# Dense dipole-dipole-coupled two-level systems in a thermal bath

Mihai A. Macovei<EMAIL_ADDRESS>Institute of Applied Physics,
Moldova State University, Academiei str. 5, MD-2028 Chişinău, Moldova

###### Abstract

The quantum dynamics of a dense and dipole-dipole coupled ensemble of two-
level emitters interacting via their environmental thermostat is investigated.
The static dipole-dipole interaction strengths are being considered strong
enough but smaller than the transition frequency. Therefore, the established
thermal equilibrium of ensemble’s quantum dynamics is described with respect
to the dipole-dipole coupling strengths. We have demonstrated the quantum
nature of the spontaneously scattered light field in this process for weaker
thermal baths as well as non-negligible dipole-dipole couplings compared to
the emitter’s transition frequency. Furthermore, the collectively emitted
photon intensity suppresses or enhances depending on the environmental thermal
baths intensities.

##  I Introduction

The mutual interactions among multiple excited two-level emitters are being
mediated by their environmental vacuum electromagnetic field reservoir.
Depending on the inter-particle separations, these interactions diminish or
can be enhanced, respectively [1, 2, 3, 4, 5, 6]. Certain of these emission-
absorption collective features were demonstrated experimentally which
resulted, for instance, in superradiance and subradiance [7, 8, 9, 10],
superabsorption [11, 12] as well as in various phase transitions phenomena
[13, 14], etc [15, 16, 17, 18, 19]. In a thermal electromagnetic field
environmental reservoir, the quantum dynamics of a small two-level ensemble
changes accordingly. Particularly, it was demonstrated that the entanglement
between two initially independent qubits can be generated if the two qubits
interact with a common heat bath in thermal equilibrium [20, 21, 22, 23].
Larger two-level ensembles, under the Dicke approximation [1, 2, 3, 4] and in
a common thermostat, obey the Bose-Einstein statistics and there are no
critical steady-state behaviours in the thermodynamic limit which are typical
to laser-driven atomic ensembles [4, 13, 14]. Moreover, quantum light features
are proper only for few atoms in these setups [24, 25]. Contrary, the
electromagnetic fields emitted on the two transitions of a three-level Dicke-
like ensemble, surrounded by the thermostat, are strongly correlated or
anticorrelated and exhibit quantum light properties also for larger numbers of
atoms. This is demonstrated via violation of the Cauchy-Schwarz inequality
[26].

The multi-qubit collective phenomena, mediated by environmental thermal baths,
have attracted an additional interest recently due to feasible applications
related to quantum thermodynamics [27, 28, 29, 30]. In this respect, the
performance of quantum heat engines [31, 32, 33, 34, 35, 36], quantum
refrigerators [37, 38] or quantum batteries [39, 40, 41] has been shown to
improve considerably due to cooperativity among their constituents.

Motivated by these advances regarding multi-qubit ensembles and their
applications, here, we investigate the quantum dynamics of a small dipole-
dipole interacting two-level ensemble in thermal equilibrium with its
surrounding thermostat. The spontaneously scattered quantum light features, in
this process, were addressed for the case when the dipole-dipole interaction
strength among the two-level radiators is commensurable, but still smaller,
than the transition frequency of a single emitter, respectively. We have
obtained the corresponding master equation describing this multi-qubit
ensemble and solved it analiticaly in the steady-state which characterizes the
established thermal equilibrium. Furthermore, we have demonstrated that the
thermal driven small two-level ensemble emits light which has sub-Poissonian
statistics for weaker thermal baths. For negligible dipole-dipole couplings,
it is known that this photon statistics occurs for few emitters only and
stronger thermal baths that may prevent its detection because of the thermal
photon background. The collectively spontaneously scattered photon intensity
enhances for stronger as well as weaker thermal baths due to dipole-dipole
interactions among the involved two-level emitters, while suppresses for
moderately intense to moderately weak intensities of the environmental heat
reservoirs, respectively.

This paper is organized as follows. In Sec. II we describe the analytical
approach and the system of interest, while in Sec. III we discuss the first-
and the second-order photon correlation functions, respectively. Sec. IV
presents and analyses the obtained results. The article concludes with a
summary given in Sec. V.

##  II Theoretical framework

We consider an ensemble of $N$ dipole-dipole coupled two-level emitters, each
having the transition frequency $\omega_{0}$, and interacting with the
environmental thermal reservoir at temperature $T$. The atomic subsystem is
being densely packed so that its linear dimensions are smaller than the photon
emission wavelength $\lambda$, i.e. within the Dicke approximation [1, 2, 3].
The static dipole-dipole coupling strength $\delta\sim d^{2}/r^{3}$ [11],
where $r$ is the mean distance among any of atomic pairs characterized by the
dipole $d$, is strong such that the ratio $\delta/\omega_{0}<1$ is not
negligible and may play a relevant role over the established thermal
equilibrium features. Hence, the Hamiltonian describing this system in the
dipole and rotating-wave approximations [1, 2, 3, 13, 4] can be represented as
follows: $H=H_{0}+H_{i}$, where

$\displaystyle
H_{0}=\sum_{k}\hbar\omega_{k}a^{\dagger}_{k}a_{k}+\hbar\bar{\omega}_{0}S_{z}-\hbar\tilde{\delta}S^{+}S^{-},$
(1)

with $\bar{\omega}_{0}=\omega_{0}+\tilde{\delta}$ and
$\tilde{\delta}=\delta/(N-1)$, while

$\displaystyle
H_{i}=i\sum_{k}\bigl{\\{}(\vec{g}_{k}\cdot\vec{d})a^{\dagger}_{k}S^{-}-H.c.\bigr{\\}}.$
(2)

The free energies of the environmental electromagnetic field (EMF) thermal
modes and atomic subsystems are represented by the first two terms of the
Hamiltonian $(\ref{h0})$, respectively. The dipole-dipole interaction
Hamiltonian is given by the third term of this Hamiltonian, i.e. $H_{0}$. Note
here that the Hamiltonian describing many two-level radiators is an additive
function, i.e. it consists from a sum of individual Hamiltonians,
characterizing separately each two-level emitters. From this reason, the
dipole-dipole coupling strength $\delta$ was divided on $N-1$, since there are
$N(N-1)$ terms describing the dipole-dipole interacting atoms. The Hamiltonian
$H_{i}$ accounts for the interaction of the whole ensemble with the
environmental thermostat. There, $\vec{g}_{k}$
=$\sqrt{2\pi\hbar\omega_{k}/V}\vec{e}_{p}$ is the coupling strength among the
few-level emitters and the thermal EMF modes. Here, $\vec{e}_{p}$ is the
photon polarization vector with $p\in\\{1,2\\}$ and $V$ is the quantization
volume, respectively.

In the above Hamiltonians, i.e. (1,2), the collective atomic operators
$S^{+}=\sum^{N}_{j=1}|e\rangle_{j}{}_{j}\langle g|$ and
$S^{-}=[S^{+}]^{\dagger}$ obey the usual commutation relations for su(2)
algebra, namely, $[S^{+},S^{-}]=2S_{z}$ and $[S_{z},S^{\pm}]=\pm S^{\pm}$,
where $S_{z}=\sum^{N}_{j=1}(|e\rangle_{j}{}_{j}\langle
e|-|g\rangle_{j}{}_{j}\langle g|)/2$ is the bare-state inversion operator.
Here, $|e\rangle_{j}$ and $|g\rangle_{j}$ are the excited and ground state of
the emitter $j$, respectively, while $a^{\dagger}_{k}$ and $a_{k}$ are the
creation and the annihilation operators of the environmental EMF thermal
reservoir which satisfy the standard bosonic commutation relations, that is,
$[a_{k},a^{\dagger}_{k^{\prime}}]=\delta_{kk^{\prime}}$, and
$[a_{k},a_{k^{\prime}}]$=$[a^{\dagger}_{k},a^{\dagger}_{k^{\prime}}]=0$ [42].

The general form of the master equation, describing the atomic subsystem alone
in the interaction picture, is given by [4]

$\displaystyle\frac{d}{dt}\rho(t)=-\frac{1}{\hbar^{2}}{\rm
Tr_{f}}\biggl{\\{}\int^{t}_{0}dt^{\prime}\bigl{[}H_{I}(t),[H_{I}(t^{\prime}),\rho(t^{\prime})]\bigr{]}\biggr{\\}},$
(3)

where

$\displaystyle H_{I}(t)=U(t)H_{i}U^{-1}(t),$ (4)

with $U(t)=e^{iH_{0}t/\hbar}$, and the notation ${\rm Tr_{f}}\\{\cdots\\}$
means the trace over the thermal EMF degrees of freedom. The explicit
expression of the interaction Hamiltonian, in the interaction picture is then

$\displaystyle
H_{I}=i\sum_{k}(\vec{g}_{k}\cdot\vec{d})a^{\dagger}_{k}e^{i(\omega_{k}-\hat{\omega})t}S^{-}+H.c.,$
(5)

where

$\displaystyle\hat{\omega}=\bar{\omega}_{0}+2\tilde{\delta}S_{z}.$ (6)

Next, one substitutes the Hamiltonian (5) in the equation (3) and performs the
trace over the thermal EMF degrees of freedom taking as well into account the
operator nature of the exponential function. Subsequently, under the Born-
Markoff approximations, one arrives at the following master equation
describing the atomic subsystem only

$\displaystyle\frac{d}{dt}\rho(t)=$ (7) $\displaystyle-$
$\displaystyle\bigl{[}S^{+},\frac{\hat{\Gamma}(\hat{\omega})}{2}\bigl{(}1+\bar{n}(\hat{\omega})\bigr{)}S^{-}\rho\bigr{]}-\bigl{[}S^{-},S^{+}\frac{\hat{\Gamma}(\hat{\omega})}{2}\bar{n}(\hat{\omega})\rho\bigr{]}$
$\displaystyle+$ $\displaystyle H.c..$

Here $\hat{\Gamma}(\hat{\omega})=2d^{2}\hat{\omega}^{3}/(3\hbar c^{3})$ is the
emitters spontaneous decay rate while
$\bar{n}(\hat{\omega})=[\exp{(\hbar\hat{\omega}/k_{B}T)}-1]^{-1}$ is the mean
thermal photon number, both at the eigenvalues of $\hat{\omega}$,
respectively. For a negligible ratio of the dipole-dipole coupling strength
over the transition frequency $\omega_{0}$, i.e. $\delta/\omega_{0}\to 0$, the
master equation (7) turns into the usual master equation describing a
collection of two-level atoms collectively interacting via their thermal
reservoir [2, 4, 24] in the Dicke limit, namely,
$\dot{\rho}(t)$=$-\Gamma(\omega_{0})\bigl{(}1+\bar{n}(\omega_{0})\bigr{)}[S^{+},S^{-}\rho]/2-\Gamma(\omega_{0})\bar{n}(\omega_{0})[S^{-},S^{+}\rho]/2+H.c.$,
where the overdot means differentiation with respect to time.

While finding the general solution of the master equation (7) is not a trivial
task at all for a many-atom sample, it possesses a steady-state solution. One
can observe, by a direct substitution in the Eq. (7), that the expression

$\displaystyle\rho_{s}=Z^{-1}e^{-\beta H_{a}},$ (8)

is a steady-state solution of it with the parameter $Z$ being the
normalization constant determined by the requirement ${\rm
Tr\\{\rho_{s}\\}=1}$. Here,

$\displaystyle
H_{a}=\hbar\bar{\omega}_{0}S_{z}-\hbar\tilde{\delta}S^{+}S^{-},$ (9)

and $\beta=(k_{B}T)^{-1}$ whereas $k_{B}$ is the Boltzmann’s constant.
Considering an atomic coherent state $|n\rangle$, denoting a symmetrized
N-atom state in which $N-n$ particles are in the lower state $|g\rangle$ and
$n$ atoms are excited to their upper state $|e\rangle$, and that
$S^{-}|n\rangle=\sqrt{n(N-n+1)}|n-1\rangle$,
$S^{+}|n\rangle=\sqrt{(N-n)(n+1)}|n+1\rangle$ and
$S_{z}|n\rangle=(n-N/2)|n\rangle$ [4, 24], one can calculate the steady-state
expectation values of any atomic correlators of interest.

Therefore, in the next Sections, we shall focus on the quantum statistics of
the spontaneously scattered photons in the process of dipole-dipole
interacting two-level emitters via their environmental thermal reservoir. In
this regard, we introduce, respectively, the first- and the second-order
photon correlation functions describing this feature.

##  III Photon correlation functions

The first- and the unnormalized second-order photon correlation functions at
position $\vec{R}$ can be represented as follows [42, 43, 44, 45]

$\displaystyle G_{1}(\vec{R},t)$ $\displaystyle=$
$\displaystyle\langle\vec{E}^{(-)}(\vec{R},t)\vec{E}^{(+)}(\vec{R},t)\rangle,$
$\displaystyle G_{2}(\vec{R},t)$ $\displaystyle=$
$\displaystyle\langle:\vec{E}^{(-)}(\vec{R},t)\vec{E}^{(+)}(\vec{R},t)\vec{E}^{(-)}(\vec{R},t)\vec{E}^{(+)}(\vec{R},t):\rangle,$

where

$\displaystyle\vec{E}^{(-)}(\vec{R},t)$ $\displaystyle=$
$\displaystyle\sum_{k}\vec{g}_{k}a^{\dagger}_{k}(t)e^{-i\vec{k}\cdot\vec{R}},$
$\displaystyle\vec{E}^{(+)}(\vec{R},t)$ $\displaystyle=$
$\displaystyle\sum_{k}\vec{g}_{k}a_{k}(t)e^{i\vec{k}\cdot\vec{R}},$ (11)

while $:f(\vec{R},t):$ indicates normal ordering. In the far-zone limit of
experimental interest $R=|\vec{R}|\gg\lambda$, one can express the first- and
second-order correlation functions via the collective atomic operators. In
this respect, one solves formally the Heisenberg equations for EMF operators
$\\{a^{\dagger}_{k}(t),a_{k}(t)$ using the Hamiltonian (5). Then, introducing
those solutions in the Exps. (LABEL:kr) and integrating over all involved
variables, see e.g. [46], one arrives finally at

$\displaystyle G_{1}(\vec{R},t)$ $\displaystyle=$ $\displaystyle\Psi(R)\langle
S^{+}(t)\hat{\omega}^{4}S^{-}(t)\rangle,$ $\displaystyle G_{2}(\vec{R},t)$
$\displaystyle=$ $\displaystyle\Psi(R)^{2}\langle
S^{+}(t)\hat{\omega}^{2}S^{+}(t)\hat{\omega}^{4}S^{-}(t)\hat{\omega}^{2}S^{-}(t)\rangle.$

Here $\Psi(R)=d^{2}(1-\cos^{2}{\xi})/(c^{4}R^{2})$, while $\xi$ is the angle
between the direction of vector $\vec{R}$ and the dipole $\vec{d}$.

Figure 1:  The steady-state second-order photon-photon correlation function
$g^{(2)}(0)$ as a function of $x=\hbar\omega_{0}/(k_{B}T)$ for a two-atom
system, $N=2$. Here the solid lines is plotted for a negligible dipole-dipole
coupling strength, i.e. $\delta/\omega_{0}\to 0$, whereas the dashed one
stands for $\delta/\omega_{0}=0.1$. The short-dashed curve is plotted based on
Exp. (17), which is valid for $x\gg 1$.

The normalized second-order photon-photon correlation function, defined in the
usual way, namely,

$\displaystyle g^{(2)}(t)=\frac{G_{2}(\vec{R},t)}{G_{1}(\vec{R},t)^{2}},$ (13)

takes the next form in the steady-state,

$\displaystyle g^{(2)}(0)=\frac{\langle
S^{+}\hat{\omega}^{2}S^{+}\hat{\omega}^{4}S^{-}\hat{\omega}^{2}S^{-}\rangle}{\langle
S^{+}\hat{\omega}^{4}S^{-}\rangle^{2}}.$ (14)

Note that, $g^{(2)}(0)<1$ characterizes sub-Poissonian, $g^{(2)}(0)>1$ super-
Poissonian, and $g^{(2)}(0)=1$ Poissonian photon statistics.

In the following Section, we shall investigate the second-order photon-photon
correlation function (14) with the help of the steady-state solution (8).
Actually, the focus will be on comparing $g^{(2)}(0)$ when
$\delta/\omega_{0}\to 0$, a case which is known in the scientific literature,
with the situation considered here, namely, when $\delta/\omega_{0}\not=0$ but
still $\delta/\omega_{0}<1$.

Figure 2:  Same as in Fig. (1), but for $N=3$.

##  IV Results and Discussion

We start our discussions by presenting some limiting cases of the second order
photon-photon correlation function given by Exp. (14). Particularly, for
$\eta\to 0$, where

$\displaystyle\eta=\frac{\delta}{\omega_{0}},$

we recover the known results (see e.g. [24]), namely,

$\displaystyle g^{(2)}(0)$ $\displaystyle=$
$\displaystyle\frac{6(N+3)(N-1)}{5N(N+2)},~{}~{}{\rm if}~{}x\to 0,$
$\displaystyle g^{(2)}(0)$ $\displaystyle=$ $\displaystyle
2-\frac{2}{N},~{}~{}{\rm when}~{}x\gg 1,$ (15)

where

$\displaystyle x=\frac{\hbar\omega_{0}}{k_{B}T}.$

However, for $\eta\not=0$ and strong thermal baths, i.e. $x\to 0$, and to the
second order in the small parameter $\eta$, we have obtained that

$\displaystyle g^{(2)}(0)$ $\displaystyle=$
$\displaystyle\frac{6(N+3)(N-1)}{5N(N+2)}$ (16) $\displaystyle+$
$\displaystyle\frac{48(N+3)(N^{2}+2N-18)\eta^{2}}{25N(N-1)(N+2)}.$

Contrary, for weaker thermal baths when $x\gg 1$, we have obtained the
following expression for the second-order photon-photon correlation function

$\displaystyle g^{(2)}(0)$ $\displaystyle=$ $\displaystyle
2\biggl{(}1-\frac{1}{N}\biggr{)}\biggl{(}\frac{N-1-(N-3)\eta}{(N-1)(1-\eta)}\biggr{)}^{4}$
(17) $\displaystyle\times$ $\displaystyle\exp{\biggl{(}-\frac{2\eta
x}{N-1}\biggr{)}}.$

Notice that Exp. (16) and (17) turn, respectively, into those given by Exp.
(15), when $\eta\to 0$. Moreover, the photon statistics, characterized by Exp.
(17) for larger values of the parameter $x$, differs considerably between the
cases $\eta=0$ and $\eta\not=0$, respectively.

Figure 3:  The log-linear plot of the steady-state second-order photon-photon
correlation function $g^{(2)}(0)$ as a function of
$x=\hbar\omega_{0}/(k_{B}T)$, for N = 7. Other involved parameters are as in
Fig. (1).

Figures (1) and (2) depict the photon statistics of the spontaneously
scattered photons by a dipole-dipole interacting few-atom system, i.e. for
$N=2$ and $N=3$, respectively, via their environmental thermostat.
Particularly, the solid lines are plotted for a negligible dipole-dipole
coupling strength, i.e. $\delta/\omega_{0}\to 0$, whereas the dashed ones
stand for $\delta/\omega_{0}=0.1$. The short-dashed curves represent the Exp.
(17), which is valid for $x\gg 1$. Interestingly, the photon statistics turns
into a quantum photon statistics, i.e. $g^{(2)}(0)<1$, when $\eta\not=0$ and
for weaker thermal baths, $x\gg 1$. This behaviour is typical for several
dipole-dipole interacting atoms, too. From this reason, in Fig. (3) we plot
the steady-state second-order photon-photon correlation function $g^{(2)}(0)$
for $N=7$ dipole-dipole interacting atoms which supports it. Thus the thermal
mediated dipole-dipole interactions among two-level emitters in a several-atom
Dicke-like sample are responsible for the sub-Poissonian photon statistics of
the spontaneously scattered photons. In the absence of the dipole-dipole
interactions, that is when $\eta\to 0$, sub-Poissonian photon statistics
occurs in this system only for $N\in\\{1,2,3\\}$ and for strong thermal baths,
$x\ll 1$, see Fig. (1) and Fig. (2). Also, one can observe a nice concordance
with the limiting cases of the second-order photon-photon correlation function
$g^{(2)}(0)$, given by the Exps. (15-17), and the plots in Figs. (1-3).

Figure 4:  The log-plot of the ratio
$G_{1}(\vec{R},\eta\not=0)/G_{1}(\vec{R},\eta=0)$ as a function of
$x=\hbar\omega_{0}/(k_{B}T)$ for $\eta=0.1$. The solid line is plotted for
$N=2$, the dashed one for $N=3$, while the short-dashed curve for $N=7$,
respectively.

While this tendency, i.e. occurring of sub-Poissonian photon statistics for
bigger values of $x=\hbar\omega_{0}/(k_{B}T)$, persists for larger atomic
ensembles, it will be less useful since the photon flux might be too weak in
this case. To clarify this issue, one represents the cooperative intensity of
spontaneously scattered EMF by the dipole-dipole interacting two-level
emitters in a weaker thermal environment, i.e. for $x\gg 1$, as follows

$\displaystyle
G_{1}(\vec{R})/\Psi(R)=N\bigl{(}1-\eta\bigr{)}^{4}\exp{\bigl{(}-x(1-\eta)\bigr{)}}.$
(18)

On the other hand, from the Exp. (17) when setting $g^{(2)}(0)=1$, one can
obtain the $\eta^{\prime}{\rm s}$ ranges in order the sub-Poissonian photon
statistics to occur for $N\gg 1$ and $x\gg 1$, namely,

$\displaystyle\eta>(N/x)\ln{(\sqrt{2})}.$ (19)

Note here that $\eta=(N/x)\ln{(\sqrt{2})}$ is the condition for Poissonian,
whereas $\eta<(N/x)\ln{(\sqrt{2})}$ for super-Poissonian photon statistics,
respectively. Although the Exp. (19) is not difficult to fulfil, the photon
intensity, given by Exp. (18), will be too low for larger ensembles and weaker
thermal baths, $x\gg 1$. The reason consists in the Bose-Einstein nature of
the atomic statistics, meaning that in a thermal environment at equilibrium,
the two-level emitters tend to reside in their ground state for $\\{N,x\\}\gg
1$. Under these circumstances, less atoms get excited and, therefore, the
spontaneously generated EMF might be weak, as well. Finally, the photon
statistics changes from sub-Poissonian to super-Poissonian if
$\delta\to-\delta$ and $x\gg 1$.

Note that, from Exp. (18) follows that for weaker thermal baths when $x\gg 1$,
$G_{1}(\vec{R},\eta\not=0)>G_{1}(\vec{R},\eta=0)$, see Fig. (4). However, from
the general expression of the intensity, i.e. Exp. (LABEL:krf), one has that
$G_{1}(\vec{R},\eta\not=0)<G_{1}(\vec{R},\eta=0)$, within moderate weak,
$x>1$, to moderate intense, $x\leq 1$ heat baths, respectively. This will turn
again to $G_{1}(\vec{R},\eta\not=0)>G_{1}(\vec{R},\eta=0)$ for stronger
thermal baths, i.e. when $x\to 0$, see Fig. (4). Particularly, for stronger
environmental thermal baths intensities when $x\to 0$, one has that

$\displaystyle\frac{G_{1}(\vec{R},\eta\not=0)}{G_{1}(\vec{R},\eta=0)}=1+72\eta^{2}+2064\eta^{4}/7.$
(20)

Generalizing, the thermal mediated dipole-dipole interactions are responsible
for sub-Poissonian statistics of the spontaneously scattered photons by a
small two-level ensemble, in a weak thermal bath and within the Dicke limit.
Furthermore, due to presence of the dipole-dipole interactions between the
two-level emitters, the collectively scattered spontaneous emission light
suppresses or enhances depending on the intensities of the environmental
thermal baths, see also [47]. In principle, the relationship among the quantum
nature of the spontaneously scattered photons in this process and the
ensemble’s quantum thermodynamics would be an interesting issue to be
addressed in future investigations.

##  V Summary

We have investigated the steady-state quantum dynamics of an ensemble of
dipole-dipole interacting two-level emitters, in the Dicke limit, which is
mediated by the environmental thermostat. Particularly, the focus was on the
quantum nature of the collective spontaneously scattered photons in this
process. We have demonstrated that the dipole-dipole interaction among the
two-level emitters is responsible for sub-Poissonian photon statistics
scattered by a small ensemble consisting from few to several atoms,
respectively. The quantum light effect occurs when the energy of a free atom
is larger than the corresponding one due to the thermal reservoir. This
quantum future survives for larger ensembles, however, the photon intensity
might be too weak in this case. Without the dipole-dipole interactions among
the two-level radiators, the sub-Poissonian statistics occurs for few atoms
and stronger thermal baths only. The collectively scattered spontaneous
emission light intensity suppresses or enhances depending on the intensities
of the environmental thermal baths.

###### Acknowledgements.

The financial support from the Moldavian Ministry of Education and Research,
via grant No. 011205, is gratefully acknowledged.

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