# Estimating phase parameters of a three-level system interacting with two
classical monochromatic fields in simultaneous and individual metrological
strategies

Nour-Eddine Abouelkhir  0000-0002-6164-0525 LPHE-Modeling and Simulation,
Faculty of Sciences, Mohammed V University in Rabat, Rabat, Morocco.  Abdallah
Slaoui  0000-0002-5284-3240 Corresponding author<EMAIL_ADDRESS>LPHE-Modeling and Simulation, Faculty of Sciences, Mohammed V University in
Rabat, Rabat, Morocco.  Centre of Physics and Mathematics, CPM, Faculty of
Sciences, Mohammed V University in Rabat, Rabat, Morocco.  Hanane El Hadfi
LPHE-Modeling and Simulation, Faculty of Sciences, Mohammed V University in
Rabat, Rabat, Morocco.  Rachid Ahl Laamara  LPHE-Modeling and Simulation,
Faculty of Sciences, Mohammed V University in Rabat, Rabat, Morocco.  Centre
of Physics and Mathematics, CPM, Faculty of Sciences, Mohammed V University in
Rabat, Rabat, Morocco.

###### Abstract

Recently, the Hilbert-Schmidt speed, as a special class of quantum statistical
speed, has been reported to improve the interferometric phase in single-
parameter quantum estimation. Here, we test this concept in the multiparameter
scenario where two laser phases are estimated in a theoretical model
consisting of a three-level atom interacting with two classical monochromatic
fields. When the atom is initially prepared in the lower bare state taking
into account the detuning parameters, we extract an exact analytical solution
of the atomic density matrix in the case of two-photon resonant transition.
Further, we compare the performance of laser phase parameters estimation in
individual and simultaneous metrological strategies, and we explore the role
of quantum coherence in improving the efficiency of unknown multi-phase shift
estimation protocols. The obtained results show that the Hilbert-Schmidt speed
detects the lower bound on the statistical estimation error as well as the
optimal estimation regions, where its maximal corresponds to the maximal
quantum Fisher information, the performance of simultaneous multiparameter
estimation with individual estimation inevitably depends on the detuning
parameters of the three-level atom, and not only the quantum entanglement, but
also the quantum coherence is a crucial resource to improve the accuracy of a
metrological protocol.

##  I Introduction

During the last two decades, the use of quantum systems to process information
has given rise to a new field of research called quantum information, which
has developed rapidly [1, 2]. At the same time, advances in the experimental
field now allow the controlled manipulation of many quantum systems [3, 4].
These two aspects have contributed to the development of new technologies,
which rely on the use of quantum resources, such as quantum entanglement [5,
6, 7] or quantum superposition [8, 9], to make the processes implemented more
efficient. On the other hand, the process of measurement, which aims to assign
a value to a physical quantity when producing an estimate, is one of the most
modern technological applications of the physical sciences. Every experimental
estimate is accompanied by an uncertainty that has an impact on the result of
the measurement, whether it is fundamental or technological. In fact, physical
laws impose fundamental restrictions on uncertainty such as those induced by
the Heisenberg relations. Additionally, the technical error is mainly
represented by an involuntary error resulting from uncontrollable defects in
the measurement procedure. Since quantum physics is the most predictive theory
and allows us to describe phenomena that have no equivalent in classical
physics, it is appropriate to study, within the framework of this theory, the
measurement process as well as the limitations of the achievable precision
[10, 11].

The quantum technology revolution, which aspires to create brand-new
technologies that take advantage of quantum phenomena, has prompted a recent
development of quantum estimation theory. It has been used to construct high-
precision measurements at the quantum frontier by providing theoretical tools
for a variety of estimation objectives. These applications include clock
synchronization [12], optimal estimation of phases [13, 14, 15], Unruh-Hawking
effect estimation [16], space-time parameters [17], reservoirs temperature
[18], the sensitivity of gravitational wave interferometry [19] and standard
frequency estimation [20]. Indeed, quantum metrology provides the tools to
obtain estimates of quantities of interest for a physical system but not
accessible by direct measurement [21, 22]. Among these quantities, there are
quantities that do not correspond to quantum observables such as the quantum
phase, the purity of a quantum state and quantum correlations. In particular,
the considerations made in this framework are quantified by calculating the
quantum Fisher information (QFI) (Fisher information (FI) in the classical
case) which allows to define the precision of the estimation of a physical
parameter present in the Hamiltonian of the system. According to the Cramer-
Rao bound (CRB) ${\rm Var}(\hat{\theta})\geq{\cal F}_{\theta}^{-1}$, where
${\rm Var}(\hat{\theta})$ is the variance of the estimator $\hat{\theta}$ and
${\cal F}_{\theta}$ denotes the FI quantity [23], the FI provides a lower
bound on the uncertainty of the parameter estimate. The QFI provides an upper
bound on the FI for any measurement and, therefore, the best possible
estimation precision is given by quantum mechanics. Besides, a small variance
provides the highest precision of the estimated parameter. Hence, the ultimate
goal of any quantum estimation process is to achieve the smallest variance
value. In this frame, it turned out that the optimal way to estimate a phase
shift in the presence of phase-diffusion and to obtain the ultimate quantum
limits of accuracy is to take into account schemes with noise as well as to
consider the environmental effects in the optical phase measurement [24].
Single-parameter unitary gates estimation, optimal probes and the role of
entanglement to improve the overall stability of the estimation scheme (i.e.
the robustness of the optimal settings with respect to fluctuations of the
probe and the measurement parameters), for qubit systems using Bayesian
inference are explored in ref [25]. Moreover, contrary to separable states,
the advantage of using entangled qubits as quantum probes to characterize the
noise induced by an external complex environment and to improve the estimation
of the correlation time has been pointed out in [26]. It is also interesting
to note that, in realistic scenarios, the improvement in quantum precision in
practical applications of phase estimation tasks is bounded by the phase
diffusion [27]. Besides, in Ref.[28] the same improvement has been proved for
a single qubit undergoing an unknown phase shift imposed by the unitary
dynamics of quantum state.

As previously mentioned, the main key to quantum estimation theory is the QCRB
which always reaches saturation in the case where a single parameter is
estimated. On the other hand, it is difficult to saturate this bound when many
parameters are estimated simultaneously, in which the variance is replaced by
covariance matrix ${\rm Cov}(\hat{\theta})$ and QFI by the QFI matrix
$F_{\theta}$ (i.e., ${\rm Cov}(\hat{\theta})\geq F_{\theta}^{-1}$). This
results from the incompatibility between the measurements of the different
estimated parameters. For this, multiparameter quantum estimation has
attracted a lot of interest and has become an important task in a variety of
diverse settings. Further, when multiple variables are estimated
simultaneously, in a way that can outperform individual estimation strategies,
simultaneous estimation can provide better precision than their individual
estimation. The myriad reasons for studying multi-parameter quantum estimation
schemes are deeply intertwined and many theoretical and experimental studies
have been conducted [29], among them decoherence parameters estimation [30],
linear and nonlinear phase shift estimation using two-mode entangled coherent
states [31], and estimating multiple-parameter unitary operators [32, 33, 34].
Moreover, the investigation of noise and dissipative effects on several open
quantum systems, in which the bosonic or fermionic character of their
constituents requires a reformulation of the standard concepts of information
theory, becomes of paramount importance. The research work in this area is not
only of theoretical but also of practical interest, as experimental
developments in the field of quantum system control have paved the way for
metrological applications in which the presence of coherence and entanglement
allows experimental accuracies in the estimation of some parameters otherwise
inaccessible [35, 36].

On the other hand, quantum optics is nowadays one of the liveliest fields of
physics; lately, it has experienced a very rapid development thanks to the
progress of experimental techniques that allow the creation and detection of
the photon, as well as the investigation of an atom inside the cavities.
Partly motivated by these advances, several theoretical and experimental works
have explored many quantum phenomena in these optical systems, such as the
enhancement of entanglement performance and the optical bistability of a two-
or even three-mode continuous-valued system, for example in a cavity
optomechanical system that studies the interplay between the light field and
the mechanical motion by the radiation pressure. Indeed, continued interests
have been focused on entanglement generation via the single-atom system and
its investigation for two moving mirrors coupled to a two-mode laser inside a
doubly resonant cavity was done in [37], and then in Ref.[38] where one-way
steering and stable entanglement in a single atom at four-levels interacting
simultaneously with two cavity modes are generated.

Emerging applications, such as ion traps, superconducting qubits, and quantum
dots, among others, are driving the exploration of the collective properties
of these quantum systems [39, 40, 41, 42]. Different strategies to describe
the interaction of these systems with the external field were proposed and one
of the most important models is the Jaynes-Cummings model (JCM) [43]. It
describes a system composed of a quantized mode interacting with a two-level
system, and was originally proposed to describe the interaction in a strongly
idealized way between a single atom and a single mode of the electromagnetic
field, both isolated from the influence of any perturbing environment. The JCM
is of great interest for atomic physics [44, 46, 47, 45], quantum optics [48]
and quantum information processing [49, 50], both theoretically and
experimentally. Furthermore, this model has been extended to additional
directions, such as adding more levels where three-level atomic systems under
different configuration; lamda ($\Lambda$), Cascade ($\Xi$) and Vee ($V$),
have been examined [51]. There is also evidence for the effects of atomic
motion [45], the Kerr-like medium [52], Stark-shift [53, 54] and multi-phonon
atomic transitions [44]. Inspired by these works, we investigate here
simultaneous and individual metrological strategies to improve the estimation
of phase parameters for a three-level atom interacting with laser fields in
the case of a two-photon resonant transition. The atomic density matrix under
the rotating wave approximation is obtained. Besides, we employ the concept of
Hilbert-Schmidt speed, as a quantum statistical speed quantifier, to detect
the accuracy of the estimated phase parameters. The roles of quantum coherence
and geometric phase in multi-parameter quantum estimation are also addressed.

This paper is arranged as follows: In Section II, we present the Hamiltonian
describing the interaction between a three-level atom and two classical
monochromatic fields. By using the Schrödinger equation, the atomic density
matrix when the atom is prepared in the lowest energy bare state is obtained.
The construction of the QFI matrix, the simultaneous and individual estimation
methods, and the bound on the joint estimation of the phase parameters are all
provided in Section III. Importantly, the QFI matrix is not diagonal and it
and the optimal bound depend on the phase parameters. In Section IV, we
devoted ourselves to the roles of the Hilbert-Schmidt speed as well as the
quantum coherence in the precision of the multiparametric quantum estimation.
We derive here their analytical expressions and compare their behaviors with
those of the QFI. Finally, conclusions and some features and comments are
given.

##  II Physical model and its atomic density matrix

The theoretical model to be considered is a idealized three-level atom
interacting with two classical monochromatic fields. As shown schematically in
Fig.(1), two levels $|l_{2}\rangle$ and $|l_{3}\rangle$ are coupled to a
single level $|l_{1}\rangle$ on two dipole-allowed transitions driven by
fields at frequencies $\Omega_{2}$ and $\Omega_{3}$, respectively. Assume that
there are two possible configurations of the bare states ($\Xi$ and
$\Lambda$-type) with two lower levels $|l_{2}\rangle$ and $|l_{3}\rangle$ are
coupled to a single upper level $|l_{1}\rangle$. Besides, the $E_{2}$ field
causing the transition $|l_{1}\rangle\to|l_{2}\rangle$ has a phase $\phi_{2}$
and is detuned from resonance by a frequency $\Delta_{2}$. The $E_{3}$ field
causing the transition $|l_{2}\rangle\to|l_{3}\rangle$ has a phase $\phi_{3}$
and is detuned by frequency $\Delta_{3}$ (see Fig.(1)).

Figure 1: The energy levels diagrams of a three-level atom subject to two
classical monochromatic fields, with detuning parameters $\Delta_{2}$ and
$\Delta_{3}$ for (a) $\Lambda$-configuration and (b) $\Xi$-configuration.

The Hamiltonian of the aforementioned model is given by
$\hat{H}=\hat{H}_{0}+\hat{V}$, where

$\hat{H}_{0}=\hbar\omega_{2}|l_{2}\rangle\langle
l_{2}|+\hbar\omega_{3}|l_{3}\rangle\langle l_{3}|,$ (1)

is the unperturbed Hamiltonian and the time-dependent perturbation is given by

$\displaystyle\hat{V}$ $\displaystyle=-\hat{\mu}_{2}E_{2}-\hat{\mu}_{3}E_{3}$
(2) $\displaystyle=-\mu_{2}E_{2}(|l_{1}\rangle\langle
l_{2}|+|l_{2}\rangle\langle l_{1}|)-\mu_{3}E_{3}\left(|l_{2}\rangle\langle
l_{3}|+|l_{3}\rangle\langle l_{2}|\right),$

with $\hat{\mu}_{2}$ (resp. $\hat{\mu}_{3}$) is the atomic dipole operator
associated with the two states $|l_{1}\rangle$ and $|l_{2}\rangle$ (resp.
$|l_{2}\rangle$ and $|l_{3}\rangle$). For a monochromatic field, the electric
fields can be written as

$E_{2}=E_{02}\cos\left(\Omega_{2}t+\phi_{2}\right)\quad\mbox{and}\quad
E_{3}=E_{03}\cos\left(\Omega_{3}t+\phi_{3}\right).$ (3)

Reporting this last equation (3) into the equation (2), this leads us to the
definition of a key parameter for atom-light interactions. The Rabi
frequencies associated with the coupling of the field modes are given by

$\vartheta_{2}=\mu_{2}E_{02}/\hbar,\quad\mbox{and}\quad\vartheta_{3}=\mu_{3}E_{03}/\hbar.$
(4)

Afterwards, using the rotating-wave approximation (RWA), the associated
Hamiltonian of the atom-field system can be written in terms of projection
operators as

$\displaystyle\hat{H}$ $\displaystyle=\hbar\omega_{2}|l_{2}\rangle\langle
l_{2}|+\hbar\omega_{3}|l_{3}\rangle\langle l_{3}|$ (5)
$\displaystyle-\frac{\hbar}{2}\vartheta_{2}\left(e^{i(\Omega_{2}t+\phi_{2})}|l_{1}\rangle\langle
l_{2}|+e^{-i(\Omega_{2}t+\phi_{2})}|l_{2}\rangle\langle l_{1}|\right)$
$\displaystyle-\frac{\hbar}{2}\vartheta_{3}\left(e^{i(\Omega_{2}t+\phi_{2})}|l_{2}\rangle\langle
l_{3}|+e^{-i(\Omega_{3}t+\phi_{3})}|l_{3}\rangle\langle l_{2}|\right).$

Since there are two laser frequencies in our system, the usual unitary
transformation to the rotating frame at $\vartheta$ does not apply. However,
it is possible to perform a unitary transformation that makes $\hat{H}$ time
independent, i.e. $|\psi(t)\rangle_{I}={\cal U}^{\dagger}|\psi(t)\rangle$
(rotating frame) where ${\cal
U}=\sum_{j=1}^{3}\exp\left[-i\gamma_{j}|l_{j}\rangle\langle l_{j}|\right]$. In
the interaction picture, the Hamiltonian takes the following form

$\hat{H}_{I}={\cal U}^{\dagger}\hat{H}{\cal U}+i\hbar\frac{\partial{\cal
U}^{\dagger}}{\partial t}{\cal U},$ (6)

Hence, the Hamiltonian can rewritten as follows

$\hat{H_{I}}=\hbar\left[\begin{array}[]{cccc}-\gamma_{1}&-\frac{1}{2}\vartheta_{2}e^{i(\Omega_{2}+\gamma_{1}-\gamma_{2})t+\phi_{2})}&0\\\
-\frac{1}{2}\vartheta_{2}e^{-i(\Omega_{2}+\gamma_{1}-\gamma_{2})t+\phi_{2})}&\omega_{2}-\gamma_{2}&-\frac{1}{2}\vartheta_{3}e^{i(\Omega_{3}-\gamma_{2}+\gamma_{3})t+\phi_{3})}\\\
0&-\frac{1}{2}\vartheta_{3}e^{-i(\Omega_{3}-\gamma_{2}+\gamma_{3})t+\phi_{3})}&\omega_{3}-\gamma_{3}\end{array}\right].$
(7)

Next, our goal is to determine the appropriate values of $\gamma_{j}$ for
making our Hamiltonian independent of time. For this purpose, we can choose
the following values for $\gamma_{j}$’s:

$\Omega_{2}+\gamma_{1}-\gamma_{2}=0,\quad\Omega_{3}-\gamma_{2}+\gamma_{3}=0,\quad\mbox{and}\quad-\gamma_{1}=0,$
(8)

from this, we can conclude that

$\gamma_{2}=\Omega_{2},\quad\gamma_{3}=\Omega_{2}-\Omega_{3},\quad\mbox{and}\quad\gamma_{1}=0.$
(9)

The time-independent Hamiltonian of the system in the rotating-wave
approximation can be written in terme of projection operators as

$\displaystyle\hat{H_{I}}=$ $\displaystyle\hbar\Delta_{2}|l_{2}\rangle\langle
l_{2}|+\hbar(\Delta_{2}+\Delta_{3})|l_{3}\rangle\langle l_{3}|$ (10)
$\displaystyle-\frac{\hbar}{2}\left(\vartheta_{2}e^{-i\phi_{2}}|l_{2}\rangle\langle
l_{1}|+\vartheta_{3}e^{i\phi_{3}}|l_{2}\rangle\langle l_{3}|+h.c\right),$

where the energy of the bare state $|l_{1}\rangle$ is zero. In our
consideration, we assume that the interaction occurs wherein the energy of the
bare state $|l_{1}\rangle$ comes to zero. The dressed states are the
eigenstates of this Hamiltonian (10) which are denoted by $|i\rangle$,
$|j\rangle$, and $|k\rangle$. These dressed states are superpositions of the
bare states $|l_{A}\rangle$, $|l_{2}\rangle$ and $|l_{3}\rangle$, as

$|a\rangle=\alpha_{a}|l_{1}\rangle+\beta_{a}|l_{2}\rangle+\gamma_{a}|l_{3}\rangle,\quad
a=i,j,\mbox{ or }k,$ (11)

where $\alpha_{a}$, $\beta_{a}$ and $\gamma_{a}$ are complex constants. We
require these states to be eigenstates satisfying the equations

$\hat{H}|a\rangle=\hbar\omega_{a}|a\rangle.$ (12)

Putting the equations (10) and (11) into (12), the eigenvalue equation can be
written as

$\displaystyle\left[-\frac{\vartheta_{2}}{2}e^{i\phi_{2}}\beta_{a}|l_{1}\rangle+\left(-\frac{\vartheta_{2}}{2}e^{-i\phi_{2}}\alpha_{a}+\Delta_{2}\beta_{a}-\frac{\vartheta_{3}}{2}e^{i\phi_{3}}\gamma_{a}\right)|l_{2}\rangle\right.$
$\displaystyle\left.+\left(-\frac{\vartheta_{3}}{2}e^{-i\phi_{3}}\beta_{a}+\left(\Delta_{2}+\Delta_{3}\right)\gamma_{a}\right)|l_{3}\rangle\right]=$
$\displaystyle\omega_{a}\left(\alpha_{a}|l_{1}\rangle+\beta_{a}|l_{2}\rangle+\gamma_{a}|l_{3}\rangle\right).$
(13)

By comparing the coefficients of the bare states, we obtain three simultaneous
equations for $\alpha_{a}$, $\beta_{a}$ and $\gamma_{a}$ which can be written
in matrix form as follows

$\left(\begin{array}[]{ccc}-\omega_{a}&-\frac{\vartheta_{2}}{2}e^{i\phi_{2}}&0\\\
-\frac{\vartheta_{2}}{2}e^{-i\phi_{2}}&\Delta_{2}-\omega_{a}&-\frac{\vartheta_{3}}{2}e^{i\phi_{3}}\\\
0&-\frac{\vartheta_{3}}{2}e^{-i\phi_{3}}&\Delta_{2}+\Delta_{3}-\omega_{a}\end{array}\right)\left(\begin{array}[]{c}\alpha_{a}\\\
\beta_{a}\\\ \gamma_{a}\end{array}\right)=0.$ (14)

Indeed, a non-trivial solution for $\alpha_{a}$, $\beta_{a}$ and $\gamma_{a}$
requires a zero determinant of the above matrix (14). This leads to

$\displaystyle\omega_{a}^{3}-\omega_{a}^{2}(2\Delta_{2}$
$\displaystyle+\Delta_{3})+\omega_{a}\left(\Delta_{2}(\Delta_{2}+\Delta_{3})-\frac{\vartheta_{2}^{2}}{4}-\frac{\vartheta_{3}^{2}}{4}\right)$
(15) $\displaystyle+\frac{\vartheta_{2}^{2}}{4}(\Delta_{2}+\Delta_{3})=0.$

Here we consider the situation in which the bare states $|l_{1}\rangle$ and
$|l_{3}\rangle$ have a two-photon resonant transition, in the sense that
$\Delta_{2}+\Delta_{3}=0$. If we write $\Delta_{2}=-\Delta_{3}=\Delta$, the
cubic equation (15) reads

$\omega_{a}^{3}-\Delta\omega_{a}^{2}-\frac{1}{4}(\vartheta_{2}^{2}+\vartheta_{3}^{2})\omega_{a}=0,$
(16)

and thus, the eigenfrequencies are provided by

$\displaystyle\omega_{i}=\frac{1}{2}\left(\Delta-\sqrt{\Delta^{2}+\vartheta_{2}^{2}+\vartheta_{3}^{2}}\right),$
(17) $\displaystyle\omega_{j}=0,$
$\displaystyle\omega_{k}=\frac{1}{2}\left(\Delta+\sqrt{\Delta^{2}+\vartheta_{2}^{2}+\vartheta_{3}^{2}}\right).$

Due to the fact that $\Delta_{2}=-\Delta_{3}$ and the selected interaction
picture, in which the bare state $|l_{1}\rangle$ has zero energy, the two-
photon resonance case has zero eigenfrequency. The normalized eigenvectors of
the matrix (14) with these eigenfrequencies can be expressed as follows

$\small|i\rangle=\frac{1}{\sqrt{4\omega_{i}^{2}+\vartheta_{2}^{2}+\vartheta_{3}^{2}}}\left(-\vartheta_{2}e^{i\phi_{2}}|l_{1}\rangle+2\omega_{i}|l_{2}\rangle-\vartheta_{3}e^{-i\phi_{3}}|l_{3}\rangle\right),$
(18)

and

$\small|k\rangle=\frac{1}{\sqrt{4\omega_{k}^{2}+\vartheta_{2}^{2}+\vartheta_{3}^{2}}}\left(-\vartheta_{2}e^{i\phi_{2}}|l_{1}\rangle+2\omega_{k}|l_{2}\rangle-\vartheta_{3}e^{-i\phi_{3}}|l_{3}\rangle\right).$
(19)

Thus, it is easier to find the eigenvector corresponding to the zero
eigenfrequency from the Hamiltonian matrix (14) with
$\omega_{a}=\omega_{j}=0$. Starting from the resultant matrix, we observe that
the normalized eigenvector is

$|j\rangle=\frac{1}{\sqrt{\vartheta_{2}^{2}+\vartheta_{3}^{2}}}\left(-\vartheta_{3}e^{i\phi_{3}}|l_{1}\rangle+\vartheta_{2}e^{-i\phi_{2}}|l_{3}\rangle\right).$
(20)

In fact, the dressed state above (20) is the eigenvector corresponding to the
zero eigenfrequency and it is a superposition of only the bare states
$|l_{1}\rangle$ and $|l_{3}\rangle$ and does not include any contribution from
the bare state $|l_{2}\rangle$. A sum over the dressed states determines the
system’s evolved state so that

$|\psi(t)>=C_{i}e^{-i\omega_{i}t}|i>+C_{j}|j>+C_{k}e^{-i\omega_{k}t}|k>,$ (21)

where $C_{a}$ are complex amplitudes calculated from the initial state of the
atom. We assume the atom to be initially in its lower bare state, so that
$|\psi(0)>=|l_{1}>$. Under such condition we have

$|l_{1}\rangle=C_{i}|i\rangle+C_{j}|j\rangle+C_{k}|k\rangle,$ (22)

with

$\displaystyle C_{i}=\langle
i|l_{1}\rangle=\frac{-1}{\sqrt{4\omega_{a}^{2}+\vartheta_{2}^{2}+\vartheta_{3}^{2}}}\vartheta_{2}e^{-i\phi_{2}},$
(23) $\displaystyle C_{j}=\langle
j|l_{1}\rangle=\frac{-1}{\sqrt{\vartheta_{2}^{2}+\vartheta_{3}^{2}}}\vartheta_{3}e^{-i\phi_{3}},$
$\displaystyle C_{k}=\langle
k|l_{1}\rangle=\frac{-1}{\sqrt{4\omega_{c}^{2}+\vartheta_{2}^{2}+\vartheta_{3}^{2}}}\vartheta_{2}e^{-i\phi_{2}}.$

In the interesting limiting case, the detuned $\Delta$ is much larger than
either $\vartheta_{2}$ or $\vartheta_{3}$. Therefore, the eigenfrequencies are
derived by expanding (17) to be

$\omega_{i}\simeq-\frac{(\vartheta_{2}^{2}+\vartheta_{3}^{2})}{4\Delta},\quad\omega_{j}=0,\quad\omega_{k}\simeq\Delta,$
(24)

with $\omega_{i}$ holds an lower order of magnitude than both $\vartheta_{2}$
and $\vartheta_{3}$, and $\omega_{j}$ holds an higher order. Moreover,
according to (23), we get

$C_{i}\simeq\frac{-\vartheta_{2}e^{-i\phi_{2}}}{\sqrt{\vartheta_{2}^{2}+\vartheta_{3}^{2}}},\hskip
28.45274ptC_{j}\simeq\frac{-\vartheta_{3}e^{-i\phi_{3}}}{\sqrt{\vartheta_{2}^{2}+\vartheta_{3}^{2}}},$
(25)

however $C_{k}$, which is of order $1/\Delta$, may be disregarded. Using the
aforementioned findings with this order of approximation, the state (21) is a
superposition of just two dressed states provided by

$|\psi(t)\rangle\simeq\frac{-1}{\sqrt{\vartheta_{2}^{2}+\vartheta_{3}^{2}}}\left[\vartheta_{2}e^{-i\phi_{2}}e^{-i\omega_{i}t}|i\rangle+\vartheta_{3}e^{-i\phi_{3}}|j\rangle\right].$
(26)

After the derivation of the atomic density matrix operator,
$\rho=\left|\psi(t)\right\rangle\left\langle\psi(t)\right|$, we can calculate
the corresponding QFI matrix to estimate the phase parameters $\phi_{2}$ and
$\phi_{3}$ as well as we will investigate the simultaneous and individual
estimation strategies for the considered system.

##  III Quantum Multiparameter Estimation Theory

Effectively, estimation theory refers to the branch of science that examines
the accuracy with which a given set of physical parameters can be estimated.
In multi-parameter estimation scenarios, multiple variables are evaluated
simultaneously and the theoretical limits on their sensitivity have become
possible through the FI matrix, which quantifies the amount of information
contained in a data set about unknown parameters, and its related Cramér-Rao
bound [22, 23]. To estimate certain parameters
$\boldsymbol{\theta}=\left\\{\theta_{1},\theta_{2},...,\theta_{n}\right\\}$
that is an open subset of ${\cal R}^{n}$ and encrypted in the dynamics
$\varphi_{\theta}$, with
$\boldsymbol{\hat{\theta}}=\left\\{\hat{\theta}_{1},\hat{\theta}_{2},...,\hat{\theta}_{n}\right\\}$
being an estimator of $\boldsymbol{\theta}$ and $\hat{\theta}_{\nu}$ the
estimator of $\theta_{\nu}$, first prepare a probe state $\rho$ and allow it
to evolve under the evolution
$\rho\xrightarrow{\varphi_{\theta}}\rho_{\theta}$ , and then, by conducting
POVM measurements $\left\\{\Pi_{x}\right\\}$ on the output state
$\rho_{\theta}$, the result of measurement $x$ with the probability
$p(x/\theta)={\rm Tr}\left(\Pi_{x}\rho_{\theta}\right)$ is obtained. According
to the Cramér-Rao theorem which asserts that for any measure $\Pi_{x}$ and an
unbiased estimator $\hat{\theta}_{\nu}$, the precision of the parameter
estimate is bounded by

$N{\rm Cov}\left(\boldsymbol{\hat{\theta}}\right)\geq{\cal
I}^{-1}\left(\boldsymbol{\theta}\right),$ (27)

where $N$ is the number of independent measurements, ${\rm Cov}(\hat{\theta})$
stands for the covariance matrix of all unbiased estimators $\hat{\theta}$
whose elements are ${\rm
Cov}(\hat{\theta})_{\mu\nu}=E\left[\theta_{\mu}\theta_{\nu}\right]-E\left[\theta_{\mu}\right]E\left[\theta_{\nu}\right]$,
$E$ being a mathematical expectation, and ${\cal
I}\left(\boldsymbol{\theta}\right)$ is the classical Fisher information matrix
with the $\mu\nu$th entry, in terms of the distribution function
$p(x/\theta)$, given by

${\cal I}\left(\boldsymbol{\theta}\right)_{\mu\nu}=\int
p(x/\theta)\left(\frac{\partial\log
p(x/\theta)}{\partial\theta_{\mu}}\frac{\partial\log
p(x/\theta)}{\partial\theta_{\nu}}\right)dx.$ (28)

Here, the attainability conditions and the optimal estimators are those where
the inequality CCRB is saturated. Moreover, the maximum estimator is found to
be optimal in the limit of a large number of measurements
$N\longmapsto\infty$. An upper quantum limit on the CFI matrix is provided for
a quantum system by the fluctuation of the unknown parameters in the system
state $\rho_{\theta}$, which allows to obtain the quantum Cramér-Rao
inequality

$N{\rm Cov}\left(\boldsymbol{\hat{\theta}}\right)\geq{\cal
I}^{-1}\left(\boldsymbol{\theta}\right)\geq\mathcal{F}^{-1}\left(\boldsymbol{\theta}\right),$
(29)

where the $\mu\nu$th entry of quantum Fisher information matrix
$\mathcal{F}^{-1}\left(\boldsymbol{\theta}\right)$ is given by

$\mathcal{F}_{\mu\nu}\left(\theta\right)=\frac{1}{2}{\rm
Tr}\left[\rho_{\theta}(\hat{L}_{\theta_{\mu}}\hat{L}_{\theta_{\nu}}+\hat{L}_{\theta_{\nu}}\hat{L}_{\theta_{\mu}})\right],$
(30)

which depends only on the final state $\rho_{\theta}$ since the measurement
$\Pi_{x}$ was automatically optimized on all POVMs, where the symmetric
logarithmic derivative (SLD) $\hat{L}_{\theta_{\mu}}$ (with $\mu=1,..,n$)
satisfies the following differential equation

$\frac{\partial\rho_{\theta}}{\partial\theta_{\mu}}=\frac{1}{2}\\{\hat{L}_{\theta_{\mu}}\rho_{\theta}+\rho_{\theta}\hat{L}_{\theta_{\mu}}\\}.$
(31)

The above equation needs to be solved to derive the SLDs
$\hat{L}_{\theta_{\mu}}$ corresponding to the various estimated parameters.
For a pure state $\rho_{\theta}^{2}\equiv\rho_{\theta}$, which interests us
here, solving the equation (31) gives us the expression of SLD as
$\hat{L}_{\theta_{\mu}}=2\partial_{\theta_{\mu}}\rho_{\theta}$ and then the
QFI matrix elements become

$\mathcal{F}_{\mu\nu}=4Re\left[\langle\partial_{\theta_{\mu}}\psi|\partial_{\theta_{\nu}}\psi\rangle-\langle\partial_{\theta_{\mu}}\psi|\psi\rangle\langle\psi|\langle\partial_{\theta_{\nu}}\psi\rangle\right].$
(32)

According to the QCR bound, one may be able to estimate the parameter
$\theta_{\mu}$ by making an appropriate measurement more precisely with a
smaller statistical error, when the QFI matrix element $\mathcal{F}_{\mu\mu}$
is larger. Typically, the quantum Cramer-Rao inequality (29) become scalar
inequalities and it is always saturable by optimizing over all valid quantum
measurements in single-parameter estimation scenarios, i.e ${\rm
Var}(\theta)\geq\mathcal{F}^{-1}_{\theta\theta}$. By using projectors on the
eigenvectors of the SLD operators $L_{\theta}$, this saturation yields the
optimal quantum measurement operators. To the contrary, the matrix Cramer-Rao
inequality in multiparameter estimation protocols, ${\rm
Cov}(\boldsymbol{\hat{\theta}})\geq\mathcal{F}^{-1}(\boldsymbol{\theta})$, is
usually not saturable. Indeed, the measurements made by the optimal operators
for various parameters may not be compatible. Hence, quantum bounds on
precision are not generally achieved. In order to saturate the bound (29), we
should be aware of the influence of this incompatibility on our estimation
problem. In fact, one can find a common eigenbasis for all SLDs in the case
where the $L_{\theta}$ operators commute. This implies that we can perform a
simultaneous measurement saturating the QCR inequality [55, 56, 57]. In the
situation where the SLDs are not commuted, the condition ${\rm
Tr}\\{\hat{\rho}[\hat{L}_{\theta_{\mu}},\hat{L}_{\theta_{\nu}}]\\}=0$ for
$\forall(\theta_{\mu},\theta_{\nu})\in\boldsymbol{\theta}$ is sufficient for
saturating the QCR bound. This commutation condition is reduced for pure
states to

${Im}\langle\partial_{\theta_{\mu}}\psi|\partial_{\theta_{\nu}}\psi\rangle=0.$
(33)

Generally speaking, the QCR bound can be attainable if a matrix ${\cal
U}_{\theta_{\nu}\theta_{\mu}}=-\frac{i}{2}{\rm
Tr}(\rho[\hat{L}_{\theta_{\mu}},\hat{L}_{\theta_{\nu}}])$ called Uhlmann
curvature matrix with elements in equation (33) vanishes, thus the influence
of the incompatibility condition on multiparameter estimation problems is
quantified by the quantity ${\cal
R}_{\boldsymbol{\theta}}:=\|2i\mathcal{F}^{-1}\left(\boldsymbol{\theta}\right){\cal
U}_{\boldsymbol{\theta}}\|_{\infty}$ referred to as quantumness, where $\|{\bf
B}\|_{\infty}$ stands for the largest eigenvalue of the matrix ${\bf B}$ [58,
59]. Importantly, the maximum incompatibility between the measures associated
with the estimated parameters is equivalent to the saturation of the upper
bound, and in this case the quantumness ${\cal R}_{\boldsymbol{\theta}}=1$.
This quantity has been used to evaluate quantitatively the critical phenomena
of the geometric phase for many-body systems interacting with critical chains
in non-equilibrium phase transitions [60]. Further, recent theoretical
contributions based on the Uhlmann curvature matrix have suggested superior
accuracy and stability performance when using quantum probes [61] and when
using coherently driven nonlinear Kerr resonators [62] over their classical
counterparts.

In our considered model described by equation (26), where we want to estimate
the phase parameters of the external fields $\phi_{2}$ and $\phi_{3}$, it is
simple to verify that the condition (33) is fulfilled by taking
$\theta_{\mu}\equiv\phi_{2}$ and $\theta_{\nu}\equiv\phi_{3}$. Instead, it is
straightforward to check that $[\hat{L}_{\phi_{2}},\hat{L}_{\phi_{3}}]=0$,
which does not imply any indeterminacy due to the quantum compatibility
arising from the quantum nature of the parameters estimation problem. Thus,
the optimal values of the two estimated parameters are mutually compatible and
then have a common optimal basis. The eigenvalues of such SLD operators
provide the optimal states and therefore their projections are the optimal
measurements. Therefore, the QCR bound is always saturable and the optimal
accuracy is obtained when the QFI of the estimated parameter is maximized.

###  III.1 Individual Metrological Strategy

We examine here the best strategies for extracting the information about the
phase parameters of two classical monochromatic fields $\phi_{2,3}$ coupled to
a three level atom. In the single-parameter protocol, where the optimal
scenario is the one that produces the minimum estimation variance, the non-
zero QFI matrix elements are found only along the diagonal which are given by

$\mathcal{F}_{\theta\theta}=4\left[\langle\partial_{\theta}\psi|\partial_{\theta}\psi\rangle-|\langle\partial_{\theta}\psi|\psi\rangle|^{2}\right].$
(34)

In this strategy in which we estimated the phase parameters individually,
these parameters are statistically independent and the accurate identification
of a single phase does not affect the precision of the other, so that
$\mathcal{F}_{\phi_{2}\phi_{3}}$. Therefore, the QCR inequalities become

${\rm Var}\left(\phi_{2}\right)_{\rm
Ind}\geq\mathcal{F}^{-1}_{\phi_{2}},\hskip 14.22636pt{\rm and}\hskip
14.22636pt{\rm Var}\left(\phi_{3}\right)_{\rm
Ind}\geq\mathcal{F}^{-1}_{\phi_{3}}.$ (35)

Equation (35) show the existence of an optimal probe state containing a
maximum amount of QFI with exact values of the estimated phase parameters, for
which the best accuracy is provided. After some straightforward algebra, the
analytical expressions of the QFI for the laser phase $\phi_{2}$ is

$\mathcal{F}_{\phi_{2}}=4\left(4\vartheta_{2}^{2}\kappa+\vartheta_{2}^{2}\vartheta_{3}^{2}\xi\right)\left[1-\left(4\vartheta_{2}^{2}\kappa+\vartheta_{2}^{2}\vartheta_{3}^{2}\xi\right)\right],$
(36)

and the QFI for the laser phase $\phi_{3}$ is written as

$\mathcal{F}_{\phi_{3}}=4\vartheta_{2}^{2}\vartheta_{3}^{2}\xi\left(1-\vartheta_{2}^{2}\vartheta_{3}^{2}\xi\right),$
(37)

with the quantities $\kappa$ and $\xi$ are given by

$\displaystyle\kappa=$
$\displaystyle\frac{\omega_{i}^{2}}{(4\omega_{i}^{2}+\vartheta_{2}^{2}+\vartheta_{3}^{2})^{2}}+\frac{\omega_{k}^{2}}{(4\omega_{k}^{2}+\vartheta_{2}^{2}+\vartheta_{3}^{2})^{2}}$
$\displaystyle+\frac{2\omega_{i}\omega_{k}\cos((\omega_{i}-\omega_{k})t)}{(4\omega_{i}^{2}+\vartheta_{2}^{2}+\vartheta_{3}^{2})(4\omega_{k}^{2}+\vartheta_{2}^{2}+\vartheta_{3}^{2})},$
(38)

and

$\displaystyle\xi$
$\displaystyle=\frac{1}{(4\omega_{i}^{2}+\vartheta_{2}^{2}+\vartheta_{3}^{2})^{2}}+\frac{1}{(4\omega_{k}^{2}+\vartheta_{2}^{2}+\vartheta_{3}^{2})^{2}}$
$\displaystyle+\frac{2\cos((\omega_{i}-\omega_{k})t)}{(4\omega_{i}^{2}+\vartheta_{2}^{2}+\vartheta_{3}^{2})(4\omega_{k}^{2}+\vartheta_{2}^{2}+\vartheta_{3}^{2})}+\frac{1}{(\vartheta_{2}^{2}+\vartheta_{3}^{2})^{2}}$
$\displaystyle\quad-\frac{2}{\vartheta_{2}^{2}+\vartheta_{3}^{2}}\left(\frac{\cos(\omega_{i}t)}{4\omega_{i}^{2}+\vartheta_{2}^{2}+\vartheta_{3}^{2}}+\frac{\cos(\omega_{k}t)}{4\omega_{k}^{2}+\vartheta_{2}^{2}+\vartheta_{3}^{2}}\right).$
(39)

###  III.2 Simultaneous Metrological Strategy

We can now address the problem of generalizing this treatment to the
simultaneous multi-parameter strategy, where we aim to measure these two
physical quantities $\phi_{2,3}$ simultaneously and seeking to get as close as
possible to the highest possible accuracy. Actually, a simultaneous estimation
strategy requires less quantum resources than the individual estimation
technique. If the estimated parameters are compatible, none of the parameters
loses accuracy but the required resources (e.g. coherence, squeezing,
entanglement or energy in the input state preparation) are thus reduced,
resulting in the greatest possible improvement. In order to describe
uncertainty of estimators
$\boldsymbol{\hat{\theta}}\equiv\left\\{\phi_{2},\phi_{3}\right\\}$ in the
simultaneous metrological schemes, the covariance matrix ${\rm
Cov}(\boldsymbol{\hat{\theta}})$ can be renewed as

${\rm
Cov}\left(\boldsymbol{\hat{\theta}}\right)=\left[\begin{array}[]{ccc}{\rm
Var}\left(\phi_{2}\right)&{\rm Cov}\left(\phi_{2},\phi_{3}\right)\\\ {\rm
Cov}\left(\phi_{3},\phi_{2}\right)&{\rm
Var}\left(\phi_{3}\right)\end{array}\right],$ (40)

and the inverse of QFI matrix can be written straightly as

$\mathcal{F}^{-1}\left(\boldsymbol{\theta}\right)=\frac{1}{{\rm
det}\left(\mathcal{F}\left(\boldsymbol{\theta}\right)\right)}\left[\begin{array}[]{ccc}\mathcal{F}_{\phi_{3}}&-\mathcal{F}_{\phi_{2}\phi_{3}}\\\
-\mathcal{F}_{\phi_{3}\phi_{2}}&\mathcal{F}_{\phi_{2}}\end{array}\right],$
(41)

with ${\rm
det}\left(\mathcal{F}\left(\boldsymbol{\theta}\right)\right)=\mathcal{F}_{\phi_{2}}\mathcal{F}_{\phi_{3}}-\mathcal{F}_{\phi_{2}\phi_{3}}^{2}$.
Replacing equations (40) and (41) in the inequality (29) which are both
positive semidefinite, and using Sylvester’s criterion [63], we obtain

${\rm Var}\left(\phi_{2}\right)_{\rm
Sim}\geq\frac{\mathcal{F}_{\phi_{3}}}{{\rm
det}\left(\mathcal{F}\left(\boldsymbol{\theta}\right)\right)},\hskip
14.22636pt{\rm Var}\left(\phi_{3}\right)_{\rm
Sim}\geq\frac{\mathcal{F}_{\phi_{2}}}{{\rm
det}\left(\mathcal{F}\left(\boldsymbol{\theta}\right)\right)},$ (42)

and the important trade-off between ${\rm Var}\left(\phi_{\mu,\nu}\right)$,
$\mathcal{F}_{\phi_{\mu,\nu}}$ and ${\rm
Cov}\left(\phi_{\mu},\phi_{\nu}\right)$ is captured by

$\displaystyle\left({\rm Var}\left(\phi_{2}\right)_{\rm
Sim}-\frac{\mathcal{F}_{\phi_{3}}}{{\rm
det}\left(\mathcal{F}\left(\boldsymbol{\theta}\right)\right)}\right)\left({\rm
Var}\left(\phi_{3}\right)_{\rm Sim}-\frac{\mathcal{F}_{\phi_{2}}}{{\rm
det}\left(\mathcal{F}\left(\boldsymbol{\theta}\right)\right)}\right)$
$\displaystyle\geq\left({\rm
Cov}\left(\phi_{2},\phi_{3}\right)-\frac{\mathcal{F}_{\phi_{2}\phi_{3}}}{{\rm
det}\left(\mathcal{F}\left(\boldsymbol{\theta}\right)\right)}\right).$ (43)

By exploiting the equation (32), one can easily show that

$\mathcal{F}_{\phi_{2}\phi_{3}}=\mathcal{F}_{\phi_{3}\phi_{2}}=4\vartheta_{2}^{2}\vartheta_{3}^{2}\xi\left[1-\left(4\vartheta_{2}^{2}\kappa+\vartheta_{2}^{2}\vartheta_{3}^{2}\xi\right)\right],$
(44)

in term of the quantities $\kappa$ (eq.38) and $\xi$ (eq.39). To compare the
performance of the estimates of the phase parameters in the individual and
simultaneous schemes, we use here a technique whereby we introduce the ratio
between the total variances in these two strategies as follows

$\gamma=\frac{\Delta_{\rm Ind}}{\Delta_{\rm Sim}},$ (45)

with the total variance in the individual case is

$\Delta_{\rm Ind}={\rm Var}\left(\phi_{2}\right)_{\min}^{\rm Ind}+{\rm
Var}\left(\phi_{3}\right)_{\min}^{\rm Ind},$ (46)

and in the simultaneous case become

$\Delta_{\rm Sim}=\frac{1}{2}\left[{\rm Var}\left(\phi_{2}\right)_{\min}^{\rm
Sim}+{\rm Var}\left(\phi_{3}\right)_{\min}^{\rm Sim}\right].$ (47)

Here it should be mentioned that the minimal values of the variances for
estimating parameter ($\phi_{2},\phi_{3}$) individually are

${\rm Var}\left(\phi_{2}\right)_{\min}^{\rm
Ind}=\mathcal{F}_{\phi_{2}}^{-1},\hskip 14.22636pt{\rm
Var}\left(\phi_{3}\right)_{\min}^{\rm Ind}=\mathcal{F}_{\phi_{3}}^{-1},$ (48)

and for the simultaneous estimation are

$\rm{Var}\left(\phi_{2}\right)_{\min}^{\rm
Sim}=\frac{\mathcal{F}_{\phi_{3}}}{{\rm
det}\left(\mathcal{F}\left(\boldsymbol{\theta}\right)\right)},\hskip
14.22636pt{\rm Var}\left(\phi_{3}\right)_{\min}^{\rm
Sim}=\frac{\mathcal{F}_{\phi_{2}}}{{\rm
det}\left(\mathcal{F}\left(\boldsymbol{\theta}\right)\right)}.$ (49)

Reporting these last equations into equation (45), the ratio $\gamma$ can then
be written rather elegantly as

$\gamma=\frac{2{\rm
det}\left(\mathcal{F}\left(\boldsymbol{\theta}\right)\right)}{\mathcal{F}_{\phi_{2}}\mathcal{F}_{\phi_{3}}}.$
(50)

In the total variance formula $\Delta_{\rm Sim}$ we inserted a factor $1/2$,
since we estimated two parameters simultaneously. This factor is necessary to
take into account the reduction in resources which showed that the
simultaneous strategy requires $2$ less resources than the individual scheme
in the multiparameter estimation procedures. As a result, $\gamma\leq 2$
generally, and when the ratio $\gamma>1$, the error limit of the simultaneous
parameter estimation scheme is smaller and offers an advantage over that of
the individual case.

$\left(a\right)$

$\left(b\right)$

$\left(c\right)$

$\left(d\right)$

Figure 2: Dynamics of the single-parameter quantum Fisher information
associated to the parameter $\phi_{2}$ and $\phi_{3}$ for different values of
the dipole matrix element $\vartheta_{3}$ when $\vartheta_{2}=1$Hz and;
$\Delta=0.04$Hz for panel ($a$) and panel ($c$), $\Delta=30$Hz for panel ($b$)
and panel ($d$).

$\left(a\right)$

$\left(b\right)$

Figure 3: Dynamics of the performance ratio $\gamma=\Delta_{\rm
Ind}/\Delta_{\rm Sim}$ of the phase parameters to be estimated $\phi_{2,3}$;
panel (a) for various values of the detuning parameter $\Delta$ with the
dipole matrix elements are fixed as $\vartheta_{2}=\vartheta_{3}=1$Hz, panel
(b) for various values of the parameter $\vartheta_{3}$ with
$\vartheta_{2}=0.5$Hz and detuning parameter is set to $\Delta=30$Hz.

The results reported in Fig.2 illustrate the dynamics behavior of single-
parameter quantum Fisher information related to the phase parameters
$\phi_{2,3}$ for various values of the dipole matrix element $\vartheta_{3}$
when the three-level atom is initially prepared in its lower bare state.
Remarkably, it is intriguing to notice that classical light fields have a high
ability to reducing the error, captured by QFI, on the estimated phase
parameters. Indeed, QFI has an oscillatory behavior with the same amplitudes
and which increases with the increasing of the elements $\vartheta$. Returning
to Eq.(4) in which $\vartheta_{2,3}\propto E_{2,3}$, we conclude that the
total variances of the estimated phase parameters are minimal, and therefore
the estimation error is reduced, when the classical monochromatic fields
increase. Moreover, the QFI vanishes for $t=0$ showing that the system evolves
to a phase-sensitive state and no information on the phase parameters can be
extracted from the three-level atom. Thereafter, the two classical
monochromatic fields leads to the generation of QFI. This generated QFI
survives during a certain time interval. Then, the amount of QFI decreases
after which it increases again until it reaches its maximum. Moreover,
increasing the dipole matrix element $\vartheta_{3}$ leads to enhancing the
amount of QFI contained in the system. This implies that the best estimate of
the phase parameters $\phi_{2,3}$ is achieved when the value of
$\vartheta_{3}$ is large. Importantly, by comparing Fig.2($a$) with Fig.2($b$)
for the estimated parameter $\phi_{2}$ and Fig.2($c$) with Fig.2($d$) for the
estimated parameter $\phi_{3}$, we observe that the oscillatory amplitudes
increase, thus the state is phase-insensitive, and a large amount of QFI could
be preserved with the increase of the detuning parameter. Therefore, the large
detuning parameters (where any three-level atom configuration includes two of
these parameters and its third one has disappeared) helps to improve the
precision of quantum estimation and it is important to use the three-level
atom with high values of $\Delta$ to obtain the best estimation efficiency of
the estimated parameters.

To evaluate the performance of metrological strategies, we display the
performance ratio between the simultaneous and individual estimation schemes
versus the time $t$ for various values of the detuning parameter in
Fig.3($a$), and for various values of the parameter $\vartheta_{3}$ in
Fig.3($b$). As shown in these Figures, the performance ratio takes its maximum
value at $t=0$, which is clear that no advantage is attained by estimating the
phase parameters individually. Also, the performance of simultaneous
estimation is better and would provide more precise result than that of
individual estimation for small values of both the parameters $\Delta$ and
$\vartheta_{3}$. As depicted in Fig.3($a$), we remark that increasing the
detuning parameter decreases the performance ratio and thus the superiority of
the individual scheme over all parameter space. In Fig.3($b$), we visualize
the influence of the dipole matrix element $\vartheta_{3}$ on the performance
ratio $\gamma$ by taking $\vartheta_{2}=0.5$ and $\vartheta_{3}=30$. Here, the
increase of the dipole matrix elements of the three-level atom leads to a
rapid decrease in the ratio between the minimal total variances, indicates
that the performance of the individual estimation is better than the
simultaneous estimation.

Our results show that the detuning and dipole matrix elements resulting from
applying the two classical monochromatic fields can improve or reduce the
precision of the parameters to be estimated. In other words, the optimal
estimation efficiency and the performance of simultaneous and individual
metrological strategies can be controlled by adjusting these two parameters.
Accordingly, with the desirable choice, one can enhance or annihilate the
estimation accuracy of the laser phases in the system.

##  IV Dynamics of relevant quantum criteria and their role in quantum
estimation

In this section, we turn our attention on the evolution of the quantum
coherence (QC) and the Hilbert-Schmidt speed (HSS) for the considered model
(10) as well as on their role in the phase estimation by comparing their
dynamical behaviors with those of the QFI.

###  IV.1 Quantum Coherence

As a significant resource in quantum information science, QC resulting from
quantum superposition is a key feature of quantum mechanics and it is crucial
for a variety of quantum effects. In fact, coherent superposition of states is
a key property of quantum systems and a necessary condition for quantum
entanglement and other quantum resources [64, 65, 66]. Hence, QC directly
affects the efficiency and reliability of quantum information processing. A
rigorous framework for QC resource theory was recently introduced by Baumgratz
and his colleagues [67] and some bona fide measures that satisfy all known
criteria for a given quantum state have been put forward [68, 69]. Several
quantifiers are significant distance-based measures between the considered
state $\rho$ and the nearest inconsistent state $\sigma$, i.e.
$C_{D}\left(\rho\right)=\min_{\sigma\in I}D\left(\rho,\sigma\right)$ with $I$
is the set of all incoherent states, and the overall form of its distance is
$D\left(\rho,\sigma\right)=\parallel\rho-\sigma\parallel$ where
$\parallel.\parallel$ being some kind of norm matrix. In this paper, we mainly
consider the $l_{2}$-norm of coherence as a measure of QC. It is defined as

$C_{l_{2}}\left(\rho\right)=\min_{\sigma\in
I}\parallel\rho-\sigma\parallel^{2}_{l_{2}}=\sum_{i,j}^{i\neq
j}|\rho_{ij}|^{2}.$ (51)

Through calculation, in our theoretical model characterized by the atomic
reduced density matrix (26), we can achieve the analytical expression of
$l_{2}$-norm of coherence as follows

$C_{l_{2}}\left(\rho\right)=2\left(|\varepsilon\varkappa|^{2}+|\varepsilon\varpi|^{2}+|\varkappa\varpi|^{2}\right),$
(52)

with the variables $\varepsilon$, $\varkappa$ and $\varpi$ are given by

$\displaystyle\varepsilon=2\vartheta_{2}\left(\omega_{i}W_{i}+W_{k}\omega_{k}\right),$
$\displaystyle\varkappa=\vartheta_{2}^{2}\left(W_{i}+W_{k}\right)+\vartheta_{3}^{2}/(\vartheta_{2}^{2}+\vartheta_{3}^{2}),$
$\displaystyle\varpi=\vartheta_{2}\vartheta_{3}\left(W_{i}+W_{k}-1/(\vartheta_{2}^{2}+\vartheta_{3}^{2})\right),$
(53)

where

$W_{i}=\frac{e^{-i\omega_{i}t}}{4\omega_{i}^{2}+\vartheta_{2}^{2}+\vartheta_{3}^{2}},\hskip
14.22636pt{\rm and}\hskip
14.22636ptW_{k}=\frac{e^{-i\omega_{k}t}}{4\omega_{k}^{2}+\vartheta_{2}^{2}+\vartheta_{3}^{2}}.$
(54)

Figure 4: Dynamics comparison of quantum coherence (solid blue curve), the
total variance in the individual (black dashed curve) and simultaneous (red
dotted curve) schemes; here, $\vartheta_{2}=1$Hz, $\vartheta_{3}=0.3$Hz and
$\Delta=0.04$Hz.

###  IV.2 Hilbert-Schmidt Speed

We analyze here the HSS [70], as a special type of quantum statistical speed,
and see how it can be related to the QFI in order to improve the estimation of
the quantum phase. Actually, HSS is recognized as an efficient method to
identifying non-Markovianity in $d$-qubit open quantum systems [71]. Consider
first the class of distance measures between the two probability distributions
$p=\\{p_{x}\\}_{x}$ and $q=\\{q_{x}\\}_{x}$ that are described as

$|d_{\alpha}(p,q)|^{\alpha}=\frac{1}{2}\sum_{x}|p_{x}-q_{x}|^{\alpha},$ (55)

where $\alpha\geq 1$. To derive the statistical speed from any given
statistical distance, we start by parameterizing the probability distribution
$p\left(\theta\right)$ and then extend it to first order in $\theta$ at
$\theta_{0}$, that is,

$p_{x}\left(\theta_{0}+\theta\right)=p_{x}\left(\theta_{0}\right)+\frac{\partial
p_{x}\left(\theta\right)}{\partial\theta}|_{\theta=\theta_{0}}\theta+{\cal
O}\left(\theta^{2}\right).$ (56)

Employing this expansion, we obtain

$d_{\alpha}\left(p\left(\theta_{0}+\theta\right),p\left(\theta_{0}\right)\right)=\left(\frac{1}{2}\sum_{x}|p^{\prime}_{x}\left(\theta_{0}\right)|^{\alpha}\right)^{\frac{1}{\alpha}}\theta+{\cal
O}\left(\theta^{2}\right),$ (57)

where
$p^{\prime}_{x}\left(\theta_{0}\right)=\partial_{\theta}p_{x}\left(\theta\right)|_{\theta=\theta_{0}}$.
The classical statistical speed associated with the classical distance (55) is
given by

$\displaystyle s_{\alpha}\left[p\left(\theta_{0}\right)\right]$
$\displaystyle=\partial_{\theta}\left[d_{\alpha}\left(p\left(\theta_{0}+\theta\right),p\left(\theta\right)\right)\right]$
$\displaystyle=\left(\frac{1}{2}\sum_{x}|p^{\prime}_{x}(\theta_{0})|^{\alpha}\right)^{\frac{1}{\alpha}}.$
(58)

When extending these classical notions to the quantum case and considering a
given pair of quantum states $\rho$ and $\sigma$, then write $p_{x}={\rm
Tr}\\{E_{x}\rho\\}$ and $q_{x}={\rm Tr}\\{E_{x}\sigma\\}$ which denote the
measurement probabilities corresponding to the POVMs. The associated quantum
distance can be obtained by maximizing the classical distance
$d_{\alpha}(p,q)$ over all possible POVM choices. It is defined as [70]

$D_{\alpha}(\rho,\sigma):=\max_{\\{E_{x}\\}}d_{\alpha}(p,q)=\left(\frac{1}{2}{\rm
Tr}|\rho-\sigma|^{\alpha}\right)^{\frac{1}{\alpha}}.$ (59)

Therefore, we get the quantum statistical speed in the following form

$S_{\alpha}\left[\rho_{\theta}\right]=\max_{\\{E_{x}\\}}s_{\alpha}\left[p\left(\theta\right)\right]=\left(\frac{1}{2}Tr|\partial_{\theta}\rho_{\theta}|^{\alpha}\right)^{\frac{1}{\alpha}},$
(60)

and for the situation where $\alpha=2$, the quantum statistical speed reduces
to the Hilbert-Schmidt speed as

$HSS\left[\rho_{\theta}\right]=HSS_{\theta}=S_{2}\left[\rho_{\theta}\right]=\sqrt{\frac{1}{2}{\rm
Tr}|\partial_{\theta}\rho_{\theta}|^{2}},$ (61)

which do not necessitate the diagonalization of
$\partial_{\theta}\rho_{\theta}$. For a three-level atom interacting with two
classical monochromatic fields, the analytical expression of the HSSs
associated with the estimated phase parameters $\phi_{2,3}$ is obtained by
applying equation (61) to equation (26). The result is

$HSS_{\phi_{2}}=\sqrt{\varLambda\left(\Theta+\Sigma\right)},\hskip
14.22636ptHSS_{\phi_{3}}=\sqrt{\Sigma\left(\varLambda+\Theta\right)},$ (62)

with the quantities $\varLambda$, $\Theta$ and $\Sigma$ are given by

$\displaystyle\varLambda=$
$\displaystyle\frac{\vartheta_{2}^{4}}{(4w_{i}^{2}+\vartheta_{2}^{2}+\vartheta_{3}^{2})^{2}}+\frac{\vartheta_{2}^{4}}{(4w_{k}^{2}+\vartheta_{2}^{2}+\vartheta_{3}^{2})^{2}}$
$\displaystyle+\frac{2\vartheta_{2}^{4}\cos((w_{i}-w_{k})t)}{(4w_{i}^{2}+\vartheta_{2}^{2}+\vartheta_{3}^{2})(4w_{k}^{2}+\vartheta_{2}^{2}+\vartheta_{3}^{2})}+\left(\frac{\vartheta_{3}^{2}}{\vartheta_{2}^{2}+\vartheta_{3}^{2}}\right)^{2}$
$\displaystyle+\frac{2\vartheta_{2}^{2}\vartheta_{3}^{2}}{\vartheta_{2}^{2}+\vartheta_{3}^{2}}\left(\frac{\cos(w_{i}t)}{4w_{i}^{2}+\vartheta_{2}^{2}+\vartheta_{3}^{2}}+\frac{\cos(w_{k}t)}{4w_{i}^{2}+\vartheta_{2}^{2}+\vartheta_{3}^{2}}\right),$
(63) $\displaystyle\Theta=$
$\displaystyle\frac{\omega_{i}^{2}}{(4\omega_{i}^{2}+\vartheta_{2}^{2}+\vartheta_{3}^{2})^{2}}+\frac{\omega_{k}^{2}}{(4\omega_{k}^{2}+\vartheta_{2}^{2}+\vartheta_{3}^{2})^{2}}$
$\displaystyle+\frac{2\omega_{i}\omega_{k}\cos((\omega_{i}-\omega_{k})t)}{(4\omega_{i}^{2}+\vartheta_{2}^{2}+\vartheta_{3}^{2})(4\omega_{k}^{2}+\vartheta_{2}^{2}+\vartheta_{3}^{2})},$
(64)

and

$\displaystyle\Sigma=$
$\displaystyle\frac{1}{(4\omega_{i}^{2}+\vartheta_{2}^{2}+\vartheta_{3}^{2})^{2}}+\frac{1}{(4\omega_{k}^{2}+\vartheta_{2}^{2}+\vartheta_{3}^{2})^{2}}$
$\displaystyle+\frac{2\cos((\omega_{i}-\omega_{k})t)}{(4\omega_{i}^{2}+\vartheta_{2}^{2}+\vartheta_{3}^{2})(4\omega_{k}^{2}+\vartheta_{2}^{2}+\vartheta_{3}^{2})}+\frac{1}{(\vartheta_{2}^{2}+\vartheta_{3}^{2})^{2}}$
$\displaystyle\quad-\frac{2}{\vartheta_{2}^{2}+\vartheta_{3}^{2}}\left(\frac{\cos(\omega_{i}t)}{4\omega_{i}^{2}+\vartheta_{2}^{2}+\vartheta_{3}^{2}}+\frac{\cos(\omega_{k}t)}{4\omega_{k}^{2}+\vartheta_{2}^{2}+\vartheta_{3}^{2}}\right).$
(65)

To see the role of quantum coherence in the quantum estimation error boundary,
the evolution of $l_{2}$-norm of coherence, total variance in the individual
and simultaneous schemes are plotted in Fig.4. It is found from Fig.4 that all
these quantities exhibit a periodic behavior during the time evolution.
Interestingly, the $l_{2}$-norm of coherence reaches the maximum possible
value when the total variances in both metrological strategies are minimal.
This implies that there is a relation between the quantum coherence and the
total invariances of the estimated parameters where the decrease of quantum
coherence corresponds to the growth of total invariances and the maximum total
invariances corresponds to the minimum quantum coherence. Based on these
results, a higher precision in all metrological strategies is achieved for the
highest value of quantum coherence; this is in accordance with the results
given by quantum entanglement [72, 73, 74] where the maximum entanglement of
the probe states constitutes a key feature to obtain an optimal multi-
parametric estimation.

$\left(a\right)$

$\left(b\right)$

$\left(c\right)$

$\left(d\right)$

Figure 5: The comparison between the dynamics of the Hilbert-Schmidt speed
(orange dashed curve) and the quantum Fisher information (solid blue curve)
associated with the estimated phase parameters $\phi_{2,3}$; for panel ($a$)
and panel ($c$) we fixed $\vartheta_{2}=1$Hz, $\vartheta_{3}=0.3$Hz and
$\Delta=0.04$Hz; for panel ($b$) and panel ($d$) we take
$\vartheta_{2}=\vartheta_{3}=0.5$Hz and $\Delta=30$Hz.

In order to quantitatively compare HSS and QFI related to the laser phase
parameters, we plot in Fig.5 their evolutions versus the time $t$ for dipole
matrix elements are set to $\vartheta_{2}=1$, $\vartheta_{3}=0.3$ and the
detuning parameter is adjusted to $\Delta=0.04$ in Fig.5(($a$) for phase
$\phi_{2}$ and ($c$) for phase $\phi_{3}$) and for
$\vartheta_{2}=\vartheta_{3}=0.5$ and $\Delta=30$ in Fig.5(($b$) for phase
$\phi_{2}$ and ($d$) for phase $\phi_{3}$). It is evident from Fig.5 that both
HSS and QFI dynamics exhibit oscillatory behaviors at the same time as well as
their minimum and maximum locations aligning precisely. These results provide
a qualitative indication that the HSS can be used to identify the precise
times at which the best estimate of the estimated phase parameters occurs,
corresponding to the maximum quantity of both HSS and QFI. Furthermore, our
analysis has shown that the HSS can be employed as a practical and effective
figure of merit to predict phase parameters, as it is a simple quantity to
calculate and has the advantage of avoiding the diagonalization of the density
matrix (26) compared to QFI. These results are completely in agreement with
the results reported in [75] which reflect that the HSS is a strong figure of
merit to improve the quantum phase estimation in a multi-qubit quantum system.
More broadly, it makes sense to investigate the links between these two
concepts since they are quantum statistical speeds related to the Bures and
Hilbert-Schmidt distances, respectively.

##  V Conclusion

To conclude, the investigation of the three-level atomic system in various
configurations interacting with a one- or two-mode field has received much
attention in quantum optics. Using the rotating wave approximation, the atom-
fields model has been extended to describe the idealized three-level atomic
system. On another hand, quantum optical metrology addresses the estimation of
an unknown phase by exploiting the quantum character of the considered input
state. Its ultimate goal is to achieve a strong sensitivity of some probe
states to small variations of external parameters as well as to find an
ultimate measurement scheme to go beyond the standard quantum limits by which
every quasi-classical estimation measurement is bounded, which opens great
opportunities to increase the resolution of interferometric measurements.

Here we summarize the important results obtained in this work. First, an exact
solution of the considered theoretical model has been achieved which describes
the interaction between a three-level atom and two classical monochromatic
fields in the case of two-photon resonant transition, when the atom is
initially prepared in the lower bare state taking into account the detuning
parameters. Then, we investigated the multiparameter estimation strategy in
the phase estimation protocol in the considered model by estimating two laser
phases. We found their associated total variances, which gives us the optimal
settings that effectively predict the values of the laser phase parameters,
using the Cramér-Rao multiparameter quantum bound for simultaneous and
individual estimation. Furthermore, the performance of the simultaneous
parameter estimation was compared with the individual estimation by entering
the ratio between the minimum amounts of total variances for each metrological
strategies. The effect of detuning parameters and dipole matrix elements are
explored, and our results suggest that the performance of simultaneous
estimation was better and provided a more accurate result than individual
estimation when both parameters had low values.

We also examine the roles of the Hilbert-Schmidt speed as well as the quantum
coherence captured by the $l_{2}$-norm of coherence, contained in the three-
level system, in improving the efficiency of quantum metrology protocols,
particularly in multi-parameter unknown phase shift estimation protocols. We
have shown that the Hilbert-Schmidt speed provides a powerful factor of merit
for improving laser phases estimation, has an analytical computational
advantage and testing its performance in large dimensional systems. Compared
to the QFI, the HSS detects the regions where the estimation is optimal and
plays the same role as the QFI in quantum metrology. Besides, quantum
coherence assumes the same role as quantum entanglement in quantum estimation,
whereby the maximum coherence coincides with the minimum variance of the
estimated parameters, meaning that quantum coherence becomes an important
resource for improving the sensitivity of quantum phase estimation. Building
on the results obtained, a question that may naturally be arisen about the
applicability of our analysis to other three-level atoms in various open-
system interactions with the environment, including the atom-optomechanical
system and many atom-cavity QED systems. In such models, the interaction
between the probing system and its environment will be taken into account and
the quantum advantage in phase detection in these real-world scenarios will be
seriously affected. We hope to report on this issue in a forthcoming work.

Lastly, we hope that the current paper can help readers to investigate the
three-level system in various tasks of quantum information theory, which may
open new perspectives on quantum optical metrology.

Disclosures: The authors declare no conflicts of interest or personal
relationships that could have appeared to influence this work, and no Data
associated in the manuscript.

## References

  * [1] D. Gabor, Communication theory and physics, Philos. Mag. Ser.7, 41 (1950) 1161-1187.
  * [2] J. Watrous, The theory of quantum information, Cambridge university press, (2018).
  * [3] A. S. Holevo, Probabilistic and Statistical Aspects of Quantum Theory, vol 1, 2011 (Berlin: Springer Science and Business Media).
  * [4] C. W. Helstrom, Quantum detection and estimation theory, Journal of Statistical Physics, 1 (1969) 231-252.
  * [5] E. Schrödinger, Die gegenwärtige Situation in der Quantenmechanik, Naturwissenschaften, 23 (1935) 807.
  * [6] A. Einstein, B. Podolsky and N. Rosen, Phys. Rev, 47 (1935) 777.
  * [7] A. El Makouri, A. Slaoui and M. Daoud, Enhancing the performance of coupled quantum Otto thermal machines without entanglement and quantum correlations, J. Phys. B: At. Mol. Opt. Phys, 56 (2023) 085501.
  * [8] R. Rouse, S. Han and J. E. Lukens, Observation of resonant tunneling between macroscopically distinct quantum levels, Phys. Rev. Lett, 75 (1995) 1614.
  * [9] Y. Nakamura, Y. A. Pashkin and J. S. Tsai, Coherent control of macroscopic quantum states in a single-Cooper-pair box, Nature, 398 (1999) 786-788.
  * [10] A. Slaoui, L. B. Drissi, E. H. Saidi and R. Ahl Laamara, Analytical techniques in single and multi-parameter quantum estimation theory: a focused review, arXiv:2204.14252.
  * [11] M. El Bakraoui, A. Slaoui, H. El Hadfi and M. Daoud, Enhancing the estimation precision of an unknown phase shift in multipartite Glauber coherent states via skew information correlations and local quantum Fisher information, J. Opt. Soc. Am. B, 39 (2022) 1297-1306.
  * [12] R. Joza, D. S. Abrams, J. P. Dowling and C.P. Williams, Quantum clock synchronization based on shared prior entanglement, Phys. Rev. Lett, 85 (2000) 2010.
  * [13] M. A. Ballester, Entanglement is not very useful for estimating multiple phases, Phys. Rev. A, 70 (2004) 032310.
  * [14] N. Abouelkhir, H. E. Hadfi, A. Slaoui and R. A. Laamara, A simple analytical expression of quantum Fisher and Skew information and their dynamics under decoherence channels, Physica A: Statistical Mechanics and its Applications, (2023) 128479.
  * [15] M. Aspachs, J. Calsamiglia, R. Muñoz Tapia and E. Bagan, Phase estimation for thermal Gaussian states, Phys. Rev. A, 79 (2009) 033834.
  * [16] M. Aspachs, G. Adesso, and I. Fuentes, Optimal quantum estimation of the Unruh-Hawking effect, Phys. Rev. Lett, 105 (2010) 151301.
  * [17] P. D. Nation, M. P. Blencowe, A. J. Rimberg and E. Buks, nalogue Hawking radiation in a dc-SQUID array transmission line, Phys. Rev. Lett, 103 (2009) 087004.
  * [18] L. A. Correa, M. Mehboudi, G. Adesso and A. Sanpera, Individual quantum probes for optimal thermometry, Phys. Rev. Lett, 114 (2015) 220405.
  * [19] R. Schnabel, N. Mavalvala, D. E. McClelland and P. K. Lam, Quantum metrology for gravitational wave astronomy, Nature Commun, 1 (2010) 121.
  * [20] J. M. Boss, K. S. Cujia, J. Zopes and C. L. Degen, Quantum sensing with arbitrary frequency resolution, Science, 356 (2017) 837-840.
  * [21] S. L. Braunstein and C. M. Caves, Statistical distance and the geometry of quantum states, Phys. Rev. Lett, 72 (1994) 3439-3443.
  * [22] M. G. A. Paris, Quantum estimation for quantum technology, Int. J. Quant. Inf, 7 (2009) 125-137.
  * [23] H. Cramér, Mathematical Methods of Statistics, Princeton University Press, 1999.
  * [24] M. G. Genoni, S. Olivares and M. G. Paris, Optical phase estimation in the presence of phase diffusion, Phys. Rev. Lett, 106 (2011) 153603.
  * [25] B. Teklu, S. Olivares and M. G. Paris, Bayesian estimation of one-parameter qubit gates. J. Phys. B: Atomic, Molecular and Optical Physics, 42 (2009) 035502.
  * [26] M. A. Rossi and M. G. Paris, Entangled quantum probes for dynamical environmental noise, Phys. Rev. A, 92 (2015) 010302.
  * [27] D. Brivio, S. Cialdi, S. Vezzoli, B. T. Gebrehiwot, M. G. Genoni, S. Olivares and M. G. Paris, Experimental estimation of one-parameter qubit gates in the presence of phase diffusion, Phys. Rev. A, 81 (2010) 012305.
  * [28] B. Teklu, M. G. Genoni, S. Olivares and M. G. Paris, Phase estimation in the presence of phase diffusion: the qubit case, Phys. Scr, 2010 (2010) 014062.
  * [29] M. Szczykulska, T. Baumgratz and A. Datta, Multi-parameter quantum metrology, Advances in Physics: X, 1 (2016) 621-639.
  * [30] P. J. Crowley, A. Datta, M. Barbieri, and I. A. Walmsley, Tradeoff in simultaneous quantum-limited phase and loss estimation in interferometry, Phys. Rev. A, 89 (2014) 023845.
  * [31] J. Cheng, Quantum metrology for simultaneously estimating the linear and nonlinear phase shifts, Phys. Rev. A, 90 (2014) 063838.
  * [32] P. Kolenderski and R. Demkowicz-Dobrzanski, Optimal state for keeping reference frames aligned and the platonic solids, Phys. Rev. A, 78 (2008) 052333.
  * [33] E. Bagan, M. Baig, and R. Muñoz-Tapia, Aligning reference frames with quantum states, Phys. Rev. Lett, 87 (2001) 257903.
  * [34] C. Vaneph, T. Tufarelli, and M. G. Genoni, Quantum estimation of a two-phase spin rotation, Quantum Measurements and Quantum Metrology, 1 (2013) 12.
  * [35] F. Benatti, R. Floreanini and U. Marzolino, Sub-shot-noise quantum metrology with entangled identical particles, Annals of Physics, 325 (2010) 924-935.
  * [36] D. Braun, G. Adesso, F. Benatti, R. Floreanini, U. Marzolino, M. W. Mitchell and S. Pirandola, Quantum-enhanced measurements without entanglement, Rev. Mod. Phys, 90 (2018) 035006.
  * [37] B. Teklu, T. Byrnes and F. S. Khan, Cavity-induced mirror-mirror entanglement in a single-atom Raman laser, Phys. Rev. A, 97 (2018) 023829.
  * [38] F. Wang, C. Wang, K. Shen and X. Hu, Stable entanglement and one-way steering via engineering of a single-atom reservoir, Optics Express, 30 (2022) 15830-15845.
  * [39] L. Allen and J.H. Eberly, Optical Resonance and Two-Level Atoms, Cambridge University, Cambridge, England, (1997).
  * [40] D.X. Li and X.Q. Shao, Rapid population transfer of a two-level system by a polychromatic driving field, Sci. Rep, 9 (2019) 9023.
  * [41] A. M. Steane, The ion trap quantum information processor, Appl. Phys. B, 64 (1997) 623.
  * [42] P. Michler, Quantum dots for quantum information technologies, Springer international publishing, Berlin (2017).
  * [43] E. T. Jaynes and F. W. Cummings, Comparison of quantum and semiclassical radiation theories with application to the beam maser, Proc. IEEE, 51 (1963) 89.
  * [44] C. V. Sukumar and B. Buck, Multi-phonon generalisation of the Jaynes-Cummings model, Phys. Lett. A, 83 (1981) 211-213.
  * [45] R. R. Schlicher, Jaynes-Cummings model with atomic motion, Optics communications, 70 (1989) 97-102.
  * [46] M. I. Shaukat, A. Slaoui, H. Terças and M. Daoud, Phonon-mediated quantum discord in dark solitons, Eur. Phys. J. Plus, 135 (2020) 1-14.
  * [47] A. Slaoui, M. I. Shaukat, M. Daoud and R. A. Laamara, Universal evolution of non-classical correlations due to collective spontaneous emission, Eur. Phys. J. Plus, 133 (2018) 413.
  * [48] J. Larson and T. K. Mavrogordatos, The Jaynes-Cummings model and its descendants, (2022) arXiv:2202.00330.
  * [49] A. Blais, S. M. Girvin and W. D. Oliver, Quantum information processing and quantum optics with circuit quantum electrodynamics, Nature Physics, 16 (2020) 247-256.
  * [50] M. H. Devoret and R. J. Schoelkopf, Superconducting circuits for quantum information: an outlook, Science, 339 (2013) 1169-1174.
  * [51] X. S. Li and D. L. Lin, Nonresonant interaction of a three-level atom with cavity fields. I. General formalism and level occupation probabilities, Phys. Rev. A, 36 (1987) 5209.
  * [52] G. S. Agarwal and R. R. Puri, Collapse and revival phenomenon in the evolution of a resonant field in a Kerr-like medium, Phys. Rev. A, 39 (1989) 2969.
  * [53] T. Nasreen and M. S. K. Razmi, Atomic emission and cavity field spectra for a two-photon Jaynes–Cummings model in the presence of the Stark shift, JOSA B, 10 (1993) 1292-1300.
  * [54] A. Slaoui, A. Salah and M. Daoud, Influence of Stark-shift on quantum coherence and non-classical correlations for two two-level atoms interacting with a single-mode cavity field, Physica A: Statistical Mechanics and its Applications, 558 (2020) 124946.
  * [55] S. Ragy, M. Jarzyna, and R. Demkowicz-Dobrzański, Compatibility in multiparameter quantum metrology, Phys. Rev. A, 94 (2016) 052108.
  * [56] L. Bakmou, A. Slaoui, M. Daoud and R. Ahl Laamara, Quantum Fisher information matrix in Heisenberg XY model, Quantum Inf Process, 18 (2019) 163.
  * [57] C. Napoli, S. Piano, R. Leach, G. Adesso, and T. Tufarelli, Towards superresolution surface metrology: quantum estimation of angular and axial separations, Phys. Rev. Lett, 122 (2019) 140505.
  * [58] A. Candeloro, M. G. Paris and M. G. Genoni, On the properties of the asymptotic incompatibility measure in multiparameter quantum estimation, J. Phy. A: Math. Theor, 54 (2021) 485301.
  * [59] F. Albarelli, M. Barbieri, M. G. Genoni and I. Gianani, A perspective on multiparameter quantum metrology: From theoretical tools to applications in quantum imaging, Phys. Lett. A, 384 (2020) 126311.
  * [60] A. Carollo, B. Spagnolo and D. Valenti, Uhlmann curvature in dissipative phase transitions, Sci. Rep, 8 (2018) 9852.
  * [61] A. Candeloro, S. Razavian, M. Piccolini, B. Teklu, S. Olivares and M. G. Paris, Quantum probes for the characterization of nonlinear media, Entropy, 23 (2021) 1353.
  * [62] M. Asjad, B. Teklu and M. G. Paris, Joint quantum estimation of loss and nonlinearity in driven-dissipative Kerr resonators, Phys. Rev. Res, 5 (2023) 013185.
  * [63] J. E. Prussing, the principal minor test for semidefinite matrices, J. Guidance Control Dyn, 9 (1986) 121-122.
  * [64] M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information, 2000 (Cambridge: Cambridge University Press).
  * [65] R. J. Glauber, The quantum theory of optical coherence, Phys. Rev, 130 (1963) 2529.
  * [66] A. Streltsov, U. Singh, H. S. Dhar, M. N. Bera and G. Adesso, Phys. Rev. Lett, 115 (2015) 020403.
  * [67] T. Baumgratz, M. Cramer, and M. B. Plenio, Quantifying coherence, Phys. Rev. Lett, 113 (2014) 140401.
  * [68] D. Girolami, Observable measure of quantum coherence in finite dimensional systems, Phys. Rev. Lett, 113 (2014) 170401.
  * [69] A. Streltsov, G. Adesso and M. B. Plenio, Colloquium: Quantum coherence as a resource, Rev. Mod. Phys, 89 (2017) 041003.
  * [70] M. Gessner and A. Smerzi, Statistical speed of quantum states: Generalized quantum Fisher information and Schatten speed, Phys. Rev. A, 97 (2018) 022109.
  * [71] H. R. Jahromi, K. Mahdavipour, M. K. Shadfar and R. L. Franco, Witnessing non-Markovian effects of quantum processes through Hilbert-Schmidt speed, Phys. Rev. A, 102 (2020) 022221.
  * [72] A. Slaoui, L. Bakmou, M. Daoud and R. A. Laamara, A comparative study of local quantum Fisher information and local quantum uncertainty in Heisenberg XY model, Phys. Lett. A, 383 (2019) 2241-2247.
  * [73] Z. Huang, C. Macchiavello and L. Maccone, Usefulness of entanglement-assisted quantum metrology, Phys. Rev. A, 94 (2016) 012101.
  * [74] A. Slaoui, M. Daoud and R.A. Laamara, The dynamics of local quantum uncertainty and trace distance discord for two-qubit X states under decoherence: a comparative study, Quantum Inf Process, 17 (2018) 178.
  * [75] H. Rangani Jahromi and R. Lo Franco, Hilbert–schmidt speed as an efficient figure of merit for quantum estimation of phase encoded into the initial state of open $n$-qubit systems, Sci. Rep, 11 (2021) 1.