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NORMAN MARSHALL VILLENEUVE
Norman Marshall Villeneuve at Rockhead's Paradise 1963
Norman Marshall Villeneuve's Canadian Jazz Messengers, 1991 - David Restivo, Mike Downes, Ken Fornetran, Joe Allan, Grant Stewart - Rick Ingleson(c)
Jon Snelson(c)
Marie Byers(c)
Norman Marshall Villeneuve is a beloved icon of Canadian Jazz. In his remarkable 8th decade in music, this legendary jazz drummer is a Félix Award winner, bandleader, recording artist, and master of bebop, is known as "Canada's Art Blakey" for his passion and iconic drum solos. His new CD, "Montreal Sessions" was named by CBC Music as one of the ten outstanding jazz albums of 2017. For decades, Norman has led jazz ensembles and bands under his banner, "Norman Marshall Villeneuve's Jazz Message". In 2017 he also began to appear under "Norman Marshall Villeneuve's Jazz Masters", bringing to the stage respected jazz icons in their own right, including Ron Di Lauro, Charles Ellison, Dave Turner, Alvin Pall, Éric Lagacé, and many others.
In December 2017, Norman made an impromptu performance at the Living Arts Centre in Mississauga. Walking on the stage alone, under a single spotlight, to join Robi Botos and Dave Young, the packed house of 1,300, welcomed Norman with a standing ovation, in a spontaneous outpouring of love and respect.
Born May 29, 1938 in Montreal, Norm grew up in St. Henri and Little Burgundy, home to Oscar Peterson, Victor Vogel and Oliver Jones, and many who became part of Montreal's beloved jazz history. Norm attended Union United Church Montreal, where he studied piano with Daisy Peterson-Sweeney, Oscar Peterson's sister.
With a natural gift for rhythm, at 8 Norm was tap dancing and singing at community gatherings, winning dance competitions at 11. He began to teach himself drums at 10 and jazz music by listening to musicians like Illinois Jacquet, Earl Bostic, Art Taylor, Philly Joe Jones, and Art Blakey, to whom he has dedicated his life's work. In 1955, at 17, Oliver Jones invited Norm to perform on CBC Radio with him and Richard Parris. This was followed by his first permanent gig at the Arcade Hotel. Montreal was swinging, and Norm played Rockhead's Paradise, Chez Parée, Penthouse, Esquire Showbar, Black Bottom and Café la Bohème. He met many jazz luminaries, including Louis Armstrong, Miles Davis, Wayne Shorter, Charlie Rouse, Booker Ervin, Cootie Williams, Cat Anderson, Tony Williams, Elvin Jones and Art Blakey.
Villeneuve has performed with jazz royalty: Duke Ellington, Oliver Jones, Junior Mance, Charles Coleman, Frank Motley, Curtis Fuller, Barry Harris, Julius Watkins, Jackie McLean, Ray Draper, Jay McShann, Charlie Rouse, Booker Ervin, Big Big Miller, "Wild Bill" Davison, Jack McDuff, Charlie Biddle, Nelson Symonds, Dave Young, Jane Bunnett and Dan Faulk. He has backed many vocalists, including Anita O'Day, Blossom Dearie, Slim Gaillard, Sheila Jordan, Ranee Lee, Michelle Sweeney and Jackie Richardson.
Throughout his long and distinguished career, Oliver Jones chose Norm as his drummer, to join him on stages throughout North and South America, and on tours with Jamaican vocalist Kenny Hamilton, Ranee Lee, Richard Ring and Dave Young. With Oliver, Norman performed for Bill Clinton and Jean Chretien, and for thousands at the Air Canada Centre, Place des Arts, Sony Centre for the Performing Arts, and even on a US Navy aircraft carrier, along with Bob Hope, Paul Anka, Red Skeleton, Jimmy Durante and others. Norm's credits include several appearances on national television and a movie with the Olsen twins.
In 1974, Norm moved to Toronto and quickly gained a reputation as a jazz master. He performed with dozens of local jazz musicians at top venues like Four Seasons Centre, Bourbon Street, George's Spaghetti House, Myers, Montreal Bistro, Top Of the Senator, Bermuda Onion, Jazz Bistro, Pilot and Rex. In 1991, Norman took his sextet on a successful cross-Canada tour. He has performed at many top venues in Canada and at jazz festivals throughout the country.
Norman has brought along and given countless performance opportunities to musicians who now form the new generation of the best of Canadian jazz, including Robi Botos, David Restivo, Mike Downes, Grant Stewart, Ken Fornetran, Eric Boucher, and Alexander Brown.
The Norman Marshall Villeneuve Scholarship, created in 2012, is awarded annually to a percussion student with high academic achievement, leadership, excellent musicianship, and a passion for jazz. Norm is a frequent guest speaker at Concordia University and participates in community events to talk about Montreal jazz history. The Norman Marshall Villeneuve Jazz Collection Fond is part of the Concordia University Archive.
In 2013 Norm returned home to Montreal, where he continues his life's work, playing on stages large and small, with the passion and excitement that is his hallmark. With an outstanding career, in his 7th decade as a professional musician, Norm remains an active and respected leader on the Canadian jazz scene. For more information, visit Norm's website at www.normanmarshallvilleneuve.com


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text:
\section{Introduction}
The description of dissipation and decoherence is an important subject of investigation,
especially in view of the current boom in quantum technologies \cite{Li20a,Li20b} (see also discussion in Ref. \cite{Mor19}).
In this field, the system of interest is a number of Qubits that are inherently coupled to one or several
environments \cite{Fey82,Bar11,Geo14}. The effect of this coupling is rather dramatic since it induces a non-unitary
evolution of the system, and ultimately tends to destroy the interesting quantum aspects, driving the system to
classical physics.  Controlling the transition from quantum to classical physics is becoming crucial
in this context. By changing the properties of the baths, one might first study the transition
from the Markovian to the non-Markovian regime. In particular, it has been recently underlined that
non-Markovian dynamics and their impact should be explored more systematically \cite{Li20a,Li20b}.

Non-Markovian effects and their description is an active
field of research in the theory of open quantum systems \cite{Riv14,Bre16,Veg17,Li18}, especially with the aim of
developing accurate and versatile approaches.  With the progress of manipulating atoms and molecules, one might
engineer systems and/or environments that can be formed of fermionic or bosonic particles.  Another example is the atomic nuclei where nucleons (fermions) act as a reservoir for the collective excitations treated as bosons \cite{Rin80}.
Treating fermions is finally of special interest in the context of quantum computing due to the one-to-one mapping between spin systems and fermions on lattices \cite{Jor28}.

Our primary goal here is to develop a unified
approach able to describe fermions or bosons coupled to a set of baths that can also be a mixture of fermions and bosons.  System coupled to several reservoirs are of special interest
in different fields of physics. To quote few of them, we mention the example of cavity
quantum electrodynamics  \cite{Maj12}, Jaynes-Cummings lattices \cite{Nun11},
photon-ion interfaces \cite{Lam11}, ion chain systems \cite{Lin11}, or
phonon-induced spin squeezing \cite{Ben13} (see also examples in Refs.~\cite{Che13,Cha14,Hov18,Mwa19}).

In the following, we first recall some recent progress we have made in the description of systems coupled
to several heat baths  \cite{Sar14,Lac15,Sar17,Hov18,Sar18,Hov19,Hov20}. We then discuss in details the extra subtleties
that appear for Fermi systems or environments compared to the Bose case. In the case of bosonic system and environments,
an exact treatment is a priori possible including fully the non-Markovian effects. When fermions are present
in the system or in the surrounding baths, some approximations are required. We describe in section \ref{sec:extfb} a specific
methodology to treat Fermi system coupled to Fermi, Bose or Fermi--Bose mixtures of baths. In this new approach, where the non-Markovian
effects are included, a special attention is paid to respect the
Pauli exclusion principle for Fermi degrees of freedom.
The approach is then illustrated for the system coupled to one or several heat baths
in section \ref{sec:result}.

\section{Method}

We follow here our previous work \cite{Sar14,Lac15,Sar17,Hov18,Sar18,Hov19,Hov20} and consider a Fermi or Bose  system coupled to one or several baths. Some of the baths could be composed of fermions and some other of bosons. In our previous studies we gradually considered  problems of increasing complexity, i.e. changing the quantum statistics of the system or bath, considering more general coupling and/or increasing the number of baths.

A single two-level system coupled to an environment is considered. The system+environment Hamiltonian is taken
as
\begin{eqnarray}
H=H_{\rm S}+H_{\rm E}+H_{\rm SE}. \label{eq:HH}
\label{ham}
\end{eqnarray}
The system and environment Hamiltonians, denoted respectively by $H_S$ and $H_E$, are given by
\begin{eqnarray}
\displaystyle H_{\rm S}=\hbar \omega_1 a^\dagger_1 a_1, ~~~\displaystyle  H_{\rm  E}=\sum_{\nu } \hbar \omega _{\nu }a_{\nu }^\dagger a_{\nu } .
\label{eq:se}
\end{eqnarray}
For the moment, we  do not specify if there is one or several baths and simply assume that the quantum nature (fermionic or bosonic)  of  each pair of creation/annihilation operators $(a^\dagger_\nu, a_\nu)$ is specified through the relation
\begin{eqnarray}
a_\nu a^\dagger_\nu = 1+ \varepsilon_\nu a^\dagger_\nu a_\nu ,
\end{eqnarray}
where $\varepsilon_\nu = +1$ ($-1$) for bosons (fermions). We also use the convention $\nu=1$ ($\nu > 1$) for the system
(for the environment).
Note that, in the present model, fermionic heat-bath is described by an infinite set of two-level systems initially at thermal equilibrium. Besides the commutation/anti-commutation rules of creation/annihilation operators, the difference between a bosonic and fermionic bath stems from the initial occupation probability
denoted by $n(\omega_\nu)$
that corresponds either to Bose-Einstein or Fermi-Dirac occupation probability. A pictorial view of the heat-bath was given in Fig. 1 of Ref. \cite{Sar14}.

In the following, we will consider
the Full--Coupling case (FC), in which the coupling between the system and environment is as follows
\begin{eqnarray}
H_{\rm SE} &=& H_{\rm FC} = \sum _{\nu>1} g_{\nu } (a^\dagger_1 +a_1 )\left(a_{\nu }^\dagger +a_{\nu }\right). \label{eq:fc}
\end{eqnarray}
The present formulation is rather flexible and includes the possibility that some environmental particles obey fermion statistics
while other obey boson statistics. It also includes the possibility that the environment can be decomposed into several baths with eventually different quantum natures and different initial temperatures.  We note that the Hamiltonian (\ref{eq:HH}) can be used
to describe a single Qubit surrounded by one or several heat-baths. In our previous
work, we used the Heisenberg representation to obtain a solution to the system+environment problem taking into account possible non--Markovian effects \cite{Sar14,Lac15,Sar17,Hov18,Sar18,Hov19,Hov20}.  The solution we provided, once the frequency is properly renormalized (see discussion below), is exact for the
FC coupling when only bosonic particles are considered for both the system and the heat-baths. The situation is more delicate in the FC case when fermions (either in the bath and/or one of the environments) are involved. In this case, the problem could not be solved exactly and a specific prescription should be made to obtain a closed form of
the equations of motion to be solved. A first solution to this problem was given in Ref. \cite{Sar17,Sar18}.  Such a solution
resulted in quite reasonable description of the Fermi systems coupled to one or several baths. Numerical applications have
recently shown however that occupation numbers of the systems with the fermionic heat bath(s) can sometimes slightly exceed $1$. This points out that some modifications of the method might be needed.
Another interesting result was the absence
of asymptotic stationary solution when the system is coupled to a mixture of fermionic and bosonic heat-baths \cite{Hov19}.
In the following, we propose an alternative solution that avoids the occurrence of non-physical occupation numbers during the evolution.

\subsection{Summary and illustration of our previous work}

We summarize here the strategy we employed previously starting from the Heisenberg equations of motion for the system and heat-bath creation operators:
\begin{eqnarray}
\displaystyle \frac{d}{dt}a^{\dag}_1 &=&i\omega_1 a^{\dag}_1 +i(1-[1-\varepsilon_1]a^{\dag}_1a_1)   \sum_{\nu} g_{\nu} \left[ a_{\nu}^{\dag}+ a_{\nu} \right],   \label{eq:eom1.1} \\
\displaystyle  \frac{d}{dt}a^{\dag}_{\alpha}&=&i\omega_{\alpha} a^{\dag}_{\alpha} +i g_{\alpha} (1-[1-\varepsilon_\alpha] a^{\dag}_{\alpha} a_{\alpha})[  a^{\dag}_1 +  a_1]. \label{eq:eom1.2}
\end{eqnarray}
When all particles are bosons, we have $[1-\varepsilon_1]=[1-\varepsilon_\alpha]=0$ for all $\alpha>1$. The equations of motion become a linear set of equations between the creation/annihilation operators. In this case, the problem is solved exactly  using the Laplace transform technique (see for instance \cite{Sar14}).
In Refs.~\cite{Lac15,Sar17}, we have also illustrated that such a problem can be accurately solved by using the discretized environment method (DEM) together with the special Bogolyubov transformation between the creation/annihilation operators (see section IV of Ref. \cite{Sar17}). An alternative solution to the present problem could be to consider directly the coupled equations of motion (CEM) for the normal  and anomalous densities, denoted respectively by $M$ and $K$, associated to the system+environment.
Using the notations \footnote{Note that for the sake of simplicity, we use slightly different convention for the indices ordering compared to the standard definition of the normal and anomalous densities in many-body systems \cite{Rin80}.}:
\begin{eqnarray}
\left\{
\begin{array} {l}
M_{\nu\alpha} =  \langle a^\dagger_\nu a_\mu \rangle , ~~~G_{\nu \alpha}(t)  =   \langle a_\nu a^\dagger_\alpha \rangle, \\
\\
K_{\nu \alpha} = \langle a^\dagger_\nu a^\dagger_\alpha  \rangle , ~~~K^*_{\nu \alpha}  = \langle a_\alpha a_\nu \rangle,
\end{array}
\right.
\label{eq:dk}
\end{eqnarray}
we obtain for the boson system coupled to bosonic environment
the set of coupled equations for the $M$ and $K$ components:
\begin{widetext}
\begin{eqnarray}
\left\{
\begin{array} {ll}
\displaystyle \frac{d M_{11}}{dt} &=  i \sum_\nu g_\nu (M_{\nu 1}  - M_{1\nu}) +  i \sum_\nu g_\nu (K^*_{1\nu} -
K_{1\nu} )   \\
\\
\displaystyle
 \frac{dM_{1\alpha} }{dt} &=      i  (\omega_1 - \omega_\alpha) M_{1\alpha}
+i  \sum_{\nu} g_{\nu} (M_{\nu\alpha} + K^*_{\alpha\nu})
-i g_\alpha (M_{11} + K_{11})  \\
\\
\\
\displaystyle \frac{d M_{\alpha \beta} }{dt} &=   i  (\omega_\alpha - \omega_\beta )M_{\alpha \beta}
+ i g_{\alpha}  (M_{1\beta} + K^*_{\beta 1})
-i g_\beta   (M_{\alpha 1} + K_{\alpha 1}) \\
\\
\displaystyle  \frac{d K_{11} }{dt} &= 2i  \omega_1 K_{11} +i \displaystyle  \sum_{\nu} g_{\nu} ( K_{\nu 1} + M_{1\nu} + G_{\nu 1} + K_{1\nu})  \\
\\
\displaystyle\frac{d K_{1\alpha}}{dt} &= i  (\omega_1 + \omega_\alpha)  K_{1\alpha}
+i    \sum_{\nu} g_{\nu} (G_{\nu \alpha} + K_{\nu\alpha})  +i  g_\alpha   (M_{11}+K_{11}) \\
\\
\displaystyle \frac{d K_{\alpha\beta}  }{dt}&= i  (\omega_\alpha + \omega_\beta)  K_{\alpha\beta}
+ i g_\alpha (G_{1 \beta} + K_{1\beta})
 + i g_\beta (M_{\alpha 1} + K_{\alpha1}).
\end{array}
\right.   \label{eq:sebosons}
\end{eqnarray}
\end{widetext}
Solving these equations
numerically, we obtain the exact solution of the problem when only bosons are presented in both the system and baths. As illustrated below, this method
is strictly equivalent to the one we used in Refs.~\cite{Sar14,Lac15}.

 \begin{figure}[!h]
    \includegraphics[width=1.0 \linewidth]{T1g01Gam12_fc_FFBBNew.pdf}
    \caption{Time evolution of the occupation probability in the system $n_1(t)=M_{11}(t)$ obtained by solving Eqs.~(\ref{eq:sebosons}) (red solid line) or by solving the set of equations  (\ref{eq:fcfulllast}) (black dashed line).
    Panels (a) and (c) correspond to the case of a bosonic system coupled
    to a bosonic bath starting from  $n_1(0) = 0$ and  $n_1(1) = 1$ respectively.
    Panels (b) and (d) correspond to the case of a fermionic system coupled to a fermionic bath again starting from $n_1(0) = 0$ and  $n_1(1) = 1$
    respectively. In all cases, the bath properties are given by $c_1=0.1$,
    $\gamma_1/\Omega=12$ and $T_1/\Omega=1$. Note that, in the B-B case, the two approaches are equivalent and the two curves displayed
    in panels (a) and (c) cannot be distinguished. }
    \label{fig:BBFFold}
\end{figure}

\subsection{Numerical solution of Eq.~(\ref{eq:sebosons}) with discretization of heat bath}
\label{sec:dem}

Here, we consider the system with frequency $\Omega$ (we use the convention $\hbar=1$ thorough the paper) and all energies,
coupling constants are given in units of $\Omega$. Time  is given in $\Omega^{-1}$ units. We assume that the system is coupled to one or several baths. Each bath, labelled by $i$ has a Lorentz-Drude spectral function given by
\begin{eqnarray}
J_i(\omega) &=&  \frac{c_i}{\pi}  \omega \frac{\gamma^2_i}{\gamma^2_i + \omega^2}. \label{eq:lorentz}
\end{eqnarray}
The two parameters $c_i$ and $\gamma_i$ determine the coupling strength with the system and the memory effect respectively.
In order to solve the CEM,
a finite discrete number of levels $\nu$ is used for each environment.  In practice, we follow the procedure proposed in Ref.  \cite{Lac15}. For a given environment, a finite set of frequencies $\omega_\nu$ is used according to:
\begin{eqnarray}
\omega_\nu & = & \Delta \omega (n + 1/2), ~~n=0,\cdots,N_{\rm max}. \label{eq:dw0}
\end{eqnarray}
Then, the coupling $g_\nu$ entering in the Hamiltonian is given by $g_\nu =   \sqrt { \Delta \omega J_i (\omega_\nu) }$. Further details and discussions can be found in Ref. \cite{Lac15}. Note that the discretization of $\omega$ can be taken non-uniform. In practice, small $\Delta \omega$ are required only in the vicinity of $\Omega$ to get a good numerical accuracy and larger $\Delta \omega$ can be used away from the system frequency. We use this property to reduce the total number of states
necessary for each bath. In the following applications,
we use $N_{\rm max}=400$ states with $\Delta \omega$ that depends on $n$ such that
$\Delta \omega_{n+1} = \lambda \Delta \omega_{n}$ with $ \Delta \omega_{0}/\Omega = 0.01$  and $\lambda=1.015$.
The choice of the values for three parameters $\Delta \omega_{0}$, $\lambda$ and $N_{\rm max}$ is critical to properly achieve good numerical accuracy. In particular, these parameters should be chosen in such a way that $\Delta \omega_{n} \ll \Omega$ in the vicinity of $\Omega$ and $T \ll \omega_{N_{ max}}$ while keeping $N_{\rm max}$ not too high in order to obtain the result in a reasonable numerical time. The choice of the parameters for the discretization has been validated for the boson system coupled to bosonic bath by comparing with the results obtained using the Laplace transform technique \cite{Sar14}.

To avoid the unphysical shift of the system frequency induced by the coupling with the baths, this frequency is renormalized
prior to the calculation as it is always done \cite{Bre02}. For
the FC case, the frequency $\omega_1$
used in Eqs.~(\ref{eq:sebosons}) is as follows
\begin{eqnarray}
\omega_1 = \Omega +4  \sum_\nu \frac{g_\nu^2}{\omega_\nu}.
\end{eqnarray}
Finally, each heat-bath is characterized by its initial temperature $T_i$ such that the initial occupation $n_\nu (0)$ of the state in the bath $i$ is given by (using the convention $k_B=1$ for the Bolztmann constant):
\begin{eqnarray}
n_\nu(0) & = & M_{\nu \nu} (0) =  \frac{1}{\exp\left(\omega_\nu / T_i  \right) - \varepsilon_\nu}.
\end{eqnarray}

The results are shown in Fig.~\ref{fig:BBFFold} for the Bose system coupled to single bosonic bath.
The time evolution of occupation probability is obtained by solving  the set of equations (\ref{eq:sebosons}) for the FC coupling using the discretization of the environment. The results perfectly match those obtained in Ref.~\cite{Sar14} using the Laplace transform technique. When we write down the creation/annihilation operators $a_1^+$/$a_1$
($a^{\dagger}_{\nu}/a_{\nu}$), we mean the creation/annihilation
operators of transition with the corresponding energy $\hbar \omega_1$
($\hbar \omega_{\nu}$). So, each $a_1$ ($a^{\dagger}_{\nu}$) and $a_1$
($a_{\nu}$) is the product of creation and annihilation operators of
particle. In our formalism, there is a conversion of excitation quanta
from the fermionic or bosonic system to the bosonic or fermionic environment
or vice versa. Since only the level associated to $a^\dagger_1$ is populated or depopulated  by the coupling, the number of particles in the system as a function of time directly identifies with the quantity $n_1(t)$.

\subsection{Direct extension of Eqs.~(\ref{eq:sebosons}) when fermions are involved}

In the following, we will systematically use the short-hand notation B-B, F-F, B-F, and F-B where the first letter refers to the system statistics (B=Bosons and F=Fermions) while the second letter refers to the bath statistics.  When more than one bath is considered, we use the convention
System-Bath$_1$-Bath$_2$-...

Equations ~(\ref{eq:sebosons}) provide an exact treatment of the B-B case.  Based on simple arguments, we previously proposed to treat the F-F case by neglecting the terms $(1-\varepsilon_1) a^\dagger_1 a_1 $  and $(1-\varepsilon_\alpha) a^\dagger_\alpha a_\alpha$  in Eqs.~(\ref{eq:eom1.1}) and (\ref{eq:eom1.2}) when the system and baths are both composed of fermions. This direct mapping from bosons to fermions has the great advantage to give linear Heisenberg equations also for the F-F case while the bath properly imposes asymptotically the Fermi statistics to the system in the weak coupling--high temperature limit.
The results of this approximate treatment are also shown in Fig.  \ref{fig:BBFFold} for the F-F case. We clearly see in this figure (panel (d)) that, while the asymptotic behavior is expected
to be properly treated, the price to pay with the simplified treatment is the occurrence of unphysical behavior
at initial time-scale with occupation numbers larger than $1$. This stems from the fact that the Pauli exclusion
principle might be broken during the time evolution. Indeed, in Eqs.~(\ref{eq:sebosons}), nothing
prevents from having $K_{11}(t) = \langle (a^\dagger_1)^2 \rangle$ non-zero during the evolution even if the system is
fermionic.

Nevertheless, the simplified treatment has additional interesting properties that have been used to overcome this difficulty.
One of them is the possibility to map exactly  Eqs.~(9) into the simple time-local diffusion equation for the occupation probability:
\begin{eqnarray}
\frac{d n_1(t)}{dt} &=& -2 \lambda_1 (t) n_1(t) + 2D_1(t) , \label{eq:diffusion}
\end{eqnarray}
including fully non-Markovian effects. Using this equation and some symmetry properties of the master
equation obtained for $n_1(t)$ with the non-linear termed, it was shown in Ref.~\cite{Sar18}   that
the F-B and B-F case can be accurately described using the diffusion equation (\ref{eq:diffusion}) with modified
transport coefficients. This approach leads to the proper asymptotic limit even though the Fermi nature of the system
or bath might be slightly broken. The results obtained with this method for the F-B and B-F case are illustrated in Fig.~\ref{fig:BFFBold}.

 \begin{figure}[!h]
    \includegraphics[width=1.0 \linewidth]{T1g01Gam12_fc_FBBFNew.pdf}
    \caption{The same as in Fig.~\ref{fig:BBFFold}, but for the mixed quantum statistics case of the Boson system coupled to Fermion bath (B-F) [(a) and (c)] and for the Fermion system coupled to Boson bath (F-B) [(b) and (d)].
 The bath properties are given by $c_1=0.1$, $\gamma_1/\Omega=12$ and $T_1/\Omega=1$. In all cases, the red solid line corresponds to the result obtained with the Laplace transform approach Ref.~\cite{Sar18}. The black dashed line corresponds to the result obtained with Eqs.~(\ref{eq:fcfulllast}).
    }
    \label{fig:BFFBold}
\end{figure}

\section{Treatment of F-F, B-F, and F-B cases enforcing the Pauli exclusion principle}
\label{sec:extfb}

In the present paper we propose a treatment that respects the Fermi nature of the particle all along the non-equilibrium evolution. This implies to explicitly account for the non-linear term in
the Heisenberg equation of motion.  The situation is similar
to the many-body problem of interacting particles where the one-body density matrix evolution
depends on the two-body density, whose evolution is itself coupled to the three-body density and so on and so forth, leading
to the so-called Bogolyubov- Born-Green-Kirkwood-Yvon hierarchy \cite{Bog46, Bor46, Kir46,Cas90,Gon90,Sch90,Bon16}.

The general form of the equations of motion, which is valid regardless of the quantum natures of the system or bath, is given as
\begin{widetext}
\begin{eqnarray}
\left\{
\begin{array}{ll}
\displaystyle
\frac{d \langle a^\dagger_1 a_1 \rangle }{dt} &=  i \sum_\nu g_\nu (\langle a^\dagger_\nu a_1 \rangle  - \langle a^\dagger_1 a_\nu \rangle) +  i \sum_\nu g_\nu ( \langle a_\nu a_1 \rangle - \langle a^\dagger_1 a^\dagger_\nu \rangle )  \\
\\
\displaystyle \frac{d \langle a^\dagger_1 a_\alpha \rangle }{dt} &=    i  (\omega_1 - \omega_\alpha) \langle a^\dagger_1 a_\alpha \rangle
+i   \sum_{\nu} g_{\nu} \langle (1-[1-\varepsilon_1]a^{\dag}_1a_1)  \left[ a_{\nu}^{\dag} a_\alpha + a_{\nu} a_\alpha \right] \rangle
-i g_{\alpha} \langle (1-[1-\varepsilon_\alpha] a^{\dag}_{\alpha} a_{\alpha}) [a^{\dag}_1 a_1 + a^{\dag}_1 a^\dagger_1]\rangle \\
\\
\displaystyle \frac{d \langle a^\dagger_\alpha a_\beta \rangle }{dt} &=   i  (\omega_\alpha - \omega_\beta )\langle a^\dagger_\alpha a_\beta \rangle
+ i g_{\alpha}\langle  (1-[1-\varepsilon_\alpha] a^{\dag}_{\alpha} a_{\alpha})[  a^{\dag}_1 a_\beta +  a_1 a_\beta ] \rangle
-i g_\beta\langle [a^\dagger_\alpha a^\dagger_1 + a^\dagger_\alpha a_1 ] (1-[1-\varepsilon_\beta] a^{\dag}_{\beta} a_{\beta}) \rangle \\
\\\displaystyle
\frac{d \langle a^\dagger_1 a^\dagger_1 \rangle}{dt} &= 2i  \omega_1  \langle a^\dagger_1 a^\dagger_1 \rangle
+i\displaystyle \frac{1+\varepsilon_1}{2}   \sum_{\nu} g_{\nu}\left( \langle  (1-[1-\varepsilon_1]a^{\dag}_1a_1) [ a_{\nu}^{\dag}a^\dagger_1 + a_{\nu} a^\dagger_1 ]\rangle
+ \langle  [ a^\dagger_1 a_{\nu}^{\dag} + a^\dagger_1a_{\nu} ] (1-[1-\varepsilon_1]a^{\dag}_1a_1) \rangle \right) \\
\\
\displaystyle
\frac{d \langle a^\dagger_1 a^\dagger_\alpha \rangle}{dt} &= i  (\omega_1 + \omega_\alpha) \langle a^\dagger_1 a^\dagger_\alpha \rangle
+i   \sum_{\nu} g_{\nu}\langle  (1-[1-\varepsilon_1]a^{\dag}_1a_1) \left[ a_{\nu}^{\dag}a^\dagger_\alpha + a_{\nu} a^\dagger_\alpha \right]\rangle
+ i g_{\alpha} \langle [a^\dagger_1  a_1 + a^\dagger_1a^\dagger_1] (1-[1-\varepsilon_\alpha] a^{\dag}_{\alpha} a_{\alpha})\rangle   \\
\\
\displaystyle
\frac{d \langle a^\dagger_\alpha a^\dagger_\alpha \rangle}{dt} &= 2i  \omega_\alpha  \langle a^\dagger_\alpha a^\dagger_\alpha \rangle
+i \displaystyle\frac{1+\varepsilon_\alpha}{2}   g_{\alpha}\left( \langle  a^{\dag}_\alpha a_1 \rangle
+ \langle  a^\dagger_\alpha a_{1}^{\dag}\rangle + \langle a_1 a^\dagger_\alpha \rangle + \langle a^{\dag}_1a^{\dag}_\alpha \rangle \right) \\
\\
\displaystyle
\frac{d \langle a^\dagger_\alpha a^\dagger_\beta  \rangle}{dt} &= i  (\omega_\alpha + \omega_\alpha)
 \langle a^\dagger_\alpha a^\dagger_\beta  \rangle + i g_{\alpha} \langle (1-[1-\varepsilon_\alpha] a^{\dag}_{\alpha} a_{\alpha})[  a^{\dag}_1 a^\dagger_\beta  +  a_1 a^\dagger_\beta] \rangle + i g_{\beta} \langle [  a^\dagger_\alpha a^{\dag}_1  +  a^\dagger_\alpha a_1]  (1-[1-\varepsilon_\beta] a^{\dag}_{\beta} a_{\beta})\rangle .
 \end{array}
\right.   \label{eq:fullexact}
\end{eqnarray}
\end{widetext}
When the system and environment contain only bosons, we have $[1-\varepsilon_1] = [1-\varepsilon_\alpha] = 0$ for all $\alpha$ and we
recover the set of equations (\ref{eq:sebosons}). The evolution of the occupation number $M_{11}$  depends on the off-diagonal elements
$M_{1\nu}$, $M_{\nu1}$, $K_{1\nu}$, and $K^{*}_{1\nu}$
of the normal and anomalous densities whose evolutions
depend on $\langle a^{\dag}_1a_1 a_{\nu}^{\dag}a_\alpha \rangle    $, $\langle a^{\dag}_1a_1 a_{\nu}a_\alpha \rangle $,
$\langle a^{\dag}_{\alpha} a_{\alpha} a^\dagger_1 a^{\dag}_1 \rangle$, $\langle a^{\dag}_{\alpha} a_{\alpha} a^\dagger_1 a_1 \rangle $, ...
These degrees of freedom are themselves coupled to higher-order moments related to higher-order quantum fluctuations.
The full problem cannot be solved exactly, due to the number of degrees of freedom
that should be followed in time when fermions are considered.

If the system is driven by the fermionic and/or bosonic harmonic potentials that destroy high-order quantum fluctuations,
then these fluctuations are presented as a product of two diagonal elements or a product of diagonal
($M_{11}$, $M_{\alpha\alpha}$, $K_{11}$, $K_{\alpha\alpha}$)
and off-diagonal ($M_{\nu \alpha}$, $K^*_{\alpha \nu}$, $K_{\nu\alpha}$, where $\alpha\ne\nu$) elements of the normal and anomalous densities,
  \begin{eqnarray}
  \left\{
  \begin{array} {l}
\displaystyle \langle a^{\dag}_1a_1 a_{\nu}^{\dag}a_\alpha \rangle
\simeq \langle a^{\dag}_1a_1  \rangle \langle a_{\nu}^{\dag}a_\alpha \rangle                = M_{11} M_{\nu \alpha}   \\
\\
\langle a^{\dag}_1a_1 a_{\nu}a_\alpha \rangle
\simeq \langle a^{\dag}_1a_1  \rangle \langle a_{\nu}a_\alpha \rangle                       = M_{11} K^*_{\alpha \nu} \\
\\
\langle a^{\dag}_{\alpha} a_{\alpha} a^\dagger_1 a^{\dag}_1 \rangle
\simeq \langle a^{\dag}_{\alpha} a_{\alpha} \rangle \langle  a^\dagger_1 a^{\dag}_1 \rangle = M_{\alpha\alpha} K_{11} \\
\\
\langle a^{\dag}_{\alpha} a_{\alpha} a^\dagger_1 a_1 \rangle
\simeq \langle a^{\dag}_{\alpha} a_{\alpha} \rangle \langle  a^\dagger_1 a_1 \rangle        = M_{\alpha\alpha} M_{11}\\
\\
\cdots
\end{array}
\right. \label{eq:moments}
\end{eqnarray}
This truncation procedure is equivalent to the linearization of the equations of motion with respect to the off-diagonal elements
$M_{\alpha\beta}$, $K_{\alpha\beta}$,  $K^{*}_{\alpha\beta}$
($\alpha\ne\beta$, including the cases of $\alpha=1$, $\beta=1$).
The right hand sides of these equations contain the terms with the off-diagonal elements
only in the first order, i.e., the
terms containing product  of two off-diagonal elements are neglected,
since   they are very small with respect
to the corresponding terms  that are proportional to a product of two diagonal elements or a product of diagonal and off-diagonal  elements.

Employing the mean-field type approximation (\ref{eq:moments}), we obtain from the
exact Eqs. (\ref{eq:fullexact}) the closed set of equations of motion:
\begin{widetext}
\begin{eqnarray}
\left\{
\begin{array} {ll}
\displaystyle \frac{d M_{11}}{dt} &=  i \sum_\nu g_\nu (M_{\nu 1}  - M_{1\nu}) +  i \sum_\nu g_\nu (K^*_{1\nu} -
K_{1\nu} )   \\
\\
\displaystyle
 \frac{dM_{1\alpha} }{dt} &=      i  (\omega_1 - \omega_\alpha) M_{1\alpha}
+i \xi_1 (t) \sum_{\nu} g_{\nu} (M_{\nu\alpha} + K^*_{\alpha\nu})
-i g_\alpha \xi_\alpha(t)  (M_{11} + K_{11})  \\
\\
\displaystyle  \frac{d M_{\alpha\alpha}}{dt} &=  i g_\alpha (M_{1\alpha}  - M_{\alpha 1} + K^*_{\alpha 1} - K_{\alpha 1}) \\
\\
\displaystyle \frac{d M_{\alpha \beta} }{dt} &=   i  (\omega_\alpha - \omega_\beta )M_{\alpha \beta}
+ i g_{\alpha} \xi_\alpha(t) (M_{1\beta} + K^*_{\beta 1})
-i g_\beta \xi_\beta(t)  (M_{\alpha 1} + K_{\alpha 1}) \\
\\
\displaystyle  \frac{d K_{11} }{dt} &= 2i  \omega_1 K_{11} +i \displaystyle \frac{(1+ \varepsilon_1)}{2} \sum_{\nu} g_{\nu} ( K_{\nu 1} + M_{1\nu} + G_{\nu 1} + K_{1\nu})  \\
\\
\displaystyle\frac{d K_{1\alpha}}{dt} &= i  (\omega_1 + \omega_\alpha)  K_{1\alpha}
+i  \xi_1(t)   \sum_{\nu} g_{\nu} (G_{\nu \alpha} + K_{\nu\alpha})  +i  g_\alpha   \xi_{\alpha} (t) (M_{11}+K_{11}) \\
\\
\displaystyle \frac{d K_{\alpha\alpha}  }{dt}&= 2 i  \omega_\alpha K_{\alpha\alpha} +\displaystyle  i \frac{(1+\varepsilon_\alpha)}{2} g_\alpha \left( M_{\alpha 1} + K_{\alpha 1} + G_{1 \alpha} + K_{\alpha 1}\right) \\
\\
\displaystyle \frac{d K_{\alpha\beta}  }{dt}&= i  (\omega_\alpha + \omega_\beta)  K_{\alpha\beta}
+ i g_\alpha \xi_\alpha(t) (G_{1 \beta} + K_{1\beta})
 + i g_\beta \xi_\beta(t) (M_{\alpha 1} + K_{\alpha 1}),
\end{array}
\right.   \label{eq:fcfulllast}
\end{eqnarray}
\end{widetext}
where $\xi_\alpha(t)  =  \langle [a_\alpha, a^\dagger_\alpha] \rangle$ (including the case of $\alpha$=1).
This set of equations is the main result of the present work.
It can be applied regardless of the quantum natures of the system or baths (fermionic or bosonic)
as well as to system coupled to several baths.  In the following, the method will be called Coupled Equations of Motion (CEM) method.

There are a number of properties of $M$, $G$, and $K$ components
which help us to solve these equations.  Assuming that each bath is composed of a set of independent two-level
systems, we have for all $\alpha$ and $\beta$:
\begin{eqnarray}
K_{\alpha \beta} &=& K_{\beta\alpha}, \nonumber \\
G_{\alpha \beta} &=& M_{\beta\alpha}, ~~~(\alpha \neq \beta)   \nonumber \\
G_{\alpha\alpha} &=& 1 + \varepsilon_\alpha M_{\alpha \alpha}.  \nonumber
\end{eqnarray}
The last relation implies that we also have $\xi_\alpha(t)  =  1 + [\varepsilon_\alpha -1] M_{\alpha\alpha}$.

When the system and baths are all composed of bosons, we have   $\xi_1(t)= 1$ and  $\xi_\alpha(t) = 1$ for all $\alpha$
and it could be easily shown that Eqs.~(\ref{eq:fcfulllast}) are reduced to Eqs.~(\ref{eq:sebosons}). Therefore we also
obtain an exact solution of the problem in the B-B case.
Another important property visible in  (\ref{eq:fcfulllast}) is that, if we assume $\alpha$ to be fermionic, we have simply $\dot K_{\alpha\alpha} = 2i\omega_\alpha K_{\alpha\alpha}$.  Since at initial time $K_{\alpha\alpha}(0)=0$ for fermions, this property is also respected at all time as it should be.

The truncation procedure leading to the set of coupled equations (\ref{eq:fcfulllast}) was tested for the F-F, F-B, and B-F cases. We
found that, for these systems, the occupation numbers $M_{11}$ calculated with (\ref{eq:fcfulllast}) and within the  Langevin
approach of Refs. \cite{Sar14,Lac15,Sar17,Hov18,Sar18,Hov19,Hov20}, taking into account the Pauli principle,  have almost the same time dependencies and asymptotic values.
This indirectly justifies our truncation procedure.

\section{Applications}
\label{sec:result}

\subsection{System coupled to single heat bath}

In the following, the results obtained for the FC Hamiltonian
using the CEM approach for a system coupled to one bath
with various quantum statistics are compared with our previous calculations in Fig. \ref{fig:BBFFold}
for the B-B and F-F and in Fig.~\ref{fig:BFFBold} for the B-F and F-B cases.

There are a number of remarks that can be
made from the comparison. Since Eqs.~(\ref{eq:fcfulllast}) are identical to Eqs.~(\ref{eq:sebosons}) in the B-B case,
we obviously observe in Fig. \ref{fig:BBFFold}  (panels (a) and (c)) that results of the two sets  of equations coincide.
In all cases, we see in Figs. \ref{fig:BBFFold}  and  \ref{fig:BFFBold}
that the time-scale to reach the asymptotic equilibrium is compatible with our previous estimates. However,
when fermions are present in the system and/or bath, the amplitudes of oscillation during the descent to equilibrium are reduced in the new
approach. This is particularly visible when the system is fermionic where the oscillations completely disappear. The asymptotic limit
is more modified  in the F-B or B-F case (Fig.  \ref{fig:BFFBold}) and to a lesser extent in the F-F case (Fig. \ref{fig:BBFFold}).
When the system is fermionic, we see by comparing  Figs.  \ref{fig:BBFFold}
and  \ref{fig:BFFBold} that the quantum nature (fermionic or bosonic) of the bath affects much
less the evolution compared to the case of a bosonic system. This will be systematically observed in all illustrations given below.
Another generic feature is the absence of unphysical values for the occupation probabilities of fermionic system when Eqs. (\ref{eq:fcfulllast})
are solved. This gives indirect indication that the fermionic nature of the system is properly accounted for.

\subsection{Toward simplified  treatments of fermionic or bosonic system coupled to single heat bath}
\label{sec:appfermi}

We propose here a method to treat the Fermi (or Bose) systems coupled to one or several heat-baths
by solving discretized versions of different heat-baths together with numerical integration of a closed set of equations
between the normal and anomalous densities of the system+environment. The discretization technique can be rather costly numerically.
With the aim to treat more elaborated systems (with many Qubits) coupled to many heat-baths, we further explore  the possibility to obtain a simpler framework compared
to Eqs.~(\ref{eq:fcfulllast}).

One possibility is to
assume that the off-diagonal matrix elements $M_{\alpha\beta}$ and $K_{\alpha\beta}$ in Eqs.~(\ref{eq:fcfulllast}) are zero when both $\alpha$ and $\beta$
belong to the bath.  This approximation, that is expected to be accurate in the weak-coupling regime, is discussed in more details in Appendix \ref{sec:approx}. In
particular, a connection with the diffusion equation (\ref{eq:diffusion}) is made.
We compare in Fig.   \ref{fig:alldiag} the full and approximate treatments for the F-F, F-B, B-F, and B-B cases. We clearly
see from this figure that the approximation leads to unphysical occupation numbers for  bosonic system.
Surprisingly enough, for Fermi systems, although not perfect, the approximation turns out to reproduce quite well the full evolution
whatever is the nature of bath. We finally also checked numerically for the F-F case that the approximation is better when the temperature increases but
degrades when the coupling strength increases.  The same conclusion can be drawn for a Fermi system coupled to several baths
independently of the quantum nature of the baths.
\begin{figure}[!h]
\includegraphics[width= 1.0\linewidth]{All_diagT2T5New.pdf}
\caption{Comparison of the evolution of  $n_1(t)$ in the FC coupling for a system coupled to a single bath
obtained  within the CEM approach (black solid lines) at $T_1/\Omega= 2$ (left panels) and $T_1/\Omega= 5$ (right panels). Results are
systematically compared to the CEM approach  assuming $K_{\alpha\beta} = 0$  and $M_{\alpha \beta}=0$ for $\alpha \neq \beta$  during the evolution (red dashed line).
Here, $\alpha$ and $\beta$ belong to the bath.
 The combinations of the system and heat-bath are indicated in each panel.
 All calculations are performed at $\gamma_1/\Omega=12$ and $c_1=0.1$. }
    \label{fig:alldiag}
\end{figure}

We explored the possibility to obtain the alternative  simplified description of bosonic system coupled to a bosonic or fermionic
systems. For the bosonic case, one cannot set only part of the anomalous density $K$ to zero. As clearly seen in Eqs.~(\ref{eq:fcfulllast}), this comes from the coupling between the diagonal and off-diagonal matrix elements. For the fermionic case, the situation is different because the diagonal part of $K$ is automatically zero.
When we neglect only part of the components of $K$ for boson system, we obtain unphysical results. As an alternative to the previous approximation for boson system, one can also set
all components of $K$ to zero together with $M_{\alpha \beta}=0$ if $\alpha$ and $\beta$ are both in the environment. We compare in Fig.
\ref{fig:alldiagBBBF}  the approximate evolution with the full CEM evolution. By setting $K=0$, the unphysical evolution observed in Fig. \ref{fig:alldiag} for bosonic system disappears.
We see that such approximation is satisfactory reproducing the asymptotic behavior but some important physics is missed
during the evolution to equilibrium. Note that, this approximation can also be applied to the Fermi system (Fig. \ref{fig:alldiagBBBF}) but the reproduction of the full CEM approach degrades compared to the red lines in Fig. \ref{fig:alldiag}.

In the present section, we discussed the possibility to simplify the description of Fermi/Boson systems coupled to an environment by neglecting  the  components of the normal $M_{\alpha\beta}$ and anomalous $K_{\alpha\beta}$ densities when $\alpha$ and $\beta$ are both in the environment in accordance with the Eigenstate Thermalization Hypothesis (ETH)  \cite{Deu91,Sre94,Rig08,Rig09,San10,Rig12,Kha12}. It would be interesting to investigate further possible connection with the  ETH and its domain of different regime of validity
  when the quantum nature of the system change from Fermions to Bosons.

\begin{figure}[!h]
\includegraphics[width= 1.0\linewidth]{All_diagT2T5NewApp.pdf}
\caption{Comparison of the evolution of  $n_1(t)$ in the FC coupling for the system coupled to single bath
obtained  within the CEM approach (black solid lines) at $T_1/\Omega= 2$ (left panels) and $T_1/\Omega= 5$ (right panels). Results are
systematically compared to the CEM approach  assuming that the anomalous density is zero ($K = 0$) together with
$M_{\alpha \beta}=0$ for $\alpha \neq \beta$  during the evolution (red dashed line). Here, $\alpha$ and $\beta$ belong to the bath.
 The combinations of the system and heat-bath are indicated in each panel.
 All calculations are performed at $\gamma_1/\Omega=12$ and $c_1=0.1$. }
    \label{fig:alldiagBBBF}
\end{figure}

\subsection{Results for system coupled with two heat-baths }

Besides the possibility to treat mixture of fermions and bosons, one of the attractive aspects
of the present method  is the possibility to treat the coexistence of several baths. Equations (\ref{eq:fcfulllast}) to be solved
are unchanged when considering  several baths. The main difference is the sizes of the $M$ and $K$ matrices
that both increase with the number of baths after discretizing each environment.  We consider below a two-level system
coupled to two baths. Each bath, discretized using the method presented in section \ref{sec:dem} (see also \cite{Lac15}),
is characterized by the parameters $c_{i}$ and $\gamma_i$ as well as its initial temperature $T_i$ ($i=1,$ $2$).

\begin{figure}[!h]
\includegraphics[width= \linewidth]{T1g01_fc_2b_bis.pdf}
    \caption{Evolution of $n_1(t)$ obtained in the FC case for the Fermi system coupled to two fermionic baths (a) (notation F-F$_1$-F$_2$) and
     for the Bose system coupled to two bosonic baths (b) (notation B-B$_1$-B$_2$) using the CEM approach. Two baths have the same temperature
     $T_1/\Omega=T_2/\Omega = 1$ and the same coupling strengths $c_1=c_2= 0.1$. The blue solid line
    corresponds to $\gamma_1/\Omega=\gamma_2/\Omega = 12$ while the red dashed line is obtained at $\gamma_1/\Omega=12$ and
    $\gamma_2/\Omega = 20$.

    }
    \label{fig:fffbbb}
\end{figure}

\subsubsection{System coupled to two heat baths of the same quantum nature}

As the first illustration, we consider the same conditions as  in  Ref. \cite{Hov18} where
the system is coupled to several heat-baths of the same quantum nature, i.e. the F-F$_1$-F$_2$ and
B-B$_1$-B$_2$ cases. This two cases are presented in Fig. \ref{fig:fffbbb} and can be compared to Fig.~3 of Ref. \cite{Hov18}.
For the B-B$_1$-B$_2$  case, we again perfectly recover our previous result. This could be considered as a numerical test
because the Laplace transform method used in Ref.~\cite{Hov18} and the present approach based on Eqs.~(\ref{eq:fcfulllast})
are strictly equivalent and exact.

For the case of fermionic system coupled to two fermionic baths, some differences are observed with \cite{Hov18}
although the global shape and asymptotic limit are similar.
In general, while the time-scale before reaching equilibrium and the asymptotic limit is globally in agreement
with Ref.~\cite{Hov18}, a difference is the absence of oscillations during the thermalization process.
We also checked numerically, as was analytically proved in our previous work, that the system coupled to two identical baths (the same $\gamma_i$ and the same temperature) can be treated as the system coupled to the single bath but with coupling strength equal to $c=c_1+c_2$.

\subsubsection{System coupled to two heat baths of mixed quantum natures}

We now analyze the change in the evolution compared to previous case when one of the bath or both baths have different quantum
natures compared to the system. Such situations are illustrated in Fig. \ref{fig:mixed}. To uncover the effect of changing Fermion into Boson or vice-versa, we compare situations where all other parameters
are unchanged, i.e. spectral function parameters, coupling strength and baths temperature remain  the same.
\begin{figure}[!h]
\includegraphics[width= \linewidth]{Fig2Vaz2bathsbis.pdf}
    \caption{Evolution of $n_1(t)$ for the F-F$_1$-F$_2$ (black solid line), B-B$_1$-B$_2$ (green dashed line), F-B$_1$-F$_2$ (red dotted line), B-F$_1$-B$_2$ cases (blue dotted-dashed line), F-B$_1$-B$_2$ (cyan long-dashed line) and B-F$_1$-F$_2$ (pink short dashed-dotted line) obtained with the CEM approach and FC coupling starting from $n_1(0) = 0$ (a) and $n_1(0)=1$ (b).   The coupling strengths, spectral properties and temperatures are set to  $c_1=c_2=0.1$,  $\gamma_1/\Omega=10$, $\gamma_2/\Omega = 15$, $T_1/\Omega=1$, and $T_2/\Omega = 2.0$. In panel (b), the results for the F-B$_1$-F$_2$, F-B$_1$-B$_2$ and F-F$_1$-F$_2$ cases are almost identical.}
    \label{fig:mixed}
\end{figure}

The first conclusion one could draw from Fig.~\ref{fig:mixed} is that, for the Fermi system, the nature of heat-bath does not affect much its evolution.
This is clearly the opposite to the bosonic system for which replacing bosonic bath by fermionic bath induces significant modification
both in the intermediate time evolution and asymptotic limit reached by the occupation probability.

Another aspect, which is visible in Fig.~\ref{fig:mixed}, is that an asymptotic stationary limit is always reached whatever are the natures of the system and heat-baths. This is in particular the case for the fermionic system coupled to two baths, one fermionic and one bosonic.
The convergence towards the stationary limit is systematically observed whatever are the properties of the baths, i.e. when changing the coupling strength, the spectral properties  and/or temperatures of two heat-baths. This conclusion is different from  the one we obtained previously for the F-B$_1$-F$_2$. Using slightly different approximation, we have shown that a stationary solution might never be reached \cite{Hov19}.  It should be noted however that both our previous prescription to the problem \cite{Sar17,Hov18,Sar18,Hov19,Hov20} and the
present one are only approximate when fermions are considered either in the system and/or baths.
The new approach proposed here has however the advantage
to account properly the Pauli principle for fermions. This could also be seen
from the occupation that remains bounded between 0 and 1. The possibility to reach or not a stationary limit when mixing baths with different
quantum natures is an interesting aspect. In particular, dedicated experiments would be interesting to clarify this issue.

 Because the bosonic systems coupled to one or several baths have been extensively investigated in our previous studies \cite{Sar14,Lac15,Sar17,Hov18,Sar18,Hov19,Hov20}, in the following, we focus on the Fermi systems and study in more details the evolutions
 obtained with the new approach proposed here.

 \subsection{Detailed study of Fermi system coupled to one or two bosonic and/or fermionic heat-baths}

 We systematically use Eqs.~(\ref{eq:fcfulllast}) to simulate the evolution of the Fermi two-level system coupled to
 one or two baths at various couplings, thermal and spectral properties. As illustrated in Fig.~\ref{fig:fffbbb} and \ref{fig:mixed},
 the evolution of $n_1(t)$ is rather simple and seems to correspond to the decay process.
 Based on this observation and with the goal to infer generic
 properties of the system evolution due to the surrounding environment, we fit the occupation number evolution $n_f(t)=n_1(t)$
 with the simple function
\begin{eqnarray}
n_f(t) = n_f(\infty) + [n_f(0) - n_f(\infty) ] e^{-\Gamma_f t}.  \label{eq:nffit}
\end{eqnarray}
In the following study, we will consider the case $n_f(0) = 1$. $n_f(\infty) $ and $\Gamma_f$ are fitted on the evolutions
and correspond respectively to the asymptotic occupation number and to the
decay time $\tau_f = 1/\Gamma_f$. Despite its simplicity, Eq.~(\ref{eq:nffit}) turns out to provide a rather precise description of the evolution for the whole range of couplings and temperatures considered.
\begin{figure}[!h]
\includegraphics[width= \linewidth]{FBFg005Decay_variousT5.pdf}
    \caption{Evolution of $n_f(\infty) $ (a) and $\Gamma_f$ (b) obtained by fitting the evolution of $n_f(t)$ with expression (\ref{eq:nffit}) for the
     Fermi system coupled to one or two baths with various initial temperatures.
    The blue circles correspond to the F-F case, where the bath properties are set to $c_1=0.1$ and
    $\gamma_1/\Omega=12$. The red squares correspond to the F-B case where the only difference with previous case is that the
    Fermi bath is replaced by the Bose bath.  The green stars correspond to the F-B$_1$-F$_2$ case with $c_1=c_2=0.05$,
    and $\gamma_1/\Omega=\gamma_2/\Omega=12$. Note the F-B$_1$-B$_2$ and F-F$_1$-F$_2$ (not shown) match exactly the F-B and F-F case with a coupling equal to $c_1+c_2$. In the multi-baths case, we assume that all baths are at the same temperature, $T_1=T_2=T$. In both panels, the inset is a focus on the low temperature limit. The green open squares corresponds
    to the result of Eq. (\ref{eq:linc}). }
    \label{fig:FBFT}
\end{figure}
The systematic evolutions of $\Gamma_f$ and $n_f(\infty)$ obtained in the presence of one or two baths are reported
in Fig.~\ref{fig:FBFT}.

\begin{figure}[!h]
\includegraphics[width= \linewidth]{GammaT5FBF.pdf}
    \caption{Evolution of $\Gamma_f$ obtained for the F-B$_1$-F$_2$ as a function of $c_1/(c_1+c_2)$ assuming that $c_1+c_2=0.1$. The structural properties of the two baths are the same as in Fig. \ref{fig:FBFT} while $T_1/\Omega = T_2/\Omega=5$. The stars
    at the extremes correspond to the reference F-F  (F-B) calculations put artificially at $c_1/(c_1+c_2) = 1$ (0). The dashed line is a linear interpolation between the two stars. }
    \label{fig:g1g2}
\end{figure}

Focusing first on the single bath case, we  observe that the F-F and F-B cases lead to rather similar properties at low temperature. The evolutions of the Fermi systems at low temperature ($T/\Omega < 1$) appear to be rather insensitive to the quantum nature of the bath and/or if one- or several baths are coupled to the system. When the
temperature of the bath(s) increases, we see  significant differences in $\Gamma_f$ and to a lesser extent in $n_f(\infty)$ depending on the quantum nature of the baths.
For $n_f(\infty)$, we see only small differences between the F-F and F-B cases. It is interesting however to mention that the asymptotic occupation number of the F-B$_1$-F$_2$ matches the one of the F-B case for all temperatures. Therefore,  for two baths with different quantum natures but with equivalent spectral properties, the
bosonic bath seems to decide the asymptotic behavior.

In the F-F case, the decay time is almost independent of the temperature while for the F-B case, $\Gamma_f$  linearly increases with temperature
at $T/\Omega > 1$. This implies that the transient time to equilibrium is much shorter if the fermionic system is coupled to  bosonic bath rather than to  fermionic bath.
We also observe in Fig.~\ref{fig:FBFT} that $\Gamma_f$ obtained for the F-B$_1$-F$_2$ case is in-between the F-F and F-B cases. More precisely, we have $\Gamma_f = (\Gamma_{\rm FF} + \Gamma_{\rm FB})/2$ where we use the notations
$\Gamma_{\rm FF}$ ($\Gamma_{\rm FB}$) for the value of $\Gamma_f$ obtained in the F-F (F-B) case.   We further
investigated this simple behavior in the  F-B$_1$-F$_2$  case by varying the couplings values of $c_1$ and $c_2$ while keeping $c_1+c_2=0.1$.
The evolution of
  $\Gamma_f$  is displayed as a function of $c_1/(c_1+c_2)$ in Fig.~\ref{fig:g1g2} at $T_1/\Omega=T_2/\Omega=5$.
  We observe in this figure that we have approximately:
 \begin{eqnarray}
\Gamma_f &\simeq& \frac{c_1}{(c_1+c_2)} \Gamma_{\rm FF} +  \frac{c_2}{(c_1+c_2)} \Gamma_{\rm FB},  \label{eq:linc}
\end{eqnarray}
that is a very simple relationship.

Note that if we change the temperature, we change the absolute value of the dependence presented in Fig.~\ref{fig:g1g2}, but  all conclusions remain valid.
Such a behavior can again be tested in the experimental observations. In particular, one might imagine by mixing several baths and changing the relative strengths of the couplings with the system to control the decay properties even if the temperatures of different baths
are kept fixed.

The approximate treatment of an open quantum Fermi system discussed in section  \ref{sec:appfermi} and Appendix \ref{sec:approx} turns out to be useful to understand qualitatively
the conclusion made from Figs.~\ref{fig:FBFT} and \ref{fig:g1g2}. We have checked numerically that the approximation is also
accurate when the Fermi system is coupled to several baths in the high temperature--weak coupling limit.
As shown in Appendix \ref{sec:approx}, the simple linear relation (\ref{eq:linc}) can be explained consistently with this simplification.
Indeed, starting from the analytical equations (\ref{eq:gamd}) for  the asymptotic occupation probability and decay time,
one can explain why the decay time is independent of temperature in the case of Fermi bath while it increases with the temperature for Bose bath.

\section{Conclusion}

We proposed here a new approach called CEM to describe the system coupled to one or several baths eventually mixing different quantum natures
of the particles (fermions or bosons). The approach is exact when only bosonic degrees of freedoms are considered and
provide an approximate solution when fermions are also present either in the system and/or in the bath. In this novel approach,
a particular attention is paid to properly account for the Pauli principle for the fermions.
The approach is illustrated for the system coupled to single bath where the system or bath can be either fermionic or bosonic.
For Fermi systems, we showed that the proper treatment of the Fermi nature is essential to obtain an accurate
treatment of the evolution.

The approach includes non-Markovian effects and is rather versatile and we do not anticipate any specific difficulty to apply it to the baths with complex structure. We  illustrated the method for a system coupled to two baths with various quantum statistics. One of the important aspects that differs from our previous solution \cite{Hov19,Hov20} to this problem is that Fermi systems coupled to two baths always reach an asymptotic stationary limit. Since in the present
work, similarly to Ref.~\cite{Hov19}, the approach we propose is not exact, the existence or not of an asymptotic time-independent solution
is an interesting debate. In particular, it would be interesting to give an experimental clarification to this aspect.
Besides the asymptotic behavior, we observe that Fermi systems have relatively simple decay properties compared to Boson systems coupled to the same baths. We showed that the decay time of the Fermi system coupled to the fermionic and bosonic baths can be easily related to the cases of the system coupled to only one bosonic bath or to only one fermionic bath.

In the present work, we focused our attention to a single Qubits  coupled to a set of environments including fully non-Markovian effects.
The theory can a priori be extended to obtained numerical simulation of an ensemble of Qubits with the price of increasing the numerical cost.
With the target goal to be able to treat eventually several hundreds  of Qubits we also explore the possibility to obtain simplified theories
of Fermi systems while not degrading the description of evolution. We show that in some regime of coupling or temperature, the
simplification can indeed be made.

\section*{Acknowledgments}
The IN2P3(France)-JINR(Dubna) Cooperation Programme is gratefully acknowledged.
This work was partially supported by  Russian Foundation for Basic Research (Moscow), N 17-52-12015 and 20-02-00176.
This project has received financial support from the CNRS through the 80Prime  program.

redpajama_set_name: RedPajamaArXiv


text:
Sense of Security in Males Addicted to the Internet
Karolina Komsta-Tokarzewska The John Paul II Catholic University of Lublin, Faculty of Theology
Keywords: sense of security; Internet addiction
The aim of the article is to present and discuss the findings of research on a group of Internet addicts in the context of a sense of security. A growing interest in the issue of a sense of security, both from the socio-cultural and psychological perspective, is also reflected in the emergence of research on a sense of security among those who are at a high risk of Internet addiction. In Poland, this type of research is virtually non-existent so far. Its implementation and dissemination, coupled with the reflection thereon seems to be of paramount importance because of its wide practical implications: in the area of a therapeutic relationship, a healing process and prevention, as well as a deeper insight into the problems of Internet addicted patients.
An interpretive analysis covered the results for 136 respondents, including: Internet addicts and those not addicted to the Internet, aged 18-39. The research involved male subjects only.
Method. The research used the Sense of Security and Personal Resiliency Questionnaire (KPB-PO) by Zenon Uchnast and the Internet Addiction Test (IAT) by Kimberly Young.
Results. There are differences in a sense of security of Internet addicts and non-addicts. Developing a sense of security can contribute to improving the functioning of people who are addicted to the Internet.
Andrzejewska, Anna. Children and Adolescents in the Net of Real and Virtual Threats. Theoretical and Empirical Aspects. Warsaw: Difin, 2014.
Armstrong, Lynette, James G. Phillips, and Lauren L. Saling. "Potential Determinants of Heavier Internet Usage." The International Journal of Human-Computer Studies 53(2000), 4: 537-50.
Brzeziński, Jerzy. Metodologiczne i psychologiczne wyznaczniki procesu badawczego w psychologii. Poznań: Wydawnictwo Uniwersytetu im. Adama Mickiewicza, 1978.
Chen, Sze Hua, and Chau Chiung. "A Review of the Research on Internet Addiction." Educational Psychology Review 17(1999), 4: 363-388.
Eichenberg, Christiane, Schott Markus, Decker Oliver, and Brigitte Sindelar. "Attachment Style and Internet Addiction: An Online Survey." Journal of Medical Internet Research 19(2017), 5: 170-78.
Horney, Karen. The Neurotic Personality of Our Time. Warsaw: Dom Wydawniczy REBIS, 1999.
Jakubik, Andrzej. "Internet Addition Syndrome." Studia Psychologica UKSW 3(2002): 133-42.
Jakubik, Andrzej, Justyna Popławska. "Internet Addition Syndrome." Studia Psychologica 4(2002): 123-31.
Kaliszewska, Katarzyna. Excessive Internet Use. Psychological Characteristics. Poznań: UAM, 2010.
Kowalski, Radosław. "Internet Addiction: Truth, Circumstantial Evidence or Fiction?" Wychowanie na co dzień 9(2005): 4-8.
Krzyżak-Szymańska, Ewa, Andrzej, Szymański. Preventing New Addictions among Children and Adolescents. Katowice: Górnośląska Wyższa Szkoła Handlowa im. Wojciecha Korfantego, 2013.
Kuss, Daria J., Olatz Lopez-Fernandez. "Internet Addiction and Problematic Internet Use: A Systematic Review of Clinical Research." World Journal of Psychiatry 6(2016), 1: 143-76.
Maslow, Abraham. Manual for the Security-Insecurity Inventory. Palo Alto: CA: Consulting Psychologists Press, 1952.
Maslow, Abraham. Toward a Psychology of Being. Warsaw: PAX, 1986.
Maslow, Abraham. "A Theory of Human Motivation." In Motivation and Personality. Edited by Abraham Maslow. Warsaw: PAX, 1990.
Poprawa, Ryszard. "Problematic Use of the Internet − Symptoms and Method of Diagnosis. Research among Adolescents," Psychology of Quality of Life 11(2012), 1: 57-82.
Reber, Arthur S. Dictionary of Psychology. London: Penguin Books, 1985.
Shubnikova, Ekaterina G., Anvar N., Khuziakhmetov, and Daria P. Khanolainen. "Internet-Addiction of Adolescents: Diagnostic Problems and Pedagogical Prevention in the Educational Environment." Eurasia Journal of Mathematics, Science and Technology Education 13(2017), 8: 5261-71.
Uchnast, Zenon. "The Concept of Man as Understood by Humanistic Psychologists." Częstochowskie Studia Teologiczne 6(1979): 461-77.
Uchnast, Zenon. "The Method of Measuring the Sense of Security." In Lectures on Psychology at the Catholic University of Lublin. Edited by Zenon Uchnast, Tomasz Witkowski. Lublin: RW KUL, 1991.
Ulman, Piotr. "Social and Family Determinants of Addictions in Children and Young People." Fides et Ratio 8(2011), 4: 74-86.
Walc, Wiesława. Addictions among Children and Young People: Selected Problems. Rzeszów: Wydawnictwo Uniwersytetu Rzeszowskiego, 2016.
Woronowicz, Bohdan T. Addictions and their Treatment: Full Disclosure. Warsaw: IPiN, 2001.
Young, Kimberly, Robert Rodgers. "The Relationship between Depression and Internet Addiction." CyberPsychology and Behavior 1(1998), 1: 25-8.
Young, Kimberly. "Psychology of Computer Use: XL. Addictive Use of the Internet: a Case that Breaks the Stereotype." Psychological Report 79(1996): 899-902.
Young, Kimberly. "Internet Addiction: A New Clinical Phenomenon and Its Consequences." American Behavioral Scientist 48(2004), 4: 402-15.
Young, Kimberly. "Technology is Changing the Nature of Problems People are Having as well as how We Treat Them". In Treatment Outcomes with Internet Addicts. Published in CyberPsychology&Behavior,2007,Vol.10,No.5;pp.671-679. Last modified April 14, 2018; http://www.netaddiction.com/articles/Treatment.pdf

redpajama_set_name: RedPajamaCommonCrawl


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The best way to get classroom arrangement ideas is to see photos of how REAL teachers have set up their rooms. On this page, you can tour all of my classrooms from 2003-2009, and also view photographs of my colleagues' classrooms and those of website visitors. You're sure to get some creative ideas for utilizing your space in a new way!
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Izzyshare is a site run by a kindergarten teacher who loves to decorate. Her name is Ashlee and she hopes to make her site the Facebook for people interested in learning environments for children. The site is a collection of classroom tours and teachers from all over have submitted their classroom photos. What a cool idea!
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text:
Chapter 7 Mapping Archaeological Landscapes in Transformation (Book chapter)
Authors: Hacıgüzeller, Piraye --- Poblome, Jeroen --- Taelman, Devi --- Vandam, Ralf --- et al. Vermeulen , Frank --- Coomans, Thomas --- Cattoor, Bieke --- De Jonge, Krista
Publisher: Leuven University Press
Subject: Archaeology --- History --- Architecture
"The relational complexity of urban and rural landscapes in space and in time.

The development of historical geographical information systems (HGIS) and other methods from the digital humanities have revolutionised historical research on cultural landscapes. Additionally, the opening up of increasingly diverse collections of source material, often incomplete and difficult to interpret, has led to methodologically innovative experiments. One of today's major challenges, however, concerns the concepts and tools to be deployed for mapping processes of transformation—that is, interpreting and imagining the relational complexity of urban and rural landscapes, both in space and in time, at micro- and macro-scale.

Mapping Landscapes in Transformation gathers experts from different disciplines, active in the fields of historical geography, urban and landscape history, archaeology and heritage conservation. They are specialised in a wide variety of space-time contexts, including regions within Europe, Asia, and the Americas, and periods from antiquity to the 21st century."
digital humanities --- historical geography --- urban history --- landscape history --- archaeology --- heritage conservation --- architecture
Leuven University Press (1)


text:
Home » Rallying for Gun Reform
Rallying for Gun Reform
Kristin Goss and Donna Dees
After the massacre of 20 Connecticut schoolchildren and six women who died trying to save them, plans are afoot for a parents' protest for stricter gun laws. More than a decade ago, we had such an event — the Million Mom March — and the lessons are instructive.
One of us devised the idea for the Mother's Day march, led the national organizing effort and remained involved as a volunteer; the other conducted a scholarly study of the march participants. Here's what we learned.
Parents will show up. Although inside-the-Beltway gun control advocates were initially skeptical of the idea, a mothers' march resonated immediately with everyday folks. Having sought a permit for 10,000 demonstrators, organizers were stunned when 75 times that number arrived — not including thousands more at 70 simultaneous events around the country.
Talking about child protection works. An analysis of surveys taken at the march and again six to eight months later found that participants motivated by child safety were significantly more likely to stay involved than were participants motivated by other reasons. These findings remind us that other social movements — such as those for tobacco and alcohol control and against abortion — also have successfully mobilized around child protection.
Marches are not enough. America has rampage shootings with numbing regularity, and lawmakers have grown accustomed to keeping their heads down, waiting for the inevitable outcry for stricter gun laws to fade. Enacting mainstream measures with majority support will take sustained, grassroots organizing in the districts where state and federal lawmakers live. In an especially polarized era, lawmakers aren't likely to lead, but they may follow.
Change happens from the ground up. Some pundits branded the Million Mom March a failure because Congress did not respond with sweeping gun control legislation. But most national policy reform happens incrementally, from the state level up. Working with their experienced allies, the moms enjoyed many early victories, including the defeat of two high-profile pro-gun lawmakers and the passage of referenda in Colorado and Oregon that closed the "gun show loophole" by requiring private sellers to conduct buyer background checks.
The National Rifle Association has perfected the "state by state" approach to nationwide policy reform, and pro-control groups have sought to emulate it. However, unlike the resourceful and well-coordinated gun lobby, state gun control groups have always been mostly volunteer-run operations, starved for sustained funding and tactical support. Hope arrived when the national Brady Campaign to Prevent Gun Violence absorbed local moms' march groups, but they lost steam as politicians came to perceive, with little evidence, that even talking about gun control was a career-ender. The recent shootings provide an opportunity to change that perception.
Movements work when old and new work together. Leaders of D.C.-based gun control organizations have been at it for decades, with little reward. But rampage shootings have catapulted new, authoritative voices onto the national stage, including Tom Mauser (whose son, Daniel, was killed at Columbine), Mindy Finkelstein and Joshua Stepakoff (shot soon thereafter at a California Jewish day camp), Colin Goddard (shot at Virginia Tech in 2007), and Lori Haas (mother of a student injured at Virginia Tech). History tells us that movements succeed when old-timers and upstarts agree to put aside their stylistic differences and share ideas and resources for common ends.
Whoever emerges to lead the post-Sandy Hook organizing would do well to listen to the veterans: Their experience in the political trenches is valuable, and they are more adaptive than were some of their colleagues a decade ago. Similarly, the veterans and their battle-weary volunteers need to respect the urgency that new activists bring, as experience tells us that the window for policy change slams fast.
After a spate of shootings and an inadequate response from lawmakers, we have reached a critical moment. Indeed, a Pew poll suggests that many Americans, especially women, see Sandy Hook not as an isolated incident but as the symptom of a broader problem. The emotion and energy flowing from the recent massacres should be channeled in a historically minded way toward policy reforms that the majority of Americans, stricken yet sensible, can embrace.
Kristin Goss is associate professor of public policy at Duke University's Sanford School of Public Policy and author of "Disarmed: The Missing Movement for Gun Control in America." Donna Dees, a lead organizer of the Million Mom March, is a television publicist and co-author of "Looking for a Few Good Moms." this commentary was originally published in The Tampa Bay Times and in The News & Observer.


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Major, Lindsey & Africa (MLA), the world's leading legal search firm, announced today that it has been voted Best National Legal Recruiter 2016 by the readers of Corporate Counsel. The inaugural "Best of Corporate Counsel" is a survey of the top providers to the in-house corporate legal marketplace, comprising 55 categories across the spectrum of legal products and services. More than 1,500 members of the in-house legal community voted in the online survey.
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text:
La Celle-sur-Morin  es una población y comuna francesa, en la región de Isla de Francia, departamento de Sena y Marne, en el distrito de Meaux y cantón de Coulommiers.

Demografía

Referencias

Enlaces externos
 INSEE

Celle Morin

redpajama_set_name: RedPajamaWikipedia


text:
Home » Mom's boyfriend tortures 3 years old girl to death: Samuel Lester Smart Biography, Wiki, Age, Net Worth, Family, Twitter, Instagram, Facebook
Mom's boyfriend tortures 3 years old girl to death: Samuel Lester Smart Biography, Wiki, Age, Net Worth, Family, Twitter, Instagram, Facebook
Samuel Lester Smart Biography, Wiki, Age, Net Worth
Samuel Lester Smart is accused of torturing 3-year-old Trinity Chandler, inside a home Saturday in Groveland Township. According to prosecutors, Smart is the boyfriend of Trinity's mother. The mother was working an overnight shift at a nursing home when the incident occurred.
Trinity Chandler is dead and police say it happened at the hands of her mother's boyfriend a day after Children's Protective Services checked up on her.
The CPS visited the house Friday and checked the boy. Social workers reportedly saw bruising at Trinity, but they decided to evict her from the house.
Ashley Brown, mother of Trinity's friends, said she reported child abuse to ABC 7 months ago and sent photos of Trinity's injuries to both the CPS and the state police.
To me, it seemed like two black eyes healing. "It was two," Brown said.
"October was when I made a report with MSP, a few months ago I saw it with scars under his eyes. It was blatant harassment. They could take him to a safe place until the investigation was over. "
Brown added that although the documented abuse dates back months ago, the CPS continues to leave Trinity with its mother, which will leave the child with Smart.
"Nothing has been done," Brown said. It was not taken. We hoped it would be, but we knew it wasn't. "
According to a Michigan State Police investigator, a video taken the day before Trinity's death showed that he was abused.
The Tri-County Times reported that Smart appeared at a trial Tuesday and his connection was denied. He is charged with torture and first degree child abuse. When an autopsy confirms Trinity's official cause of death, the charges may be raised.
"There are many CPS studies and reports between Mr. Smart and 3-year-old Trinity Chandler", Det. Sergeant. Said Walden of the MSP.
"There is an active and open investigation of child abuse by soldiers at the Metro North Post, and the CPS actually appeared on Friday to do a welfare check with Trinity. chief suspect. "
Smart gets behind bars in Oakland County Prison. Check back for updates.


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Listed by Margery Westerbeck, Long & Foster Real Estate, Inc.


text:
Irma aftermath: Lee schools knocked by strong winds, rain
More comprehensive damage reports are going to be put together by the district to share with the state DOE and federal agencies.
Irma aftermath: Lee schools knocked by strong winds, rain More comprehensive damage reports are going to be put together by the district to share with the state DOE and federal agencies. Check out this story on news-press.com: http://newspr.es/2w1XGRY
PAMELA McCABE, PMCCABE@NEWS-PRESS.COM Published 7:42 p.m. ET Sept. 11, 2017 | Updated 6:11 p.m. ET Sept. 12, 2017
Books and paper are strewn about a classroom at Lehigh Senior High School, where wind from Hurricane Irma broke a window.(Photo: Provided by Lee County schools, leeschools.net)
For more coverage of Hurricane Irma and the storm's aftermath visit: news-press.com/hurricane.
Busted-in classroom windows with books strewn around the floor. An empty portable flipped upside down, and crushed like an old can. Flooded parking lots surrounding the front entrance of a school campus.
These images, and many more, show the damage Hurricane Irma wrecked on campuses throughout the Lee County school system Sunday.
An empty portable being stored at a bus depot in Buckingham was flipped over and crushed by the wind of Hurricane Irma. (Photo: Provided by Lee County schools, leeschools.net)
Teams of employees, which are part of a "Rapid Assessment Team," went out to the public school system's 95 sites — not including its 26 charter sites — to see what was left in the storm's wake.
"Over the next few days, we will send out larger teams to put together a more comprehensive report of damage information to share with the Florida Department of Education and other federal agencies," reported Lauren Stillwell, the district's spokeswoman.
Here's some of the photos the district provided The News-Press:
Trees down at the entrance of Three Oaks Elementary:
Trees are down at the entrance to Three Oaks Elementary School. (Photo: Provided by Lee County schools, leeschools.net)
Damage to a portable classroom at Three Oaks Middle:
The portables at Three Oaks Middle School took a beating in Hurricane Irma. (Photo: Provided by Lee County schools, leeschools.net)
Broken classroom window at Lehigh Senior High School:
A classroom window is blown in by the wind from Hurricane Irma at Lehigh Senior High School. (Photo: Provided by Lee County schools, leeschools.net)
Flooding around Gulf Middle School:
Gulf Middle School is surrounded by water after Hurricane Irma. (Photo: Provided by Lee County schools, leeschools.net)
A flipped — but empty — portable that was being stored at the Buckingham Bus Compound:
Earlier today, the school system announced there would be no classes until Monday, Sept. 18, which would give staff and students who evacuated the area time to get back to Lee County.
This would also give everyone time to start repairs to their own homes, not to mention the schools that were damaged.
Because of safety issues, the district urges people to stay off local school campuses until the return of class Monday.
Bonita High construction site 'locked down' for Hurricane Irma
Connect with this reporter on Twitter @NP_pstaik.
Read or Share this story: http://newspr.es/2w1XGRY
After four busy hurricane seasons, will 2020 give us a break?
A bumpy hurricane decade comes to a close
Freeze will reach Florida as snow, record-breaking cold sweeps nation
Potential Tropical Cyclone 16 will likely become a tropical storm later today
Tropical storm warnings issued for Panhandle
Want to help the Bahamas recover from Hurricane Dorian? Don't stay away


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Here's a video with all of the popular ball python morhps.
Prices range from hundreds of dollars to thousands, so for those of you who plan on breeding ball pythons, it's good to start reading up on what type produces what.
My personal favorite ball python morph is the black eyed leucistic.


First, a disclaimer. I am very good at a lot of things. I remain calm in crisis, am willing to try new things (albeit grudgingly) and can play the piano. I have a degree in Art and English and I know a lot about baking cookies. I'm even up to date on my family scrapbooking! My point here is that I can hold my own in many different situations.
I am bad at directions. I believe I was born this way. I believe diligent work can help improve my everyday directional life, but I know to my very core that my directional intuition will always be suspect. I don't know why directions get me so mixed up – I am a visual person, I am a smart person. And yet, as soon as something involves directions I will 100% make wacky weird wrong assumptions.
I once got lost going to my local market. There are three parallel N/S streets I knew it must be one of them. And yet, though I tried two of the several times I refused to believe it was on that one that it actually was on and never made it there – went to a market slightly farther away.
No matter how much mobile phones/tablets destroy society I will always love them because without the maps app on my phone I would probably have long since become permanently lost; forced to restart my life twenty minutes from home, never knowing how to get back.
Which is why it is beyond ironic when one of our beta readers suggested we draw a little map of Bath to insert into our book, The Particular Charm of Miss Jane Austen , that I was the one most qualified to do it.
Is Cass bad at directions you ask? NO. Not at all. But she's unfamiliar with photoshop which is where we knew we'd end up needing to make it all work together considering our timeline and the lateness of the hour that the idea was given to us.
I shouldn't get all the credit however as, despite the internet's help and Cass sending me three emails, I could not for the life of me get the 4 Sydney Place map marker in the right spot. At one point, Cass (who KNOWS of my directional disability) was wondering if I was doing it on purpose to get out of doing the map at all! Hint: I wasn't.
SO yes, this map of Bath will be in our book, highlighting some of the sights of Bath and some of the locations important to our characters. Some of these references are relevant only in the book, so don't take anything too seriously. Also, do remember it was made by me (a person that when I am in Bath cannot move more than a window's length before getting lost since all the buildings look the same and I don't want to spend a million dollars on my cellphone data plan so I don't have my maps app open!) so don't count on it's accuracy in a life or death situation.
We are happy to be sharing it online since it will likely be too small to read in the actual printed copy!!
Next > Cover Reveal and Upcoming Blog Posts!
There seem to be a lot of 'refreshment places' on your map-I must try them next time I'm in Bath.