequently

$\partial_{t}\rho_{p,p^{\prime}}(t)=\partial_{t}\langle\bm{{\rm
p}}|\rho|\bm{{\rm p}}^{\prime}\rangle=\langle\bm{{\rm
p}}|\partial_{t}\rho|\bm{{\rm p}}^{\prime}\rangle~{}.$ (39)

Therefore, by using the master equation, i.e., Eq. (30), we get

$\partial_{t}\rho_{p,p^{\prime}}(t)=-\frac{i}{\hbar}\langle\bm{{\rm
p}}|[{H}_{0},{\rho}]|\bm{{\rm p}}^{\prime}\rangle-2m\sigma\langle\bm{{\rm
p}}|\left[{H}^{3/2}_{0},[{H}^{3/2}_{0},{\rho}]\right]|\bm{{\rm
p}}^{\prime}\rangle~{}.$ (40)

By opening the commutators, one can easily check that

$\displaystyle\partial_{t}\rho_{p,p^{\prime}}(t)$ $\displaystyle=$
$\displaystyle-\frac{i}{\hbar}\left(\langle\bm{{\rm p}}|{H}_{0}{\rho}|\bm{{\rm
p}}^{\prime}\rangle-\langle\bm{{\rm p}}|{\rho}{H}_{0}|\bm{{\rm
p}}^{\prime}\rangle\right)$ (41) $\displaystyle-2m\sigma\left(\langle\bm{{\rm
p}}|{H}^{3}_{0}{\rho}|\bm{{\rm p}}^{\prime}\rangle-\langle\bm{{\rm
p}}|{H}^{3/2}_{0}{\rho}{H}^{3/2}_{0}|\bm{{\rm
p}}^{\prime}\rangle-\langle\bm{{\rm
p}}|{H}^{3/2}_{0}{\rho}{H}^{3/2}_{0}|\bm{{\rm
p}}^{\prime}\rangle+\langle\bm{{\rm p}}|{\rho}{H}^{3}_{0}|\bm{{\rm
p}}^{\prime}\rangle\right).$

In the momentum representation, we know that ${H}(p)|\bm{{\rm
p}}\rangle=E(p)|\bm{{\rm p}}\rangle$. Moreover, by using the fact that the
Hamiltonian is a Hermitian operator, we obtain

$\displaystyle\partial_{t}\rho_{p,p^{\prime}}(t)$ $\displaystyle=$
$\displaystyle-\frac{i}{\hbar}\left(E(p)\langle\bm{{\rm p}}|{\rho}|\bm{{\rm
p}}^{\prime}\rangle-E(p^{\prime})\langle\bm{{\rm p}}|{\rho}|\bm{{\rm
p}}^{\prime}\rangle\right)$ (42)
$\displaystyle-2m\sigma\left(E^{3}(p)\langle\bm{{\rm p}}|{\rho}|\bm{{\rm
p}}^{\prime}\rangle-2E^{3/2}(p)E^{3/2}(p^{\prime})\langle\bm{{\rm
p}}|{\rho}|\bm{{\rm p}}^{\prime}\rangle+E^{3}(p^{\prime})\langle\bm{{\rm
p}}|{\rho}|\bm{{\rm p}}^{\prime}\rangle\right).$

By writing the density matrix elements as in Eq. (38), the previous equation
reads

$\partial_{t}\rho_{p,p^{\prime}}(t)=-\frac{i}{\hbar}\left(E(p)-E(p^{\prime})\right)\rho_{p,p^{\prime}}-2m\sigma\left(E^{3}(p)-2E^{3/2}(p)\
E^{3/2}(p^{\prime})+E^{3}(p^{\prime})\right)\rho_{p,p^{\prime}}$ (43)

and introducing the notation $\Delta E^{3/2}\equiv
E^{3/2}(p)-E^{3/2}(p^{\prime})$, we get

$\partial_{t}\rho_{p,p^{\prime}}=\left[-\frac{i}{\hbar}\left(E(p)-E(p^{\prime})\right)-2m\sigma(\Delta
E^{3/2})^{2}\right]\rho_{p,p^{\prime}}.$ (44)

It is easily seen that Eq. (44) is a separable differential equation which can
be solved to yield

$\rho_{p,p^{\prime}}(t)={\rm
exp}\left[-\frac{i\left(E(p)-E(p^{\prime})\right)t}{\hbar}-2m\sigma\left(\Delta
E^{3/2}\right)^{2}t\right]\rho_{p,p^{\prime}}(0)$ (45)

from which it is possible to deduce the decoherence time as being equal to

$\tau_{D}=\frac{1}{2m\sigma\left(\Delta
E^{3/2}\right)^{2}}=\frac{\hbar^{4}}{8mt_{p}\ell_{p}^{2}\left(\Delta
E^{3/2}\right)^{2}}~{}.$ (46)

As expected, as long as the diagonal part, i.e., $\Delta E^{3/2}=0$, is
concerned, there is no time evolution. However, when considering the off-
diagonal terms (for which $\Delta E^{3/2}\neq 0$), we can recognize an
exponential damping that is a typical signature of the decoherence mechanism
with a characteristic time which precisely equals the decoherence time. In
order to better emphasize the behavior of $\tau_{D}$, in Fig. 1 we plot the
decoherence time as a function of the energy and the mass of a given quantum
system to show that it gets smaller and smaller as the macroscopic scale is
approached.

Figure 1: Plot of $\log_{10}(\tau_{D})$ as a function of mass and energy.

It is noteworthy to mention that in Ref. Petruzziello:2020wkd , the
decoherence time obtained from considerations involving a GUP of the form
shown in Eq. (3) (which is only quadratic in the momentum) has the following
form:

$\tau_{D}=\frac{\hbar^{6}}{16m^{2}t_{p}\ell_{p}^{4}\left(\Delta
E^{2}\right)^{2}},\qquad\Delta E^{2}\equiv E^{2}(p)-E^{2}(p^{\prime}).$ (47)

In Fig. 2, we exhibit the comparison between the two forms of the decoherence
times. As the picture clearly conveys, the decoherence time obtained from
LQGUP arguments is shorter than the one obtained in Ref. Petruzziello:2020wkd
in the mesoscopic regime below the Planck size but it is longer beyond the
same scale.

Figure 2: Comparison between decoherence times as a function of energy derived
from LQGUP with both linear and quadratic momentum terms (present paper, solid
line) and from GUP with only quadratic momentum terms (Ref.
Petruzziello:2020wkd , dashed line)

##  IV Conclusions

In this paper, we have shown that decoherence phenomena arising from a GUP
with linear and quadratic terms in momentum can be deemed as a viable
candidate for the quantum-to-classical transition. This can be achieved by
assuming a stochastic nature for the GUP parameter $\alpha$; indeed, we have
regarded it as a Gaussian white noise with constant mean and sharp auto-
correlation. The justification of the above requirement can be found in the
assumption of a fluctuating minimal length, which in turn is legitimated by
space-time fluctuations that led to the stochastic treatment of $\alpha$. This
very consideration has allowed us to derive a Lindblad-type master equation,
from which we have been able to deduce several physical implications of the
GUP-induced gravitational decoherence mechanism, such as the entropy variation
and decoherence time. Specifically, we have seen how the linear entropy
monotonically increases with time, thus leading to a continuous increase of
the state mixedness and to an irreversible evolution process. On the other
hand, the decoherence time was shown to decrease for increasing energy and
mass of the system. This occurs at a rate faster than the decoherence due to
GUP without linear momentum term (as derived in Ref. Petruzziello:2020wkd )
for mesoscopic regimes beyond the Planck size, whilst the same trend is
inverted below the Planck scale. Such an awareness entails that there is a
strong bond between the form of GUP and the ensuing decoherence time.

As a matter of fact, the above deep connection can be employed to model
phenomenologically meaningful uncertainty relations, capable of explaining the
quantum-to-classical transition and quantum gravitational effects. Since the
decoherence mechanism stemming from the GUP strictly depends on the deviation
from the HUP, the next future high-precision laboratory experiments may be
able to progress and provide valuable pieces of evidence that are essential
for the construction of a consistent and detectable quantum gravity model.
Even if such experiments are not attained, we could potentially set strong
bounds on the plethora of generalized uncertainty principles available at
present, so as to eventually reach the purported correct description of
natural manifestations occurring at scales which are still inaccessible via
direct experimental probes. Clearly, more work is inevitably required along
this line and is currently being undertaken.

##  V Acknowledgement

This work was supported by the Natural Sciences and Engineering Research
Council of Canada. SD and ECV would like to acknowledge networking support by
the COST Action CA18108.

##  Appendix A Second-order perturbation of $\alpha$ in the Hamiltonian

In this Appendix, we show how the main outcome of the current work is left
untouched even if we push our analysis beyond the first-order approximation.
Indeed, if we consider second-order approximations in $\alpha$ for $H_{1}$ in
the Hamiltonian of Eq. (10), we get

$i\hbar\partial_{t}|\psi\rangle=H|\psi\rangle=\left(H_{0}+H_{1}+\mathcal{O}(\alpha^{3})\right)|\psi\rangle,$
(48)

with

$H_{0}=\frac{p^{2}_{0}}{2m}+V,\ \
H_{1}=-\frac{\alpha}{m}p_{0}^{3}+\frac{5\alpha^{2}}{2m}p_{0}^{4}~{},$ (49)

where we ignore terms that go like $\mathcal{O}(\alpha^{3})$.

We note that the addition of the second order $\alpha$ term will begin to have
a significant effect only in correspondence of Eq. (24), which can be written
as

$\partial_{t}\tilde{\rho}(t)=-\frac{1}{\hbar^{2}}\int_{0}^{t}\left\langle\tilde{H}_{1}(t)\tilde{H}_{1}(t^{\prime})\tilde{\varrho}(t)-\tilde{H}_{1}(t)\tilde{\varrho}(t)\tilde{H}_{1}(t^{\prime})-\tilde{H}_{1}(t^{\prime})\tilde{\varrho}(t)\tilde{H}_{1}(t)+\tilde{\varrho}(t)\tilde{H}_{1}(t^{\prime})\tilde{H}_{1}(t)\right\rangle
dt^{\prime}.$ (50)

In this case, if we substitute the new value for $H_{1}$ we obtain

$\displaystyle\partial_{t}\tilde{\rho}(t)$ $\displaystyle=$
$\displaystyle-\frac{1}{\hbar^{2}}\int_{0}^{t}\bigg{\langle}\left(\frac{\alpha(t)}{m}\tilde{p}_{0}^{3}(t)+\frac{5\alpha^{2}(t)}{2m}\tilde{p}_{0}^{4}(t)\right)\left(\frac{\alpha(t^{\prime})}{m}\tilde{p}_{0}^{3}(t^{\prime})+\frac{5\alpha^{2}(t^{\prime})}{2m}\tilde{p}_{0}^{4}(t^{\prime})\right)\tilde{\varrho}(t)$
(51)
$\displaystyle-\left(\frac{\alpha(t)}{m}\tilde{p}_{0}^{3}(t)+\frac{5\alpha^{2}(t)}{2m}\tilde{p}_{0}^{4}(t)\right)\tilde{\varrho}(t)\left(\frac{\alpha(t^{\prime})}{m}\tilde{p}_{0}^{3}(t^{\prime})+\frac{5\alpha^{2}(t^{\prime})}{2m}\tilde{p}_{0}^{4}(t^{\prime})\right)$
$\displaystyle-\left(\frac{\alpha(t^{\prime})}{m}\tilde{p}_{0}^{3}(t^{\prime})+\frac{5\alpha^{2}(t^{\prime})}{2m}\tilde{p}_{0}^{4}(t^{\prime})\right)\tilde{\varrho}(t)\left(\frac{\alpha(t)}{m}\tilde{p}_{0}^{3}(t)+\frac{5\alpha^{2}(t)}{2m}\tilde{p}_{0}^{4}(t)\right)$
$\displaystyle+\tilde{\varrho}(t)\left(\frac{\alpha(t^{\prime})}{m}\tilde{p}_{0}^{3}(t^{\prime})+\frac{5\alpha^{2}(t^{\prime})}{2m}\tilde{p}_{0}^{4}(t^{\prime})\right)\left(\frac{\alpha(t)}{m}\tilde{p}_{0}^{3}(t)+\frac{5\alpha^{2}(t)}{2m}\tilde{p}_{0}^{4}(t)\right)\bigg{\rangle}dt^{\prime}.$

If we keep the terms up to $\mathcal{O}(\alpha^{2})$, we are left with

$\partial_{t}\tilde{\rho}(t)=-\frac{4}{\hbar^{2}}\int_{0}^{t}\left\langle\alpha(t)\alpha(t^{\prime})\left(\frac{\tilde{p}^{3}_{0}(t)}{2m}\frac{\tilde{p}^{3}_{0}(t^{\prime})}{2m}\tilde{\varrho}(t)-\frac{\tilde{p}^{3}_{0}(t)}{2m}\tilde{\varrho}(t)\frac{\tilde{p}^{3}_{0}(t^{\prime})}{2m}-\frac{\tilde{p}^{3}_{0}(t^{\prime})}{2m}\tilde{\varrho}(t)\frac{\tilde{p}^{3}_{0}(t)}{2m}+\tilde{\varrho}(t)\frac{\tilde{p}^{3}_{0}(t^{\prime})}{2m}\frac{\tilde{p}^{3}_{0}(t)}{2m}\right)\right\rangle
dt^{\prime}$ (52)
$\partial_{t}\tilde{\rho}(t)=-\frac{4}{\hbar^{2}}\int_{0}^{t}\langle\alpha(t)\alpha(t^{\prime})\rangle\left[\frac{\tilde{p}^{3}_{0}(t)}{2m},[\frac{\tilde{p}^{3}_{0}(t^{\prime})}{2m},\tilde{\rho}(t)]\right]dt^{\prime}$
(53)

that is the same integral obtained in Eq. (25) in which we only considered
terms up to $\mathcal{O}(\alpha)$. Therefore, by following the same steps and
recalling that
$\alpha(t)={\alpha_{0}\ell_{p}}/{\hbar}={\ell_{p}\sqrt{t_{p}}\chi(t)}/{\hbar}$,
we obtain

$\partial_{t}\tilde{\rho}(t)=-\sigma\int_{0}^{t}\left\langle\chi_{0}(t)\chi_{0}(t^{\prime})\right\rangle\left[\frac{\tilde{p}^{3}_{0}(t)}{2m},\left[\frac{\tilde{p}^{3}_{0}(t^{\prime})}{2m},\tilde{\rho}(t)\right]\right]dt^{\prime},$
(54)

which becomes

$\partial_{t}\tilde{\rho}(t)=-\sigma\int_{0}^{t}\delta(t-t^{\prime})\left[\frac{\tilde{p}^{3}_{0}(t)}{2m},\left[\frac{\tilde{p}^{3}_{0}(t^{\prime})}{2m},\tilde{\rho}(t)\right]\right]dt^{\prime}~{}.$
(55)

Finally, we derive the expression

$\partial_{t}\tilde{\rho}(t)=-\sigma\left[\frac{\tilde{p}^{3}_{0}(t)}{2m},\left[\frac{\tilde{p}^{3}_{0}(t)}{2m},\tilde{\rho}(t)\right]\right]$
(56)

which is precisely Eq. (27).

By recovering the Schrödinger representation and going back to the Liouville-
Von Neumann equation, we can retrieve the Lindblad-type master equation, i.e.,
Eq. (29), by vanishing the external potential

$\partial_{t}\rho(t)=-\frac{i}{\hbar}\left[\frac{p_{0}^{2}}{2m},\rho(t)\right]-\sigma\left[\frac{\tilde{p}^{3}_{0}}{2m},\left[\frac{\tilde{p}^{3}_{0}}{2m},\rho(t)\right]\right].$
(57)

Because of the appearance of a double Hamiltonian in Eq. (50), it is clear
that we can safely neglect all higher-order terms above $\mathcal{O}(\alpha)$,
which by the way are extremely suppressed due to the smallness of the
corrections compared to the unmodified scenario.

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