# Multi-Goal Optimal Route Planning Using the Cell Mapping Technique

Athanasios Karagounis  School of Science
Department of Digital Industry Technologies
GR34400, Psachna, Greece
<EMAIL_ADDRESS>
###### Abstract

This manuscript explores the complexities of multi-objective path planning,
aiming to optimize routes against a backdrop of conflicting performance
criteria. The study integrates the cell mapping approach as its foundational
concept. A two-pronged search strategy is introduced; initially, the cell
mapping technique is utilized to develop a comprehensive database,
encompassing all cells within the specified area. This database records the
performance metrics for the most efficient routes from each cell to the
designated target. The second phase involves analyzing this database to
pinpoint the extent and count of all Pareto optimal routes from a selected
starting cell to the target. This analysis contributes to solving the
overarching multi-objective optimization challenge inherent in path planning.
To validate this approach, case studies are included, and the results are
benchmarked against the well-established multi-objective A* (MOA*) method. The
study discovers that while the cell mapping method achieves similar outcomes
to the MOA* method for routes originating from a single point, it demonstrates
superior computational benefits, particularly when the starting and ending
points are in separate, non-overlapping areas.

###### Index Terms:

multi-objective, path planning, cell mapping, robotics

##  I Introduction

Path planning is a crucial aspect of research in mobile robotics. This
challenge is typically defined as the process of identifying feasible and safe
routes from an initial point to a target location within a workspace.
Concurrently, the identified paths must adhere to certain optimal criteria,
such as minimal distance, reduced travel duration, and least energy
consumption. Path planning must take into account two key elements: 1) the
nature of the environment, which could be either static or dynamic, and 2) the
robot’s understanding of its surroundings. When a robot possesses complete
environmental knowledge, the task is classified as global path planning.
Conversely, when the robot’s understanding of its surroundings is limited, the
task falls under local path planning. This study introduces an effective
approach to address the global multi-objective optimal path planning issue in
static settings.

Numerous studies have been conducted to devise strategies for solving the
global path planning problem [1]. These include various approaches like the
roadmap method, the grid-based A* algorithm, hierarchical cell decomposition
techniques, graph algorithms, and the artificial potential field method. In an
effort to enhance the efficiency of these traditional methodologies,
probabilistic algorithms have been introduced. Notable among these are the
probabilistic roadmap method and the rapidly-exploring random trees. For a
comprehensive understanding of both classical and probabilistic approaches,
one can refer to specific studies. It’s important to note that the majority of
the existing path planning methods are primarily focused on addressing single-
objective optimization challenges.

In practical scenarios, optimal path finding seldom involves a single
objective. Real-world applications necessitate addressing multiple, and often
conflicting, objectives at once. To address the multi-objective nature of path
planning, several heuristic methods have been proposed [2]. The multi-
objective extension of the A* algorithm, or MOA*, is discussed in certain
studies [3]. The Firefly algorithm has also been examined. Other popular
methodologies include Genetic Algorithms (GA) [4] and Particle Swarm
Optimization (PSO) [5], which have been extensively explored. Additional
methods that have garnered significant attention include neural networks,
memetic algorithms, simulated annealing (SA) [6], tabu search (TS), and
cellular automata. Moreover, hybrid path planning algorithms have been
developed, such as the integration of particle swarm optimization with the
probabilistic roadmap method.

The cell-to-cell mapping technique, initially proposed by Hsu [7] for the
comprehensive analysis of nonlinear dynamic systems, has recently shown
promising results in uncovering global Pareto fronts with detailed structures
for low and moderate dimensional multi-objective optimization problems (MOPs).
The objective of our research is to investigate the capability of the cell
mapping method in efficiently identifying all Pareto optimal paths within the
context of multi-objective optimal path planning.

This study delves into the application of the cell mapping method to resolve
multi-objective optimization problem (MOP) path planning challenges in static
environments. It elaborates on the development of a cell space-based search
algorithm, designed to pinpoint Pareto optimal paths from all cells to the
goal cell. Additionally, the paper discusses the creation of a database system
and a subsequent post-processing algorithm. These tools are instrumental in
determining the coverage area of all Pareto optimal paths originating from a
specific start cell to the target. The database also facilitates the
extraction of the number of Pareto optimal paths along with their respective
objective values. This process enables the construction of the Pareto front, a
feature that is notably scarce in existing path planning literature.

##  II Multi-variate Optimal Path Planning

The problem of multi-objective optimization in the context of global path
planning is formulated as:

$\min_{k\in Q_{s,g}}F(k)=\min_{k\in Q_{s,g}}[f_{1}(k),\ldots,f_{n}(k)],$ (1)

where each $f_{i}:\mathbb{R}^{n}\to\mathbb{R}$, with $i=1,2,\ldots,n$, denotes
an individual objective function that is subject to simultaneous minimization.
The domain $Q_{s,g}$ represents the set of all viable routes linking the start
point $s$ and the target $g$. For this Multi-Objective Optimization Problem
(MOP), optimal solutions are determined based on the principle of dominance,
as outlined in subsequent definitions.

  1. (a)

For paths $i$ and $j\in Q_{s,g}$, path $j$ is said to be dominated by path $i$
with respect to the MOP (1) if $F(i)\preceq F(j)$ and $F(i)\neq F(j)$. The
relation $F(i)\preceq F(j)$ signifies that every element of the vector $F(i)$
is less than or equal to the corresponding element of $F(j)$. If this is not
the case, then $j$ is considered non-dominated by $i$.

  2. (b)

A path $i\in Q_{s,g}$ is termed Pareto optimal for the MOP (1) if there exists
no path $j\in Q_{s,g}$ that dominates $i$.

  3. (c)

The collection $P$ of all Pareto optimal solutions is referred to as the
Pareto set, denoted by

$P:=i\in Q_{s,g}$ $i$ is a Pareto optimal solution of the MOP (1).

  4. (d)

The $F(P)$ of $P$ is known as the Pareto front.

##  III Cell Mapping

The cell mapping method discretizes the continuous point-wise space into a
finite collection of discrete cells, each with integer coordinates. The total
number of cells is finite, allowing them to be indexed by a single integer.
The motion within a cell is represented by the movement of its central point,
a concept known as the simple cell mapping [8, 9].

To apply this method, divide the continuous 2D physical space into
$N=N_{1}\times N_{2}$ cells, assigning the objective vectors evaluated at the
center of each cell. Finding a path involves identifying a sequence of cells
that connects the start position cell to the target position cell. In a two-
dimensional space, eight neighboring cells of any given cell are identified.

In this work, we consider two objective functions or vectors, defined as:

$f_{1}(k)=\sum_{i,j\in P_{k}}\delta z(i,j);\quad f_{2}(k)=\sum_{i\in
P_{k}}\delta\theta(i),$ (2)

where $F(k)=[f_{1}(k),f_{2}(k)]$, $P_{k}$ denotes the sequence of cells on the
$k$th path, and $i$ and $j$ are neighboring cells in $P_{k}$. The function
$\delta z(i,j)$ is given by:

$\delta z(i,j)=\sqrt{(x_{i}-x_{j})^{2}+(y_{i}-y_{j})^{2}}.$ (3)

The function $\delta\theta(i)$ indicates the cost associated with navigating
through cell $i$. The term $\delta z(i,j)$ is used to quantify the direct
distance between cell $i$ and its adjacent cell $j$, where $(x_{i},y_{i})$
represent the coordinates of the center of cell $i$. For simplicity in
calculations, we define $\delta z(i,j)$ based on the relative positioning of
the cells: $\delta z(i,j)=10$ when cell $i$ and cell $j$ lie along a
horizontal or vertical axis, and $\delta z(i,j)=14$ in cases where the cells
are diagonally aligned.

The cell mapping technique facilitates the generation of a database
encompassing all optimal paths and their corresponding objective values, which
connect starting cells to the goal. This database, represented as $B_{k}$, is
constructed iteratively. The algorithm for database construction is outlined
as follows:

$B_{k}:=\\{[n_{i},F_{n_{i}\to G}]\mid 1\leq n_{i}\leq N\\},$ (5) $F_{n_{i}\to
G}:=\\{F^{r}_{n_{i}\to G}\mid 1\leq r\leq m(n_{i})\\},$ (4) $F_{n_{i}\to
G}\leftarrow\text{dom\\_chk}(F_{n_{i}\to G}),$ (5)

Termination Condition: $B_{k-1}=B_{k}$.

In this formulation, the database $B_{k}$ stores information about free cells
and their objective vectors leading to the goal cell, denoted as $G$. The
function $m(n_{i})$ denotes the count of distinct objective vectors $F$ from
cell $n_{i}$ to the goal $G$. It’s important to note that the number of
potential paths from cell $n_{i}$ to $G$ may differ from $m(n_{i})$.

In the database $B_{k}$, $n_{i}$ represents the index of cell $n_{i}$. The
superscript $r$ in $F^{r}_{n_{i}\to G}$ signifies the $r$th set of objective
vectors $F$ from cell $n_{i}$ to $G$. The objective vectors $F^{r}_{n_{i}\to
G}$ are calculated using the equation:

$F^{r}_{n_{i}\to G}=F^{p}_{n_{j}\to G}+\Delta F_{n_{i}\to n_{j}},$ (6)

where $1\leq p\leq m(n_{j})$. The incremental objective vectors $\Delta
F_{n_{i}\to n_{j}}$ are computed as per Equation (3).

The function $\text{dom\\_chk}(F_{n_{i}\to G})$ refers to the dominance check
operation applied to the objective vectors from cell $n_{i}$ to $G$. Through
this operation, the non-dominated objective vectors are identified and
selected. Subsequently, we elaborate on the steps involved in the cell space
stratification mapping algorithm in a detailed manner.

The stratification mapping, incremental search $\Delta F$, and dominance
checking strategies are pivotal in ensuring that the database holds truly non-
dominated objective vectors $F_{n_{i}\to G}$ from every starting cell $n_{i}$
to the goal $G$. This database is comprehensive, containing information on
multi-objective optimal paths from all initial conditions within the domain to
the goal $G$, which denotes the global nature of the proposed method.

Furthermore, the process of layer-wise updating and dominance checking results
in Pareto optimal vectors that adhere to Bellman’s principle of optimality.
This principle asserts: ”An optimal policy has the property that whatever the
initial state and initial decision are, the remaining decisions must
constitute an optimal policy with regard to the state resulting from the first
decision” [10]. For instance, consider a non-dominated path
$P^{*}=(S,n_{1},\ldots,n_{i},n_{i+1},\ldots,G)$ originating from the set $S$
with objective vectors $F_{P^{*}_{S\to G}}\subseteq B_{k}$, and its subpath
$P^{*}_{i}=(n_{i},n_{i+1},\ldots,G)$ with objective vectors
$F_{P^{*}_{i_{n_{i}\to G}}}\subseteq B_{k}$. The algorithm confirms that
$F_{P^{*}_{i_{n_{i}\to G}}}\subseteq F_{n_{i}\to G}$ and $P^{*}_{i}$ continue
to be non-dominated and hence, remain Pareto optimal.

Additionally, the computational complexity of the cell mapping algorithm does
not vary with the number of cells $N_{G}$ in $G$. It’s also noteworthy that
while all optimal paths from the set $S$ are likely to converge at the goal
$G$, they may not encompass the entire set comprehensively.

##  IV Experimental Results

We illustrate the application of the cell mapping method in identifying Pareto
optimal paths and constructing the Pareto front for specified start and goal
cells. The outcomes of the Pareto front and the subsequent post-processing
utilizing the cell mapping method are presented.

A space is segmented into $117\times 117$ cells. We designate a cell as a
starting cell, and a cell as the goal. The height of the terrain is considered
as the terrain cost. The Pareto set, representing the coverage area of optimal
paths from the start cell to the goal using the cell mapping method with the
Pareto front for optimal paths determined by the cell mapping algorithm. This
approach yields 255 non-dominated objective vectors (forming the Pareto front)
and 15089 optimal paths. Notably, the Pareto front demonstrates a locally non-
convex characteristic, underscoring the complexity inherent in the underlying
multi-objective optimal path planning problem.

In the forthcoming examples, we aim to evaluate the computational efficiency
and the accuracy of the solutions derived through the cell mapping method, in
relation to the MOA* method. A blue cell is designated as the goal $G$. The
terrain cost $\delta\theta$ is set to 1 for magenta cells and 0 for others.
Through this straightforward example, the global solution capability of the
cell mapping method is demonstrated.

The directions of optimal paths from all cells to $G$, as encoded in the
database $B_{k}$. Consequently, the database can furnish optimal paths from
any initial cell to the goal. For instance, if we select the green cell as the
start cell $S$, it is feasible to extract all Pareto optimal paths connecting
$S$ to $G$. It is important to note that an identical set of Pareto optimal
paths from the start cell $S$ to the goal $G$ could be obtained using the MOA*
algorithm within a comparable computational timeframe. The solution represents
a subset. To achieve the same global solution with the MOA* algorithm, it
would be necessary to run the algorithm $N-1$ times, where $N$ denotes the
total number of cells in the domain.

Our examination of the computational efficiency of the cell mapping method,
alongside a comparison of solution accuracy with the MOA* method [11],
involved 100 complex random maps with varying divisions of the domain. We
discovered that both methods yield identical sets of Pareto optimal paths for
a given start cell to the same goal. The CPU time required by the cell mapping
method to create a database representing global solutions is approximately
equivalent to that needed by the MOA* algorithm to generate Pareto optimal
paths for one start cell, and can be run in edge devices [12].

It is also noteworthy that in scenarios where Pareto optimal paths are needed
from a multitude of starting locations converging to a common target, the cell
mapping method exhibits its full potential. This is particularly true for
applications requiring multi-objective path planning to connect one or more
disjoint areas. In such cases, the cell mapping method provides a substantial
advantage over the MOA* algorithm and other methods. Next, we will present a
map to plan multi-optimal paths connecting disjoint goal areas.

##  V Conclusions

This paper introduced an efficient approach using the cell mapping method for
addressing multi-objective optimal path planning challenges. Central to the
study were objectives like minimizing path length and reducing terrain cost. A
comprehensive global database, containing the optimal objective vectors
(Pareto front) for routes from all domain cells to the target, was developed
employing this method. Additionally, the extent of the Pareto optimal paths
(Pareto set) and the total count of all feasible paths from a selected start
cell $S$ to the destination $G$ were derived through a specialized post-
processing algorithm applied to the database. The efficacy of this innovative
method was underscored through comparative analyses with the established MOA*
method.

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# Dynamics of Quantum Coherence and Quantum Fisher Information of a V-type
Atom in Isotropic Photonic Crystal

Ghafar Ahmadi  Institute for Advanced Studies in Basic Sciences (IASBS)
Shahpoor Saeidian<EMAIL_ADDRESS>Institute for Advanced Studies in
Basic Sciences (IASBS)  Ghasem Naeimi<EMAIL_ADDRESS>Department of
Physics, Qazvin Branch, Islamic Azad University, Qazvin, Iran

###### Abstract

The time evolution of quantum Fisher information, quantum coherence, and non-
Markovianity of a V-type three-level atom embedded in free space or a photonic
band gap crystal have been investigated. It has been demonstrated that the
photonic band gap crystal, as a structured environment, significantly
influences the preservation and enhancement of these quantum features.
Additionally, we observe that by manipulating the initial relative phase
values encoded in the atomic state and the relative positions of the upper
levels within the forbidden gap, control over the dynamics of quantum features
can be achieved. These findings highlight the potential benefits of utilizing
photonic band gap crystals in quantum systems, offering improved preservation
and manipulation of quantum information. The ability to control quantum
features opens new avenues for applications in quantum information processing
and related technologies.

###### pacs:

03.67.Pp, 03.65.Yz, 03.67.Mn, 03.65.Ud

##  I Introduction

In the field of quantum mechanics, estimation plays a crucial role in
extracting valuable information from quantum systems. One of the fundamental
challenges in quantum physics is accurately determining the values of unknown
parameters that characterize a quantum system, such as the precise value of a
physical quantity or the state of a quantum system. Parameter estimation is
essential in quantum metrology Dowling (1998); Pirandola et al. (2015); Zhang
et al. (2013); Demkowicz-Dobrzański et al. (2015); Najafi et al. (2023); ALB
(2020); Naeimi et al. (2016). Quantum metrology aims to utilize of quantum
features and effects to enhance the sensitivity in estimating physical
parameters. Quantum Estimation Theory (QET) has emerged as a powerful
framework to address the problem of parameter estimation in quantum systems
Helstrom (1969); Suzuki (2019); Paris (2009); Helstrom (1976). QET provides a
mathematical formalism to analyze and optimize measurement schemes for
extracting unknown parameters with high precision. At the core of QET is the
concept of Quantum Fisher Information (QFI), which quantifies the information
content contained in a given quantum state and determines the fundamental
limits of parameter estimation. A larger value of QFI indicates a higher
precision in parameter estimation. QFI has been widely investigated
theoretically and experimentally for different systems Szczykulska et al.
(2016); Mitchell et al. (2004); Knott et al. (2014); Ren et al. (2022); Zhang
et al. (2013); Yu et al. (2021); Nikdel Yousefi et al. (2022). A key factor
that affects estimation in quantum systems is the interaction between the open
quantum system and its surrounding environment Breuer and Petruccione (2002);
Lidar (2020). This interaction can lead to decoherence, where the fragile
quantum states lose their coherence and become entangled with the environment.
Decoherence causes a loss of information and degrades the accuracy of
parameter estimation. Therefore, understanding the effects of the environment
and mitigating decoherence is vital for precise estimation in quantum
mechanics. One way to characterize the effects of the environment on quantum
coherence and parameter estimation is through the distinction between
Markovian and non-Markovian dynamics. In a Markovian environment, the memory
of past interactions between the quantum system and the environment is lost,
and the dynamics are memoryless. This simplifies the analysis but limits the
control over the environment’s effects on the quantum system. On the other
hand, non-Markovian dynamics in open quantum systems exhibit memory effects.
The system’s evolution depends not only on its current state but also on its
past history, leading to time-dependent dynamics with memory, revivals, or
oscillations Li et al. (2018); Breuer et al. (2009); Luo et al. (2012);
Jahromi et al. (2020); Mahdavipour et al. (2022); Nikdel Yousefi et al.
(2022). By using structured environments or engineering the system-environment
interactions, non-Markovian dynamics can be utilized to enhance the accuracy
and robustness of estimation in quantum mechanics Bennink and Lougovski
(2019); Chin et al. (2012); Escher et al. (2011); Alipour et al. (2014); Xiao
et al. (2014). One fascinating approach to controlling the effects of the
environment on quantum systems is through the concept of photonic band gap
(PBG) crystal. PBG refer to ranges of frequencies or wavelengths in which the
propagation of light is forbidden within a material or a photonic crystal
structure. In such structures, the interference of waves creates regions where
specific wavelengths of light cannot propagate, leading to the confinement of
light within certain regions John (1987); Woldeyohannes and John (1999); Singh
et al. (2011). By engineering PBG, researchers can create environments that
influence the interaction of light with quantum systems. PBG materials can be
utilized to control the propagation and emission of photons, which are
essential for various quantum technologies such as waveguides D’Orazio et al.
(2003); Kaniber et al. (2007), optical memory devices Sprague et al. (2014);
Quang et al. (1997), and single-photon sources Kaniber et al. (2008); Laucht
et al. (2012). Recently, It has been shown that PBG materials can help confine
and protect quantum states, minimizing decoherence effects and enhancing the
precision of parameter estimation Nikdel Yousefi et al. (2022). Furthermore,
atom–photon entanglement and the control of the spontaneous emission of atoms
within photonic band gap materials have been investigated Woldeyohannes and
Shutayfi (2019); Jiang et al. (2013); Yang and Zhu (2000); Huang et al.
(2011); Entezar (2010); Nikdel Yousefi et al. (2022). In this paper, we study
the time evolution of quantum Fisher information, quantum coherence, and non-
Markovianity of a V-type three-level atom embedded in free space or in a PBG
crystal. We assess the influence of the initial relative phase values encoded
in the atomic state and the relative positions of the upper levels within the
forbidden gap over the dynamics of quantum features.

The structure of the paper is as follows: In Sec. II, we describe the model
system and obtain the analytical expression for probability amplitudes of the
atomic system when the atom embedded in PBG and free space. In Sec. III, The
dynamical behavior of quantum Fisher information and parameter estimation for
both cases is studied. In Sec. IV, we discuss the dynamical behavior of
quantum coherence for both situation. In Sec. V, we also delve into the
dynamical behavior of non-Markovianity based on Hilbert Schmidt Speed (HSS).
Finally, in Sec. VI, we provide a short conclusion.

##  II Model and Equations

Figure 1: Schematic diagram of a three-level atomic system in the V
configuration

We consider a V-type three-level atom with two upper levels $|a_{3}\rangle$,
$|a_{2}\rangle$, and a lower level $|a_{1}\rangle$ as shown in Fig.1. The
upper levels $|a_{3}\rangle$, $|a_{2}\rangle$ are coupled by the same vacuum
mode to the lower level $|a_{1}\rangle$, and the transition between upper
levels $|a_{3}\rangle$, $|a_{2}\rangle$ is forbidden due to symmetry
consideration. The transition frequencies between exited state
$|a_{3}\rangle$, $|a_{2}\rangle$ and the ground state $|a_{1}\rangle$ are
$\omega_{31}=\omega_{3}-\omega_{1}$ and $\omega_{21}=\omega_{2}-\omega_{1}$,
respectively. The Hamiltonian of this system with the rotating-wave
approximation and the electric-dipole approximation can be written as

$\hat{H}=\hat{H_{0}}+\hat{H_{I}},$ (1)

where

$\hat{H_{0}}=\hslash\omega_{3}|a_{3}\rangle\langle
a_{3}|+\hslash\omega_{2}|a_{2}\rangle\langle
a_{2}|+\sum_{\lambda=1}^{2}\sum_{k}\hslash\omega_{k}\hat{a}_{k\lambda}^{\dagger}\hat{a}_{k\lambda},$
(2)

$\displaystyle\hat{H_{I}}=i\hslash\sum_{k\lambda}\left[g_{k,\lambda}^{(31)}\hat{a}_{k\lambda}^{\dagger}|a_{1}\rangle\langle
a_{3}|+g_{k,\lambda}^{(21)}\hat{a}_{k\lambda}^{\dagger}|a_{1}\rangle\langle
a_{2}|\right]+H.C.$ (3)

Where $\hat{a}_{k\lambda}$ and $\hat{a}_{k\lambda}^{\dagger}$ are the
annihilation and creation operators for the $k^{th}$ electromagnetic mode with
frequency $\omega_{k}$ , and $\hbar k$ and $\lambda$ indicate momentum and two
transverse polarization of electromagnetic mode, respectively. The coupling
constants between the $k^{th}$ electromagnetic mode and the atomic transitions
$|a_{i}\rangle\hskip 2.84526pt(i=2,3)$ are

$g_{k,\lambda}^{(i1)}=\frac{\omega_{i}d_{i}}{\hslash}\left(\frac{\hslash}{2\epsilon_{0}\omega_{k}V}\right)^{1/2}\hat{e}_{\mathrm{k\lambda}}.\hat{u}_{\mathrm{i}},$
(4)

that are assumed to be real. $d_{i}$ and $\hat{u_{i}}_{\mathrm{}}$ are the
magnitude and unit vector of transition dipole moment of the atom, $V$ is the
sample volume , $\hat{e}_{k,\lambda}$ are the two transverse unit vectors, and
$\epsilon_{0}$ is the vacuum permittivity. We assume that the photon reservoir
is initially in the vacuum state $|0\rangle$ and the atomic system is prepared
in a pure superposition of the two upper levels $|a_{3}\rangle$ and
$|a_{2}\rangle$. Therefore, for the initial vector state of the system we have

$|\psi(0)\rangle=\cos\left(\frac{\theta}{2}\right)|a_{3},0\rangle+e^{i\phi}\sin\left(\frac{\theta}{2}\right)|a_{2},0\rangle,$
(5)

the parameter $\theta$ measures the degree of superposition of the two upper
levels $|a_{3}\rangle$, $|a_{2}\rangle$, and the factor $e^{i\phi}$ gives the
relative phase between the expansion coefficients of $|a_{3}\rangle$ and
$|a_{2}\rangle$. The state vector of the system at arbitrary time t can be
written as

$\displaystyle|\psi(t)\rangle$ $\displaystyle=$ $\displaystyle
A_{3}(t)e^{-i\omega_{3}t}|a_{3},0\rangle+A_{2}(t)e^{-i\omega_{2}t}|a_{2},0\rangle$
(6) $\displaystyle+$
$\displaystyle\sum_{k\lambda}A_{k\lambda}(t)e^{-i\omega_{k}t}|a_{1},1_{k\lambda}\rangle,$

Above, the state vectors $|a_{3},0\rangle$ and $|a_{2},0\rangle$ indicate the
atom in its excited states $|a_{3}\rangle$ and $|a_{2}\rangle$ respectively,
without any photons in the reservoir modes, and the state vector
$|a_{1},1_{k\lambda}\rangle$ indicates the atom in its ground state
$|a_{1}\rangle$, with a single photon in the $k^{th}$ electromagnetic mode.

Substituting the Hamiltonian (1) and the state vector (6) into the Schrödinger
equation, we can obtain the coupled equations for the amplitudes as follows:

${\dot{A}_{3}(t)}=-\sum_{k\lambda}g_{k,\lambda}^{(31)}A_{k\lambda}(t)e^{-i\delta_{31}t},$
(7a)
${\dot{A}_{2}(t)}=-\sum_{k\lambda}g_{k,\lambda}^{(21)}A_{k\lambda}(t)e^{-i\delta_{21}t},$
(7b)
${\dot{A}_{k\lambda}(t)}=g_{k,\lambda}^{(21)}A_{2}(t)e^{i\delta_{21}t}+g_{k,\lambda}^{(31)}A_{3}(t)e^{i\delta_{31}t},$
(7c)

where $\delta_{ij}=\omega_{k}-\omega_{ij}$ is the detuning of the radiation
mode frequency $\omega_{k}$ from the atomic transition frequency
$\omega_{ij}$. Performing a simple time integration of Eq. (7c) and
substituting the result into Eqs. (7a) and (7b), we obtain

$\displaystyle{\dot{A}_{3}(t)}$ $\displaystyle=$
$\displaystyle-\int_{0}^{t}G_{33}(t-t^{\prime})A_{3}(t^{\prime})dt^{\prime}$
(8)
$\displaystyle-e^{i\omega_{32}t}\int_{0}^{t}G_{32}(t-t^{\prime})A_{2}(t^{\prime})dt^{\prime},$

$\displaystyle{\dot{A}_{2}(t)}$ $\displaystyle=$
$\displaystyle-\int_{0}^{t}G_{22}(t-t^{\prime})A_{2}(t^{\prime})dt^{\prime}$
(9)
$\displaystyle-e^{-i\omega_{32}t}\int_{0}^{t}G_{23}(t-t^{\prime})A_{2}(t^{\prime})dt^{\prime},$

where
$G_{ij}(t-t^{\prime})=\sum_{k,\lambda}g^{i1}_{k,\lambda}g^{j1}_{k,\lambda}e^{-i\delta_{j1}(t-t^{\prime})},(i,j=2,3)$
are the delay Green’s functions. The resulting memory kernel strongly depends
on the photon density of states of the field in the reservoirWoldeyohannes and
John (1999). In the following, the solutions of the time-dependent amplitudes
for the cases of isotropic photonic band gap and free space are given.

###  II.1 An atom in PBG Material

Let’s consider a three-level atom is embedded in an isotropic photonic
crystalJohn and Quang (1994); Yang and Zhu (2000). The resonant transition
frequencies $\omega_{31}$ and $\omega_{21}$ are assumed to be near the band
gap, so there exist a strong quantum interference between the transitions from
the upper levels to the lower levelYang and Zhu (2000). The dispersion
relation of an isotropic photonic crystal near the band edge $\omega_{c}$
could be approximately expressed by

$\omega_{k}=\omega_{c}+A(k-k_{0})^{2},$ (10)

where $A=\omega_{c}/k_{0}^{2}$ is the curvature near $\omega_{c}$. This
corresponds to a density of states of the form,
$\rho(\omega)\propto\Theta(\omega-\omega_{c})(\omega-\omega_{c})^{-1/2}$ and
the Heaviside step function $\Theta(\omega-\omega_{c})$ characterizing the
cut-off behaviorEntezar (2010).

Using the isotropic dispersion relation and assuming that the dipole moments
are parallel to each other and
$g^{21}_{k,\lambda}=g^{31}_{k,\lambda}=g_{k,\lambda}$ for simplicity, we
arrive at the following solutions for $A_{3}(t)$ and $A_{2}(t)$ Yang and Zhu
(2000)

$A_{3}(t)=\sum_{j}\frac{f_{1}(x^{1}_{j})}{Z^{\prime}(x_{j}^{1})}e^{ix_{j}^{1}t}+\sum_{j}\frac{f_{2}(x^{2}_{j})}{H^{\prime}(x_{j}^{2})}e^{ix_{j}^{2}t}-R_{3}(t),$
(11)

$A_{2}(t)=e^{-i\omega_{12}t}\left[\sum_{j}\frac{f_{3}(x^{1}_{j})}{Z^{\prime}(x_{j}^{1})}e^{ix_{j}^{1}t}+\sum_{j}\frac{f_{4}(x^{2}_{j})}{H^{\prime}(x_{j}^{2})}e^{ix_{j}^{2}t}\right]-R_{2}(t).$
(12)

Refer to Appendix A for more details.

###  II.2 An atom in Free Space

In this subsection, we study the case when the three-level atom is in free
space with the photon dispersion relation $\omega_{k}=ck$Woldeyohannes and
John (1999). The free space case will be helpful to compare and interpret the
results of the PBG case, which will be discussed in the following sections.
For such a photon dispersion relation, the Green’s function is given by

$G_{ij}(t-t^{\prime})=\sqrt{\gamma_{i1}\gamma_{j1}}\delta(t-t^{\prime}),$ (13)

whit $\gamma_{ij}=\omega_{ij}^{3}d_{ij}^{2}/6\pi\epsilon_{0}\hslash
c^{3},(i,j=2,3)$. By substituting Eq. (13) into Eq. (8) and Eq. (9), the time
dependence of the amplitudes can be obtained asWoldeyohannes and John (2003)

$A_{3}(t)=e^{-\gamma_{31}t}\sum_{j=1}^{2}C_{j}e^{q_{j}t},$ (14)

$A_{2}(t)=e^{-(\gamma_{31}+i\omega_{32})t}\sum_{j=1}^{2}B_{j}e^{q_{j}t},$ (15)

where,

$q_{1,2}=\frac{\lambda}{2}\pm\sqrt{\left(\frac{\lambda}{2}\right)^{2}+{(\bar{\gamma})}^{2}},$
(16)

$\lambda=\gamma_{31}-\gamma_{21}+i\omega_{32},\hskip
14.22636pt\bar{\gamma}=\sqrt{\gamma_{31}\gamma_{21}},$ (17)

$C_{j}=\frac{q_{k}\cos(\frac{\theta}{2})+\bar{\gamma}e^{i\phi}\sin(\frac{\theta}{2})}{q_{k}-q_{j}},\hskip
14.22636pt(k=1,2;k\neq j),$ (18)

$B_{j}=-q_{j}C_{j}/\bar{\gamma}.$ (19)

By using the Eq. (6), the density operator of the atom-field system is given
by $\rho_{af}=|\psi\rangle\langle\psi|$, and the reduced density matrix of the
atom can be obtained as

$\rho_{a}(t)=Tr_{f}\\{{\rho_{af}(t)}\\}=\begin{pmatrix}\rho_{33}&\rho_{32}&0\\\
\rho_{23}&\rho_{22}&0\\\ 0&0&\rho_{11},\end{pmatrix}$ (20)

where

$\rho_{33}=|A_{3}(t)|^{2},\rho_{22}=|A_{2}(t)|^{2},\rho_{32}=\rho_{23}^{*}=A_{3}(t){A_{2}(t)}^{*}.$
(21)

In the following, we study the dynamics of the quantum properties of the
three-level atom in PBG compared to free space.

##  III Quantum Fisher Information

Multiparameter quantum estimation theory is a branch of quantum metrology that
deals with the estimation of multiple parameters in quantum systems. It
provides a framework for understanding and quantifying the precision with
which these parameters can be measuredSzczykulska et al. (2016). In this
section, we are interested in studying the dynamic variation in the estimation
precision of parameters $\theta$ and $\phi$ encoded into the initial state of
the three-level atom. We want to understand how the initial values of the
relative phase $\phi$ and the different relative positions of the upper levels
from the forbidden gap affect the sensitivity in the measurement of these
parameters. To this end, let us briefly review the notion of multiparameter
quantum estimation theory.

We use the quantum Fisher information matrix (QFIM) to investigate the optimal
precision for estimating the parameters $\theta$ and $\phi$ which is given
byLiu et al. (2020)

$F(\theta,\phi)=\begin{pmatrix}\mathcal{F}_{\theta\theta}(t)&\mathcal{F}_{\theta\phi}(t)\\\
\mathcal{F}_{\phi\theta}(t)&\mathcal{F}_{\phi\phi}(t)\\\ \end{pmatrix},$ (22)

where the coefficients are defined as

$\mathcal{F}_{ij}:=Tr(\rho(t)L_{i}L_{j}+\rho(t)L_{j}L_{i}),$ (23)

with $i,j\in{\theta,\phi}$, and $L_{i}(L_{j})$ is the symmetric logarithmic
derivative (SLD), which is determined by the equation

$\partial_{i}\rho(t)=\frac{1}{2}(\rho(t)L_{i}+L_{i}\rho(t)).$ (24)

Since the SLD operator is a Hermitian operator, the QFIM is Hermitian. The
QFIM is indeed a crucial tool for understanding the fundamental limits of
precision in parameters estimation in quantum systems. It can be used to
derive the quantum Cramér-Rao bound, which provides a lower bound on the
achievable variance of any unbiased estimator. It satisfies the following
inequality

$\Sigma(\theta,\phi)\geq F^{-1}(\theta,\phi)$ (25)

where $\Sigma$ is the covariance matrix for the parameters $\theta$ and
$\phi$, and $F^{-1}(\theta,\phi)$ is the inverse matrix of the QFIM
$F(\theta,\phi)$. The corresponding quantum Cramér-Rao bounds for independent
estimations of the parameters $\phi$ and $\theta$ can be obtained as Helstrom
(1976); Holevo (1982)

$\Delta\phi\geq\frac{1}{\sqrt{F_{\phi}}},\hskip
14.22636pt\Delta\theta\geq\frac{1}{\sqrt{F_{\theta}}}$ (26)

where $\Delta\theta$ and $\Delta\phi$ represent the minimum achievable
uncertainties in the estimation of $\phi$ and $\theta$, respectively. Based on
Eq. (23), the quantities $F_{\theta}=\mathcal{F}_{\theta\theta}$ and
$F_{\phi}=\mathcal{F}_{\phi\phi}$ are the diagonal elements of the QFIM,
computed as $F_{\theta}=Tr(\rho(t){L_{\theta}}^{2})$ and
$F_{\phi}=Tr(\rho(t){L_{\phi}}^{2})$. As Eq. (26) shows, in order to achieve
better measurement precision, it is indeed desirable to maximize the QFI
during the time evolution.

Figure 2: Dynamical behavior of quantum Fisher information for different
initial relative phase values $\phi$, with $\theta=\pi/2$. The left and right
panels correspond to the atom being located in free space and the PBG with
$\omega_{3c}=-1\beta$, respectively. Figure 3: Dynamical behavior of quantum
Fisher information for different values of $\omega_{3c}$ with $\theta=\pi/2$
and $\phi=0$.

Figure 2 shows the dynamical behavior of $F_{\phi}$ and $F_{\theta}$ as a
function of the scaled time for different initial relative phase values
$\phi$, with $\theta=\pi/2$. The left and right panels correspond to the atom
being located in free space and the photonic band gap with
$\omega_{3c}=-1\beta$, respectively. As expected, both QFIs quickly approach
zero in free space, regardless of the initial relative phase value (see Figs.
2(a) and 2(c)). In contrast, $F_{\phi}$ and $F_{\theta}$ exhibit different
behavior when the atom is located in the photonic band gap. Despite variations
in the behavior of the QFIs across different initial relative phase values,
all of them display a monotonic oscillatory pattern within a range close to 1
and can be effectively protected (see Figs. 2(b) and 2(d)). It is worth noting
that setting the initial relative phase to $\phi=0$ leads to the maximum value
of the QFIs during the time evolution within the photonic band gap. Therefore,
the selection of an appropriate relative phase difference for the initial
states significantly influences the quantum Fisher information.

Figure 3 depicts the dynamical behavior of $F_{\phi}$ and $F_{\theta}$ as a
function of the scaled time for the different relative positions of the upper
levels from the forbidden gap, with $\phi=0$ and $\theta=\pi/2$. When the
different relative position of the upper level is $\omega_{3c}=0.9\beta$, the
QFIs exhibit damped oscillations and eventually decay to zero as time passes.
This implies that the QFIs cannot be preserved in this case. As the relative
position of the upper level from the forbidden gap decreases to
$\omega_{3c}=0.2\beta$ and $\omega_{3c}=-1\beta$, the process of losing
information slows down, and more photons are absorbed back into the quantum
system (qutrit). As a result, spontaneous emission is mainly suppressed, and
the quantum Fisher information (QFI) exhibits periodic oscillatory behavior
with constant amplitude. When the atomic transition frequency is well inside
the band gap with $\omega_{3c}=-1\beta$, the QFIs are closer to their maximum
value of 1, compared to the case of $\omega_{3c}=0.2\beta$. So one can find
that in a strongly non-Markovian environment, the decay of the QFI can be
suppressed, which means that the information about the qutrit can be preserved
for a longer time.

Figure 4: Density plot of $F_{\phi}$ and $F_{\theta}$ as a function of $\phi$
and $\theta$ in photonic band gap with $\omega_{3c}=-1\beta$.

Based on the previous plots, we expect that changing the value of $\theta$
while keeping $\phi$ constant has a symmetric effect on the evolution of QFIs.
To gain a broader perspective on this characteristic, in Fig. 4, we present a
contour plot that displays the values of $F_{\phi}$ and $F_{\theta}$ over a
wide range of initial state parameters $\phi$ and $\theta$ with
$\omega_{3c}=-1\beta$. By choosing appropriate initial state parameters
$\theta$ and $\phi$, we can obtain a high value for $F_{\phi}$ and
$F_{\theta}$.

Figure 5: Dynamical behavior of $\Sigma_{min}$ for different valuse of $\phi$
with $\theta=\pi/2$. The left and right panels correspond to the atom being
located in free space and the PBG with $\omega_{3c}=-1\beta$, respectively.
Figure 6: Dynamical behavior of $\Sigma_{min}$ for different valuse of
$\omega_{3c}$ with $\theta=\pi/2$ and $\phi=0$.

In the following, we used the QFIM approach to calculate the QCRB for the
simultaneous estimation of both parameters $\phi$ and $\theta$. Figure 5
illustrates the dynamical behavior of
$\Sigma_{min}=\min{(\Sigma(\theta,\phi))}$ as a function of scaled time for
different initial relative phase values $\phi$, with $\theta=\pi/2$. The left
and right panels correspond to the atom being located in free space and the
photonic band gap with $\omega_{3c}=-1\beta$, respectively.

In free space, $\Sigma_{min}$ increases dramatically for all initial relative
phases $\phi$ (see Fig. 5(a)). This could be due to decoherence, and the
absence of any enhancement mechanisms to preserve the information. In
contrast, in the photonic band gap, $\Sigma_{min}$ exhibits a relatively
stable behavior for all initial relative phases $\phi$ (see Fig. 5(b)). The
oscillatory behavior around an initial constant value with a small amplitude
indicates that $\Sigma_{min}$ is relatively well-preserved over time, so there
is a potential for measuring both parameters simultaneously in the photonic
band gap. It is evident that the optimal two-parameter estimation can be
obtained for $\phi=0$.

Figure 6 shows the time evolution of $\Sigma_{min}$ as a function of scaled
time for the different relative positions of the upper levels from the
forbidden gap, with $\phi=0$ and $\theta=\pi/2$. As anticipated, when the
relative position $\omega_{3c}=0.9\beta$, $\Sigma_{min}$ rapidly increases
over time. Furthermore, $\Sigma_{min}$ exhibits an oscillatory pattern with a
consistent amplitude for both $\omega_{3c}=0.2\beta$ and
$\omega_{3c}=-1\beta$. However, the amplitude of oscillation is quite small
when $\omega_{3c}=-1\beta$. Overall, for attaining the highest level of
optimal sensitivity in the two-parameter measurement, the photonic band gap
material with $\omega_{3c}=-1\beta$, is suitable.

##  IV Quantum Coherence

One of the fundamental features that separate quantum physics from classical
physics is the idea of quantum superposition, which leads to quantum
entanglement and quantum coherence. Quantum coherence is one of the most
important signatures of quantum theory and can be considered as a necessary
condition for quantum correlationYao et al. (2015); Streltsov et al. (2015);
Radhakrishnan et al. (2016). From the application point of view, it is a key
resource for quantum information processingPan et al. (2017); Mortezapour et
al. (2018) and quantum metrologyGiovannetti et al. (2011). We now investigate,
the effect of the relative phase of the initial state and the different
relative positions of the upper levels from the forbidden gap on the time
evolution of the quantum coherence in our three-level atom.

Many measures have been proposed for quantum coherence [1]. Among them we
focus here on the $l_{1}$ norm of coherence $C_{l_{1}}$. For an arbitrary
quantum state $\rho$, $C_{l_{1}}$ is quantified byBaumgratz et al. (2014)

Figure 7: Dynamical behavior of quantum coherence for different values of
$\phi$ with $\theta=\pi/2$. The left and right panels correspond to the atom
being located in free space and the PBG with $\omega_{3c}=-1\beta$,
respectively.

$C_{l_{1}}(\rho(t))=\sum_{i\neq j}|\rho_{ij}(t)|$ (27)

where $\rho_{ij}(i\neq j)$ are the off-diagonal elements of the density matrix
$\rho(t)$.

Figure 8: Dynamical behavior of quantum coherence for different values of
$\omega_{3c}$ with $\theta=\pi/2$ and $\phi=0$.

In Fig. 7, we plot the dynamical behaviors of the quantum coherence versus
scaled time for different values of the initial relative phase $\phi$, with
$\theta=\pi/2$. Figures 7(a) and 7(b), respectively, correspond to the case
when the atom is located in free space and photonic band gap material with
$\omega_{3c}=-1\beta$. The initial relative phase $\phi$ strongly affects the
dynamical behavior of quantum coherence in both cases. Figure 7(a) reveals
that although in free space (Markovian regime) the quantum coherence initially
strongly depends on the relative phase of the initial states, it tends
monotonously to zero in a short time for all values of the relative phase
$\phi$. Losing the information from an open quantum system to the environment
causes the quantum coherence to generally decrease over time. It is noticed
that for the case where the atom is in photonic band gap with
$\omega_{3c}=-1\beta$, the quantum coherence has a regular oscillation
behavior for all values of relative phase $\phi$. The amplitudes of the
periodic oscillations do not decrease during the time evolution. In addition,
when $\phi=\pi$ the quantum coherence oscillates with no decaying (see Fig.
7(b)).

Figure 8 depicts the time evolution of the quantum coherence for the different
relative positions of the upper levels from the forbidden gap with $\phi=0$
and $\theta=\pi/2$. It is obvious that when the relative position of the level
$|a_{3}\rangle$ from the forbidden gap is $\omega_{3c}=0.9\beta$, the
oscillation amplitude of the quantum coherence decays faster and reaches its
steady-state value. Moreover, when the relative position of the upper level
from the forbidden gap decreases to $\omega_{3c}=0.2\beta$ or
$\omega_{3c}=-1\beta$, the quantum coherence displays an oscillatory behavior
with a constant amplitude.

Physically, the oscillatory behavior of the quantum coherence in photonic band
gap material is interpreted as memory effects of the non-Markovian
environment. In addition, by choosing the appropriate relative phase of the
initial state and the position of the photon band gap, the most optimal state
can be obtained that maintains the highest amount of quantum coherence in the
system.

##  V Non-Markovianity

In this section, we are interested in studying the influences of system-
environment interaction on the dynamics of the open system. Usually the
interaction between an open quantum system and its environment results in
information and coherence loss. However, it can be observed that in a photonic
band gap, the non-Markovian effect can sustain the quantum Fisher information
and quantum coherence, preventing their rapid loss. These memory effects can
lead to richer dynamics, characterized by oscillations, revivals, or even
information backflow. In this paper, the Hilbert-Schmidt speed measure (HSS)
Jahromi et al. (2020); Mahdavipour et al. (2022) is utilized to quantify non-
Markovianity, which refers to the departure of a quantum system’s evolution
from a Markovian or memoryless process.

$HSS(\rho(\phi))=\sqrt{\frac{1}{2}Tr\left[\left(\frac{d\rho(\phi)}{d\phi}\right)^{2}\right]}$
(28)

It quantifies how much information about the future dynamics of the system can
be gained by observing the environment. HSS, as an efficient tool in quantum
metrology, helps in understanding the flow of information between a system and
its environment. Generally for an initial state as

$|\psi\rangle=\frac{1}{\sqrt{n}}(e^{i\phi}|\psi_{1}\rangle+...+|\psi_{n}\rangle)$
(29)

(here $\phi$ is an unknown phase shift and {$|\psi_{i}\rangle\\}$ with
$i=1,...,n$ is a complete and orthonormal basis of the Hilbert space) the HSS
witness of non-Markovianity is obtained by

$\chi(t)=\frac{dHSS(\rho(\phi))}{dt}>0$ (30)

in which $\rho(\phi)$ denotes the evolved state of the system.

Figure 9: Dynamical behavior of HSS for different values of $\phi$ with
$\theta=\pi/2$. The left and right panels correspond to the atom being located
in free space and the PBG with $\omega_{3c}=-1\beta$, respectively.

Figure 9 displays the dynamical behavior of HSS as a function of the scaled
time for different initial relative phase values $\phi$ , with $\theta=\pi/2$.
The left and right panels correspond to the atom being located in free space
and photonic band gap with $\omega_{3c}=-1\beta$, respectively. It is observed
that both situation show the same dynamical behavior of the HSS for the pairs
$\phi=0,\pi$ and $\phi=\pi/2,3\pi/2$. As is evident in the case of free space,
HSS exhibits Markovian behavior for all values of $\phi$ and shows no
significant changes in its time derivative $\chi(t)$ (see Fig. 9(a)). This
means that in free space, information always flows to the environment. On the
other hand, in the case of PBG, the time evolution of HSS exhibits
fluctuations for all values of $\phi$. The dynamics in this case are highly
non-Markovian, and the time derivative of $\chi(t)$ intermittently becomes
positive (see Fig. 9(b)). This indicates that during these times, the Hilbert-
Schmidt speed increases and leads to the backflow of information from the
environment to the system.

The dynamical behavior of HSS as a function of scaled time is illustrated in
Fig. 10 for different relative positions of the upper levels from the
forbidden gap, with $\phi=0$ and $\theta=\pi/2$. It is evident that when the
relative position of the upper level from the forbidden gap decreases to
$\omega_{3c}=0.2\beta$ or $\omega_{3c}=-1\beta$, the dynamics of HSS exhibit
oscillatory patterns. These oscillations indicate that the system’s
interaction with its environment has led to feedback or memory effects,
resulting in the flow of information to the environment and backflow to the
system. When the relative position of the upper level is
$\omega_{3c}=0.9\beta$, the HSS demonstrates damped oscillations and gradually
decays to zero as time increases. This implies the flow of information to the
environment and, occasionally, the backflow of information from the
environment to the system. Ultimately, over an extended duration, the
information flows completely into the environment. This analysis shows that
structured reservoirs, such as photonic band gap materials, enhance the non-
Markovianity of the system’s dynamics.

Figure 10: Dynamical behavior of HSS for different values of $\omega_{3c}$
with $\theta=\pi/2$ and $\phi=0$.

## CONCLUSION

We have analyzed the dynamical behavior of quantum Fisher information, quantum
coherence, and non-Markovianity of a V-type three-level atom embedded in free
space or a photonic band gap crystal. Our results imply that the photonic band
gap crystal, as a structured environment, significantly influences the
preservation and enhancement of these quantum features. Moreover, we have
examined in detail the effect of the initial relative phase values encoded in
the atomic state and the relative positions of the upper levels within the
forbidden gap on the evolution of quantum Fisher information, quantum
coherence, and non-Markovianity.

It is shown that when the atomic transition frequency is well inside the band
gap with $\omega_{3c}=-1\beta$ and $\phi=0$ , $F_{\phi}$ and $F_{\theta}$ are
closer to their maximum value of 1. This indicates that in a strongly non-
Markovian environment, the decay of the quantum Fisher information can be
suppressed, allowing for longer preservation of information about the qutrit.
Additionally, we have observed that the optimal two-parameter estimation can
be obtained for $\phi=0$ in the photonic band gap with $\omega_{3c}=-1\beta$.

It is noticed that for the case where the atom is in a photonic band gap with
$\omega_{3c}=-1\beta$, the quantum coherence exhibits an oscillating behavior
for all values of the relative phase $\phi$, corresponding to non-Markovian
evolution. The amplitudes of the periodic oscillations do not decrease during
the time evolution. The dynamical behavior of non-Markovianity measured by HSS
has confirmed our analysis of the dynamics of quantum coherence.

## Appendix

Using the isotropic dispersion relation of Eq. (10) we can evaluate the
corresponding Green’s function asWoldeyohannes and John (2003)

$G_{ij}(t-t^{\prime})=\beta^{3/2}_{j}\frac{e^{i[\delta_{j1}(t-t^{\prime})-\pi/4]}}{\sqrt{\pi(t-t^{\prime})}};(i,j=2,3),$
(A.1)

where
$\beta_{j}^{3/2}=[(\omega_{j}d_{j})^{2}/6\pi\epsilon_{0}\hslash](k_{0}^{3}/\omega_{c}^{3/2})$
and $(j=2,3)$. By taking the Laplace transform of Eqs. (8) and (9), we obtain.

$\displaystyle A_{3}(s$ $\displaystyle)=$ (A.2)
$\displaystyle\frac{\cos(\frac{\theta}{2})(s-i\omega_{32}+G_{22}(s-i\omega_{32}))-e^{i\phi}\sin(\frac{\theta}{2})G_{32}(s-i\omega_{32})}{[s+G_{33}(s)][(s-i\omega_{32})+G_{22}(s-i\omega_{32})]-G_{23}(s)G_{32}(s-i\omega_{32})},$

$\displaystyle A_{2}(s$ $\displaystyle-i\omega_{32})=$ (A.3)
$\displaystyle\frac{e^{i\phi}\sin(\frac{\theta}{2})[s+G_{33}(s)]-\cos(\frac{\theta}{2})G_{23}(s)}{[s+G_{33}(s)][(s-i\omega_{32})+G_{22}(s-i\omega_{32})]-G_{23}(s)G_{32}(s-i\omega_{32})},$

here G(s) is the Laplas transform of the Green’s function in Eq. (A.1), which
can be derived in the following form,

$G_{33}(s)=-\frac{\beta_{3}^{3/2}}{\sqrt{is+\omega_{3c}}},$ (A.4)

$G_{22}(s-i\omega_{32})=-\frac{\beta_{2}^{3/2}}{\sqrt{is+\omega_{3c}}},$ (A.5)

$G_{32}(s-i\omega_{32})=G_{23}(s)=-\frac{\beta_{3}^{3/4}\beta_{2}^{3/4}}{\sqrt{is+\omega_{3c}}},$
(A.6)

where $\omega_{3c}=\omega_{3}-\omega_{c}$. In the following discussion, we
consider that the two dipole moments are parallel to each other and for the
sake of simplicity $g^{21}_{k,\lambda}=g^{31}_{k,\lambda}=g_{k,\lambda}$ so,
$G_{33}(s)=G_{22}(s-i\omega_{32})=G_{23}(s)=G(s)$ and
$\beta_{3}=\beta_{2}=\beta$. In this case, by using the inverse Laplace
transform the amplitudes $A_{3}(t)$ and $A_{2}(t)$ can be obtained as Yang and
Zhu (2000).

$A_{3}(t)=\sum_{j}\frac{f_{1}(x^{1}_{j})}{Z^{\prime}(x_{j}^{1})}e^{ix_{j}^{1}t}+\sum_{j}\frac{f_{2}(x^{2}_{j})}{H^{\prime}(x_{j}^{2})}e^{ix_{j}^{2}t}-R_{3}(t),$
(A.7)

$A_{2}(t)=e^{-i\omega_{12}t}\left[\sum_{j}\frac{f_{3}(x^{1}_{j})}{Z^{\prime}(x_{j}^{1})}e^{ix_{j}^{1}t}+\sum_{j}\frac{f_{4}(x^{2}_{j})}{H^{\prime}(x_{j}^{2})}e^{ix_{j}^{2}t}\right]-R_{2}(t),$
(A.8)

where

$R_{3}(t)=\frac{\beta^{3/2}}{\pi\sqrt{i}}\int_{0}^{\infty}\frac{e^{-bt}\sqrt{x}a[\cos(\frac{\theta}{2})a+e^{i\phi}\sin(\frac{\theta}{2})b]}{xb^{2}a^{2}-i[2x-i(\omega_{3c}+\omega_{2c})]^{2}\beta^{3}}dx,$
(A.9)

$R_{2}(t)=\frac{\beta^{3/2}}{\pi\sqrt{i}}\int_{0}^{\infty}\frac{e^{-at}\sqrt{x}b[\cos(\frac{\theta}{2})a+e^{i\phi}\sin(\frac{\theta}{2})b]}{xb^{2}a^{2}-i[2x-i(\omega_{3c}+\omega_{2c})]^{2}\beta^{3}}dx.$
(A.10)

with $a=x-i\omega_{2c}$ , $b=x-i\omega_{3c}$ and the functions are defined as
follows

$f_{1}(x)=\cos(\frac{\theta}{2})(x-i\omega_{32})-\frac{[\cos(\frac{\theta}{2})-e^{i\phi}\sin(\frac{\theta}{2})]\beta^{3/2}}{\sqrt{ix+\omega_{3c}}},$
(A.11)

$f_{2}(x)=\cos(\frac{\theta}{2})(x-i\omega_{32})+\frac{[\cos(\frac{\theta}{2})-e^{i\phi}\sin(\frac{\theta}{2})]\beta^{3/2}}{\sqrt{ix+\omega_{3c}}},$
(A.12)

$f_{3}(x)=e^{i\phi}\sin(\frac{\theta}{2})x-\frac{[e^{i\phi}\sin(\frac{\theta}{2})-\cos(\frac{\theta}{2})]\beta^{3/2}}{\sqrt{ix+\omega_{3c}}},$
(A.13)

$f_{4}(x)=e^{i\phi}\sin(\frac{\theta}{2})x+\frac{[e^{i\phi}\sin(\frac{\theta}{2})-\cos(\frac{\theta}{2})]\beta^{3/2}}{\sqrt{ix+\omega_{3c}}},$
(A.14)

$Z(x)=x(x-i\omega_{32})-\frac{[2x-i\omega_{32}]\beta^{3/2}}{\sqrt{ix+\omega_{3c}}},$
(A.15)

$H(x)=x(x-i\omega_{32})+\frac{[2x-i\omega_{32}]\beta^{3/2}}{\sqrt{ix+\omega_{3c}}},$
(A.16)

where $Z^{\prime}(x)=\frac{dZ(x)}{dx}$ and $H^{\prime}(x)=\frac{dH(x)}{dx}$.
In Eqs. (A.7) and (A.8), $x_{j}^{1}$ are the root of $Z(x)=0$ in the region
$Im(x_{j}^{1})>\omega_{3c}$ or $Re(x_{j}^{1})>0$; $x_{j}^{2}$ are the root of
$H(x)=0$ in the region $Im(x_{j}^{2})<\omega_{3c}$ and $Re(x_{j}^{2})<0$.

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# A Solution to Galileo’s Enigma “Mostro Son Io”

G. Busetto and A. De Angelis
Dipartimento di Fisica e Astronomia “Galileo Galilei” dell’Università di
Padova

###### Abstract

Galileo Galilei was a skilled writer and explored several genres, from the
well-known scientific writings (often in the form of dialogs) to theater and
poetry. His last published poem, Mostro son io (A Monster am I), is a riddle
written in the form of a sonnet. We suggest that the solution to Galileo’s
riddle is the Zodiac.

To the second part of La Sfinge, a collection of riddling sonnets by Antonio
Malatesti published in Firenze in 1643 [1] one year after Galileo’s death, a
sonnet entitled Enimma (Enigma, or Riddle) by Galileo himself was added in the
beginning of the book. Malatesti had received the sonnet by Galileo as a sign
of appreciation: in the first part of La Sfinge, published in Venice in 1640
[2], the fifth sonnet was indeed dedicated to “Galileo’s telescope”. Enimma
was published with no solution. Galileo was a skilled writer and explored
several genres, from scientific writing (often in the form of dialog) to
theater and poetry [3, 4] and this is probably his last poem.

The original text of the Enimma is reported here:

> 1 Mostro son io più strano e più diforme
> 5 Che l’arpía, la sirena o la chimera;
> Né in terra, in aria, in acqua è alcuna fiera,
> Ch’abbia di membra così varie forme.
> Parte a parte non ho che sia conforme,
> più che s’una sia bianca e l’altra nera;
> Spesso di cacciator dietro ho una schiera,
> Che de’ miei piè van rintracciando l’orme.
> 9 Nelle tenebre oscure è il mio soggiorno;
> Ché se dall’ombre al chiaro lume passo,
> Tosto l’alma da me sen fugge, come
> 12 Sen fugge il sogno all’apparir del giorno;
> E le mie membra disunite lasso,
> E l’esser perdo, con la vita, e ’l nome.

that we translate as:

> 1 A monster am I, stranger in shape and form
> 5 Than a Harpy, a Siren or a Chimera;
> Nor has on Earth, in air, in water, any beast
> Such a varied forms of limbs.
>
>
> No part of mine to another does conform
> More than if one were white and the other black;
> Often of hunters behind me I have a host,
> Tracing the footprints of my feet.
> 9 In the obscure darkness is my sojourn:
> If from the shadows to clear light I pass,
> Quickly my soul from me flees out,
> 12
>
> Just as a dream flees out at break of day:
> And disjoined my limbs I leave,
> And my being I lose, my life and name.

Despite various suggestions had been proposed since then for who could be the
“monster” of the riddle, there is no agreement on the solution to this day.

In the introduction to his Galileo biography [5], Camerota suggests that it is
Galileo himself, although he finds this solution not completely persuasive.
Another possible solution, suggested by Bignami, is ‘a comet’ [6]; however
Enimma does not refer to, or suggest, anything contained in Galileo’s
extensive writing about comets [7]. Recently Peterson suggested the solution
to be the Ptolemaic system of the Universe [7]: a very brave proposal, but
unpersuasive to us, and somehow forced. Daniele [8] suggests a gambit: the
solution could be the telescope – the same solution as for Malatesti’s sonnet
dedicated to Galilei. Another suggestion, by Bartezzaghi, is [9] that the
solution is the enigma itself: also a gambit, facinating but also, to us,
questionable.

After a discussion, one of us (G.B.) had an intuition, that was elaborated by
the other (A.D.A.), and then together. We present it here: we are convinced
that the solution is the Zodiac.

According to the most diffuse ethimological interpetation [10], the word
Zodiac means “[circle of] little animals”, and originates from the Greek
zoidiakós, from zoidion (‘small animal’ or ‘sculptured animal figure’,
diminutive of zoion, ‘animal’). The name reflects the prominence of animals
(and mythological hybrids) among the twelve signs. A discussion on the
etymology is present in a book well known to Galilei: Kepler’s De Stella Nova
[11]. The first 6 verses seem to refer to this. In addition, the signs of the
Zodiac were integral to understanding how the human body functioned, and each
one was mapping a different part of the body in medical astrology, a “science”
having its roots in Greek astrology, though it fully bloomed in the Middle
Age. The correspondence, called melothesía, was discussd by Marcus Manilius
(1st century AD) in his epic poem Astronomica, well known to Galilei [12] and
to Kepler [11].

Verses 7 and 8 seem to refer to the constellation of Orion, the giant hunter.
Orion is not on the path of the ecliptic, and is “behind” the Zodiac.
According to a Greek cosmogonical myth reported for example by Gaius Iulius
Higinus (64 BC – 17 AD) in his Poeticon Astronomicon, Gaia, the goddess of
Earth and protector of the animals, was angered by Orion who was chasing all
animals, and asked Scorpio, a giant scorpion, to kill him before he could harm
them. Scorpio accomplished his mission and chased Orion on the opposit side of
the Zodiac. Notice that the word ‘cacciator’ (‘hunter’) in the original
sonnet, in Italian, is truncated with the effect that it can be singular or
plural – although the word “schiera” (host) inclines to the plural.

The last 6 verses are quite clear. In particular, verse 13 seems to refer to
the visual disappearance of the constellations in the morning which causes
losing track of the images associated to them (the stars with greater
megnitude disappear later).

Of course the riddle might refer to the only constellation. Scorpio is a
possible candidate. Another is the Ophiucus, the serpent bearer (on the
zodiacal circle but not listed among the Zodiac signs for
astrological/hystorical reasons), together with the serpent itself: Orion’s
stars “follow” Ophiucus’ feet (verse 8), and Galilei studied in detail the
1604 stella nova in Ophiucus’ left foot [13]. However these hypotheses appears
less immediate. And in any case, in the riddling game, the concept of “right
answer” contains some degree of arbitrariness.

## Acknowledgement –

We thank our colleague Maria Teresa Musacchio for check and suggestions on the
English translation of Galilei’s sonnet, and Alessandro Bettini for comments.

## References

  * [1] La Sfinge, Enimmi del Sig. Antonio Malatesti, Parte seconda, Stamperia di Sua Altezza Serenissima, Firenze 1643
  * [2] La Sfinge, Enimmi del Sig. Antonio Malatesti, Sarsina, Venezia 1640
  * [3] Le Opere di Galileo Galilei, Edizione Nazionale, Antonio Favaro ed., vol. I-XX, Barbera, Firenze 1890–1909
  * [4] A. De Angelis, Galileo and a lost poem on the 1604 supernova, arXiv: 2204.04001, April 2022
  * [5] M. Camerota, Galileo Galilei, Salerno, Roma 2004
  * [6] G. Bignami, in G. Galilei, Against the Donning of the Gown; Enigma, Moon Books, London 2000
  * [7] M. Peterson, Academia Letters, Article 2758 (2021)
  * [8] A. Daniele, “Galileo Letterato”, Padova 2009, in Intorno a Galileo, CLEUP, Padova 2022
  * [9] S. Bartezzaghi, Incontri con la sfinge, Einaudi, Torino 2004
  * [10] Merriam-Webster Thesaurus, https://www.merriam-webster.com
  * [11] J. Kepler, De stella nova in pede serpentarii, Prag 1606, Chapter 5
  * [12] C. Hall, Galileo’s Reading, Cambridge University Press, 2013
  * [13] G. Galilei, Lecture Notes on the stella nova appeared in 1604, in [3], vol. II