# Schrödinger cats coupled with cavities losses: the effect of finite and
structured reservoirs.

J. Lira<EMAIL_ADDRESS>Colegiado de Engenharia, Unex, 45020-510,
Vitória da Conquista-BA, Brazil  L. Sanz<EMAIL_ADDRESS>Instituto de Física,
Universidade Federal de Uberlândia,38400-902, Minas Gerais, Brazil

###### Abstract

We discuss the generation of a Schrödinger cat in a nanocavity created by the
coupling of an electromagnetic mode with an exciton in a quantum dot
considering the dispersive limit of the Jaynes-Cummings model. More than the
generation itself, we focus on the effects of the environment over the bosonic
state in the nanocavity, which has losses simulated by coupling with two
different kind of reservoirs. In the first case, the interaction between the
system with a finite reservoir shows that fragments of different sizes of the
reservoir deliver the same amount of information about the physical system in
the dynamics of the birth and death of the Schrödinger cat. The second case
considers a structured reservoir, whose spectral density varies significantly
with frequency. This situation becomes relevant in solid-state devices where
quantum channels are embedded, as memory effects generally cannot be
neglected. Under these circumstances, it is observed that the dynamics can
differ substantially from the Markovian, presenting oscillations related to
the average number of photons. These oscillations influence the information
flow between the system and the environment, evidenced here by the measurement
of non-Markovianity.

##  I Introduction

Since the beginning of the quantum theory, the Schrödinger cat state [1, 2, 3,
4] has been pointed out as one of the crossing points between both, the
classical and the quantum worlds. Successful experimental implementations have
included several studies on cavity quantum electrodynamics involving Rydberg
atoms passing through superconducting cavities [5], cold ions trapped within
cavities [6], and even a transmon qubit coupled to cavities in a circuit
quantum electrodynamics system [7, 8]. In the initial experimental setup, for
example, the state of the cavity must be probed through the subsequent passage
of new atoms, while the cavity may simultaneously lose photons. The
experiments also offer the possibility of reconstructing the Wigner function
[9, 10, 11, 5]. A particular instance of these cat states, specifically the
one proposed by Yurke and Stoler [12], has proven difficult to implement
experimentally due to its precise requirement for a relative phase ($\pm i$)
between the superposition states. The average number of atoms in these
cavities is low (approximately 4), hence the term “kittens” rather than “cats”
is used to describe the state of the cavity.

Regarding the boundary between quantum and classical mechanics, it is
interesting to explore how a quantum state, once generated, evolves under the
influence of the decoherence process [13, 14, 15]. Recent studies have
demonstrated that quantum information reaching the environment is not always
irretrievably lost [16, 17, 18]; sometimes, it can be recovered, at least
temporarily. For instance, a team from the Federal University of Rio de
Janeiro - UFRJ conducted an experiment with photons, demonstrating that under
certain conditions, some information is preserved and might even be
retrievable [19]. Consistent with this finding, an entropic measure has been
established as an indicator of the correlation between non-Markovian behavior
and the information exchange between the system and its environment [20]. In
this particular case, a two-level system undergoing a relaxation process was
examined, utilizing an experimental optical setup that provides complete
access to the environmental state. To enhance our comprehension of how a
system interacts with its environment, we can “engineer” the environment. This
engineered approach allows us to characterize, control, and possibly reduce
the adverse effects on the system [21].

Some researchers contend that merely stating decoherence drives quantum
behavior, thus rendering it classical to an observer, is insufficient to
explain the emergence of classical, objective reality [22]. There must be a
consensus among multiple observers regarding the properties of quantum
systems. To achieve agreement on a measured value (a hallmark of classicality)
numerous replications are necessary. For instance, if ten observers measure
the position of a speck of dust, they will find it in the same location
because each observer has access to a distinct replica of the information
[15]. In this perspective, monitoring a small portion of the environment is
sufficient to obtain most of the available information, and observing more
than a fraction yields diminishing returns. A prominent concept within this
theoretical framework is that the defined properties of objects, which we
associate with classical physics, emerge from a range of quantum possibilities
through a process somewhat akin to natural selection in evolution. The
properties that persist are, in a sense, the fittest.

In this study, we examine the interaction between a two-level system
illustrated by an exciton in a quantum dot and a bosonic mode within a cavity,
similar to the nanocavities that interact with excitons as described in Ref.
[23]. This interaction experiences energy dissipation and is characterized by
a Hamiltonian that models the Jaynes-Cummings interaction under the dispersive
approximation. The dissipating cavity mode is depicted as a harmonic
oscillator linked linearly to $N$ oscillators, which represent the reservoir
[24, 25]. In the field of condensed-matter physics, the production of quantum
light using quantum dots within nanocavities has been extensively discussed
[26, 27, 28, 29, 30]. However, the feasibility of producing a Schrödinger cat
state with the specific system has not been studied, although the key element,
the dispersive interaction between a two-level system and a boson state, is
present in this experimental context. We will focus our attention on the
reservoir, analyzing the system interaction with two types of reservoirs, one
finite and the other structured, conducting an analysis of dynamics of the
normalized average mutual information, entanglement measures and non-
Markovianity in our system.

This paper is structured as follows: Section II introduces the model of our
physical system, along with some essential definitions and mathematical tools
utilized in our calculations. Section III is divided into two subsections; the
first offers a general discussion on the quantum dynamics of the system, and
the second details our principal findings concerning the effects of two types
of reservoirs, finite and structured, and the behavior of quantum information
in these non-trivial scenarios. Lastly, Section IV provides a summary of our
work.

##  II Physical System, model and mathematical tools.

Our system of interest is composed of a two-level particle or quasi-particle
interacting with bosons, as modeled by the Jaynes-Cummings model. We consider
an exciton in a quantum dot, which acts as the two-level system, encoding a
qubit, while the electromagnetic mode represents the single mode of a photonic
nanocavity. To encode the qubit, we define $\lvert 0\rangle$ as the state
without an exciton and $\lvert 1\rangle$ for the direct exciton state.
Conversely, to generate and sustain a coherent state within the nanocavity,
one can employ a continuous laser resonant with the cavity mode as described
in Ref. [23]. Initially, for qualitative analysis, we will omit this source of
cavity pumping.

The system’s Hamiltonian is expressed as follows: ($\hbar=1$)

$\hat{H}=\omega_{x}\hat{\sigma}_{+}\hat{\sigma}_{-}+\omega_{c}\hat{a}^{\dagger}\hat{a}+g(\hat{\sigma}_{+}\hat{a}+\hat{\sigma}_{-}\hat{a}^{\dagger})\
,$ (1)

where $\hat{\sigma}_{\pm}$ are the pseudospin operators that describe the
exciton, and $\hat{a}^{\dagger}$ ($\hat{a}$) is the creation (annihilation)
operator of the cavity mode. The frequencies of the exciton and cavity mode
are denoted by $\omega_{x}$ and $\omega_{c}$, respectively, and $g$ signifies
the exciton-cavity coupling strength.

When the frequency of the exciton is significantly different from that of the
cavity mode, the system is considered to be in a dispersive regime. The
interaction between an exciton and a cavity mode under the dispersive
approximation is described by an effective Hamiltonian. This effective
Hamiltonian, as detailed in Ref. [31], is presented as follows:

$\hat{H}_{\mathrm{eff}}=\omega_{x}\hat{\sigma}_{+}\hat{\sigma}_{-}+\omega_{c}\hat{a}^{\dagger}\hat{a}+\omega\hat{a}^{\dagger}\hat{a}\sigma_{z}\
.$ (2)

At this point, we are ready to make some considerations about the losses in
the cavity. Instead of consider the usual master equation considering Lindblad
operators, it is our interest to consider that the cavity mode suffers
dissipation by a bath of $N$ coupled linear oscilators. In that way, we want
to evaluate the effects of reservoirs over the dynamics of our now open
quantum system. The Hamiltonian reads as:

$\displaystyle
H_{t}=\hat{H}_{\mathrm{eff}}+\sum_{k}\omega_{k}\hat{b}_{k}^{\dagger}\hat{b}_{k}+\sum_{k}\gamma_{k}(\hat{a}^{\dagger}\hat{b}_{k}+\hat{a}\hat{b}_{k}^{\dagger}),$
(3)

where $\hat{b}_{k}$ is the $k-$th operator on the bosonic bath with frequency
$\omega_{k}$ and $\gamma_{k}$ is the coupling constant between the mode in the
nanocavity and the same mode. In principle, the oscilators in the bath are not
interacting. Still, it is expected an effective interaction mediated by the
nanocavity.

Once we are considering the whole system, the dynamics is governed by the
Schrödinger equation:

$\displaystyle H\lvert\psi(t)\rangle=i\frac{d}{dt}\lvert\psi(t)\rangle.$ (4)

A mathematical tool that we will use in some of our calculations is the
encoding of a superposition state of coherent states using a basis of two
bosonic qubits. Consider a generic quantum state as given by:

$\lvert\psi\rangle=a\lvert\alpha,\lambda\rangle+b\lvert\beta,\chi\rangle,$ (5)

where $\left\\{\lvert\alpha\rangle,\lvert\beta\rangle\right\\}$ belongs to a
subsystem $A$ and $\left\\{\lvert\lambda\rangle,\lvert\chi\rangle\right\\}$
belongs to a subsystem $B$. The objective is to express this state as a
generic superposition within the bosonic basis $\left\\{\lvert 0\rangle,\lvert
1\rangle\right\\}_{A}\otimes\left\\{\lvert 0\rangle,\lvert
1\rangle\right\\}_{B}$ written as

$\lvert\psi\rangle=p_{+}\lvert 11\rangle_{AB}+q_{+}\lvert
10\rangle_{AB}+q_{-}\lvert 01\rangle_{AB}+p_{-}\lvert 00\rangle_{AB},$ (6)

where one need to find the expressions for the four unknown parameters
$p_{\pm}$ and $q_{\pm}$. The reader is invited to verify that this is
accomplished by applying the subsequent transformations:

$\displaystyle\lvert\alpha\rangle$ $\displaystyle=$ $\displaystyle S_{+}\lvert
1\rangle_{A}-S_{-}\lvert 0\rangle_{A},$ $\displaystyle\lvert\beta\rangle$
$\displaystyle=$ $\displaystyle e^{-i\theta}\left(S_{+}\lvert
1\rangle_{A}+S_{-}\lvert 0\rangle_{A}\right),$
$\displaystyle\lvert\lambda\rangle$ $\displaystyle=$ $\displaystyle
R_{+}\lvert 1\rangle_{B}-R_{-}\lvert 0\rangle_{B},$
$\displaystyle\lvert\chi\rangle$ $\displaystyle=$ $\displaystyle
e^{-i\phi}\left(R_{+}\lvert 1\rangle_{B}+R_{-}\lvert 0\rangle_{B}\right),$ (7)

where we define

$S_{\pm}=\sqrt{\frac{1\pm\lvert\langle\alpha|\beta\rangle\rvert}{2}},\,R_{\pm}=\sqrt{\frac{1\pm\lvert\langle\lambda|\chi\rangle\rvert}{2}},\,e^{i\theta}=\frac{\langle\beta|\alpha\rangle}{\lvert\langle\alpha|\beta\rangle\rvert},\,e^{i\phi}=\frac{\langle\chi|\lambda\rangle}{\lvert\langle\lambda|\chi\rangle\rvert}.$
(8)

resulting in the coefficients in Eq. (6) given by:

$p_{\pm}=\left[a+be^{-i\left(\theta+\phi\right)}\right]S_{\pm}R_{\pm},q_{\pm}=\left[-a+be^{-i\left(\theta+\phi\right)}\right]S_{\pm}R_{\mp}$
(9)

##  III Results

###  III.1 Dynamics of the physical system

Consider the physical system set in the initial state given by

$\lvert\psi(0)\rangle=(\sin\varphi\lvert 1\rangle+\cos\varphi\lvert
0\rangle)\otimes\lvert\alpha\rangle\otimes\prod_{k}\lvert 0_{k}\rangle\ ,$
(10)

At this point, we can assert that the hamiltonian (3) evolves this initial
state into a superposition of coherent states in both the cavity and the
reservoirs with undetermined amplitudes $\alpha(t)$, $\beta(t)$, $\lambda(t)$,
and $\chi(t)$ as follows:

$\lvert\psi(t)\rangle=\sin\varphi\lvert
1\rangle\otimes\lvert\alpha(t)\rangle\otimes\lvert\lambda(t)\rangle+\cos\varphi\lvert
0\rangle\otimes\lvert\beta(t)\rangle\otimes\lvert\chi(t)\rangle=\lvert\psi(t)\rangle_{1}+\lvert\psi(t)\rangle_{2},$
(11)

where $\lvert\lambda(t)\rangle=\prod_{k}\lvert\lambda_{k}(t)\rangle$ e
$\lvert\chi(t)\rangle=\prod_{k}\lvert\chi_{k}(t)\rangle$, with $\alpha(t)$
($\beta(t)$) and $\lambda(t)$ ($\chi(t)$) being the amplitudes of the coherent
states of the nanocavity and the reservoir, respectively. The state
$\lvert\psi(t)\rangle$ must be a solution of the Schrödinger equation (4)
which implies that we will obtain a set of coupled differential equations for
the amplitudes $\alpha(t)$ and $\lambda(t)$, associated with the first term in
Eq.(11) as well as $\beta(t)$ and $\chi(t)$, corresponding to the second term.

Let us work with the first part. We start with the equation:

$\displaystyle\frac{d}{dt}\lvert\psi(t)\rangle_{1}=\frac{d}{dt}(\sin\varphi\lvert
1\rangle\otimes\lvert\alpha(t)\rangle\otimes\lvert\lambda(t)\rangle).$ (12)

For a generic coherent state $\lvert\xi\rangle$, it is true that

$\frac{d}{dt}\lvert\xi(t)\rangle=\frac{d}{dt}\Big{(}e^{-|\xi(t)|^{2}/2}\sum_{n=0}^{\infty}\frac{\xi(t)^{n}}{\sqrt{n!}}\lvert
n\rangle\Big{)}=-\frac{1}{2}\Big{(}\frac{d}{dt}\lvert\xi(t)\rvert^{2}\Big{)}\lvert\xi(t)\rangle+\dot{\xi}(t)\hat{a}^{\dagger}\lvert\xi(t)\rangle\,$
(13)

thus we can rewrite Eq.(12) as

$\displaystyle\frac{d}{dt}\lvert\psi(t)\rangle_{1}=\bigg{[}-\frac{1}{2}\frac{d}{dt}\Big{(}|\alpha(t)|^{2}+\sum_{k}|\lambda_{k}(t)|^{2}\Big{)}+\dot{\alpha}(t)\hat{a}^{\dagger}+\sum_{k}\dot{\lambda}_{k}(t)\hat{b}_{k}^{\dagger}\bigg{]}\lvert\psi(t)\rangle_{1}\
,$ (14)

where the total excitation number is a constant given by
$|\alpha(t)|^{2}+\sum_{k}|\lambda_{k}(t)|^{2}$. Applying this result in
Eq.(12) we obtain

$\displaystyle\frac{d}{dt}\lvert\psi(t)\rangle_{1}=\Big{[}\dot{\alpha}(t)\hat{a}^{\dagger}+\sum_{k}\dot{\lambda}_{k}(t)\hat{b}_{k}^{\dagger}\Big{]}\lvert\psi(t)\rangle_{1}\
.$ (15)

On the other hand

$H_{t}\lvert\psi(t)\rangle_{1}=\Big{\\{}\omega_{x}+\Big{[}\big{(}\omega_{c}+\omega\big{)}\alpha(t)+\sum_{k}\gamma_{k}\lambda_{k}(t)\Big{]}\hat{a}^{\dagger}+\sum_{k}\Big{(}\omega_{k}\lambda_{k}(t)+\gamma_{k}\alpha(t)\Big{)}\hat{b}_{k}^{\dagger}\Big{\\}}\lvert\psi(t)\rangle_{1}\
.$ (16)

Using Eq.(15) and Eq.(16) on the Schrödinger equation, we obtain the following
system of coupled equations for the amplitudes of the coherent states

$\displaystyle i\dot{\alpha}(t)$ $\displaystyle=$
$\displaystyle\big{(}\omega_{c}+\omega\big{)}\alpha(t)+\sum_{k}\gamma_{k}\lambda_{k}(t)\
,$ (17) $\displaystyle i\dot{\lambda}_{k}(t)$ $\displaystyle=$
$\displaystyle\omega_{k}\lambda_{k}(t)+\gamma_{k}\alpha(t)\ .$ (18)

By making the same calculation considering the second part of Eq.(11), we
obtain a similar set of coupled equations for $\beta(t)$ and $\chi(t)$ given
by:

$\displaystyle i\dot{\beta}(t)$ $\displaystyle=$
$\displaystyle\big{(}\omega_{c}-\omega\big{)}\beta(t)+\sum_{k}\gamma_{k}\chi_{k}(t)\
,$ (19) $\displaystyle i\dot{\chi}_{k}(t)$ $\displaystyle=$
$\displaystyle\omega_{k}\chi_{k}(t)+\gamma_{k}\beta(t)\ .$ (20)

At this point, it is important to notice that the evolved state of the
Eq.(11), which is a result of the dispersive dynamics, is the initial input to
generate Schrödinger cats inside the nanocavity. Consider the coupling of the
two-level system with an external probe:

$\hat{H}_{\mathrm{probe}}=\frac{\Omega}{2}\left[e^{-i(\omega_{p}-\omega_{x})\tau}\hat{\sigma}_{+}+e^{i(\omega_{p}-\omega_{x})\tau}\hat{\sigma}_{-}\right],$
(21)

where $\Omega$ represents the Rabi frequency describing the interaction of the
laser with the qubit, and $\omega_{p}$ is the frequency of the pulsed laser.
In order to eliminate the temporal dependence of the Hamiltonian, we will
consider $\omega_{p}=\omega_{x}$, that is, a resonance condition between the
external source and the qubit. If a pulse has a duration given by
$\tau=\pi/2\Omega$ the two-level system is rotated, placing the field states
in a quantum superposition.

$\displaystyle\lvert\psi(t+\pi/2\Omega)\rangle$ $\displaystyle=$
$\displaystyle\left(\frac{\sin(\varphi)\lvert\alpha(t)\rangle\otimes\lvert\lambda(t)\rangle-i\cos(\varphi)\lvert\beta(t)\rangle\otimes\lvert\chi(t)\rangle}{\sqrt{2}}\right)\lvert
1\rangle$ (22)
$\displaystyle+\left(\frac{-i\sin(\varphi)\lvert\alpha(t)\rangle\otimes\lvert\lambda(t)\rangle+\cos(\varphi)\lvert\beta(t)\rangle\otimes\lvert\chi(t)\rangle}{\sqrt{2}}\right)\lvert
0\rangle\ .$

As can be seen, the net effect of the action of the pulse is to couple the
excitonic states $\lvert 1\rangle$ and $\lvert 0\rangle$ in a mesoscopic
superposition in the bosonic cavity, with terms in agreement with the form of
Eq.(11).

###  III.2 Entanglement properties of system and reservoir

In this section, we explore the characteristics of quantum information
pertaining to two types of reservoirs coupled with the cavity. To examine the
properties of quantum information in the interaction between the physical
system and a reservoir, we employ the concept of quantum mutual information,
which is defined as

$I_{A:B}\equiv S(\rho_{A})+S(\rho_{B})-S(\rho_{AB}),$ (23)

where $\rho_{i}$ is the density matrix of the system or subsystem and
$S(\rho_{i})$ is the von Neumann entropy.

About the reservoirs, the first type is a finite one, from which we extract a
fragment to search for a quantifier of the information redundancy concerning
the physical system in the environment. The second type is a structured
reservoir that emerges in the context of solid-state devices, where
interactions with the quantum system under study are not uniform across all
frequencies. Instead, the environmental influence on the system varies
significantly with frequency, which is modeled through a spectral density.
This leads to nontrivial effects on the dynamics of the quantum system,
particularly non-Markovian effects.

####  III.2.1 Finite Reservoir

If the qubit is in the state $\lvert 1\rangle$ with initial phase
$\varphi=\pi/4$, The evolved state of the field and the reservoir will be in a
superposition state, written as:

$\displaystyle\lvert\psi(t)\rangle_{1}=\frac{\lvert\alpha(t)\rangle\otimes\lvert\lambda(t)\rangle-i\lvert\beta(t)\rangle\otimes\lvert\chi(t)\rangle}{2}\
.$ (24)

Therefore, the density operator for the combined field and reservoir system
is:

$\displaystyle\hat{\rho}_{1}(t)$ $\displaystyle=$
$\displaystyle\frac{1}{4}\Big{\\{}\lvert\alpha(t)\rangle\langle\alpha(t)\rvert\otimes\lvert\lambda(t)\rangle\langle\lambda(t)\rvert+\lvert\beta(t)\rangle\langle\beta(t)\rvert\otimes\lvert\chi(t)\rangle\langle\chi(t)\rvert$
(25)
$\displaystyle+\big{[}i\lvert\alpha(t)\rangle\langle\beta(t)\rvert\otimes\lvert\lambda(t)\rangle\langle\chi(t)\rvert+\mathrm{h.c.}\big{]}\Big{\\}}\
.$

Now, it is possible to calculate the reduced density operator belonging to
each subsystem. For example, for the field inside the cavity is written as:

$\displaystyle\hat{\rho}_{c}(t)$ $\displaystyle=$
$\displaystyle\mathrm{tr}_{R}[\hat{\rho}_{1}(t)]$ (26) $\displaystyle=$
$\displaystyle\frac{1}{4}\Big{\\{}\lvert\alpha(t)\rangle\langle\alpha(t)\rvert+\lvert\beta(t)\rangle\langle\beta(t)\rvert+\big{[}i\langle\chi(t)|\lambda(t)\rangle\lvert\alpha(t)\rangle\langle\beta(t)\rvert+\mathrm{h.c.}\big{]}\Big{\\}}\
.$

As initially proposed by Blume-Kohout and Zurek [32], we will focus on a
specific region within the reservoir, referred to as the $F$ fragment.
Considering that the system interacts with the degrees of freedom of $F$, for
such a fragment of the reservoir $R$, we have:

$\displaystyle\hat{\rho}_{F}(t)$ $\displaystyle=$
$\displaystyle\mathrm{tr}_{c,R-F}[\hat{\rho}_{1}(t)]$ (27) $\displaystyle=$
$\displaystyle\frac{1}{4}\left\\{{\lvert\lambda(t)\rangle}{\langle\lambda(t)\rvert_{F}}+{\lvert\chi(t)\rangle}\langle\chi(t)\rvert_{F}\right.$
$\displaystyle\left.+\left[i\langle\beta(t)|\alpha(t)\rangle_{R-F}{\langle\chi(t)|\lambda(t)\rangle}_{R-F}\lvert\lambda(t)\rangle\langle\chi(t)\rvert_{F}+\mathrm{h.c.}\right]\right\\}\
.$

Finally, the density operator for the combined cavity and fragment system is
given by:

$\displaystyle\hat{\rho}_{cF}(t)$ $\displaystyle=$
$\displaystyle\mathrm{tr}_{R-F}[\hat{\rho}_{1}(t)]$ $\displaystyle=$
$\displaystyle\frac{1}{4}\left\\{\lvert\alpha(t)\rangle\langle\alpha(t)\rvert\otimes{\lvert\lambda(t)\rangle}{\langle\lambda(t)\rvert}_{F}+\lvert\beta(t)\rangle\langle\beta(t)\rvert\otimes{\lvert\chi(t)\rangle}{\langle\chi(t)\rvert_{F}}\right.$
$\displaystyle\left.+\left[i{\langle\chi(t)|\lambda(t)\rangle}_{R-F}\lvert\alpha(t)\rangle\langle\beta(t)\rvert\otimes\lvert\lambda(t)\rangle\langle\chi(t)\rvert_{F}+\mathrm{h.c.}\right]\right\\}\
.$

where $\lvert\lambda(t)\rangle=\prod_{k}\lvert\lambda_{k}(t)\rangle$,
$\lvert\chi(t)\rangle=\prod_{k}\lvert\chi_{k}(t)\rangle$, and the scalar
product is given by
$\langle\lambda(t)|\chi(t)\rangle=\prod_{k}\langle\lambda_{k}(t)|\chi_{k}(t)\rangle=\prod_{k}e^{-\frac{1}{2}(\lvert\lambda_{k}\rvert^{2}+\lvert\chi_{k}\rvert^{2})+\chi_{k}\lambda_{k}^{*}}=e^{\sum_{k}\\{-\frac{1}{2}(\lvert\lambda_{k}\rvert^{2}+\lvert\chi_{k}\rvert^{2})+\chi_{k}\lambda_{k}^{*}\\}}$.

Quantum Mutual Information (QMI), $I_{A:F}$, as defined in Eq.(23), can be
used to ascertain what is known about the system by the environment fragment.
When calculating the von-Neumann entropies $S(A)$, $S(F)$ and $S(AF)$ of the
system $A$, the fragment $F$ and the set $A+F$, respectively, $I_{A:F}$ can be
obtained. Note that $I_{A:F}=0$, if and only if,
$\hat{\rho}_{AF}=\hat{\rho}_{A}\otimes\hat{\rho}_{F}$, that is, the subsystems
are uncorrelated. When $\hat{\rho}_{AF}$ is a pure state, $I_{A:F}=2S(A)$,
since $S(F)=S(A)$ and $S(AF)=0$. To identify redundant correlations, we
examine the partial information provided about all random fragments $F$ that
contain a size fraction $f$ of $R$, such that $F=fN$, where $0\leq f\leq 1$ is
the fraction of the environment that is taken. Therefore, the average mutual
information over every fragment $F$ of size $f$ can be written as [33]

$\displaystyle\bar{I}(f)=\left\langle
I_{A:F}\right\rangle_{F}=\mathrm{avg}_{\mathrm{All\,}F\mathrm{\,of}\,\mathrm{size\,}f}(I_{A:F})\
.$ (28)

In Fig. 1, the graphs of the Wigner function illustrate the formation and
decay of a Schrödinger cat-like state in phase space, as shown in panels
(a-d). The Wigner functions are associated with the coherent superposition of
states of the qubit $\lvert 1\rangle$ and $\lvert 0\rangle$ from Eq.(22). We
also present the normalized average mutual information (NAMI),
$\bar{I}(f)//2S(A)$, which quantifies the information redundancy of the
current system in the environment, panels (e-h). For our simulations, we
consider $N=900$ as the number of oscillators in the reservoir. Within this
limit, the dynamics of the finite reservoir align with our numerical
simulations, where the frequencies $\omega_{k}$ vary from $0$ to $10$, and the
coupling constant is given by $\gamma_{k}=\omega/8$. The initial average
number of photons is, $\lvert\alpha\rvert^{2}=\lvert\beta\rvert^{2}=10$ and
and each simulation is averaged over 100 realizations.

The Wigner function of the initial state of the nanocavity, as shown in Fig.
1(a), exhibits the characteristics of a Gaussian package with minimum
uncertainty, indicative of a coherent state. At this moment, since all parts
of the physical system are uncorrelated, the NAMI curve, Fig. 1(e), maintains
a consistent value across any fraction size $f$. Upon evolution, the state in
the cavity displays quantum signatures, evidenced by the emergence of
squeezing in Fig. 1(b) at $t=0.05\pi/\omega$. Concurrently, if we look at Fig.
1(f), is observed an abrupt increment on the NAMI value. This behavior
indicates that from the moment we switch on the interaction of the system with
the reservoir, the mutual information increases, showing that the whole system
(qubit, nanocavity and reservoir) moves from an uncorrelated state to a pure
quantum state.

In Fig.1(c) and Fig.1(d), we observe the creation and disappearance of a
coherent superposition of two mesoscopic states, representing the life and
death of a Schrödinger cat-like state, respectively. Although the Wigner
function in Fig.1(c) exhibits a negative part, the NAMI at this instant, as
shown in Fig.1(g), indicates a correlation between the system’s loss of
coherence and the emergence of a plateau in mutual information. Over time, as
more parts of the environment interact with the quantum system, it begins to
lose coherence to the reservoir, leading to the formation of a plateau in
mutual information. This plateau becomes increasingly pronounced when the cat
state transitions into a statistical mixture, as depicted in Fig.1(d) with
NAMI approaching $\bar{I}(f)/2S(A)=0.5$ corresponding to an increase in the
system’s loss of coherence, as seen in Fig.1(h).

This behavior can be interpreted as a transition point to classical behavior,
where the quantum system begins to exhibit more deterministic characteristics
and becomes less dependent on its quantum nature. Different sizes of fragments
have access to the same amount of information about the system, so it is not
necessary to monitor much of the environment to gather most of the information
about the system. However, accessing such information becomes difficult since
it is a large and complex system.

Figure 1: Comparison between the behavior of the Wigner function of the field
inside the cavity, with the normalized average mutual information,
$\bar{I}(f)/2S(A)$, for different instants of time $t$, with average number of
photons $\lvert\alpha\rvert^{2}=10$. Here $f$ is the size fraction of the
reservoir.

####  III.2.2 Structured Reservoir

We now return to Eqs.(17) and making the substitution
$\tilde{\alpha}(t)=\alpha(t)e^{i\Delta\omega t}$ and
$\tilde{\lambda}_{k}(t)=\lambda_{k}(t)e^{i\omega_{k}t}$, where
$\Delta\omega=\omega_{c}+\omega$, we write

$\displaystyle i\dot{\tilde{\alpha}}(t)$ $\displaystyle=$
$\displaystyle\sum_{k}\gamma_{k}\tilde{\lambda}_{k}(t)e^{i(\Delta\omega-\omega_{k})t}$
(29) $\displaystyle i\dot{\tilde{\lambda}}_{k}(t)$ $\displaystyle=$
$\displaystyle\gamma_{k}\tilde{\alpha}(t)e^{-i(\Delta\omega-\omega_{k})t}\ .$
(30)

Integrating (30) and substituting in (29), we obtain

$\dot{\tilde{\alpha}}(t)=-\sum_{k}\gamma_{k}^{2}\int_{0}^{t}\tilde{\alpha}(t^{\prime})e^{i(\Delta\omega-\omega_{k})(t-t^{\prime})}dt^{\prime}=-\int_{0}^{t}dt^{\prime}f(t-t^{\prime})\tilde{\alpha}(t^{\prime})\
,$ (31)

where we define Kernel memory

$\displaystyle
f(t-t^{\prime})=\sum_{k}\gamma_{k}^{2}e^{i(\Delta\omega-\omega_{k})(t-t^{\prime})}\
.$ (32)

The non-Markovian character of the problem is evident in the non-local time
integral of the equation (31). If the number of oscillators tends to infinity
and a constant spectral density is assumed, the Kernel memory is approximately

$\displaystyle f(t-t^{\prime})\approx 2\kappa\delta(t-t^{\prime})\ ,$ (33)

where $\kappa$ is a constant. In this case

$\displaystyle\dot{\tilde{\alpha}}(t)=-\int_{0}^{t}dt^{\prime}2\kappa\delta(t-t^{\prime})\tilde{\alpha}(t^{\prime})=-2\kappa\tilde{\alpha}(t)\
,$ (34)

this equation has as a solution

$\displaystyle\tilde{\alpha}(t)=\tilde{\alpha}(0)e^{-\kappa t}\ .$ (35)

This is the well-known solution of the time evolution of the master equation
in the Markov approximation, of the mode of a cavity that is initially in a
given coherent state $\tilde{\alpha}(0)$. In terms of continuous limits of the
ambient frequency, the correlation function, i.e., Kernel memory becomes

$\displaystyle
f(t-t^{\prime})=\int_{0}^{\infty}J(\omega_{k})e^{i(\Delta\omega-\omega_{k})(t-t^{\prime})}d\omega_{k}\
,$ (36)

Here, $J(\omega_{k})$ represents the spectral density of reservoir modes. We
assume a Lorentzian spectral density, which is typical of a structured cavity
[13], whose shape is

$\displaystyle
J(\omega_{k})=\frac{1}{2\pi}\frac{\gamma\Lambda^{2}}{(\omega_{c}-\omega_{k})^{2}+\Lambda^{2}},$
(37)

where parameter $\gamma$ is related to the microscopic system-reservoir
coupling constant, while $\Lambda$ defines the spectral width of the cavity
modes. It is worth mentioning that the parameters $\gamma$ and $\Lambda$ are
related, respectively, to the correlation time of the reservoir $\tau_{r}$ and
the relaxation time of the system $\tau_{q}$, with $\tau_{r}=\Lambda^{-1}$ and
$\tau_{q}\approx\gamma^{-1}$ [13]. The weak qubit-cavity coupling limit occurs
for $\Lambda>\gamma$ ($\tau_{r}<\tau_{q}$). The inverse corresponds to the
strong coupling limit $\Lambda<\gamma$ ($\tau_{r}>\tau_{q}$). The higher the
cavity quality factor, the smaller the $\Lambda$ spectral width and,
consequently, the photon decay rate. With this spectral density and applying
the residue theorem [34], the Kernel memory integral of Eq. (36) can be
solved, obtaining

$\displaystyle
f(t-t^{\prime})=\frac{\gamma\Lambda}{2}e^{-(\Lambda-i\omega)(t-t^{\prime})}\
.$ (38)

Substituting the result into the equation (31) and solving using the Laplace
transform, we obtain

$\displaystyle\tilde{\alpha}(t)=\tilde{\alpha}(0)\bigg{[}\cosh(\eta
t/2)+\frac{M\sinh(\eta t/2)}{\eta}\bigg{]}e^{-\frac{1}{2}Mt}\ ,$ (39)

where $M=\Lambda-i\omega$ and $\eta=\sqrt{-2\Lambda\gamma+M^{2}}$. Thus,
substituting (39) in (30) and integrating, we find the amplitude of the state
belonging to the reservoir

$\displaystyle\tilde{\lambda}_{k}(t)$ $\displaystyle=$
$\displaystyle\frac{i\gamma_{k}\tilde{\alpha}(0)}{-\eta^{3}+4\eta
x^{2}}\Big{\\{}2\eta[\cosh(xt)+\sinh(xt)](M-2x)\cosh(\eta t/2)$ (40)
$\displaystyle-4(Mx-\eta^{2}/2)[\cosh(xt)+\sinh(xt)]\sinh(\eta
t/2)-2\eta(M-2x)\Big{\\}}\ ,$

where $x=-(M/2+i(\Delta\omega-\omega_{k}))$. Thus, making the inverse
transformations $\alpha(t)=\tilde{\alpha}(t)e^{-i\Delta\omega t}$ and
$\lambda_{k}(t)=\tilde{\lambda}_{k}(t)e^{-i\omega_{k}t}$, and performing
analogous operations for $\beta(t)$ and $\chi(t)$, which are solutions of
Eqs.(19), we obtain the exact form of the evolved state
$\left|\psi(t)\right\rangle$, Eq.(11).

In Fig. 2 we present in (a) the behavior of the average number of photons
$\left\langle\hat{n}(t)\right\rangle$ inside the cavity and in (b) the
idempotency defect $\Gamma(t)=1-\mathrm{Tr}\rho^{2}_{c}$ of the same, both as
a function of time, for an average number of photons
$|\alpha|^{2}=|\beta|^{2}=10$. Two limits are compared with the Markovian case
(represented with the orange line in the figures): the strong coupling limit
considering $\Lambda=0.01\gamma$, dashed blue line in both panels, and the
weak coupling limit with $\Lambda=3\gamma$, solid green line. From the
analysis of the figures, we found that in the strong coupling regime, both the
average number of photons and the idempotency defect present oscillations.
Oscillations in the number of photons are related to the exchange of
excitations between the system and the reservoir. In the case of weak
coupling, since the reservoir correlation time is negligible compared to the
system relaxation time, the average number of photons quickly goes to zero.
The oscillations on the idempotency defect shows that the system and reservoir
interact strongly, exchanging information along the dynamics.

Figure 2: (a) Average number of photons $\left\langle\hat{n}(t)\right\rangle$
and (b) idempotency defect $\Gamma(t)$, as a function of time $\gamma t$. For
average numbers of photons $|\alpha|^{2}=|\beta|^{2}=10$. The limits are taken
for analysis: $\Lambda=0.01\gamma$ in dashed blue line, equivalent to the
strong coupling limit and $\Lambda=3\gamma$ in solid green line, modeling the
weak coupling limit. For comparison purposes, the Markov limit is shown in the
orange line.

A special type of quantum correlation present in the system is entanglement,
which has two sides: its notable side presents itself as a great resource for
the development of quantum technologies, but it also has its villainous side,
motivated through the decoherence process. Concerning our problem, at this
point it becomes interesting to quantify the entanglement so we calculate the
concurrence $C(\rho)$, to explore the degree of entanglement of the mesoscopic
superposition state. With this goal, we use the transformation of the state of
the products of coherent states into a base of two qubits, as described in
Sec.II, Eqs.(II). On this new basis, the concurrence calculation can be
carried out without major problems. We present our results in Fig. 3 for the
strong coupling limit $\Lambda=0.01\gamma$, where panel (a) shows the behavior
of concurrence as a function of time for different values of the phase
$\varphi$ considering a fixed initial average number of photons
$|\alpha|^{2}=|\beta|^{2}=10$. In Fig. 3(b) the behavior of concurrence for
different average numbers of photons $|\alpha|^{2}=|\beta|^{2}$ is studied,
where it is evident that the entanglement-type correlations of the system is
directly related to its size: where the larger it is, the more easily it
becomes entangled with its environment. In contrast, the more entangled the
system is with its environment, the faster it loses coherence.

Figure 3: Concurrence dynamics $C(\rho)$ of the mesoscopic superposition
state, Eq.(22), in the strong coupling limit $\Lambda=0.01\gamma$. In (a) for
different weights between wave packets, broken down by the initial phase
$\varphi$, with average initial photon number $|\alpha|^{2}=|\beta|^{2}=10$.
On the other hand, in graph (b) we observe the concurrence behavior for
different average numbers of initial photons $|\alpha|^{2}=|\beta|^{2}$, with
initial phase $\varphi=\pi/4$.

As discussed, when the system is in the strong coupling limit the timescale of
the system is comparable to that of the reservoir, so the reservoir is not
fast enough to return to equilibrium and some memory effects accumulate. Such
effects have a non-Markovian character and are taken into account by the non-
local kernel time integral, Eq.(32).

The commonly used approach for quantifying quantum non-Markovianity is based
on the idea that memory effects in the dynamics of open systems are linked to
the exchange of information between the open system and its environment. In
this way, quantum non-Markovianity is associated with a notion of quantum
memory, that is, information that was transferred to the environment, arising
from system-environment correlations or changes in environmental states, and
is subsequently recovered by the system. Based on this concept, we can
introduce a well-established measure for the degree of memory effects [35, 17]

$\displaystyle\mathcal{N}=\max_{\rho^{1,2}_{S}}\int_{\sigma>0}dt\sigma(t)\ ,$
(41)

where

$\displaystyle\sigma(t)\equiv\frac{d}{dt}D[\rho^{1}(t),\rho^{2}(t)]$ (42)

denotes the time derivative of the trace distance111It is an average of the
distance between two quantum states $\rho_{1}(t)$ and $\rho_{2}(t)$, that is,
similarity, defined as
$D[\rho_{1}(t),\rho_{2}(t)]=(1/2)\mathrm{tr}|\rho_{1}(t)-\rho_{2}(t)|$, where
$|X|=\sqrt{X^{\dagger}X}$. of the pair of evolved states, which can be
interpreted as characterizing a flow of information between the system and the
environment. By construction, the integral is done over every interval in
which $\sigma(t)>0$ and the maximization is done over all pairs of possible
initial states. Therefore, $\mathcal{N}=0$, if and only if, the process is
Markovian. It was shown that the pairs of optimal initial states
$\rho^{1,2}_{S}$ lie on the boundary of the state space and, in particular,
are always orthogonal [37]. In Fig. 4 the measurement of $\mathcal{N}$ of the
mesoscopic superposition state of two coherent wave packets (22) is presented
as a function of the radius $\Lambda/\gamma$ for different average number of
initial photons $|\alpha|^{2}=|\beta|^{2}$. Note that, unlike the concurrence,
as the average number of photons increases the degree of non-Markovianity of
the system decreases, which is something of interest and requires a more
rigorous investigation of this dynamic. Furthermore, large average numbers of
photons have a lower degree of non-Markovianity initially, but they lose it
more slowly compared to those with small average numbers of photons. However,
as expected, they all always tend to zero in the weak coupling limit. These
non-Markov effects will be investigated in more detail in future work.

Figure 4: Measure of non-Markovianity of the system $\mathcal{N}$ as a
function of $\Lambda/\gamma$, for different average numbers of initial photons
$|\alpha|^{2}=|\beta|^{2}$.

##  IV Summary

In this work, we propose the formation of a Schrödinger cat state within a
system composed of an excitonic state coupled to a nanocavity, accounting for
losses through two distinct types of reservoirs. The first, a finite
reservoir, exhibits a plateau in the normalized average mutual information,
signifying that reservoir fragments of varying sizes convey equivalent
information about the physical system of interest. By transforming the
reservoir into a structured one (an infinite number of oscillators with
Lorentzian spectral density), we investigate non-Markovian effects on the
cavity’s photon count, idempotency defect, and calculate the concurrence
between the reservoir and the system (via mapping to a two-qubit model), along
with assessing memory effects. We observe that the dynamics in this latter
scenario deviate significantly from the Markovian limit, exhibiting
oscillations not only in the average photon count but also in the idempotency
defect. These oscillations affect the information flow between the system and
its environment, as evidenced by the non-Markovianity measures calculated.

###### Acknowledgements.

This work was supported by CAPES, the Brazilian National Institute of Science
and Technology of Quantum Information (INCT-IQ), grant 465469/2014-0/CNPq, and
CNPq, grant 422350/2021-4.

## References

  * Schrödinger [1935] E. Schrödinger, Die gegenwärtige situation in der quantenmechanik, Naturwissenschaften 23, 807 (1935).
  * Gerry and Knight [1997] C. Gerry and P. L. Knight, Quantum superpositions and schrödinger cat states in quantum optics, Am. J. Phys. 65, 964 (1997).
  * Haroche [2013] S. Haroche, Nobel lecture: Controlling photons in a box and exploring the quantum to classical boundary, Rev. Mod. Phys. 85, 1083 (2013).
  * Wineland [2013] D. J. Wineland, Nobel lecture: Superposition, entanglement, and raising schrödinger’s cat, Rev. Mod. Phys. 85, 1103 (2013).
  * Deleglise  _et al._ [2008] S. Deleglise, I. Dotsenko, C. Sayrin, J. Bernu, M. Brune, J.-M. Raimond, and S. Haroche, Reconstruction of non-classical cavity field states with snapshots of their decoherence, Nature 455, 510 (2008).
  * Monroe  _et al._ [1996] C. Monroe, D. M. Meekhof, B. E. King, and D. J. Wineland, A “schrödinger cat” superposition state of an atom, Science 272, 1131 (1996), http://science.sciencemag.org/content/272/5265/1131.full.pdf .
  * Haroche  _et al._ [2020] S. Haroche, M. Brune, and J. M. Raimond, From cavity to circuit quantum electrodynamics, Nature Physics 16, 243 (2020).
  * Sun  _et al._ [2014] L. Sun, A. Petrenko, Z. Leghtas, B. Vlastakis, G. Kirchmair, K. M. Sliwa, A. Narla, M. Hatridge, S. Shankar, J. Blumoff, L. Frunzio, M. Mirrahimi, M. H. Devoret, and R. J. Schoelkopf, Tracking photon jumps with repeated quantum non-demolition parity measurements, Nature 511, 444 (2014).
  * Dowling  _et al._ [1994] J. P. Dowling, G. S. Agarwal, and W. P. Schleich, Wigner distribution of a general angular-momentum state: Applications to a collection of two-level atoms, Phys. Rev. A 49, 4101 (1994).
  * Hillery  _et al._ [1984] M. Hillery, R. O’Connell, M. Scully, and E. Wigner, Distribution functions in physics: Fundamentals, Physics Reports 106, 121 (1984).
  * Bertet  _et al._ [2002] P. Bertet, A. Auffeves, P. Maioli, S. Osnaghi, T. Meunier, M. Brune, J. M. Raimond, and S. Haroche, Direct measurement of the wigner function of a one-photon fock state in a cavity, Physical Review Letters 89, 200402 (2002).
  * Yurke and Stoler [1986] B. Yurke and D. Stoler, Generating quantum mechanical superpositions of macroscopically distinguishable states via amplitude dispersion, Phys. Rev. Lett. 57, 13 (1986).
  * Breuer and Petruccione [2007] H.-P. Breuer and F. Petruccione,  _The Theory of Open Quantum Systems_ , 1st ed. (Oxford University Press, Oxford, 2007).
  * Shishkov  _et al._ [2019] V. Y. Shishkov, E. S. Andrianov, A. A. Pukhov, A. P. Vinogradov, and A. A. Lisyansky, Relaxation of interacting open quantum systems, Physics-Uspekhi 62, 510 (2019).
  * Zurek [2003] W. H. Zurek, Decoherence, einselection, and the quantum origins of the classical, Reviews of modern physics 75, 715 (2003).
  * Rivas  _et al._ [2014] Á. Rivas, S. F. Huelga, and M. B. Plenio, Quantum non-markovianity: characterization, quantification and detection, Reports on Progress in Physics 77, 094001 (2014).
  * Breuer  _et al._ [2016] H.-P. Breuer, E.-M. Laine, J. Piilo, and B. Vacchini, Colloquium: Non-markovian dynamics in open quantum systems, Reviews of Modern Physics 88, 021002 (2016).
  * De Vega and Alonso [2017] I. De Vega and D. Alonso, Dynamics of non-markovian open quantum systems, Reviews of Modern Physics 89, 015001 (2017).
  * Farías  _et al._ [2012] O. J. Farías, G. Aguilar, A. Valdés-Hernández, P. S. Ribeiro, L. Davidovich, and S. P. Walborn, Observation of the emergence of multipartite entanglement between a bipartite system and its environment, Physical Review Letters 109, 150403 (2012).
  * Fanchini  _et al._ [2014] F. F. Fanchini, G. Karpat, B. Çakmak, L. Castelano, G. Aguilar, O. J. Farías, S. Walborn, P. S. Ribeiro, and M. De Oliveira, Non-Markovianity through accessible information, Physical Review Letters 112, 210402 (2014).
  * Harrington  _et al._ [2022] P. M. Harrington, E. J. Mueller, and K. W. Murch, Engineered dissipation for quantum information science, Nature Reviews Physics 4, 660 (2022).
  * Ball [2019] P. Ball, Quantum Darwinism, an Idea to Explain Objective Reality, Passes First Tests, Quantum Magazine (2019).
  * de Freitas  _et al._ [2017] A. de Freitas, L. Sanz, and J. M. Villas-Bôas, Coherent control of the dynamics of a single quantum-dot exciton qubit in a cavity, Phys. Rev. B 95, 115110 (2017).
  * de Paula Jr  _et al._ [2014] A. de Paula Jr, J. de Oliveira Jr, J. P. de Faria, D. S. Freitas, and M. Nemes, Entanglement dynamics of many-body systems: Analytical results, Physical Review A 89, 022303 (2014).
  * Oliveira  _et al._ [2019] S. Oliveira, A. de Paula Jr, and R. Drumond, Quantum Darwinism and non-Markovianity in a model of quantum harmonic oscillators, Physical Review A 100, 052110 (2019).
  * Müller  _et al._ [2015] K. Müller, A. Rundquist, K. A. Fischer, T. Sarmiento, K. G. Lagoudakis, Y. A. Kelaita, C. Sánchez Muñoz, E. del Valle, F. P. Laussy, and J. Vučković, Coherent generation of nonclassical light on chip via detuned photon blockade, Phys. Rev. Lett. 114, 233601 (2015).
  * Heinze  _et al._ [2015] D. Heinze, D. Breddermann, A. Zrenner, and S. Schumacher, A quantum dot single-photon source with on-the-fly all-optical polarization control and timed emission, Nat Commun 6, 8473 (2015).
  * Muñoz  _et al._ [2014] C. S. Muñoz, E. Del Valle, A. G. Tudela, K. Müller, S. Lichtmannecker, M. Kaniber, C. Tejedor, J. Finley, and F. Laussy, Emitters of n-photon bundles, Nature Photonics 8, 550 (2014).
  * Farrow  _et al._ [2008] T. Farrow, P. See, A. Bennett, M. Ward, P. Atkinson, K. Cooper, D. Ellis, D. Unitt, D. Ritchie, and A. Shields, Single-photon emitting diode based on a quantum dot in a micro-pillar, Nanotechnology 19, 345401 (2008).
  * Kiraz  _et al._ [2004] A. Kiraz, M. Atatüre, and A. Imamoǧlu, Quantum-dot single-photon sources: Prospects for applications in linear optics quantum-information processing, Phys. Rev. A 69, 032305 (2004).
  * Peixoto de Faria and Nemes [1999] J. G. Peixoto de Faria and M. C. Nemes, Dissipative dynamics of the jaynes-cummings model in the dispersive approximation: Analytical results, Phys. Rev. A 59, 3918 (1999).
  * Blume-Kohout and Zurek [2005] R. Blume-Kohout and W. H. Zurek, A simple example of quantum darwinism: Redundant information storage in many-spin environments, Foundations of Physics 35, 1857 (2005).
  * Blume-Kohout and Zurek [2008] R. Blume-Kohout and W. H. Zurek, Quantum Darwinism in quantum Brownian motion, Physical Review Letters 101, 240405 (2008).
  * Butkov [1988] E. Butkov,  _Física matemática_ (Livros Técnicos e Científicos, 1988).
  * Breuer  _et al._ [2009] H.-P. Breuer, E.-M. Laine, and J. Piilo, Measure for the degree of non-Markovian behavior of quantum processes in open systems, Physical Review Letters 103, 210401 (2009).
  * Note [1] It is an average of the distance between two quantum states $\rho_{1}(t)$ and $\rho_{2}(t)$, that is, similarity, defined as $D[\rho_{1}(t),\rho_{2}(t)]=(1/2)\mathrm{tr}|\rho_{1}(t)-\rho_{2}(t)|$, where $|X|=\sqrt{X^{\dagger}X}$.
  * Wißmann  _et al._ [2012] S. Wißmann, A. Karlsson, E.-M. Laine, J. Piilo, and H.-P. Breuer, Optimal state pairs for non-Markovian quantum dynamics, Physical Review A 86, 062108 (2012).