Timescales for decoherence and dissipation: where does the success of master equations come from?
A perturbative treatment of reduced density operators of quantum subsystems is implemented in the same spirit as Fermi Golden Rule for scattering. Analytic expressions for linear entropy (a measure of purity loss, and in some cases of coherence loss) and for subsystem's energy variations (dissipation) are given. Application to electromagnetic field superposition states in a dissipative cavity is performed. Evaluation of typical field and reservoir time scales ($\tau _{F}$ and $\tau _{R}$) show that even for small temperatures they are very different. We also indicate the condition on the cutoff temperature above which the decoupling assumption is quantitatively justified.
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\begin{document}


\title{Timescales for decoherence and dissipation: where does the success 
of master equations come from?}
\author{M.\ O.\ Terra Cunha\thanks{%
tcunha@mat.ufmg.br}}
\affiliation{Departamento de Matem\'atica, Universidade Federal de Minas Gerais, C.P. 702,
Belo Horizonte, MG, 30123-970, Brazil}
\affiliation{Departamento de F\'{\i}sica, Universidade Federal de Minas Gerais, C.P. 702,
Belo Horizonte, MG, 30123-970, Brazil}
\author{S. Geraij Mokarzel}
\affiliation{Instituto de F\'{\i}sica, Universidade de S\~ao Paulo, C.P. 20516,
S\~ao Paulo, SP, 01317-970, Brazil}
\author{J.\ G.\ Peixoto de Faria}
\affiliation{Departamento de Ci\^{e}ncias Exatas e Tecnol\'{o}gicas,
Universidade Estadual de Santa Cruz, Ilh\'{e}us, BA, 45662-000, Brazil}
\author{M.\ C.\ Nemes}
\affiliation{Departamento de F\'{\i}sica, Universidade Federal de Minas Gerais, C.P. 702,
Belo Horizonte, MG, 30123-970, Brazil}
\date{\today}


\begin{abstract}
A perturbative treatment of reduced density operators of quantum subsystems
is implemented in the same spirit as Fermi Golden Rule for scattering.
Analytic expressions for linear entropy (a measure of purity loss, and in some
cases of coherence loss) and for subsystem's energy variations (dissipation)
are given. Application to electromagnetic field superposition states in a
dissipative cavity is performed. Evaluation of typical field and reservoir
time scales ($\tau _{F}$ and $\tau _{R}$) show that even for small
temperatures they are very different. We also indicate the condition on the cutoff temperature 
above which the decoupling assumption is quantitatively justified. 
\end{abstract}

\pacs{03.65.-w, 03.65.Bz, 03.65.Ca  }
\maketitle

Since 1970 many techniques for controlling single quantum systems have been developed, 
\textit{e.g.}, trapping a single ion in what is called an ``ion trap''
\cite{ionTraps}, or storing a single mode of electromagnetic field in
high quality cavities \cite{cavity},
trying to isolate them from the rest of 
the world and thus allowing us to probe many different aspects of its behavior with 
amazing precision. However, the control over properties of quantum systems, the 
essential ingredient  for the technological implementation of quantum computers 
(as \textit{e.g.} entanglement among many qubits), is strongly hindered by deleterious 
environmental effects. The first experiment which clearly exhibits the effect 
of the environmental dynamics on a mesoscopic quantum superposition has been 
performed at the Laboratoire Kastler-Brossel in Paris \cite{paris}.

From the theoretical point of view several independent ideas have been developed to 
treat open quantum systems \cite{openQuSys}. The best known and used is the approach with 
master equations. Its starting point consists of a system of interest ($S$) 
linearly coupled to a 
``reservoir'' ($R$) usually simulated by a large number of harmonic oscillators 
(for a complete derivation see \cite{mod, cohen, Mte}). The trace 
over the reservoir degrees of freedom leads to an effective equation for the system 
of interest containing nonunitary operators, responsible for coherence and energy loss, where
the reservoir appears only through some mean values, just like in Thermodynamics.
Several approximations are necessary in order to obtain such linear, markovian equation 
for $S$. One approximation which needs physical support to rest on \cite{Amir}
is the one which considers the reservoir density
uncoupled from $S$ \emph{for all times}\footnote{It is usual to state only that
there are no correlation between $R$ and $S$ at the initial time, use it to
derive the master equation, and then apply it to any subsequent time. This
is equivalent to remake the same factorizability assumption at any time.}. It
can not be rigorously true, since decoherence of $S$ system can only come from
correlations with $R$. As discussed by Cohen-Tannoudji and co-workers \cite{cohen},
this is easy to accept it provided the temperature is large enough so that the typical time 
scales are very different. This tends to make any previous correlation between $R$ and $S$
irrelevant after a short time. However, we know from Schmidt's decomposition
that if we have
any pure bipartite system, the evolution of correlations is exactly the same for both 
subsystems. In this case, a ``cold'' reservoir consisting of all oscillators in the 
ground state belongs to this category and therefore the argument breaks down. In spite 
of this fact, master equations usually work extremely well also for zero temperature
\cite{lowTemp}. Why? This is one of the long standing questions in the area 
of open quantum systems.

The objective of the present contribution is to work out the typical time scales 
for variation of coherence and energy in any two coupled quantum systems. When 
one of the systems is a reservoir we obtain a limiting temperature for the 
validity of the factorization assumption holds.
For this purpose we next implement a perturbative treatment of reduced matrices 
of quantum systems in the same spirit as the Golden Rule for scattering.

Consider a Dyson like expansion for reduced operators which obey the following Hamiltonian dynamics
\be
H = H_{1} + H_{2} + H_C \equiv H_0 + H_C
\ee
where $H_0$ contains the autonomous evolution of the systems in question and $H_C$ 
represents the coupling between them. We will obtain perturbative equations for the reduced densities 
\be 
\rho_i = \Tr_j\rho(t)\,,
\ee 
where $\rho(t)$ is the total density, $i=1,2$, $j=1,2$ and $i\neq j$ are subsystem's indexes. We get
\be\begin{split}
\rho_i(t) & = \rho_i(0) - i\int_0^t\dif t' \Tr_j\bigl[H_C(t'),\rho(0)\bigr] \\
& \qquad -\int_0^t\dif t' \int_0^{t'}\dif t'' \Tr_j\bigl[H_C(t'),\bigl[H_C(t''),\rho(0)\bigr]\bigr] + 
\dotsb\quad (i\neq j) \\
& \equiv \rho_i^{(0)} + \rho_i^{(1)} +\rho_i^{(2)} + \dotsb\,,
\label{born}
\end{split}\ee
where $H_C(t) \equiv \e^{iH_0t}H_C\e^{-iH_0t}$ (the interaction representation is applied). Note the symmetry in the indexes $i,j$, 
which will allow us to study the short time dynamics of any of the two subsystems. Instead 
of proceeding formally and rewriting the above equation in terms of matrix elements, we prefer 
to illustrate the procedure for obtaining dissipation and decoherence time scales by mean of 
an example. Let ($\hbar = 1$)
\be
H_0 \equiv \omega a^\dag a + \sum_j \Omega_j b_j^\dag b_j,
\ee
which represents, \textit{e.g.}, the free Hamiltonian of a given cavity mode plus that of the environment 
simulated by oscillators as usual. $\omega$, $\Omega_j$, $a(a^\dag)$ and $b_j(b_j^\dag)$ 
are the natural frequencies, annihilation (creation) operators of the mode and the reservoir, 
respectively. We assume that the coupling between the two systems is given by
\be
H_C = \sum_j \gamma_j \bigl(a^\dag b_j + ab_j^\dag\bigr)\,,
\ee
where $\gamma_j$ are the coupling constants assumed to be real.

\emph{Subsystems dissipation timescales}: Let us define a subsystem energy by 
\be
E_i(t) = \Tr \left[H_{i}\rho_{i}(t)\right]
\label{eq5}
\ee
and assume that we initially have the state of the composed system given by
\be
\rho(0) = \rho_1(0)\otimes\frac{\e^{-\beta H_{2}}}{\Tr\bigl[\e^{-\beta H_{2}}\bigr]}\,,
\label{initial}
\ee
where $\beta = 1/(k_B T)$. Inserting Eq.~\eqref{born} in \eqref{eq5} we get, up 
to second order for the cavity mode labeled 1, 
\be
E_1(t) = E_1^{(0)} + E_1^{(1)}(t) + E_1^{(2)}(t) + \dotsb\,,
\ee
where the upper indexes indicate the order of the approximation. We get
\begin{align}
E_1^{(0)} & = \Tr_1\bigl[\omega a^\dag a \rho_1(0)\bigr] = 
\omega \langle a^\dag a\rangle_{0} = E_1(0), 
\label{E0th} \\
E_1^{(1)}(t) & = -i\omega\int_0^t\dif t' 
\Tr\bigl\{a^\dag a[H_C(t'),\rho(0)]\bigr\} = 0, 
\label{E1st} \\
E_1^{(2)}(t) & = -\omega \int_0^t\dif t'\int_0^{t'}\dif t'' 
\Tr\bigl\{a^\dag a[H_C(t'),[H_c(t''),\rho(0)]]\bigr\} \nonumber \\
& =  \omega\sum_j \left(\frac{\gamma_j \sin\Delta_j t}{\Delta_j}\right)^2
\bigl(\bar n_j - \langle a^\dag a\rangle_{0}\bigr)\, ,
\label{E2nd} 
\end{align}
where $\Delta_j = \left(\Omega_j-\omega\right)/2$, 
$\bar n_j = \left(e^{\beta \Omega_j} - 1\right)^{-1}$ is the average 
occupation number of the 
$j$-th reservoir state at temperature $T$, and $\langle a^\dag a\rangle_{0} $
is the average initial excitation number of the subsystem 1. 
Note in the above equation the basic ingredients 
of Fermi's Golden Rule: each reservoir mode contributes to the system energy initially in a way proportional to its coupling constant and to the difference in occupation numbers between the mode and the system. As time goes by, the
$\left( \sin\Delta_j t/\Delta_j\right) ^2$ function selects only those modes with frequency very close to $\omega$. In scattering language, this is the ``phase space'' available for the process.

In order to gain deeper physical insight into the energy and coherence loss
processes, let us consider the subsystem 1 initially in the so called even 
coherent state \cite{DMM74} with amplitude $\alpha $, 
$\rho _{1}\left(0\right)=\left| e\alpha \right\rangle \left\langle e\alpha \right|$, where 
\[
|e\alpha\rangle = \frac1{2\bigl(1+\e^{-|\alpha|^2}\bigr)}
\bigl[|\alpha\rangle + |-\alpha\rangle\bigr]\,.
\]
Here, $\left| \alpha
\right\rangle $ is Glauber's coherent state \cite{Gla63}.
The average initial excitation number is given by
\be 
\langle a^\dag a \rangle_{0} = |\alpha|^2 \tanh |\alpha|^2\,.
\label{eq:aa}
\ee
To proceed with the calculation we approximate (as usual) the summation over $j$ 
by an integral in Eq. \eqref{E2nd}
\be\begin{split}
\sum_j F_j \left(\frac{\sin\Delta_j t}{\Delta_j}\right)^2 & 
\rightarrow \int_0^\infty\dif\Omega\,g(\Omega)F(\Omega) 
\left(\frac{\sin\Delta t}{\Delta }\right)^2\\
& \approx \int_{\omega-1/(2t)}^{\omega+1/(2t)} \dif\Omega\,g(\Omega)F(\Omega) 
\left(\frac{\sin\Delta t}{\Delta }\right)^2 \approx t^2 \int_{-1/(2t)}^{1/(2t)} 
\dif\Delta\,g(\omega+\Delta)F(\omega+\Delta) = \frac{t}{\tau_F}\,.
\end{split}\ee
where $g(\Omega)$ is the density of modes of the reservoir at frequency 
$\Omega$ and $\Delta = \left(\Omega -\omega\right)/2 $. The first approximation is like a stationary phase argument and the second comes from a linear approximation to the sine function close to $\Delta = 0$. 
This equation defines the time scale $\tau_F$, provided $F$ has frequency squared dimension. 

For the expansion in energy of subsystem 1, we get
\begin{eqnarray}
E_{1}\left( t\right) - E_{1}\left( 0\right) &\simeq& 
\omega t^{2}\int_{0}^{\infty }\mathrm{d}\Omega g\left( \Omega \right) 
\bar{n}\left( \Omega \right) \left[ \gamma \left( \Omega \right) \frac{%
\sin \left( \Delta t\right) }{\Delta t}\right] ^{2} \nonumber \\
&& -\omega t^{2}\left| \alpha
\right| ^{2}\tanh \left| \alpha \right| ^{2}\int_{0}^{\infty }\mathrm{d}%
\Omega g\left( \Omega \right) \left[ \gamma \left( \Omega \right) \frac{\sin
\left( \Delta t\right) }{\Delta t}\right] ^{2} . \label{E2ndcat}
\end{eqnarray}
% This equation estimates the energy of the subsystem 1 in the markovian limit
%\be\begin{split}
%E_1(t) \approx E_1(0) - |\alpha|^2\tanh |\alpha|^2 g(\omega)\gamma^2(\omega) 
%\omega t + \bar n(\omega)g(\omega)\gamma^2(\omega)\omega t.
%\end{split}\ee
Note that in the above equation, the second term on the r.h.s. 
is independent of temperature and therefore defines a dissipation time 
scale, whereas the first term, on contrary, depends on temperature and allows 
us to define a thermal timescale. These timescales are determined through the ratio
$\left|E_{1}\left( t\right) -E_{1}\left( 0\right) \right| /E_{1}\left( 0\right) $,
and reads
\be
\tau _{{\rm dis}}^{-1} = t 
\int_{-1/\left( 2t\right) }^{1/\left( 2t\right) }{\rm d}\Delta \
g\left( \omega +\Delta \right) \gamma ^{2}\left( \omega +\Delta \right) , 
\ee
\be
\tau _{{\rm th}}^{-1} = \frac{t}{\left| \alpha \right| ^{2} 
\tanh\left| \alpha \right| ^{2}}  
\int_{-1/\left(2t\right) }^{1/\left( 2t\right) }{\rm d}\Delta \ 
g\left( \omega +\Delta \right) \bar{n} \left(\omega + \Delta\right)
\gamma ^{2}\left( \omega +\Delta \right) .
\ee
In the asymptotic case $t\rightarrow \infty $, 
which corresponds to the Markovian approximation, we obtain 
\be
\tau_\text{dis}^{-1} = g(\omega)\gamma^2(\omega)\,
\label{tauDiss}
\ee
and
\be
\tau_\text{th}^{-1} = \frac{\bar n(\omega)g(\omega)\gamma^2(\omega)}
{\left| \alpha \right| ^{2} \tanh\left| \alpha \right| ^{2}},
\label{tauTh}
\ee
which give us the general timescale relation 
\be
\frac{\tau _{\mathrm{th}}}{\tau _{\mathrm{dis}}} = \frac{\left| \alpha \right|
^{2}\tanh \left| \alpha \right| ^{2}}{\bar{n}\left( \omega \right) }.
\label{relthdis}
\ee
In the large temperature regime $\beta
\omega \ll 1$ we have $\tau _{\mathrm{th}}/\tau _{\mathrm{dis}}
\rightarrow \beta \omega \left| \alpha \right| ^{2}
\tanh \left| \alpha\right| ^{2}$ and Eq. (\ref{relthdis}) has a simple semiclassical
interpretation: as $\left| \alpha \right| ^{2} \tanh \left| \alpha \right| ^{2}$ is proportional to
oscillator's mean energy, the larger $\left| \alpha \right| ^{2}$ is, the
faster dissipation occurs; accordingly, as $\bar{n}\left( \omega \right)
=\exp \left( {-\beta \omega }\right) /\left[ 1-\exp \left( {-\beta \omega }%
\right) \right] $ gives the mean number of quanta of frequency $\omega $ on
thermal reservoir at temperature $T$, the larger $\bar{n}\left( \omega
\right) $, the faster thermal effects appear. Zero temperature is thus characterized by the absence of thermal effects.

\emph{Subsystems decoherence timescales}:
The expression for the idempotency deficit \cite{Ji} of the subsystem $i$, up 
to second order, is given by
\be
\delta _{i}\left( t\right) = 1 - \Tr_{i} \rho_{i}^{2}\left(t\right)
\approx \delta _{i}^{\left( 0\right) }\left( t\right)
+\delta _{i}^{\left( 1\right) }\left( t\right) +\delta _{i}^{\left( 2\right)
}\left( t\right)  , 
\label{expandLinEnt}
\ee
where $\delta _{i}^{\left( 0\right) }\left( t\right) =
\delta _{i}\left( 0\right)$,
$\delta _{i}^{\left( 1\right) }\left( t\right) = -2\mathop{\rm Tr}
\nolimits_{i}\left[ \rho
_{i}\left( 0\right) \rho _{i}^{\left( 1\right) }\left( t\right) \right] ,
\label{deltaS_1}$ and 
$\delta _{i}^{\left( 2\right) }\left( t\right) = 
-\mathop{\rm Tr}\nolimits_{i}\left[ \rho _{i}^{\left( 1\right) }\left(
t\right) \right] ^{2}-2\mathop{\rm Tr}\nolimits_{i}\left[ \rho _{i}\left(
0\right) \rho _{i}^{\left( 2\right) }\left( t\right) \right]$ .
In the last expressions, the operator $\rho _{i}^{\left( n\right) }\left(
t\right) $ represents the  $n$-th term of the expansion  
(\ref{born}). 
For the state given by Eq. (\ref{initial}), we have
$\delta _{1}^{\left( 1\right) }\left( t\right) =0$ and
$\delta _{1}^{\left(2\right) }\left( t\right) = 
-2\mathop{\rm Tr}\nolimits_{1}\left[ \rho
_{1}\left( 0\right) \rho _{1}^{\left( 2\right) }\left( t\right) \right]$. 
Assuming that the subsystem 1 was prepared in the initial state
$\rho _{1}\left( 0\right) =\left| e\alpha \right\rangle \left\langle
e\alpha \right| $, the evaluation of $\delta _{1}^{\left(2\right) }\left( t\right)$ yields,
in the continuum limit, 
\be
\delta _{1}\left( t\right) \simeq 2 \int_{0}^{\infty }\mathrm{d}\Omega g\left(
\Omega \right) \left[ \gamma \left( \Omega \right) \frac{\sin \left( \Delta
t\right) }{\Delta }\right] ^{2}\left\{ \bar{n}\left( \Omega \right) 
\left( \left| \alpha \right| ^{2}\tanh
\left| \alpha \right| ^{2}+1\right)+
\left[ \bar{n}\left( \Omega
\right) +1\right] \left| \alpha \right| ^{2}\tanh \left| \alpha \right| ^{2}
\right\} 
\label{delta2ndcat}. 
\ee
We can also employ the same reasoning that allows us to determinate the
dissipation timescales in order to define the timescale $\tau _{{\rm dec}}$. In the
present case, purity loss and decoherence timescales coincide. Therefore,
within the Markovian approximation, we get 
\be
\tau _{\mathrm{dec}}^{-1}=2 g\left( \omega \right) \gamma ^{2}\left( \omega
\right) \left\{ \bar{n}\left( \omega \right) \left( \left| \alpha \right| ^{2}
\tanh \left| \alpha \right|^{2}+1\right)
+ \left[ \bar{n}\left( \omega \right) +1\right] \left|
\alpha \right| ^{2}\tanh \left| \alpha \right| ^{2} \right\} . 
\label{tauDec}
\ee

As is clear, in the expressions (\ref{E2ndcat}) and (\ref{delta2ndcat}) the
ingredients are the same, but the results are quite distinct. The greater
difference is the relative sign and it is easy to understand it in this
example. In both cases, the first term is the temperature contribution and
the second is still there even for zero temperature. The first term tends to
increase $E_1$ (it is a ``hot'' term), while the second to decrease it (it
is a ``cold'' one), but both add to decoherence. This difference in sign
reflects in the difference in behavior and timescales of these processes.

In case $T=0$ only ``cold'' terms contribute, and we obtain the relation 
\begin{equation}
\frac{\tau _{\mathrm{dis}}}{\tau _{\mathrm{dec}}}=2\left| \alpha \right|
^{2}\tanh \left| \alpha \right| ^{2}  
\label{timescalesT=0}
\end{equation}
and there is no thermalization in the sense which we are using this word. In
the limit $\left| \alpha \right|^{2} \gg 1$,
Eq. (\ref{timescalesT=0}) clearly shows the dependence of the purity loss 
timescale with the separation of the states that form the superposition in
the initial state $\rho_1 \left(0 \right)$. In fact, the more distinguishable 
these states, the smaller the purity loss timescale. 
%This is
%essentially the relation obtained in \cite[Eq. (2.15)]{DBRH96}, with the
%extra term $+1$ given by the zero point energy. 
Such relations of time
scales are usually considered the answer to the impossibility of naturally
finding the great majority of quantum states for macroscopic systems (e.g.:
Schr\"{o}dinger cat states) \cite{DBRH96}.

For temperature $T$ we have 
\begin{equation}
\frac{\tau _{{\rm th }}}{\tau _{{\rm dec }}}=\frac{2\left| \alpha \right|
^{2}\tanh \left| \alpha \right| ^{2}}{\bar{n}\left(\omega\right)}
\left\{ \left[ \bar{n}\left( \omega \right) +1\right] \left|
\alpha \right| ^{2}\tanh \left| \alpha \right| ^{2}+\bar{n}\left(
\omega \right) \left( \left| \alpha \right| ^{2}\tanh \left| \alpha \right|
^{2}+1\right) \right\}  ,
\label{ratthdec}
\end{equation}
which stresses the fact that decoherence is even faster than thermalization.
%unless $\alpha =0$, in which case they are equal. 
In the same way 
\begin{equation}
\frac{\tau _{{\rm dis }}}{\tau _{{\rm dec }}}=2
\left\{ \left[ \bar{n}\left( \omega \right) +1\right] \left|
\alpha \right| ^{2}\tanh \left| \alpha \right| ^{2}+\bar{n}\left(
\omega \right) \left( \left| \alpha \right| ^{2}\tanh \left| \alpha \right|
^{2}+1\right) \right\}  ,
\label{ratdisdec}
\end{equation}
where $\left| \alpha \right| $ plays its crucial role, and for large
temperatures we obtain the expected behavior of linear increasing of
decoherence rate with respect to temperature, in complete agreement with
expression (20) of \cite{JK98} and also with \cite{KB92}, but obtained in a
very different framework. To authors knowledge, it is the first time general
timescale relations as Eqs.\ (\ref{ratthdec}) and (\ref{ratdisdec}) appears
for this important example.

\emph{Master equation -- a quantitative indicative of separability}: 
The time scale of the reservoir can be straightforwardly evaluated by using 
the expansion Eq. \eqref{expandLinEnt} now for the linear entropy of the reservoir 
$\delta_2(t) = 1 - \Tr_2\rho_2(t)^2$. If we have 
$\Tr_2[\rho_2^2(0)-\rho_2^2(t)]/\Tr_2\rho_2^2(0) \approx 0$, we can consider 
that $\rho_2(t) \approx \rho_2(0)$. However, up to second order we get 
\be
1-\frac{\Tr_2 \rho^2_2 (t)}{\Tr_2 \rho^2_2 (0)} = t^{2}\sum_{k=1}^{N}
\gamma_{k}^2 \frac{\sin ^{2}\left( \Delta _{k}t\right) }
{\left( \Delta _{k}t\right) ^{2}}\left\{ 4\left( 2\left\langle a^{\dagger}a\right\rangle +1\right) 
\left[ \tanh \left( \frac{\beta \Omega _{k}}{2}\right) -1\right] 
+ 2\left\langle a^{\dagger }a\right\rangle -\left\langle a\right\rangle 
\left\langle a^{\dagger }\right\rangle \tanh \left( \frac{\beta \Omega _{k}}{2}\right) \right\} . 
\ee
Using the same procedure as before one can obtain the characteristic 
coherence change for the reservoir as 
\be
\tau_{2,\text{dec}}^{-1} = \frac{g(\omega)\gamma^2(\omega)}{2\bar n(\omega)+1}
\left| 2\left\langle a^{\dagger }a\right\rangle \left[2 \bar n (\omega)+ 1 \right] 
-8\bar n (\omega) \left(2 \langle a^{\dagger }a\rangle+ 1 \right)- \langle a\rangle 
\langle a^{\dagger }\rangle \right|. 
\label{tauDec2}
\ee

A reasonable condition for the separability of the timescales of systems 1 and 2 is 
\be \tau _{2,\text{dec}} \gg \omega^{-1}\,, \label{eq:tau} \ee 
where $\omega^{-1}$ is the characteristic time of the free evolution of system 1. 
The worst case we have is when the temperature is zero and therefore 
$\bar n(\omega)=0$. In this regime we can guarantee the use of the approximation 
$\rho_2(t) \approx \rho_2(0)$ if we have 
\be
g(\omega) \gamma^2 (\omega)\left\{2 \langle a^{\dagger }a\rangle 
- \langle a\rangle \langle a^{\dagger }\rangle \right\} \ll \omega.
\ee
Just to have some magnitudes to work with, we took from Ref. \cite{paris} the values $
\langle a^{\dagger }a\rangle \approx 9.5$, $\bar n(\omega) = .05$,
$\omega/2 \pi = 51 \, \mathrm{GHz}$, and $\tau _{\mathrm{dis}} = 160 \, \mathrm{\mu s}$,
and obtain $\tau _{2,\mathrm{dec}} \approx 13 \, \mathrm{\mu s} \gg \omega ^{-1}
\approx 20 \, \mathrm{ps}$, as one would expect.

If we want to have a condition which allows for the determination 
of a cutoff temperature in terms of dynamical ingredients and initial 
conditions we should use the condition \eqref{eq:tau} in Eq. \eqref{tauDec2}. 

To summarize we have provided a quantitative support for the factorization 
assumption between the system of interest and the reservoir density matrices 
which is widely used in the decoherence literature. Moreover we give analytical 
expressions for typical correlation and relaxation timescales which depend only 
on general ingredients such as the Hamiltonian and the initial state.

M. C. Nemes acknowledges the support of CNPq. J. G. Peixoto de Faria
thanks to PROPP--UESC for financial support under grants 220.1300.324.

\begin{thebibliography}{8}
\bibitem{ionTraps} For review on ion trapping, see D. Liebfried,
R. Blatt, C. Monroe, and D. Wineland, \rmp {\bf 75}, 281 
(2003); D. Wineland {\it et al.}, J. Res. Natl. Stand.
Technol. {\bf 103}, 259 (1998).

\bibitem{cavity} M. Brune {\it et al.}, \prl {\bf 65}, 976 (1990); 
M. Brune {\it et al.}, \pra {\bf 45}, 5193 (1992); G. Nogues 
{\it et al.}, Nature {\bf 400}, 239 (1999).

\bibitem{paris}  M. Brune {\it et al.}, \prl {\bf 77}, 4887
(1996).

\bibitem{openQuSys} There are several approaches to
quantum open systems and we cite a few. 
\\
Master equations: besides the references \cite{mod, cohen, Mte, lowTemp}
listed below, for a rigorous treatment, see E. B. Davies,
Comm. Math. Phys. {\bf 39}, 91 (1974);
G. Lindblad, {\it id.} {\bf 48}, 199 (1976).
\\
Quantum trajectories and stochastic extensions of Schr\"{o}dinger
equation: see H. Carmichael, {\it An Open Systems
Approach to Quantum Optics} (Springer, Berlin, 1993); for review, 
see also A. Bassi, and G. Ghirardi, Phys. Rep. {\bf 379}, 257 (2004).  
\\
Decoherent histories: for review, see R. Omn\`{e}s, \rmp
{\bf 64}, 339 (1992).


\bibitem{mod} A. O. Caldeira, and A. J. Legget, {\rm Ann. Phys. (NY)} {\bf %
149}, 374 (1983); W. H. Zurek, {\rm Phys. Rev. D} {\bf 26}, 1862 (1982);
K. M. Fonseca Romero, and M. C. Nemes, {\rm Phys. Lett. A} {\bf 235}, 432
(1997).

\bibitem{cohen}  C. Cohen-Tannoudji, J. Dupont-Roc, and G. Grynberg, 
{\it Atom-Photon Interactions -- Basic Processes and Applications} 
(John Wiley \& Sons, New York, 1992). 

\bibitem{Mte}  R. Omn\`{e}s, {\rm Phys. Rev. A} {\bf 56}, 3383 (1997).

\bibitem{Amir} C. Morais Smith, and A. O. Caldeira, \pra {\bf{36}}, 3509 (1987).

\bibitem{lowTemp}  An interesting pioner work is L. Di\'osi,
{\rm Europhys. Lett.} {\bf 30}, 63 (1995), also available as gr-qc/9403046.
An exact master equation for a particle coupled to a general environment at arbitrary temperature is deduced in
B. L. Hu, J. P. Paz, and Y. Zhang, \prd {\bf 45}, 2843 (1992). This result is used 
for a study of decoherence of a harmonic oscillator coupled
to an ohmic environment at zero temperature in S. Sinha, Phys. Lett. A {\bf 228}, 1 (1997). 
A master equation
in the Lindblad form valid at high and low temperatures is obtained in
S. Gao, \prl {\bf 79}, 3101 (1997).

\bibitem{DMM74}  V. V. Dodonov, I. A. Malkin, and V. I. Man'ko, {\rm Physica}
{\bf 72}, 597 (1974).

\bibitem{Gla63}  R. Glauber, {\rm Phys. Rev.} {\bf 131}, 2766 (1963).

\bibitem{Ji}  J. I. Kim, M. C. Nemes, A. F. R. de Toledo Piza, and H. E.
Borges, {\rm Phys. Rev. Lett.} {\bf 77}, 207 (1996).


%\bibitem{WZ}  J. A. Wheeler and W. H. Zurek (orgs.), {\it Quantum Theory and
%Measurement}, Princeton (1983).

%\bibitem{Paris}  S. Haroche, {\rm Phys. Today} {\bf 51}(7), 36, (July 1998).

%\bibitem{qco}  See special issue of {\rm Physics World}, (March 1998).


%\bibitem{Giuetal96}  D. Giulini {\it et al.}, {\it Decoherence and the
%Appearence of a Classical World in Quantum Theory}, Springer (1996).

%\bibitem{PZ99}  J. P. Paz and W. H. Zurek, {\rm Phys. Rev. Lett.} {\bf 82},
%5181 (1999).

\bibitem{DBRH96}  L. Davidovich, M. Brune, J. M. Raimond, and S. Haroche, 
{\rm Phys. Rev. A} {\bf 53}, 1295 (1996).

\bibitem{JK98}  C. Kiefer and E. Joos, {\it Decoherence: Concepts and
Examples} in P. Blanchard and A. Jadczyk (eds.) {\it Quantum Future}
(Springer, Berlin, 1998). Also available as {\tt quant-ph/9803052} (and
references therein).


\bibitem{KB92}  M. S. Kim and V. Buzek, {\rm Phys. Rev. A} {\bf 46}, 4239
(1992).
\end{thebibliography}




\end{document}

%%%%%%%%%%%%%%%%%%%%% VELHO %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%5
Theoretical description of parties of large systems, aside from a short list
of exactly solvable models \cite{mod}, is basically given in terms of master
equations \cite{Mte}, which preclude the description of short time regimes,
where the dynamics of quantum correlations plays a decisive role. A few
years ago, a short time perturbative scheme was implemented to describe such
(very) short time scales \cite{Ji}. The purpose of present contribution is a
generalization of that work which extends its validity in order to explore
more realistic (observable) time scales.

An interesting characteristic of the expansion proposed in Ref. \cite{Ji} is
its symmetry in what concerns the subsystems involved in the problem. It is
therefore possible to study time scales of both the system of interest and
the environment with which it is interacting and to provide analytic
expressions for those time scales. The present study contributes to the solution of
a long standing problem, namely, a quantitative justification  to the commonly
 used approximations regarding the properties of the ``heat bath'' and how
it affects the dynamics of the system of interest. In fact, if $\rho
_{S}\left( t\right) $ and $\rho _{R}\left( t\right) $ are the reduced
densities of the system of interest and of the reservoir respectively, the usual 
derivations of the master equations assume that the total
density at all times can be written as $\rho \left( t\right) =\rho
_{S}\left( t\right) \otimes \rho _{R}\left( t\right) $. Qualitative
arguments to the validity of such approximation for non-null temperatures
have been given \cite{mod,cohen}. However, master equations are known to
yield excellent quantitative results even for very low temperatures \cite[lowTemp]. Of
course, in the extreme case of zero temperature the entanglement time scale
for both systems $S$ and $R$ is the same (assuming initial pure states). So, we aim at showing that the key
physical ingredient which justifies the approximation at low temperatures is
the large number of degrees of freedom of the reservoir, $N$ (which is the 
true characteristic of macroscopicity). In fact, provided the
temperature is not absolute zero (which is always the case) the
characteristic reservoir time scale $\tau _{R}$ is exponentially larger than
that of the system $S$, $\tau _{S}$, {\it{i.e.}}, $\tau _{S}/\tau _{R}\sim \alpha
^{N}$, where $\alpha $ is a constant smaller than one. The fact that $\alpha
<1$ is guaranteed by the finite, nonzero temperature, however small it might
be.

In order to generalize the derivation in Ref. \cite{Ji}, we work out a Dyson
series expansion for reduced density operators and obtain the analog of Fermi Golden 
Rule for the reduced densities and consequently for purity
and energy losses. Let us consider a general Hamiltonian (with discrete
spectrum) of the form 
\[
H=H_{0}+H_{12},\nonumber
\]
where $H_{0}=H_{1}+H_{2}$, with 
$H_{1} = \sum_{n}\left| n\right\rangle \varepsilon _{n}^{\left( 1\right)
}\left\langle n\right| \otimes 1_{2}$, 
$H_{2} = 1_{1}\otimes \sum_{s}\left| s\right\rangle \varepsilon _{s}^{\left(
2\right) }\left\langle s\right|$ , and 
$H_{12}=\sum_{m,n,r,s}\gamma _{mrns}\left| m\right\rangle \left\langle
n\right| \otimes \left| r\right\rangle \left\langle s\right|$.
In the above expressions $\varepsilon _{n}^{\left( i\right) }$ stands for
the eigenenergies of system $i$ whose autonomous dynamics is governed by the
Hamiltonian $H_{i}$, and $\gamma _{mrns}$ are the interaction matrix
elements in the tensor product basis. We further assume that the time scales
associated with $H_{12}$ are larger than subsystems characteristic times
({\it{i.e.}}: small coupling limit, $\gamma \ll \varepsilon _{i}$) and the initial
state is a factorized one 
$\rho \left( 0\right) =\rho _{1}\left( 0\right) \otimes \rho _{2}\left(
0\right) $. 
In the interaction picture, the density operator is given by (we use $\hbar =1$) 
$\rho _{I}\left( t\right) =\exp \left( iH_{0}t\right) \rho \left( t\right)
\exp \left( -iH_{0}t\right)$ , 
and the time dependent interaction Hamiltonian becomes 
\begin{eqnarray*}
H_{12}\left( t\right) &\equiv &\exp \left( iH_{0}t\right) H_{12}\exp \left(
-iH_{0}t\right) \\
&=&\sum \gamma _{12}\left( t\right) \left| m\right\rangle \left\langle
n\right| \otimes \left| r\right\rangle \left\langle s\right| ,
\end{eqnarray*}
where $\gamma _{12}\left( t\right) =\gamma _{mrns}\exp \left[ i\left( \varepsilon
_{m}^{\left( 1\right) }-\varepsilon _{n}^{\left( 1\right) }+\varepsilon
_{r}^{\left( 2\right) }-\varepsilon _{s}^{\left( 2\right) }\right) t\right]$ .
For later convenience we also define the symbol 
$\gamma _{12}^{\prime }\left( t\right) =\gamma _{m^{\prime }r^{\prime
}n^{\prime }s^{\prime }}\exp \left[ i\left( \varepsilon _{m^{\prime
}}^{\left( 1\right) }-\varepsilon _{n^{\prime }}^{\left( 1\right)
}+\varepsilon _{r^{\prime }}^{\left( 2\right) }-\varepsilon _{s^{\prime
}}^{\left( 2\right) }\right) t\right]$ . 
One should note that $\gamma _{12}$ and $\gamma _{12}^{\prime}$ are convenient symbols, 
but they have a hidden dependency on the labels $m,r,n,s$.

Von Neumann's equation takes the form 
\[
i\dot{\rho}_{I}\left( t\right) =\left[ H_{12}\left( t\right) ,\rho
_{I}\left( t\right) \right] . 
\]
From now on, we drop the subscript $I$, but interaction picture is implied.

A Born like expansion is given as 
\[
\rho \left( t\right) =\rho \left( 0\right) -i\int_{0}^{t}dt^{\prime }\left[
H_{12}\left( t^{\prime }\right) ,\rho \left( 0\right) \right]
-\int_{0}^{t}dt^{\prime }\int_{0}^{t^{\prime }}dt^{\prime \prime }\left[
H_{12}\left( t^{\prime }\right) ,\left[ H_{12}\left( t^{\prime \prime
}\right) ,\rho \left( 0\right) \right] \right] +... 
\]
where we identify 
$\rho =\rho ^{0th}+\rho ^{1st}+\rho ^{2nd}+... $
In this sense we will obtain up to second order (in interaction) expressions
for the reduced density operators 
$\rho _{i}={\rm Tr}_{j}\rho ,\ i\neq j$, 
from which we can obtain linear entropy (or idempotency
deficit) for each subsystem 
\begin{equation}
\delta _{i}=1-{\rm Tr}\left( \rho _{i}^{2}\right) ,  \label{deltaS}
\end{equation}
which indicates how far from pure states the density operators $\rho _{1}$ and $%
\rho _{2}$ are, and gives an idea of how entangled $1$ and $2$ are (in case
the whole $1-2$ system is in a pure state). Also subsystem's energies 
\begin{equation}
E_{i}={\rm Tr}\left( H_{i}\rho _{i}\right) ,  \label{ES}
\end{equation}
can be obtained. From these quantities typical time scales will be
extracted. Straightforward calculation gives 
\[
\rho _{j}^{1st}\left( t\right) =i\sum_{m,n,r,s}\left\langle s\right| \rho
_{i}^{0th}\left| r\right\rangle \left[ \left| m\right\rangle \left\langle
n\right| ,\rho _{j}^{0th}\right] \int_{0}^{t}d\tau \gamma _{12}\left( \tau
\right) ,\ i\neq j, 
\]
and 
\begin{eqnarray}
\rho _{j}^{2nd}\left( t\right) &=&\sum \left\{ \delta _{r^{\prime
}s}\left\langle s^{\prime }\right| \rho _{i}^{0th}\left| r\right\rangle 
\left[ \left| m\right\rangle \left\langle n\right| ,\left| m^{\prime
}\right\rangle \left\langle n^{\prime }\right| \rho _{j}^{0th}\right]
-\delta _{rs^{\prime }}\left\langle s\right| \rho _{i}^{0th}\left| r^{\prime
}\right\rangle \left[ \left| m\right\rangle \left\langle n\right| ,\rho
_{j}^{0th}\left| m^{\prime }\right\rangle \left\langle n^{\prime }\right| %
\right] \right\} \times \nonumber \\
&&\times \int_{0}^{t}d\tau \int_{0}^{\tau }d\tau ^{\prime }\gamma
_{12}\left( \tau \right) \gamma _{12}^{\prime }\left( \tau ^{\prime }\right)
\label{second_order},
\end{eqnarray}
where the sum is over $m,n,r,s,m^{\prime },n^{\prime },r^{\prime },$ and $%
s^{\prime }$. At this point the main ingredients of the subsystem's dynamics
up to second order become apparent: first notice the manifest symmetry in
the subsystem's indexes. Moreover, it is easy to see that results will be
strongly depedent on initial condition.  

General expressions for first and second order contributions in $\delta
_i\left( t\right) $ and $E_i\left( t\right) $ can be written as follows 
\[
\delta _i^{1st}\left( t\right) =0, 
\]
since it involves the trace of a commutator, 
\[
\delta _i^{2nd}\left( t\right) =-{\rm Tr}\left( \rho _i^{1st}\left( t\right)
\right) ^2-2{\rm Tr}\left( \rho _i^{0th}\left( t\right) \rho _i^{2nd}\left(
t\right) \right) , 
\]
\[
E_i^{1st}\left( t\right) ={\rm Tr}\left( H_i\rho _i^{1st}\left( t\right)
\right) , 
\]
\[
E_i^{2nd}\left( t\right) ={\rm Tr}\left( H_i\rho _i^{2nd}\left( t\right)
\right) . 
\]

In order to gain deeper physical insight into the energy and coherence loss
processes we work out an important example in the context of quantum optics.
Let us consider two very different subsystems: the first, which we now call $%
S$ is a harmonic oscillator ({\it{i.e.}} a radiation field mode) of frequency $%
\omega $ initially in the so called even coherent state \cite{DMM74} with
amplitude $\alpha $, 
\[
\rho _{S}^{0th}=\left| e\alpha \right\rangle \left\langle e\alpha \right| , 
\]
where $\left| e\alpha \right\rangle =N\left( \alpha \right) \left( \left|
\alpha \right\rangle +\left| -\alpha \right\rangle \right) $, $N\left(
\alpha \right) $ being a normalization constant and $\left| \alpha
\right\rangle $ Glauber's coherent state \cite{Gla63}. The other subsystem
will represent a dissipative cavity, as modelled by a set of independent
harmonic oscillators of frequency $\Omega_{j}$ in thermal equilibrium 
\[
\rho _{R}^{0th}=\bigotimes_{j}\rho _{j}^{0th}, 
\]
where $\rho _{j}^{0th}=Z_{j}^{-1}\exp \left( -\beta H_{j}\right) $, $\beta
=\left( k_{B}T\right) ^{-1}$, $k_{B}$ being Boltzmann's constant, $T$ the
temperature and $Z_{j}=Z_{j}\left( \beta \right) $ partition function of the 
$j$th harmonic oscillator.

For the coupling Hamiltonian we consider the Rotating Wave Approximation
(RWA) for the interaction between the main oscillator and each $j$
oscillator. In our notation it means the following explicit
form for the matrix elements 
\[
\gamma _{mrns}^j=\gamma ^j\left( \delta _{r,s+1}\delta _{m+1,n}\sqrt{rn}%
+\delta _{r+1,s}\delta _{m,n+1}\sqrt{sm}\right) . 
\]

In this case first order corrections to energy vanish due to commutativity
between $\rho _{R}^{0th}$ and $H_{R}$. Up to second order we have 
\begin{eqnarray}
E_{S}\left( t\right) &=&{\rm Tr}\left[ H_{S}\rho \left( 0\right) %
\right] +\sum_{j}\omega \left( \gamma ^{j}\right) ^{2}\left( \bar{n}%
_{j}-\left\langle a^{\dagger }a\right\rangle \right) \frac{\sin ^{2}\left[
\Delta _{j}t\right] }{\Delta _{j}^{2}}  \nonumber \\
&=&\omega \left( 1+e^{-2\left| \alpha \right| ^{2}}\right) \sum_{j}\frac{%
e^{-\beta \Omega _{j}}}{1-e^{-\beta \Omega _{j}}}\left( \gamma ^{j}\right)
^{2}\left\{ \frac{\sin \left[ \Delta _{j}t\right] }{\Delta _{j}}\right\}
^{2}+  \nonumber \\
&&-\omega \left| \alpha \right| ^{2}\left( 1-e^{-2\left| \alpha \right|
^{2}}\right) \sum_{j}\left( \gamma ^{j}\right) ^{2}\left\{ \frac{\sin \left[
\Delta _{j}t\right] }{\Delta _{j}}\right\} ^{2}  \nonumber \\
&\equiv &E_{{\rm th}}\left( t\right) -E_{{\rm dis}}\left(
t\right) .  \label{E2ndcat}
\end{eqnarray}
Here, $a^{\dagger }$ and $a$ stand for the creation and annihilation
operators of the harmonic oscillator, respectively. In the above expression
all the essential ingredients are present:
the terms involving the
integrals in Eq. (\ref{second_order} will be responsible for the contribution of the 
particularly chosen
interaction and for the available ``phase space'' just as in Fermi's Golden
Rule for scattering. This will become more concrete in the example below.
The initial states of both subsystems (through amplitude $\alpha $ and temperature $T$),
the coupling strength $\left( \gamma ^{j}\right) ^{2}$ and 
$\left\{ \sin \left[ \Delta _{j}t\right] / \Delta _{j} \right\} ^{2}$, 
where $\Delta _{j}=\left( \Omega _{j}-\omega \right) /2$, which teach us
something about very short times when correlations are established. Note
that the time-energy uncertainty relation appears in this calculation in the
following way: in a time $t$ uncertainty in energy must be of order $t^{-1}$
and all modes with $\Delta _{j}$ up to this order give significant
contribution in the summations involved in Eq.~(\ref{E2ndcat}). As times grows a
narrower set of oscillators become significative and energy conservation
(suitably modified) is achieved. Asymptotically in time, the energy transfer
between the system and the reservoir depends just on the latter's density of
states at frequency $\omega $. This result can be viewed as a translation of
Fermi's Golden Rule to dissipation and decoherence context.

We have separated the energy variation of subsystem $S$ into two parts: one
which is temperature dependent $E_{{\rm th}}$ (for {\it {thermal}}) and the
other which will be present even in the $T=0$ case. We called this the
``dissipative'' part of $E_S$. The reason for this will become clear
in the sequel.

In the above expression, if we further assume a continuum smooth spectrum
for the reservoir and a sufficiently smooth $j$ dependence of $\gamma ^{j}$
we can convert summations into integrals, change variables to frequency
integrals by using the density of modes $g\left( \Omega \right) $ and the
following approximation to such integrals 
\begin{eqnarray*}
\int_{0}^{\infty }{\rm d}\Omega \ g\left( \Omega \right) F\left( \Omega \right)\left\{ \frac{\sin \left( \Delta t\right) }{\Delta }\right\} ^{2} &\simeq &\int_{\omega
-1/\left( 2t\right) }^{\omega +1/\left( 2t\right) }{\rm d}\Omega \ g\left(
\Omega \right) F\left( \Omega \right)\left\{ \frac{\sin \left( \Delta t\right) }{\Delta }\right\}^{2} \\
&\simeq &t^{2}\int_{-1/\left( 2t\right) }^{1/\left( 2t\right) }{\rm d}\Delta
\ g\left( \omega +\Delta \right) F\left( \omega +\Delta \right) =\frac{t}{%
\tau _{F}},
\end{eqnarray*}
where, the first approximation follows from the above discussion on
time-energy uncertainty relation and the last one from the required
smoothness of $g\left( \Omega \right) $ and $\gamma \left( \Omega \right) $.
It is important to stress that the sense of smoothness we are using for $%
g\left( \Omega \right) $ and $\gamma \left( \Omega \right) $ is subtle. For
now it means that there is no ubiquitous behavior like the absence of states
in the interval of width $t^{-1}$ around $\omega $. In particular we are not
considering {\it {adiabatic environments}} in the sense of Ref. \cite{PZ99}.
This allows us to consider each term in eq.~(\ref{E2ndcat}) as defining time
scales $\tau _{{\rm th}}$ and $\tau _{{\rm dis}}$: 
\begin{eqnarray*}
\tau _{{\rm dis}}^{-1} &=&\frac{\left| \alpha \right| ^{2}\left(
1-e^{-2\left| \alpha \right| ^{2}}\right) }{\left| \alpha \right| ^{2}+\frac{%
1}{2}}t\int_{-1/\left( 2t\right) }^{1/\left( 2t\right) }{\rm d}\Delta \
g\left( \omega +\Delta \right) \gamma ^{2}\left( \omega +\Delta \right) , \\
\tau _{{\rm th}}^{-1} &=&\frac{\left( 1+e^{-2\left| \alpha \right|
^{2}}\right) }{\left| \alpha \right| ^{2}+\frac{1}{2}}t\int_{-1/\left(
2t\right) }^{1/\left( 2t\right) }{\rm d}\Delta \ g\left( \omega +\Delta
\right) \frac{e^{-\beta \left( \omega +\Delta \right) }}{1-e^{-\beta \left(
\omega +\Delta \right) }}\gamma ^{2}\left( \omega +\Delta \right) .
\end{eqnarray*}
But we must drop out the dependence on $t$ on the right hand side of these
equations. There seems to be two ways of doing it: one is to consider both
as selfconsistent equations, substituting $t$ by the appropriated $\tau $
and solving; the other is to consider asymptotic case $t\rightarrow \infty $%
, which corresponds to the Markovian approximation. The first way would be a
nice way to go beyond Markovian approximation, necessary in cases in which $%
g $ or $\gamma $ vary drastically around the frequency $\omega $. We will
adopt the second, which allows us to obtain 
\begin{eqnarray*}
\tau _{{\rm dis,M}}^{-1} &=&\frac{\left| \alpha \right| ^{2}\left(
1-e^{-2\left| \alpha \right| ^{2}}\right) }{\left| \alpha \right| ^{2}+\frac{%
1}{2}}g\left( \omega \right) \gamma ^{2}\left( \omega \right) , \\
\tau _{{\rm th,M}}^{-1} &=&\frac{\left( 1+e^{-2\left| \alpha \right|
^{2}}\right) }{\left| \alpha \right| ^{2}+\frac{1}{2}}g\left( \omega \right) 
\frac{e^{-\beta \omega }}{1-e^{-\beta \omega }}\gamma ^{2}\left( \omega
\right) ,
\end{eqnarray*}
which give us the general time scale relation 
\begin{eqnarray}
\frac{\tau _{{\rm th,M}}}{\tau _{{\rm dis,M}}} &=&\left| \alpha \right|
^{2}\tanh \left[ \left| \alpha \right| ^{2}\left( \frac{e^{-\beta \omega }}{%
1-e^{-\beta \omega }}\right) ^{-1}\right]  \nonumber \\
&\rightarrow &\beta \omega \left| \alpha \right| ^{2}\tanh \left| \alpha
\right| ^{2},  \label{relthdis}
\end{eqnarray}
where the last expression is valid for the large temperature regime $\beta
\omega \ll 1$. Eq. (\ref{relthdis}) has a simple semiclassical
interpretation: as $\left| \alpha \right| ^{2}$ is proportional to
oscillator's mean energy, the larger $\left| \alpha \right| ^{2}$ is, the
faster dissipation occurs; accordingly, as $n\left( \beta ,\omega \right)
=\exp \left( {-\beta \omega }\right) /\left[ 1-\exp \left( {-\beta \omega }%
\right) \right] $ gives the mean number of quanta of frequency $\omega $ on
thermal reservoir at temperature $T$, the larger $n\left( \beta ,\omega
\right) $, the faster thermal effects appear.

The expression for the idempotency deficit of $S$, up to second order, is given by 
\begin{eqnarray}
\delta _{S}\left( t\right) &=&2\left( 1+e^{-2\left| \alpha \right|
^{2}}\right) \sum_{j}\frac{e^{-\beta \Omega_{j}}}{1-e^{-\beta \Omega_{j}}}%
\left( \gamma ^{j}\right) ^{2}\left\{ \frac{\sin \left( {\Delta_{j}}t\right) 
}{\Delta_{j}}\right\} ^{2}+  \label{delta2ndcat} \\
&&2\left| \alpha \right| ^{2}\left( 1-e^{-2\left| \alpha \right|
^{2}}\right) \sum_{j}\frac{1+e^{-\beta \Omega_{j}}}{1-e^{-\beta \Omega_{j}}}%
\left( \gamma ^{j}\right) ^{2}\left\{ \frac{\sin \left( {\Delta_{j}}t\right) 
}{\Delta_{j}}\right\} ^{2}.  \nonumber
\end{eqnarray}
and the same reasoning defines the time scale $\tau _{{\rm dec}}$. In the
present case purity loss and decoherence time scales coincide. Therefore,
within the Markovian approximation, we get 
\[
\tau _{{\rm dec,M}}^{-1}=2\left[ \left( 1+e^{-2\left| \alpha \right|
^{2}}\right) \frac{e^{-\beta \omega }}{1-e^{-\beta \omega }}+\left| \alpha
\right| ^{2}\left( 1-e^{-2\left| \alpha \right| ^{2}}\right) \frac{%
1+e^{-\beta \omega }}{1-e^{-\beta \omega }}\right] g\left( \omega \right)
\gamma ^{2}\left( \omega \right) 
\]

As is clear, in the expressions (\ref{E2ndcat}) and (\ref{delta2ndcat}) the
ingredients are the same, but the results are quite distinct. The greater
difference is the relative sign and it is easy to understand it in this
example. In both cases, the first term is the temperature contribution and
the second is still there even for zero temperature. The first term tends to
increase $E_S$ (it is a ``hot'' term), while the second to decrease it (it
is a ``cold'' one), but both add to decoherence. This difference in sign
reflects in the difference in behavior and time scales of these processes.

In case $T=0$ only ``cold'' terms contribute, and we obtain the relation 
\begin{equation}
\frac{\tau _{{\rm dis,M}}}{\tau _{{\rm dec,M}}}=2\left| \alpha \right|
^{2}+1,  \label{timescalesT=0}
\end{equation}
and there is no thermalization in the sense which we are using this word. In
this case, the time scales are the same for small $\left| \alpha \right| $,
but can be very different for large $\left| \alpha \right| $. This is
essentially the relation obtained in \cite[Eq. (2.15)]{DBRH96}, with the
extra term $+1$ given by the zero point energy. Such relations of time
scales are usually considered the answer to the impossibility of naturally
finding the great majority of quantum states for macroscopic systems (e.g.:
Schr\"{o}dinger cat states).

For temperature $T$ we have 
\begin{equation}
\frac{\tau _{{\rm th,M}}}{\tau _{{\rm dec,M}}}=\left( 2\left| \alpha \right|
^{2}+1\right) \left[ 1+\left| \alpha \right| ^{2}\tanh \left( \left| \alpha
\right| ^{2}\frac{1+e^{-\beta \omega }}{e^{-\beta \omega }}\right) \right] ,
\label{ratthdec}
\end{equation}
which stresses the fact that decoherence is even faster than thermalization,
unless $\alpha =0$, in which case they are equal. In the same way 
\begin{equation}
\frac{\tau _{{\rm dis,M}}}{\tau _{{\rm dec,M}}}=\left( 2\left| \alpha
\right| ^{2}+1\right) \left[ \frac{1}{\left| \alpha \right| ^{2}\tanh \left|
\alpha \right| ^{2}}\frac{e^{-\beta \omega }}{1-e^{-\beta \omega }}+\frac{%
1+e^{-\beta \omega }}{1-e^{-\beta \omega }}\right] ,  \label{ratdisdec}
\end{equation}
where $\left| \alpha \right| $ plays its crucial role, and for large
temperatures we obtain the expected behavior of linear increasing of
decoherence rate with respect to temperature, in complete agreement with
expression (20) of \cite{JK98} and also with \cite{KB92}, but obtained in a
very different framework. To authors knowledge, it is the first time general
time scale relations as Eqs.\ (\ref{ratthdec}) and (\ref{ratdisdec}) appears
for this important example.

Some general consequences of this framework must be pointed. Expressions (%
\ref{ES}) are also linear in $\rho $ and this has important consequences:
subsystem's energy may decrease or may increase, as discussed in the
example. In this sense, dissipation is not an adequate term, because a low
excited particle in contact with a ``hot'' reservoir will increase its
energy, by the same process that an excited particle will decay. We prefer
to call it thermalization. So, by the linearity of (\ref{ES}), interaction
will allow energy transfer among subsystems, but not necessarily subsystem's
energy variation. On the other hand, expression (\ref{deltaS}) is nonlinear
in $\rho $. Even more, it has a minimum for pure states. This minimum is
generally very unstable. General interactions take $\rho $ out of pure
states. Again in the sense of energy-time uncertainty relation, the equal
population of all attainable levels in times extremely short can make energy
increase or decrease, but all this terms add to decoherence, if we start in
a pure state.

Let us now consider the dynamics of the reservoir. We get, to second order, 
\begin{eqnarray*}
E_{R}\left( t\right) &=&E_{R}^{\left( 0\right) }\left( t\right)
+E_{R}^{\left( 2\right) }\left( t\right) \\
&=&{\rm Tr}\left[ H_{R}\rho _{R}\left( 0\right) \right] -\sum_{j=1}^{N}%
\Omega _{j}\gamma ^{j}\left( \bar{n}_{j}-\left\langle a^{\dagger
}a\right\rangle \right) \frac{\sin ^{2}\left[ \Delta _{j}t\right] }{\Delta
_{j}^{2}}
\end{eqnarray*}
and 
\begin{eqnarray*}
\delta _{R}\left( t\right) &=&1-\left[ \prod_{j=1}^{N}\tanh \left( 
\frac{\beta \Omega _{j}}{2}\right) \right] \\
&&\times \left\{ 1-\sum_{k=1}^{N}\gamma ^{k}\frac{\sin ^{2}\left[ \Delta
_{k}t\right] }{\Delta _{k}^{2}}\left\{ 4\left( 2\left\langle a^{\dagger
}a\right\rangle +1\right) \left[ \tanh \left( \frac{\beta \Omega _{k}}{2}%
\right) -1\right] \right. \right. \\
&&\left. \left. +2\left\langle a^{\dagger }a\right\rangle -\left\langle
a\right\rangle \left\langle a^{\dagger }\right\rangle \tanh \left( \frac{%
\beta \Omega _{k}}{2}\right) \right\} \right\} .
\end{eqnarray*}
The fundamental ingredient which is responsible for the major quantitative
differences between $\tau _{{\rm dec,S}}$ and $\tau _{{\rm dec,R}}$ is $%
\prod_{j=1}^{N}\tanh \left( \beta \Omega _{j}/2\right) $. In realistic
situations, this term is close to zero, even in the low temperature limit, $%
\beta \Omega _{j}\gg 1$, provide that the number of available modes of the
reservoir is huge. In this limit, we have a product with a lot of factors
smaller than the unity that tends to zero. On the other hand, if the
temperature is high, the factors are close to zero and the product also
vanishes.

In order to investigate the behavior of $\delta _{R}\left( t\right) $
in the low temperature limit, we perform the following approximation, valid
if the condition $\beta \Omega _{j}\gg 1$ is satisfied for all modes $\Omega
_{j}$:

\begin{equation}
\prod_{j=1}^{N}\tanh \left( \frac{\beta \Omega _{j}}{2}\right) \approx \exp
\left( -2\sum_{j=1}^{N}e^{-\beta \Omega _{j}}\right) .  \label{approx}
\end{equation}
Let us again assume continuum smooth spectrum for the reservoir with a
sufficiently smooth density of modes $g\left( \Omega \right) $. The
summation in the above expression takes the form 
\[
\sum_{j=1}^{N}e^{-\beta \Omega _{j}}\rightarrow \int_{0}^{\infty }{\rm d}%
\Omega \ g\left( \Omega \right) e^{-\beta \Omega }. 
\]
The product of $\tanh $ will be small if  the summation in (\ref{approx})
is equal or greater than unity. This condition suggests a rough estimate
of a lower bound for the temperature of the reservoir, so that for
temperatures greater than this limit, the correlations between the system of
interest and the reservoir can be certainly neglected. In other words, the
working hypothesis $\rho \left( t\right) \simeq \rho _{S}\left( t\right)
\otimes \rho _{R}\left( 0\right) $, largely used in master equation
derivations, appear justifiable if the condition 
\begin{equation}
\sum_{j=1}^{N}e^{-\beta \Omega _{j}}\gtrsim 1,  \label{condition}
\end{equation}
is satisfied.

%\begin{quotation}
%Colocar o exemplo com $g\left( \Omega \right) $ exibindo ``sharp cutoff'' em 
%$\omega _{D}$, onde $\omega _{D}$ \'{e} a frequ\^{e}ncia de Debye.
%D\'{u}vida: ``sharp cutoff'' viola a condi\c{c}\~{a}o de ``$g\left( \Omega
%\right) $ sufficiently smooth''? In the case that one of the subsystems is a
%reservoir, the number of levels (density of states, in the continuum limit)
%allowed in the prekinetic regime grows, and decoherence and thermalization
%occur even faster. As recurrence times also grows, both processes attain
%their characteristic irreversibility.
%
%Just as a final word, we point out that it is very striking that
%entanglement be at the same time at the root of most quantum mysteries, such
%as EPR ``paradox'', Schr\"{o}dinger cats, Bell inequalities and Quantum
%Communication, and also in the way how classicality emerges from Quantum
%Mechanics when we insist in treating a part of a whole system as a system
%itself.
%\end{quotation}

In synthesis, we worked out an approximation scheme for subsystems study without the use o master equations. It allows us to obtain timescales for the processes of decoherence, dissipation and thermalization when a harmonic oscillator interacts with a reservoir. Timescales for the reservoir could also be obtained, and, for non-null temperature, the essential feature of macroscopicity, {\it{i.e.}} the huge number of degrees of freedom, justifies quantitatively the one hardly justifiable approximation usually made in master equation derivations.

M. C. Nemes acknowledges the support of CNPq.

\begin{thebibliography}{8}
\bibitem{mod}  A. O. Caldeira and A. J. Legget, {\rm Ann. Phys. (NY)} {\bf %
149}, 374 (1983); W. H. Zurek, {\rm Phys. Rev. D} {\bf 26}(8), 1862 (1982);
K. M. Fonseca Romero and M. C. Nemes, {\rm Phys. Lett. A} {\bf 235}, 432
(1997).

\bibitem{Mte}  R. Omn\'{e}s, {\rm Phys. Rev. A} {\bf 56}(5), 3383 (1997).

\bibitem{Ji}  J. I. Kim, M. C. Nemes, A. F. R. de Toledo Piza, and H. E.
Borges, {\rm Phys. Rev. Lett.} {\bf 77}(2), 207 (1996).

\bibitem{cohen}  Cohen \'{A}tomos e f\'{o}tons.

\bibitem{lowTemp}  Sobre temperaturas nulas.

%\bibitem{WZ}  J. A. Wheeler and W. H. Zurek (orgs.), {\it Quantum Theory and
%Measurement}, Princeton (1983).

%\bibitem{Paris}  S. Haroche, {\rm Phys. Today} {\bf 51}(7), 36, (July 1998).

%\bibitem{qco}  See special issue of {\rm Physics World}, (March 1998).

\bibitem{DMM74}  V. V. Dodonov, I. A. Malkin, and V. I. Man'ko, {\rm Physica}
{\bf 72}, 597 (1974).

\bibitem{Gla63}  R. Glauber, {\rm Phys. Rev.} {\bf 131}, 2766 (1963).


%\bibitem{Giuetal96}  D. Giulini {\it et al.}, {\it Decoherence and the
%Appearence of a Classical World in Quantum Theory}, Springer (1996).

\bibitem{PZ99}  J. P. Paz and W. H. Zurek, {\rm Phys. Rev. Lett.} {\bf 82},
5181 (1999).

\bibitem{DBRH96}  L. Davidovich, M. Brune, J. M. Raimond, and S. Haroche, 
{\rm Phys. Rev. A} {\bf 53}(3), 1295 (1996).

\bibitem{JK98}  C. Kiefer and E. Joos, {\it Decoherence: Concepts and
Examples} in P. Blanchard and A. Jadczyk (eds.) {\it Quantum Future}
(Springer, Berlin, 1998). Also available as {\tt quant-ph/9803052} (and
references therein).


\bibitem{KB92}  M. S. Kim and V. Buzek, {\rm Phys. Rev. A} {\bf 46}(7), 4239
(1992).
\end{thebibliography}


quant-ph0409061.extracted_bib
\begin{thebibliography}{8}
\bibitem{ionTraps} For review on ion trapping, see D. Liebfried,
R. Blatt, C. Monroe, and D. Wineland, \rmp {\bf 75}, 281 
(2003); D. Wineland {\it et al.}, J. Res. Natl. Stand.
Technol. {\bf 103}, 259 (1998).

\bibitem{cavity} M. Brune {\it et al.}, \prl {\bf 65}, 976 (1990); 
M. Brune {\it et al.}, \pra {\bf 45}, 5193 (1992); G. Nogues 
{\it et al.}, Nature {\bf 400}, 239 (1999).

\bibitem{paris}  M. Brune {\it et al.}, \prl {\bf 77}, 4887
(1996).

\bibitem{openQuSys} There are several approaches to
quantum open systems and we cite a few. 
\\
Master equations: besides the references \cite{mod, cohen, Mte, lowTemp}
listed below, for a rigorous treatment, see E. B. Davies,
Comm. Math. Phys. {\bf 39}, 91 (1974);
G. Lindblad, {\it id.} {\bf 48}, 199 (1976).
\\
Quantum trajectories and stochastic extensions of Schr\"{o}dinger
equation: see H. Carmichael, {\it An Open Systems
Approach to Quantum Optics} (Springer, Berlin, 1993); for review, 
see also A. Bassi, and G. Ghirardi, Phys. Rep. {\bf 379}, 257 (2004).  
\\
Decoherent histories: for review, see R. Omn\`{e}s, \rmp
{\bf 64}, 339 (1992).


\bibitem{mod} A. O. Caldeira, and A. J. Legget, {\rm Ann. Phys. (NY)} {\bf %
149}, 374 (1983); W. H. Zurek, {\rm Phys. Rev. D} {\bf 26}, 1862 (1982);
K. M. Fonseca Romero, and M. C. Nemes, {\rm Phys. Lett. A} {\bf 235}, 432
(1997).

\bibitem{cohen}  C. Cohen-Tannoudji, J. Dupont-Roc, and G. Grynberg, 
{\it Atom-Photon Interactions -- Basic Processes and Applications} 
(John Wiley \& Sons, New York, 1992). 

\bibitem{Mte}  R. Omn\`{e}s, {\rm Phys. Rev. A} {\bf 56}, 3383 (1997).

\bibitem{Amir} C. Morais Smith, and A. O. Caldeira, \pra {\bf{36}}, 3509 (1987).

\bibitem{lowTemp}  An interesting pioner work is L. Di\'osi,
{\rm Europhys. Lett.} {\bf 30}, 63 (1995), also available as gr-qc/9403046.
An exact master equation for a particle coupled to a general environment at arbitrary temperature is deduced in
B. L. Hu, J. P. Paz, and Y. Zhang, \prd {\bf 45}, 2843 (1992). This result is used 
for a study of decoherence of a harmonic oscillator coupled
to an ohmic environment at zero temperature in S. Sinha, Phys. Lett. A {\bf 228}, 1 (1997). 
A master equation
in the Lindblad form valid at high and low temperatures is obtained in
S. Gao, \prl {\bf 79}, 3101 (1997).

\bibitem{DMM74}  V. V. Dodonov, I. A. Malkin, and V. I. Man'ko, {\rm Physica}
{\bf 72}, 597 (1974).

\bibitem{Gla63}  R. Glauber, {\rm Phys. Rev.} {\bf 131}, 2766 (1963).

\bibitem{Ji}  J. I. Kim, M. C. Nemes, A. F. R. de Toledo Piza, and H. E.
Borges, {\rm Phys. Rev. Lett.} {\bf 77}, 207 (1996).


%\bibitem{WZ}  J. A. Wheeler and W. H. Zurek (orgs.), {\it Quantum Theory and
%Measurement}, Princeton (1983).

%\bibitem{Paris}  S. Haroche, {\rm Phys. Today} {\bf 51}(7), 36, (July 1998).

%\bibitem{qco}  See special issue of {\rm Physics World}, (March 1998).


%\bibitem{Giuetal96}  D. Giulini {\it et al.}, {\it Decoherence and the
%Appearence of a Classical World in Quantum Theory}, Springer (1996).

%\bibitem{PZ99}  J. P. Paz and W. H. Zurek, {\rm Phys. Rev. Lett.} {\bf 82},
%5181 (1999).

\bibitem{DBRH96}  L. Davidovich, M. Brune, J. M. Raimond, and S. Haroche, 
{\rm Phys. Rev. A} {\bf 53}, 1295 (1996).

\bibitem{JK98}  C. Kiefer and E. Joos, {\it Decoherence: Concepts and
Examples} in P. Blanchard and A. Jadczyk (eds.) {\it Quantum Future}
(Springer, Berlin, 1998). Also available as {\tt quant-ph/9803052} (and
references therein).


\bibitem{KB92}  M. S. Kim and V. Buzek, {\rm Phys. Rev. A} {\bf 46}, 4239
(1992).
\end{thebibliography}