text:
\section{Introduction}
Quantum transport in small tunneling structures (quantum dots)
have attracted great attention due to the possibility of
investigating single-electron effects in the electric current\cite{likh}.
Until now research has been mostly concentrated on single dots, but
the rapid progress in microfabrication technology has made it possible the
extension to coupled dot systems with aligned levels\cite{haug,vdr,wbm}.
An example of such a system is shown schematically in Fig. 1, where
the coupled quantum dot (coupled wells) are
connected with two separate reservoirs (the leads).
In contrast with a single dot, the electron wave function inside
a coupled dot structure is a superposition of electron
states localized in each of the dots. As a result, the effects of
quantum coherence would appear in electron current flowing through
such a system. Usually these effects are treated in the framework
of single electron approach. Although this approach is an appropriate
one for coupled-well semiconductor heterostructures \cite{bry,g,kal},
it cannot be applied for coupled dots. The reason is
the dominant role of the Coulomb interaction that leads to
the Hubbard-type Hamiltonian for a description of
these systems\cite{sarma}.
\vskip1cm
\begin{minipage}{13cm}
\begin{center}
\leavevmode
\epsfxsize=13cm
\epsffile{ratef1.ps}
\end{center}
{\begin{small}
Fig.~1. Resonant transport through a coupled-dot structure.
$\Gamma_{L,R}$ denote the corresponding tunneling rate from (to) the
left (right) reservoirs, and $\Omega_0$ is the interdot hopping
amplitude. $E_F^{L,R}$ denote the Fermi level in the left (right)
reservoirs. Only those energy levels inside the dots that carry
electron current are shown.
\end{small}}
\end{minipage} \\ \\

The Coulomb blockade effects in electron transport through
a {\em single} dot can be taken into account in the most simple way by
using the ``classical'' rate equations\cite{glaz,ak,been,Davies4603}.
These equations describe the current through the dot
in terms of balance between incoming and outgoing rates
from (to) the leads. Basically the same classical
approach has been used to calculate the conductance
of coupled multi-dot systems\cite{klim,chen}. In these works the
whole array of quantum dots has been treated as a single quantum
system, where its many-body eigenstates were calculated by exact (numerical)
diagonalization of the Hubbard Hamiltonian. Then the leads (reservoirs)
were incorporated through the rate equations for single dots\cite{been}.
Yet, such a procedure is correct only for small coupling with
the leads. If the corresponding rates $\Gamma$ are of the order
of the interdot transitions $\Omega$ (Fig. 1), they affect
the diagonalization of the Hubbard Hamiltonian\cite{g}.
Besides, the approach\cite{klim,chen} is mainly numerical
and the inelastic scattering within the array is accounted
for phenomenologically.

A different way to treat the quantum transport in multi-dot
systems is to use the Bloch-type rate
equations\cite{naz,glp1,gp,g1} instead of the classical rate equations.
In contrast with previous treatments\cite{klim,chen}
this approach is valid also for strong
coupling with the leads and it accounts the inelastic
scattering effects inside the multi-dot system\cite{gp}.
It proves to be much simpler than the other approaches,
so that in many cases the analytical treatment of the problem is
possible\cite{glp1,gp}. It is also important to note that this approach
is not a phenomenological one: new generalized Bloch-type rate equations
were {\em derived} from the many-body Schr\"odinger equation
with the Hubbard Hamiltonian by integrating out the continuum
states of the reservoirs\cite{gp}.

An important advantage of the Bloch-type equations is a clear
distinction between coherent(``quantum'') and non-coherent
(``classical'') terms.
The quantum coherence in the Bloch equations manifests itself
in the non-diagonal density-matrix elements (``coherences''), coupled
with the diagonal density-matrix elements
(``probabilities'')\cite{bloch}.
Such a coupling always takes place, whenever a carrier
jumps between {\em isolated} states inside the system\cite{gp}.
Otherwise, the diagonal and non-diagonal matrix elements
are decoupled and the Bloch equations turn into
the {\em classical} rate equation.

Actually Bloch equations describe quantum motion between
isolated (non-orthogonal) states, which are directly coupled
(as in the double-well potential). In general, these states can be
separated by a medium with high density of states.
An example of such a system is shown in Fig. 2, where two dots
with aligned levels are separated by the
ballistic channel. In this case
a carrier transport from the left to the right reservoir
proceeds through continuum states in the
channel. Usually transitions from discrete to
continuum states lead to dephasing, i.e. to destruction
of quantum coherence. Yet, some coherent effects may survive.
It is therefore important to establish of whether quantum coherence
still affects the transport through separated dots, or
more precisely: are the diagonal
density-matrix elements still coupled with the non-diagonal matrix
elements in the corresponding rate equations?
\vskip1cm
\begin{minipage}{13cm}
\begin{center}
\leavevmode
\epsfxsize=13cm
\epsffile{ratef2.ps}
\end{center}
{\begin{small}
Fig.~2. Resonant transport through two quantum dots separated
by a ballistic channel. Here $\Omega_l$ and $\Omega_r$ denote the coupling of
the left and right dots dot with the levels $E_l$ and $E_r$ in the
left and in the right leads. $\Omega_m$ and $\bar\Omega_m$ denote
the coupling of the left and the right dots with the level $E_m$ in
the ballistic channel.
\end{small}}
\end{minipage} \\ \\

This question leads us to a more general problem, of how to modify
the Bloch rate equations for transport in coupled-dots for a
general case of arbitrary distributed quantum dots. Similar to
the Bloch equations one can anticipate that that the new rate
equations would incorporate the classical and
quantum descriptions in a very efficient way.
Therefore these equations can provide most useful account of
quantum coherence effects in different processes
in a comparison with usual approaches\cite{rammer}.

In this paper we study the above mentioned problems
taking as a basis the microscopic many-body Schr\"odinger equation with
the Hubbard-type tunneling Hamiltonian, describing the entire system of the
reservoirs and the quantum dots.
The plan of the paper is the following. In Sect. 2 we describe
the previously derived Bloch-type rate equations for
coupled dots and some particular features of the coherent transport
in these systems.  In Sect. 3 we present the rate equations
for separated dots and outline their
derivation from the microscopic Schr\"odinger equation.
The details of the derivation are described in Appendix.
A comparison of coherent effects in quantum transport
through coupled and separated dots is given in Sect. 4. The quantum rate
equations for a general configuration of quantum dots are
presented in Sect. 5. The last section is a summary.

\section{Rate equations and coherent effects in coupled quantum-dots}

We start with a review of the quantum rate equations for coupled multi-dot
systems, connected with two reservoirs
(the ``emitter'' and the ``collector'') and interacting with phonon reservoir.
The entire system is described by the Hubbard-type tunneling Hamiltonian
where the electron-electron interaction is taken into account
by introducing the corresponding electrostatic charging
energy. In case of large voltage bias one can reduce
the many-body Schr\"odinger equation to the system of quantum rate equations
by integrating out continuum reservoir states\cite{gp}. As a result
the following Bloch-type equations for the density matrix of
the multi-dot system $\{\sigma_{\alpha\beta}\}$ are obtained,
where $\alpha ,\beta$ denote the
isolated (non-orthogonal) states of the system:
\begin{mathletters}
\label{c3}
\begin{eqnarray}
\dot\sigma_{\alpha\alpha} & = &
 i\sum_{\beta}\Omega_{\alpha\to\beta}
(\sigma_{\alpha\beta}-\sigma_{\beta\alpha})
-\sigma_{\alpha\alpha} \sum_{\gamma}
\Gamma_{\alpha \rightarrow \gamma} +
 \sum_{\delta} \sigma_{\delta\delta}
\Gamma_{\delta \rightarrow \alpha}\;,
 \label{c3a}\\
\dot\sigma_{\alpha\beta} & = & i(E_\beta - E_\alpha) \sigma_{\alpha\beta} +
i\left (\sum_{\gamma}\sigma_{\alpha\gamma}\Omega_{\gamma\to\beta}
-\sum_{\delta}\Omega_{\alpha\to\delta}
\sigma_{\delta\beta}\right )
\nonumber\\
    &  &~~~~~~~~~~~~~~~~~~~~~~~~~~~~ -\frac{1}{2}\sigma_{\alpha\beta}
   \sum_{\delta}\left (\Gamma_{\alpha \rightarrow\delta}
    +\Gamma_{\beta \rightarrow \delta}\right )
                  +\sum_{\gamma\delta\neq \alpha\beta}
\sigma_{\gamma\delta}
        \Gamma_{\gamma\delta\rightarrow \alpha\beta}.
\label{c3b}
\end{eqnarray}
\end{mathletters}
Here $\Omega_{\alpha\to\beta}$ is the amplitude of one-electron hopping
that results in the transition between the states $\alpha$ and $\beta$.
The width $\Gamma_{\alpha\to \gamma}=2\pi\rho |\Omega_{\alpha\to \gamma} |^2$
is the probability per unit time for the system to make a transition from
the state $|\alpha\rangle$ to
the state $|\gamma\rangle$ of the device due to the tunneling to (or from) the
reservoirs, or due to interaction with the phonon bath, or any other
interaction generated by a continuum state medium with the density of states
$\rho$. Notice that Eq.~(\ref{c3a}) for diagonal elements has
a form of classical rate equations,
except for the first term. This term is generated by an electron hopping
between the {\em isolated} levels, which results in the coupling
with non-diagonal density-matrix elements. Therefore it
is responsible for coherent quantum effects in electron transport.
The nondiagonal matrix elements are described by Eq.~(\ref{c3b}).
The last term in Eq.~(\ref{c3b}) appears
only for systems with the number of isolated
states participating in the transport is more than two.
It describes the simultaneous conversion of the states
$\gamma\to\alpha$ and $\delta\to\beta$,
generated by the same one-electron decay to (from) continuum.

The current flowing through the system is given by
\begin{equation}
I(t)=\sum_\gamma\sigma_{\gamma\gamma}(t)\Gamma^{(\gamma)}_R,
\label{c4}
\end{equation}
where the sum is extended over states $|\gamma\rangle$ in which the dot
adjacent to the collector is occupied (we consider the electron charge $e=1$).
$\Gamma_R^{(\gamma)}$ is the partial width of the state $|\gamma\rangle$
due to tunneling to the collector (the right reservoir).

It follows from Eqs.~(\ref{c3}) that the coherent effects
do appear in the quantum transport whenever a carrier
jumps from one to another {\em isolated} state inside the device.
In the absence of such transition as, for
instance, in resonant tunneling through a single dot,
the diagonal and non-diagonal matrix elements
are {\em decoupled} and the evolution of diagonal density-matrix elements is
described by the {\em classical} rate equation.

For an example we consider the quantum transport through
a double-dot system at zero temperature\cite{foot1},
shown in Fig. 1. In order to diminish the number of equations
we assume that the Coulomb repulsion does not allow for
two electrons to occupy the same dot, i.e. $E_i+U_{ii}\gg E_F^L$,
where $i=1,2$ and $U_{ii}$ is the corresponding
charging energy. Yet, the
interdot Coulomb repulsion $U_{12}$ is much smaller, so the
both levels $E_{1,2}$ can be occupied simultaneously.
In this case there are four available states of the double-dot system
(for simplicity we neglected the spin):
$|a\rangle$ -- the levels $E_{1,2}$ are empty,
$|b\rangle$ -- the level $E_1$ is occupied,
$|c\rangle$ -- the level $E_2$ is occupied,
$|d\rangle$ -- the both level $E_{1,2}$ are occupied.
Using Eqs.~(\ref{c3}) with $\alpha ,\beta,\ldots =\{ a,b,c,d\}$
we find the following equations describing the time evolution
of the corresponding density-matrix elements
\begin{mathletters}
\label{c1}
\begin{eqnarray}
\dot\sigma_{aa} & = & -\Gamma_L\sigma_{aa}
+\Gamma_R\sigma_{cc}\;,
\label{c1a}\\
\dot\sigma_{bb} & = & \Gamma_L\sigma_{aa}
+\Gamma'_R\sigma_{dd}+i\Omega_0(\sigma_{bc}-\sigma_{cb})\;,
\label{c1b}\\
\dot\sigma_{cc} & = & -\Gamma_R\sigma_{cc}
-\Gamma'_L\sigma_{cc}-i\Omega_0(\sigma_{bc}-\sigma_{cb})\;,
\label{c1c}\\
\dot\sigma_{dd} & = & -\Gamma'_R\sigma_{dd}
+\Gamma'_L\sigma_{cc}\;,
\label{c1d}\\
\dot\sigma_{bc} & = & i(E_2-E_1)\sigma_{bc}+
i\Omega_0(\sigma_{bb}-\sigma_{cc})
-\frac{1}{2}(\Gamma'_L+\Gamma_R)\sigma_{bc}\;,
\label{c1e}
\end{eqnarray}
\end{mathletters}
which are supplemented with the probability conservation condition:
$\sum_i\sigma_{ii}(t)=1$. Note that the interdot Coulomb repulsion results in a variation
of the corresponding transition rate whenever
both levels are occupied  ($\Gamma_{L,R}$ is replaced by $\Gamma'_{L,R}$,
evaluated at the energy $E_{1,2}+U_{12}$ respectively\cite{gp}).

The dc current is
$I= \Gamma_R\sigma_{cc}(t\to\infty )+ \Gamma'_R \sigma_{dd}(t\to\infty )$,
Eq.~(\ref{c4}).  In order to shorten the final expression
we assume that the rates are weakly dependent on the energy,
$\Gamma'_{L,R}=\Gamma_{L,R}$.
Then solving Eqs.~(\ref{c1}) in the limit $t\to\infty$ we obtain
\begin{equation}
I=\left (\frac{\Gamma_L\Gamma_R}{\Gamma_L+\Gamma_R}\right )
\frac{\Omega_0^2}{\epsilon^2\Gamma_L\Gamma_R/
(\Gamma_L+\Gamma_R)^2+\Omega_0^2+\Gamma_L\Gamma_R/4},
\label{c2}
\end{equation}
where $\epsilon =E_1-E_2$.

It is interesting to compare Eqs.~(\ref{c1}) with the corresponding
classical rate equations where only transitions between
the {\em diagonal} density-matrix elements take place.
In this case the probabilities $\sigma_{bb}$ and $\sigma_{cc}$
of finding an electron in the first and the second dot are
coupled by the some rate $\Gamma_M$.
One easily finds that Eqs.~(\ref{c1}) become
\begin{mathletters}
\label{c5}
\begin{eqnarray}
\dot\sigma_{aa} & = & -\Gamma_L\sigma_{aa}
+\Gamma_R\sigma_{cc}\;,
\label{c5a}\\
\dot\sigma_{bb} & = & \Gamma_L\sigma_{aa}
+\Gamma_R\sigma_{dd}-\Gamma_M(\sigma_{bb}-\sigma_{cc})\;,
\label{c5b}\\
\dot\sigma_{cc} & = & -\Gamma_R\sigma_{cc}
-\Gamma_L\sigma_{cc}+\Gamma_M(\sigma_{bb}-\sigma_{cc})\;,
\label{c5c}\\
\dot\sigma_{dd} & = & -\Gamma_R\sigma_{dd}
+\Gamma_L\sigma_{cc}\;.
\label{c5d}
\end{eqnarray}
\end{mathletters}
(Cf. with the analogues rate equations in\cite{bjorn,gm}).
Solving these classical rate equations in the limit $t\to\infty$ we
obtain for the (classical) dc current
\begin{equation}
I_{cl}=\frac{\Gamma_L\Gamma_R\Gamma_M}
{\Gamma_L\Gamma_R+\Gamma_L\Gamma_M+\Gamma_R\Gamma_M}.
\label{c6}
\end{equation}

Comparing Eq.~({\ref{c6}) with its ``quantum'' counterpart,
Eq.~(\ref{c2}), we find that the quantum-mechanical nature of the
dc current displays itself in
the Lorentzian-shape resonance as a function
of the levels disalignment, $\epsilon =E_2-E_1$.
Such resonances were already observed in
the coupled-dot systems\cite{vdr}. Yet, the existence of the resonance
is not necessarily related to the quantum coherence effect. The later
manifests itself in a peculiar and even a counter-intuitive
dependence of the resonant current on coupling with the reservoirs.
It follows from Eq.~(\ref{c2}) that
the current {\em decreases} when the coupling with the reservoirs
{\em increases}. For instance, the peak value of the resonant current
$I_{\epsilon=0}\propto \Gamma_L/\Gamma_R\to 0$ for
$\Gamma_R\gg\Gamma_L$. It implies that by
{\em increasing} the penetrability of the barrier, connecting the second
dot with the collector we {\em diminish} the total current.
As an example we show in Fig.~3 the peak value of the resonant current
as a function of $\Gamma_R$ for constant $\Gamma_L$ and $\Omega_0$
(the solid line). This effect can be observed in coupled-dot structures by
changing the corresponding gate voltage.
\vskip1cm
\begin{minipage}{13cm}
\begin{center}
\leavevmode
\epsfxsize=10cm
\epsffile{ratef3.ps}
\end{center}
{\begin{small}
Fig.~3. Current through the coupled-dot structure, Fig. 1, as
a function of $\Gamma_R/\Gamma_L$.
The solid line corresponds to Eq.~(2.4) for $\epsilon=0$ and
$\Gamma_L^2/4\Omega_0^2=5$. The dashed line is the result of
the classical rate equations, Eq.~(2.6), for $\Gamma_M=\Gamma_L$.
\end{small}}
\end{minipage} \\ \\
For a comparison the corresponding ``classical'' current
$I_{cl}$, Eq.~(\ref{c6}), is shown in Fig. 3 by the dashed line.

The decrease of the resonant current shown in Fig. 3
is a result of destructive quantum interference during
decay of a coherent superposition to continuum in the regime
of strong coupling with the reservoirs\cite{gm,gbj}. A similar effect
had been observed in electron decay from coupled quantum wells
to continuum\cite{bg}. It is quite interesting that
such a counter-intuitive behavior of the resonant current
looks as a manifestation of the Zeno effect\cite{zeno}.
It tells us that the quantum transitions between different states
slow down when one of the states is continuously observed. Indeed,
the increase of the right barrier penetrability leads to
immediate tunneling of an electron to the collector whenever it arrives
to the second dot.
It can be considered as a continuous observation of the second dot state,
which results in an effective electron localization in the first
dot\cite{kor}.

\section{Rate equations for two separated dots}

Now we are going to derive rate equations for quantum transport
in mesoscopic systems with arbitrary configuration
of quantum dots. As a generic example
we consider quantum transport through two quantum dots separated by
a ballistic channel, Fig. 2.
The dots contain only isolated levels, whereas
the density of states in the ballistic channel and in the emitter and the
detector is very high (continuum).
This system can be described by the tunneling Hamiltonian
\begin{eqnarray}
{\cal H} &=&\sum_l E_{l}a^{\dagger}_{l}a_{l} +
E_1 a_1^{\dagger}a_{1}+
\sum_m E_{m}a^{\dagger}_{m}a_{m} +
E_2 a_2^{\dagger}a_{2}\sum_r a_{r}a^{\dagger}_{r}a_{r}
+\sum_{i,j=1,2}U_{ij}n_in_j
\nonumber\\
&+& \left\{ \sum_l \Omega_{l}a^{\dagger}_{1}a_l
+ \sum_m \Omega_{m}a^{\dagger}_ma_{1}
+ \sum_m \bar\Omega_{m}a^{\dagger}_{2}a_m
+ \sum_r \Omega_{r}a_r^{\dagger}a_2 + {\rm H.c.}\right\}
\label{a1}
\end{eqnarray}
Here the subscripts $l$, $m$ and
$r$ enumerate correspondingly the levels in the
left reservoir, in the (middle)
ballistic channel and in the right reservoir.
The spin degrees of freedom were omitted. In order to
simplify the derivation we assumed that the intradot charging
energy $U_{ii}$ is large, $E_{1,2}+U_{ii}\gg E_F^L$. Thus
only one electron can occupy each of the dots.
However, the interdot charging energy $U_{12}$
is much smaller, so it does not prevent simultaneous occupation
of the two dots. The same is assumed for
the Coulomb repulsion between electrons inside the dots and
the ballistic channel. Although we did not include this interaction
in the Hamiltonian (\ref{a1}), it can be treated in the same way
as the interdot interaction $U_{12}$.
As in the previous case we restrict ourselves to the zero temperature case,
even though the results are valid for a finite temperature, as
would be clear from the derivation.

Let us assume that all the levels in the emitter, in the
ballistic channel and in the collector are initially filled up to the Fermi
energies $E_F^L$, $E_F^M$ and $E_F^R$ respectively. We call
it as the ``vacuum'' state, $|0\rangle$. (In the following we consider
the case of large bias, so that $E_F^L\gg E_F^M, E_F^R$).
This vacuum state is unstable: the Hamiltonian
Eq.~(\ref{a1}) requires it to decay exponentially to continuum states
having the form $a_{1}^{\dagger}a_{l}|0\rangle$ with an electron in
the level $E_1$ and a hole in the emitter continuum,
$a_{m}^{\dagger}a_{l}|0\rangle$ with an electron in
the level $E_m$ in the ballistic channel and a hole in the emitter, and so on.
The many-body wave function describing this system can be written
in the occupation number representation as
\begin{eqnarray}
|\Psi (t)\rangle & = & \left [ b_0(t) + \sum_l b_{1l}(t)
     a_{1}^{\dagger}a_{l} + \sum_{l,m} b_{lm}(t)a_{m}^{\dagger}a_{l}
   +\sum_l b_{2l}(t)a_{2}^{\dagger}a_{l}  \right.
      \nonumber\\
 &+& \left. \sum_{l,r} b_{lr}(t)a_{r}^{\dagger}a_{l}
           +\sum_{l<l'} b_{12ll'}(t)a_{1}^{\dagger}a_{2}^{\dagger}a_{l}a_{l'}
           +\sum_{l<l',r} b_{1ll'r}(t)
           a_{1}^{\dagger}a_{r}^{\dagger}a_{l}a_{l'}
           + \ldots \right ] |0\rangle,
\label{a2}
\end{eqnarray}
where $b(t)$ are the time-dependent probability amplitudes to
find the system in the corresponding states described above.
These amplitudes are obtained from the Shr\"odinger equation
$i|\dot\Psi (t)\rangle ={\cal H}|\Psi (t)\rangle$, supplemented
with the initial condition
($b_0(0)=1$, and all the other $b(0)$'s being zeros). Using the amplitudes
$b(t)$ we can find the density-matrix of the quantum dots,
$\sigma_{ij}^{(k,n)}(t)$, by tracing out the continuum
states of the reservoirs and the ballistic channel. Here the subscript
indices in $\sigma$
denote four states of the dots:  $i,j=\{a,b,c,d\}$, where
$|a\rangle$ -- the levels $E_{1,2}$ are empty,
$|b\rangle$ -- the level $E_1$ is occupied,
$|c\rangle$ -- the level $E_2$ is occupied,
$|d\rangle$ -- the both level $E_{1,2}$ are occupied,
and the superscript indices $k,n$ denote the number of
electrons accumulated in the ballistic channel and in the collector
respectively at time $t$, Fig. 4.
\vskip1cm
\begin{minipage}{13cm}
\begin{center}
\leavevmode
\epsfxsize=11cm
\epsffile{ratef4.ps}
\end{center}
{\begin{small}
Fig.~4. Electron states of the two separated dot structure, shown in Fig. 2.
$\Gamma_{L,R}$, $\Gamma_M$ and $\bar\Gamma_M$ are the tunneling rates
between the dots and the reservoirs, and between the dots and the
ballistic channel. The indices $k$ and $n$ denote the number of electrons
penetrating to the ballistic channel and to the collector at time $t$.
\end{small}}
\end{minipage} \\ \\
One finds
\begin{mathletters}
\label{a7}
\begin{eqnarray}
&&\sigma_{aa}^{(0,0)}(t) = |b_{0}(t)|^2,~~~
\sigma_{aa}^{(1,0)}(t)= \sum_{l,m} |b_{lm}(t)|^2,~~~
\sigma_{aa}^{(1,1)}(t)=\sum_{l<l',m,r} |b_{ll'mr}(t)|^2,~~~ \ldots
\label{a7a}\\
&&\sigma_{bb}^{(0,0)}(t) =\sum_l |b_{1l}(t)|^2,~~~
\sigma_{bb}^{(1,0)}(t)=\sum_{l<l',m} |b_{1ll'm}(t)|^2,~~~
\nonumber\\
&&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
\sigma_{bb}^{(1,1)}(t)=\sum_{l<l'<l'',m,r} |b_{1ll'l''mr}(t)|^2,~~~ \ldots
\label{a7b}\\
&&\cdots\cdots\cdots\cdots\cdots
\nonumber\\
&&\sigma_{bc}^{(0,0)}(t) = \sum_l b_{1l}(t)b^*_{2l}(t),~~~
\sigma_{bc}^{(1,0)}(t)= \sum_{l<l',m} b_{1ll'm}(t)b^*_{2ll'm}(t),~~~
\nonumber\\
&&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
\sigma_{bc}^{(1,1)}(t)=\sum_{l<l'<l'',m,r} b_{1ll'l''mr}(t)b^*_{2ll'l''mr}(t),
~~~\ldots
\label{a7c}
\end{eqnarray}
\end{mathletters}
\par
The rate of electrons arriving to the collector determines the electron
current in the system. Therefore the current operator is
$\hat I=i[{\cal H},\hat N_R]$, where
$\hat N_R=\sum_ra^{\dagger}_ra_r$ is the operator for
the total number of electrons accumulated in the right reservoir.
Using Eqs.~(\ref{a1}), (\ref{a2}), (\ref{a7}) we find that the
current $I(t)$ flowing through the system is
\begin{equation}
I(t)=\langle\Psi (t)|\hat I|\Psi (t)\rangle =
\sum_{k,n} n\left [\dot\sigma_{aa}^{(k,n)}(t)+
\dot\sigma_{bb}^{(k,n)}(t)+
\dot\sigma_{cc}^{(k,n)}(t)+
\dot\sigma_{dd}^{(k,n)}(t)\right ]
\label{a8}
\end{equation}
As expected, $I(t)$ is the time derivative of the total charge
accumulated in the collector.

It follows from Eq.~(\ref{a8}) that the current $I(t)$ flowing
through this system is expressed in terms of the diagonal
elements of the density-matrix $\sigma (t)$.
In order find the differential equations for
$\sigma (t)$ we need to sum over the states of
the reservoirs and the ballistic channel, Eqs.~(\ref{a7}). It can be done
analytically by using the procedure similar to that in Refs.\cite{gp,g1},
providing that the levels are not close to Fermi levels,
$|E_i-E_F|\gg\Gamma$, Fig. 4. The details of derivation are
described in Appendix.
As a result we obtain the following Bloch-type rate equations for the matrix
elements of the density-submatrix $\sigma (t)$:
\begin{mathletters}
\label{a13}
\begin{eqnarray}
&&\dot{\sigma}^{(k,n)}_{aa} = - \Gamma_L \sigma_{aa}^{(k,n)}
+\Gamma_M\sigma_{bb}^{(k-1,n)}+\bar\Gamma_M\sigma_{cc}^{(k-1,n)}
+\Gamma_R\sigma_{cc}^{(k,n-1)}\nonumber\\
&&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
+2\pi\rho_M\Omega_M\bar\Omega_M(\sigma^{(k-1,n)}_{bc}+\sigma^{(k-1,n)}_{cb})
\label{a13a}\\
&&\dot{\sigma}^{(k,n)}_{bb} = - \Gamma_M \sigma_{bb}^{(k,n)}
+\Gamma_L\sigma_{aa}^{(k,n)}+\bar\Gamma_M\sigma_{dd}^{(k-1,n)}
+\Gamma_R\sigma_{dd}^{(k,n-1)}\nonumber\\
&&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
-\pi\rho_M\Omega_M\bar\Omega_M(\sigma^{(k,n)}_{bc}+\sigma^{(k,n)}_{cb})
\label{a13b}\\
&&\dot{\sigma}^{(k,n)}_{cc} = -(\Gamma_L +\bar\Gamma_M +\Gamma_R)
\sigma_{cc}^{(k,n)}
+\Gamma_M\sigma_{dd}^{(k-1,n)}
-\pi\rho_M\Omega_M\bar\Omega_M(\sigma^{(k,n)}_{bc}+\sigma^{(k,n)}_{cb})
\label{a13c}\\
&&\dot{\sigma}^{(k,n)}_{dd} = -(\Gamma_M +\bar\Gamma_M +\Gamma_R)
\sigma_{dd}^{(k,n)}+\Gamma_L\sigma_{cc}^{(k,n)}
\label{a13d}\\
&&\dot{\sigma}^{(k,n)}_{bc} = i(E_2-E_1)\sigma^{(k,n)}_{bc}
-\pi\rho_M\Omega_M\bar\Omega_M(\sigma^{(k,n)}_{bb}+\sigma^{(k,n)}_{cc})
+2\pi\rho_M\Omega_M\bar\Omega_M\sigma_{dd}^{(k-1,n)}\nonumber\\
&&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
-\frac{1}{2}(\Gamma_L+\Gamma_M +\bar\Gamma_M +\Gamma_R)
\sigma_{bc}^{(k,n)}\, .
\label{a13e}
\end{eqnarray}
\end{mathletters}
Here $\Gamma$ denote the tunneling widths to the leads and the ballistic
channel ($\Gamma =2\pi\rho |\Omega |^2$,
where $\rho $ is the corresponding density of states), and
$\Omega_M$, $\bar\Omega_M$ are the hopping
amplitudes between the left and the right dots and
the states $E_m=E_{1,2}$ of the ballistic channel.

Equations (\ref{a13}) have clear physical interpretation. Consider
for instance Eq.~(\ref{a13a}) for the probability rate of finding
the system in the state $a$ with
$k$ electrons in the ballistic channel and $n$ electrons
in the right reservoir (Fig. 4a). This state decays with the rate $\Gamma_L$
into the state $b$ (Fig. 4b) whenever an electron enters the first
dot from the left reservoir. This process is
described by the first term in Eq.~(\ref{a13a}). On the other hand,
the states $b$ and $c$ (Figs.~4b, 4c) with $k-1$ electrons
in the ballistic channel
decay into the state $a$ with $k$ electrons in the ballistic channel.
It takes place due to one-electron tunneling from the quantum dots
into the ballistic channel with the rates $\Gamma_M$ and
$\bar\Gamma_M$ respectively. This process is described by
the second and the third terms in Eq.~(\ref{a13a}). Also
the state $c$ (Fig. 4c) with $n-1$ electrons in the
right reservoir can decay into the state $a$ due to tunneling
to the right reservoir with the rate $\Gamma_R$
(the fourth term in Eq.~(\ref{a13a})).  The last term in this
equation describes the decay of coherent superposition of the
states $b$ and $c$ into the state $a$. It takes place due to single
electron tunneling from the first and the second dots into the same state
of the ballistic channel with the amplitudes $\Omega_M$ and $\bar\Omega_M$,
respectively. Obviously, this process has no classical analogy,
since classical particle cannot simultaneously occupy two dots.

Equations (\ref{a13b}), (\ref{a13c}) and (\ref{a13d}) describe the
probability rate of finding the system in the states
where one of the dots or both dots are occupied. In the first case
an electron can jump into unoccupied dot via continuum states of the
ballistic channel. As a result, the states $b$ and $c$ can decay into
linear superposition of the states $b$ and $c$. This process is
described by the last terms in Eqs.~(\ref{a13b}), (\ref{a13c}).
Obviously, if the both dots are occupied, such a process cannot
take a place. Therefore
$\sigma_{dd}$ is not coupled with the nondiagonal density-matrix
elements, Eq.~(\ref{a13d}). The last equation, (\ref{a13e}) describes
the time-dependence of the nondiagonal density matrix element.
It has the same interpretation as all previous equations.

Equations (\ref{a13}) for the reduced density matrix $\sigma(t)$
were derived starting from the wave function $|\Psi (t)\rangle$,
Eq.~(\ref{a2}), instead of using the density matrix for the entire
system. Of course, it makes no difference if the
entire system is initially in the pure state, or all the levels of
the reservoirs are occupied up to the corresponding Fermi energies.
At finite temperature, however, the system is not initially in a pure
state. Then one needs to perform the derivation using
the Liouville (Landau--von Neumann) equation for
the density-matrix. Yet, if energy levels $E_{1,2}$ of the
dots are far away from the Fermi-levels
($T\ll E_F^L-E_{1,2},\; E_{1,2}-E_F^R$), then the reservoir levels
that carry the current ($|E_{l}-E_{1,2}|\lesssim \Gamma$)
are deeply inside the Fermi sea. In this case one can consider these
levels as fully occupied. Then neglecting the relaxation processes in the
reservoir and assuming zero temperature inside
the dots, we would arrive to the same Eqs.~(\ref{a13}), derived from
the pure state.

Using Eqs.~(\ref{a13}) one finds for the total current,
Eq.~(\ref{a8})
\begin{equation}
I(t)=\Gamma_R[\sigma_{cc}(t)+\sigma_{dd}(t)],
\label{a14}
\end{equation}
where $\sigma_{ii}=\sum_{k,n}\sigma^{(k,n)}_{ii}$ are the total
``probabilities''. We can easily understand this result
by taken into account that  $\sigma_{cc}+\sigma_{dd}$ is the
total probability for occupation of the second dot and
$\Gamma_R$ is the rate of electron transitions from this
dot to the (adjacent) right reservoir, Eq.~(\ref{c4}).

In order to find differential equations for $\sigma_{ij}$ we sum
over $k,n$ in Eqs.~(\ref{a13}). Then we obtain the following
Bloch-type equations, which describe the time-dependence of the
density-matrix for separated dots
\begin{mathletters}
\label{a15}
\begin{eqnarray}
&&\dot{\sigma}_{aa} = - \Gamma_L \sigma_{aa}
+\Gamma_M\sigma_{bb}+\bar\Gamma_M\sigma_{cc}
+\Gamma_R\sigma_{cc}
+2\pi\rho_M\Omega_M\bar\Omega_M(\sigma_{bc}+\sigma_{cb})
\label{a15a}\\
&&\dot{\sigma}_{bb} = - \Gamma_M \sigma_{bb}
+\Gamma_L\sigma_{aa}+\bar\Gamma_M\sigma_{dd}
+\Gamma_R\sigma_{dd}
-\pi\rho_M\Omega_M\bar\Omega_M(\sigma_{bc}+\sigma_{cb})
\label{a15b}\\
&&\dot{\sigma}_{cc} = -(\Gamma_L +\bar\Gamma_M +\Gamma_R)
\sigma_{cc}
+\Gamma_M\sigma_{dd}
-\pi\rho_M\Omega_M\bar\Omega_M(\sigma_{bc}+\sigma_{cb})
\label{a15c}\\
&&\dot{\sigma}_{dd} = -(\Gamma_M +\bar\Gamma_M +\Gamma_R)
\sigma_{dd}+\Gamma_L\sigma_{cc}
\label{a15d}\\
&&\dot{\sigma}_{bc} = i(E_2-E_1)\sigma_{bc}
-\pi\rho_M\Omega_M\bar\Omega_M(\sigma_{bb}+\sigma_{cc})
+2\pi\rho_M\Omega_M\bar\Omega_M\sigma_{dd}-\frac{1}{2}
\Gamma_{tot}\sigma_{bc},
\label{a15e}
\end{eqnarray}
\end{mathletters}
where $\Gamma_{tot}=\Gamma_L+\Gamma_M +\bar\Gamma_M +\Gamma_R$.

The stationary (dc) current $I=I(t\to\infty )$
Eq.~(\ref{a14}) can be easily obtained from Eqs.~(\ref{a15}) by taken
into account that $\dot\sigma_{ij}\to 0$ for $t\to\infty$. Then
Eqs.~(\ref{a15}) turn into a system of linear algebraic equations,
supplemented by a probability conservation condition
$\sigma_{aa}+\sigma_{bb}+\sigma_{cc}+\sigma_{dd}=1$. Consider, for example,
the case of the same partial widths of the levels $E_{1,2}$, i.e.
$\Gamma_L=\Gamma_R=\Gamma_0$ and $\Gamma_M=\bar\Gamma_M$. The latter
implies $2\pi\rho_M\Omega_M\bar\Omega_M=\pm \Gamma_M$, since the amplitudes
$\Omega_M$, $\bar\Omega_M$ can be of the opposite signs. Solving
Eqs.~(\ref{a14}), (\ref{a15}) one finds for dc current
\begin{equation}
I=\frac{\Gamma_0^2\Gamma_M^2}{2\epsilon^2(\Gamma_0+\Gamma_M)
+2\Gamma_0(\Gamma_0^2+3\Gamma_0\Gamma_M+3\Gamma_M^2)},
\label{a16}
\end{equation}
where $\epsilon =E_1-E_2$.

Similar to the coupled-dot case, Eq.~(\ref{c2}), dc current
in separated dots displays the Lorentzian shape resonance as
a function of $\epsilon$ and the same peculiar dependence on the
coupling with the leads. The latter manifests the quantum-coherence
effects in separated dot systems. We discuss these effects
in the next section.

\section{Coherent effects in two-dot systems}
It follows from the rate equations (\ref{a15})
that the diagonal density-matrix elements
are coupled with the non-diagonal terms  ($\sigma_{bc}$, $\sigma_{cb}$),
similar to Eqs.~(\ref{c1}) for a coupled-dot system. It means that the
coherent effects should survive even in separated quantum dots.
Indeed, let us consider the dependence of the resonant current on
a coupling with the reservoirs.
As an example, we show in Fig. 5a the peak value of the resonant current
($\epsilon =0$) obtained from Eqs.~(\ref{a14}), (\ref{a15})
as a function of $\Gamma_R$ at fixed $\Gamma_L=\Gamma_M$.
One finds that the current through the dots separated
by a ballistic channel displays
the same peculiar behavior with $\Gamma_R$, as in the case
of coupled dots, Fig. 3. Namely, the current {\em decreases}
when the penetrability of the right barrier {\em increases}.
Such a behavior of the resonant
current is a clear manifestation of the quantum coherence
effects, as explained in Sect. II (see also\cite{gm,gbj,bg}).
It also can be verified experimentally by changing the voltage
on the gate, connecting the right dot with the collector.

On the other hand, the dependence of the resonant current on the width
$\Gamma_M$ is quite different, Fig. 5b. One finds that the current
increases with $\Gamma_M$. It is not surprising, since $\Gamma_M$
plays a role of coupling between the two dots. In fact, a similar
behavior with the coupling $\Omega_0$ displays the resonant current
in the case of coupled dots, Eq.~(\ref{c2}).

Despite the analogues quantum coherence effects,
the rate equations (\ref{a15}) are different from their counterpart
for the coupled-dots, Eqs.~(\ref{c1}). One finds that the diagonal
matrix elements in Eqs.~(\ref{a15}) are coupled with
the {\em real} part of the non-diagonal
matrix element,
\vskip1cm
\begin{minipage}{13cm}
\begin{center}
\leavevmode
\epsfxsize=13cm
\epsffile{ratef5.ps}
\end{center}
{\begin{small}
Fig.~5. DC current through the separated dot structure, Fig. 2, as
a function of $\Gamma_R/\Gamma_L$ for $\Gamma_M=\Gamma_L$ (a), and
as a function of $\Gamma_M/\Gamma_L$ for $\Gamma_R=\Gamma_L$ (b).
\end{small}}
\end{minipage} \\ \\
whereas the corresponding terms in Eqs.~(\ref{c1}),
are  coupled with the {\em imaginary} part
of the non-diagonal matrix element. Notice also that the coupling
is determined by the hopping amplitude
$\Omega$ in Eqs.~(\ref{c1}), while the corresponding coupling
in Eqs.~(\ref{a15}) is proportional to $\Omega^2\rho$

It is important to point out that the coupling with the
non-diagonal density-matrix elements in Eqs.~(\ref{a15})
does not decrease with the separation distance between the dots,
although the hopping amplitudes $\Omega_M,\,\bar\Omega_M$
do decrease. Indeed, using semi-classical expressions one finds
$|\Omega_M|=(1/\sqrt{\tau_1\tau_M})\exp (-S)$ and
$|\bar\Omega_M|=(1/\sqrt{\tau_2\tau_M})\exp (-\bar S)$, where
$S(\bar S)$ is the action under the barrier, separating the first
(second) dot from the ballistic channel. It can be written as
$S=\int_{x_i}^{x_f}|p(x)|dx$, where $|p(x)|=\sqrt{2m[V(x)-E]}$
and $x_{i,f}$, are the classical turning points.
$\tau_{1,2}$ are the classical periods of motion
in the first (second) quantum well
and $\tau_M$ is the classical periods of motion
in the ballistic channel. It implies that $\tau_M\to\infty$ when the length
of the ballistic channel increases.  Yet, $\tau_M$ is canceled
out from the coupling between the diagonal and non-diagonal matrix elements,
since the ballistic channel is effectively
one-dimensional so that the density of states
is $\rho_M=\tau_M/2\pi$. Finally we obtain
\begin{equation}
2\pi\rho_M\Omega_M\bar\Omega_M=\pm
\frac{1}{2\pi\sqrt{\tau_1\tau_2}}\exp [-(S+\bar S)].
\label{b1}
\end{equation}
Therefore, the quantum coherence effects in dc current could
survive even for long ballistic channels.
Obviously, it is true only for ideal one-dimensional channels, where
all the flux coming from the first dot arrives the second dot.
It should be also pointed out that electron-electron
scattering inside the ballistic channel (not taken into account in
these calculations) may reduce the interference effects. Yet, we
expect that such a process would not diminish the coupling
$(\ref{b1})$, but increase the ``dissipation''
width $\Gamma_{tot}$ in Eq.~(\ref{a15e}).
This problem, however, needs a special investigation.

It is interesting to make a comparison between peak values
of the dc current in coupled and separated dots.
In both cases the current reaches its maximal value
for $\Gamma_L=\Gamma_R=\Gamma_0$ and $\epsilon=0$, Eqs.~(\ref{c2}),
(\ref{a16}).  One finds  for the coupled dots
\begin{equation}
I_{max}=\frac{\Gamma_0}{2}
\frac{\Omega_0^2}{\Omega_0^2+\Gamma_0^2/4}\rightarrow
\frac{\Gamma_0}{2},\quad\quad {\mbox{for}}\quad\quad \Omega_0\gg \Gamma_0,
\label{b2}
\end{equation}
while for the separated dots
\begin{equation}
I_{max}=\frac{\Gamma_0}{2}
\frac{\Gamma_M^2}{3\Gamma_M^2+3\Gamma_0\Gamma_M
+\Gamma_0^2}\rightarrow
\frac{\Gamma_0}{6},\quad\quad {\mbox{for}}\quad\quad \Gamma_M\gg \Gamma_0,
\label{b3}
\end{equation}
Thus the peak value of dc current
in the coupled dots is three times larger than that
in the separated dots.

It is quite natural to associate quantum coherence effects in double-well
systems with quantum oscillations of an electron between the wells.
Indeed, an electron in a coupled-dot system, detached from the
emitter and the collector, oscillates between the states (a) and (b),
Fig. 6. These oscillations are reproduced by the same rate equations
that describe quantum transport in coupled-dots, Eqs.~(\ref{c1}),
but now with $\Gamma_L=\Gamma_R=0$. We then obtain for
the probability of finding an electron in the states (a) and (b),
$\sigma_{aa}(t)$ and $\sigma_{bb}(t)$,  Fig.6:
\begin{mathletters}
\label{b33}
\begin{eqnarray}
&&\dot\sigma_{aa}=i\Omega_0(\sigma_{ab}-\sigma_{ba})
\label{b33a}\\
&&\dot\sigma_{bb}=-i\Omega_0(\sigma_{ab}-\sigma_{ba})
\label{b33b}\\
&&\dot\sigma_{ab}=i(E_2-E_1)\sigma_{ab}
+i\Omega_0(\sigma_{aa}-\sigma_{bb}).
\label{b33c}
\end{eqnarray}
\end {mathletters}
\vskip1cm
\begin{minipage}{13cm}
\begin{center}
\leavevmode
\epsfxsize=6cm
\epsffile{ratef6.ps}
\end{center}
{\begin{small}
Fig.~6. Electron states of the coupled-dot structure of Fig. 1,
isolated from the leads.
\end{small}}
\end{minipage} \\ \\
Solving this equations with the initial condition
$\sigma_{aa}(0)=1$ and $\sigma_{bb}(0)=0$, we obtain
\begin{equation}
\sigma_{aa}(t)=\frac{\Omega_0^2\cos^2(\omega_1t)+\epsilon^2/4}
{\Omega_0^2+\epsilon^2/4}
\label{b4}
\end{equation}
where $\omega_1=(1/2)\sqrt{4\Omega_0^2+\epsilon^2}$.
As expected, an electron
oscillates between the dots with an amplitude $1/(1+\epsilon^2/4\Omega_0^2)$.

The situation is different, however, if the dots are separated
by a ballistic channel, Fig.~7. Indeed, the corresponding rate
equations for the probabilities
$\sigma_{aa}(t)$ and $\sigma_{bb}(t)$
of finding an electron in the states (a) and (b) of Fig.~7 are obtained from
Eqs.~(\ref{a15}) for $\Gamma_L=\Gamma_R=0$. Consider
for simplicity the case of $\Omega_M=\bar\Omega_M$. Then
Eqs.~(\ref{a15}) become
\begin{mathletters}
\label{b44}
\begin{eqnarray}
&&\dot\sigma_{aa}=-\Gamma_M\sigma_{aa}-\frac{1}{2}\Gamma_M
(\sigma_{ab}+\sigma_{ba})
\label{b44a}\\
&&\dot\sigma_{bb}=-\Gamma_M\sigma_{bb}-\frac{1}{2}\Gamma_M
(\sigma_{ab}+\sigma_{ba})
\label{b44b}\\
&&\dot\sigma_{ab}=i(E_2-E_1)\sigma_{ab}-\frac{1}{2}\Gamma_M
(\sigma_{aa}+\sigma_{bb})-\Gamma_M\sigma_{ab}
\label{b44c}
\end{eqnarray}
\end {mathletters}
Solving  these equations with
the initial condition $\sigma_{aa}(0)=1$ and $\sigma_{bb}(0)=0$, we obtain
\begin{equation}
\sigma_{aa}(t)=\frac{\Gamma_M^2\cosh^2(\omega_2t)
-\epsilon^2}{\Gamma_M^2-\epsilon^2}\exp (-\Gamma_Mt),
\label{b5}
\end{equation}
where $\omega_2=(1/2)\sqrt{\Gamma_M^2-\epsilon^2}$.
It follows from Eq~(\ref{b5}) that $\sigma_{aa}(t)$ does not display any
\vskip1cm
\begin{minipage}{13cm}
\begin{center}
\leavevmode
\epsfxsize=12cm
\epsffile{ratef7.ps}
\end{center}
{\begin{small}
Fig.~7. Electron states of the two separate dot structure of Fig. 2,
isolated from the leads.
\end{small}}
\end{minipage} \\ \\
oscillations. As an example, we show in Fig. 8 the probability
of finding an electron in
the left dot as a function of time in the case of coupled and separated
dots (Figs.~6, 7) for  $\epsilon =\Gamma_M =\Omega_0$.
The solid line corresponds to Eq.~(\ref{b4}), and
the dashed line to Eq.~(\ref{b5}).
Notice that $\sigma_{aa}=\sigma_{bb}\to 1/4$
for $t\to\infty$ when the levels are aligned ($\epsilon=0$), and
$\sigma_{aa}=\sigma_{bb}\to 0$ for $t\to\infty$ when $\epsilon\not = 0$.
It means that an electron decays into the ballistic channel.

\vskip1cm
\begin{minipage}{13cm}
\begin{center}
\leavevmode
\epsfxsize=7cm
\epsffile{ratef8.ps}
\end{center}
{\begin{small}
Fig.~8. The occupation of the left dot in Figs. 6 and 7
as a function of time for $\epsilon =\Gamma_M =\Omega_0$. The
solid line corresponds to coupled dots, and the
dashed line to separated dots.
\end{small}}
\end{minipage} \\ \\

The reason for a dissipative behavior of $\sigma_{aa}$
in the case separated dots is a coupling of
the dots with continuum states of the ballistic
channel. The latter leads to dephasing that destroys
quantum oscillations. It is thus rather remarkable that the
quantum coherence effects would ``reappear'' when
the same system is connected with the leads, Fig.~2.

\section{General case}
The rate equations (\ref{a15}), describing electron
transport in separated dots can be extended to any multi-dot system.
By applying the same technique of integrating out the reservoir
states as in Ref.\cite{gp} and in Appendix, we
arrive to the rate equations for the density-matrix $\sigma_{\alpha\beta}$
of the multi-dot system. These equations can be written as
\begin{eqnarray}
\dot\sigma_{\alpha\beta} &=&  i(E_\beta - E_\alpha) \sigma_{\alpha\beta} +
i\left (\sum_{\gamma}\sigma_{\alpha\gamma}
\tilde\Omega_{\gamma\to\beta}
-\sum_{\gamma}\tilde\Omega_{\alpha\to\gamma}
\sigma_{\gamma\beta}\right )\nonumber\\
&-&{\sum_{\gamma,\delta}}^{\prime}
\pi\rho(\sigma_{\alpha\gamma}\Omega_{\gamma\to\delta}\Omega_{\delta\to\beta}
+\sigma_{\gamma\beta}\Omega_{\gamma\to\delta}\Omega_{\delta\to\alpha})
+\sum_{\gamma,\delta}
\pi\rho\,(\Omega_{\gamma\to\alpha}\Omega_{\delta\to\beta}+
\Omega_{\gamma\to\beta}\Omega_{\delta\to\alpha})
\sigma_{\gamma\delta}\, .
\label{d1}
\end{eqnarray}
Here $|\alpha\rangle,\,  |\beta\rangle,\ldots$ are the states of
the multi-dot system in the occupation number representation, and
$\Omega_{\alpha\to\beta}$ denotes one-electron
hopping amplitude that generates $\alpha\to\beta$-transition. We distinguish
between the amplitudes $\tilde\Omega$ and $\Omega$ of
one-electron hopping among isolated states and among isolated
and continuum states, respectively. The latter transitions are
of the second order in the hopping amplitude $\sim\Omega^2$.
These transition are produced by two consecutive hoppings of an electron with
the same energy across continuum states with the density of states $\rho$.
The symbol $\sum'$ means that only those transitions
$\gamma\to\delta\to (\alpha ,\beta)$ are accounted for
in the sum, where the number of electrons in continuum
states (in the reservoirs and ballistic channels) stays the
same in the initial and final states (c.f. Eqs.~(\ref{a13})).

It is rather easy to verify that Eqs.~(\ref{d1}) coincide with
Eqs.~({\ref{a15}) for $\alpha,\beta,\ldots = \{a,b,c,d\}$,
which are the states of the separated dot system, shown in Fig.~5.
Let us compare now Eqs.~(\ref{d1}) with the Bloch rate
equations (\ref{c3}) for quantum
transport in coupled dots. We begin with the
equations for diagonal density-matrix elements, $\beta=\alpha$.
The first term in Eq.~(\ref{d1}) is zero, since $E_\alpha=E_\beta$.
The first term in  Eq.~(\ref{c3a}) and the second term in
Eq.~(\ref{d1}) have the same
form and describe the coupling with non-diagonal matrix elements
generated by one-electron hopping between isolated states.
The remaining terms in both equations look differently.
However, one can easily realize that in the case of coupled dots the only
possible transitions for diagonal matrix elements
are those corresponding to $\gamma =\alpha$.
Then the third term of Eq.~ (\ref{d1}) becomes
\begin{equation}
\sum_{\delta}
2\pi\rho\Omega_{\alpha\to\delta}\Omega_{\delta\to\alpha}
\sigma_{\alpha\alpha}=
\sigma_{\alpha\alpha}\sum_{\delta}\Gamma_{\alpha\to\delta}.
\label{d2}
\end{equation}
It coincides  with the second (``dissipative'') term
of the Bloch rate equation (\ref{c3a}).
The last term in Eq.~(\ref{d1}) describes the reverse process, i.e.
all possible conversions of the states $\gamma$,  $\delta$
into the state $\alpha$. Again, this term turns into the third term of
Eq.~(\ref{c3a}) for   $\gamma =\delta$.

The same relation can be traced for the non-diagonal
density-matrix elements. We find that the third term of Eq.~(\ref{d1}), which
describe all possible decays of the states $\alpha$
and $\beta$, reproduce the third term in Eq.~(\ref{c3b})
for $\gamma=\beta$ and $\delta=\alpha$, respectively. The fourth term
in Eq.~(\ref{d1}) describes the reverse process, i.e.
the conversion of $\sigma_{\gamma\delta}\to\sigma_{\alpha\beta}$
produced by two consecutive hoppings through continuum state
media.  In particular, if $\Omega_{\gamma\to\alpha}=\Omega_{\delta\to\beta}$,
these terms reproduce the last term of Eq.~(\ref{c3b}), where
$\Gamma_{\gamma\delta\rightarrow \alpha\beta}=
2\pi\rho\,\Omega_{\gamma\to\alpha}\Omega_{\delta\to\beta}$.

\section{summary}
In this paper we derived quantum rate equations, which provide
the most simple and transparent way for a description of the both
coherent and incoherent electron transport in quantum dot systems.
In the beginning we considered
a system of two quantum dots linked by a ballistic channel and
connected with the emitter and collector reservoirs.
Starting with the many-particle wave function
and integrating out the continuum states,
we have obtained the equations of motion for the density
submatrix of the two-dot system in the occupation number representation.
In spite of the quantum dots might be away one from another,
we found that the diagonal density-matrix elements are still
coupled with the non-diagonal density-matrix elements, similar
to Bloch equations for double-well  systems.
The essential difference, however, is that
the diagonal matrix elements in the Bloch equations are
coupled with the {\em imaginary} part
of the non-diagonal density-matrix elements, while
for the separated wells the corresponding terms
are coupled with the {\em real} part of the non-diagonal
density-matrix elements.

As expected, the coupling between diagonal and
non-diagonal matrix elements generated quantum coherent effects
in electron transport. For instance, we found that due to
destructive quantum interference the dc current in separated dots
is reduced when the barrier penetrability increases.
This effect would survive even if the dots are largely separated,
providing that they are linked
by ideal one-dimensional ballistic channels.

In spite of the quantum coherence effects found in dc current, the same
system of separated dot, detached from the emitter and
collector reservoirs does not show any quantum oscillations.
It is quite different from a
coupled-dot system with aligned levels,
in which an electron oscillates between the dots.

The rate equations for two separated quantum dots
are extended to a general case of multi-dot system,
in which the dots are either directly coupled, or interconnected
via ballistic channels.
These new rate equations generalize the
well-known Bloch equations, describing time-evolution of the
density-matrix of coupled
multi-well systems in the presence of a dissipative media.
We thus expect that the applicability of
our generalized rate equations is not restricted by
quantum transport in multi-dot system only,
but these equations would be very useful for
various physical problems where quantum coherence and classical
dissipation effects do interplay.
\section{Acknowledgments}
I thank Yu. Nazarov for valuable discussions.

redpajama_set_name: RedPajamaArXiv


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Chris Goulding (Launceston, Tasmania, 24 de octubre de 1988) es un jugador de baloncesto australiano que actualmente juega en el Melbourne United.

Biografía
Goulding nació y creció en Launceston, Tasmania pero se mudó a Queensland cuando era aun un adolescente, donde era alumno del Brisbane State High School.

Carrera profesional

NBL
Goulding empezó su carrera en la NBL como un jugador de desarrollo en los Brisbane Bullets durante la temporada 2006–07 NBL season. En 2008, él se unió a los Perth Wildcats para la temporada 2008–09.

En junio de 2009, Goulding firmó con los Gold Coast Blaze y se reunió con su entrenador en los Bullets, Joey Wright.

En agosto de 2012, siguiendo la decisión de los Gold Coast Blaze's de retirarse de la NBL debido a los problemas económicos que atravesaba el club, Goulding firmó con los Melbourne Tigers para la temporada 2012–13. En el 2012 NBL All-Star game disputado en el Adelaide Arena el 22 de diciembre de 2012, Goulding fue el máximo anotador del partido con 24 puntos. Esto, unido al resto de sus estadísticas del partido, le valieron el premio de MVP del partido.

El 19 de abril de 2013, Goulding renovó su contrato con los Tigers por otros dos años.

En julio de 2013, Goulding se unió a los Cleveland Cavaliers para la NBA Summer League.

El 9 de marzo de 2014, Goulding marcó su récord personal con 50 puntos en un partido, tras jugar los 40 minutos completos de la victoria de su equipo 92-82 frente a los Sydney Kings. Esto lo convirtió en el primer jugador en lograr los 50 puntos en un partido jugando los 40 minutos.

En julio de 2014, Goulding se unió a los Dallas Mavericks para la NBA Summer League.

Europa
El 22 de julio de 2014, Goudling firmó un contrato de un años de duración con el CAI Zaragoza de la Liga ACB.

Selección nacional
En verano de 2021, fue parte de la selección absoluta australiana que participó en los Juegos Olímpicos de Tokio 2020, que ganó la medalla de bronce.

Referencias

Enlaces externos
Perfil en Eurobasket.com
Ficha en ACB.COM

Baloncestistas de Australia
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NEW CHAMBER MUSIC
ABHANDENGEKOMMEN
Fantasia Circolare for Saxophone Quartet
Instrumentation: SATB or AATB Saxophone quartet
Duration: 9'
Written in February 2014 in Karlsruhe, Germany; commissioned by Dr. Alan Theisen, director of the Mars Hill Saxophone Quartet.
The Saxophone was my first serious instrument, beginning in the 5th grade. I, at least three of my siblings, and two of my nephews all learned on the same Bundy II Alto that still lives at my brother's house in Virginia. The Saxophone is part of my musical identity - when I sight sing, I still unconsciously do Sax fingerings. My earliest compositions were for Sax quartet. Written shortly after graduating from high school, I arranged show tunes and hymns, weaving them all together into some kind of strange, lovely, and super-serious Sax-mash.
This is the first work for Saxes since those early experiments. Prominent are extended techniques including key clicks and mixed, wafting multiphonics. For large sections of the work, however, I fell deep into the parts of the horn that I love - rich, organ-like homophonic passages, crisp, crackling compound time writing, and weaving, seemingly unending lyricism.
The title reflects the form of the piece. Relying on three broad, contrasting passages, the piece moves in a circular fashion. Each sequence is revisited in an exact order, one leading to another, with slight variation throughout. It is possible to begin anywhere in the work, play to the printed end, go back to the printed beginning and finish where you originally started. Each passage stands on its own and leads to the next in a never-ending circle.
© Copyright 2020 Frank Pesci

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Delegation of AltSTU represented by rector Andrei Markov, vice-rector for strategic development, research and international cooperation Evgeny Zryumov and head of Russian Language Chair Natalia Baryshnikova visited Chinese Universities.
The purpose of the visit is the prolongation of cooperation agreements with Dalian Neusoft University of Information (Dalian) and Yanshan University (Qinhuangdao).
According to the rector AltSTU Andrei Markov, universities have a long-standing mutually beneficial cooperation.


 Book Coblentz Inn Boutique Hotel (Port Of Spain) - 2019 PRICES FROM A$153!
Featuring a restaurant, Coblentz Inn Boutique Hotel offers accommodations in Port-of-Spain. Free WiFi access is available. Rooms all offer a unique experience, with different themes of the rich history and culture of Trinidad. They are all fitted with a private bathroom with free toiletries, air conditioning, a desk, a safety deposit box, a TV with cable channels, a mini-bar, a microwave and a coffee maker. The Inn, beautifully decorated with local elements, offers a 24-hour front desk, a terrace and a bar. Free parking is available and continental breakfast is offered. Piarco Airport is 12 mi away; shuttle service can be arranged for an extra fee.


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Case studies, Latest news / blogs
Rich history, exciting future
Kate Swade
Shared Assets recently designed and delivered a workshop for senior National Trust and Derbyshire County Council officers about the future of Elvaston Castle and Country Park. Read more about this fascinating site and our approach here.
The tale of Elvaston Castle and Country Park is perhaps a familiar one to many local authorities. The very definition of a complicated heritage asset, Elvaston is 130 hectares of Country Park, with a grade II* listed castle at its centre, and a number of associated buildings throughout the Park and listed gardens, as well as housing a local nature reserve.
Elvaston has a rich history and glamourous associations; dating from 1633, and extended in the early 19th century. Home of a celebrated love story (the fourth Earl of Harrington created a private "gothic paradise" for his actress wife), it was later part of the film set for Ken Russell's Women in Love.
The gardens were designed and laid out by William Barron over 20 years from 1830, and were revolutionary at the time, and remain significant today.
Derbyshire County Council bought the site from the Harrington family in 1969 and opened up the gardens as a Country Park the following year.
Fast-forward 44 years, and Elvaston Country Park is now a much loved local open space, full of biodiversity, heritage and very happy dogs on long and energetic walks. It is the closest open space to the east side of Derby, and has around 350,000 visits a year.
It is not currently a sustainable place, though. The castle is all but closed to the public, with a significant backlog of repairs. The visitor infrastructure is tired, the drainage in need of substantial investment. Derbyshire County Council, like other local authorities, is feeling the effect of substantial budget cuts and can no longer continue to invest in Elvaston to the extent it needs.
Active community groups, including the Friends of Elvaston Castle, have made it very clear that selling the park and grounds to the private sector is unacceptable, and highlighted the value and sense of ownership that local people have over this substantial and important green space.
The council has been working with the National Trust to develop a revised vision for the site and has agreed that setting up a new "single management body" for the estate could offer the best future. The Trust and the council have done substantial work, including lots of community consultation, and developed six "guiding principles" for any new management body, including the protection of public access, a focus on stewardship and financial sustainability, and community involvement. Crucially, the county council will retain the freehold.
But how to move from the current situation to the future vision? What should this new body look like? Who should be in control, and why? These are complicated questions, and are likely to be answered differently by different people depending on where they currently sit, what responsibilities they have and where they see potential. Just the kind of complicated questions that we like!
Shared Assets was brought in to facilitate a workshop for senior members of the council and National Trust, along with some local experts. Our brief was to help them inform the development of this new body for Elvaston.
Shared Assets brought a wealth of knowledge of different models for managing 'community' assets, and the thinking and steps that are needed to get there. They developed and led a well-structured workshop, with plenty of room for debate and discussion. Nick Sellwood, National Trust "Future Elvaston" Project Manager.
It was brilliant to work with such an informed group of people, all coming from different perspectives, but united in a desire to see what was best for Elvaston. We designed an interactive and participative afternoon, taking things back to first principles and building up from there a picture of what functions any new organisation would be taking on, what skills and attributes it needed and how it might be organised to fulfil these functions the best.
We spent some time considering the difference between the management and the governance of the site, and the roles of democracy, accountability and enterprise in the future of the site. We had pre-prepared some sample organisational structures for participants to consider and deconstruct, and finished off the day with an open debate and discussion on the next steps.
These next steps are now for the council to consider, but momentum is gathering and we will be watching this space to see what happens next!


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mother Tag
Jamie's Success
by	Brian Covey
Education/Training, FSET Successes, Jobs, Other, Success Stories, Wisconsin Works Successes
Jamie came to FSC looking for some support building up her employable skills, and ultimately to help her reach her goals of working in...
Cecilia's Motivation for Success
With a baby at home and another on the way, Cecilia couldn't pull 12-hour shifts with mandatory overtime anymore. She'd worked in production for...
News, Success Stories, Wisconsin Works Successes
When Latoyia learned that she was about to become a mom, she went straight into prep mode. She had a job that barely paid...
Amanda Pursues Her Passions
Education/Training, News, Success Stories, Wisconsin Works Successes
Congratulations to Amanda, who recently started her new job at Americollect! We know she's going to do great work, because we've seen her...
Buckling Down to Work Up
"I did everything Angie said, and I got the job!" Dawanna announced to us the minute she was hired at Spectrum. As soon as she...
Julia's Success
News, Success Stories, WIOA Successes, Wisconsin Works Successes
Please join us in congratulating Julia, a young mother who won't let anything stop her from giving her newborn the best start possible. This...
Necole Finds Family Here
For Necole, family is one of the most precious things in the world. Her mother has been her key ally throughout life, and she...
Anyelika's Dreams Now Possible
"I swear this girl was not letting anything stop her," says Christa, a case manager with Wisconsin Works. She's talking about Anyelika, the amazing...
Kate's Pride and Joy
In July 2017, Kate was sitting in her parent's living room, rocking her newborn baby, and thinking about her future. She had already started...
Rosa's Leading the Way
As the mother of seven children, Rosa has been a full-time chef, nurse, coach, driver, personal secretary, teacher, and supervisor. When she decided to...
Off to a Running Start
Harmony knows that a strong support system is key to thriving as a parent, particularly one who's going for her degree. Her family has...
Danima's Success
With three young boys to love, Danima had no time to lose. She came to FSC to jump out of unemployment and get her...
Jacole's Unrestricted Success
"It means a lot to me to have a good job with a great company!" exclaims Jacole. After a back injury left her out...
Makhia's Expecting Great Things
Imagine studying for your finals with one youngster at home and another on the way. That would be enough to make anyone gulp, but...
With five children at home, Tranace is a pro at management. Even so, when she decided to switch careers, she needed some help juggling...
Lifeline to Lasting Success
When complications left Gabrielle unable to work during pregnancy, Wisconsin Works became a lifeline of support. Gabrielle spent the next few years at home to...
A Little Energy, A Lot of Love
With a young child at home and a baby girl on the way, Samantha knew her future was going to take a lot of...
From Compassion to Career
LaToya is a soft-spoken, caring mother who always puts her family first. When she enrolled in Wisconsin Works, she was just about ready to...
Amanda's First Year as a Mom
"Amanda and I love to share all our mommy highs and lows each time we meet," says Der Lo, a Career Planner with WIOA....
Unexpected Change Yields Big Rewards
"School shopping for the kids this year has been worry free," says Debra with relief. " I am able to actually pay bills and...
Karen Graduates with Honors
Once her children reached school-age, Karen was ready to go back to school herself. "My husband had a good job," she says, "but we...
Roseatta Inspires Success
"I have a fight inside me like no other," says Roseatta. Homeless and struggling to overcome a traumatic upbringing in foster care, she can...
Jenni Finds Her Hope
Jenni loved being a server, but when she fell sick, her doctor said she'd need to find a new career. Thankfully, she knew exactly where to...
Gabby Plans to Save Lives
"Being a Nursing Assistant is something important to me," says Gabby. "It is such a rewarding feeling." Before joining Wisconsin Works, Gabby did not...
Andrea's Inspiration
"As a single mother, it was scary losing my job," Andrea admits. Months behind on rent, she felt like her world had been turned...
Super-Nurse
Caring full-time for her newborn gave Jessica time to imagine a wonderful future for her family. In it, she helped people face life-and-death battles...
A Great Future
Family has always been Allyssa's top priority. Studying full-time, maintaining the household, or financial concerns would wait until the children were asleep. She knew...
Cassie's In the Game
Cassie has always been the mom you can depend on. She had been steadily employed until last October, when suddenly she found herself looking...
Sierra wanted to make the best life possible for her newborn girl when she joined Wisconsin Works. Determined to get back to school and...
Career Change Success
Lisa is a young mother with big plans for her future. Capable and driven, she became her family's breadwinner while still earning her high...
Bobbie's Course of Action
Bobbie's passion for helping others has kept her going through the obstacle course of life these past few years. She became a skilled juggler...
Believing in Herself
If you're a trendy shopper, you might see Tiana at her new job at Forward 21. Confident and happy, she says that this job...
This is Wendy's Year
Every strong woman has a story about the weights she had to lift from her shoulders. This year, Wendy has lifted four of life's greatest...
Kami's Story
There are options for people in every circumstance, and knowing what they are can make all the difference. A single mom, Kami's job barely...


I have always been rather fascinated with the ruins scattered around the southwest, built a thousand or more years ago by a culture we know as the Ancestral Puebloans (once called the Anasazis). You almost can't hike a canyon in southeastern Utah or southwestern Colorado without finding some ruin tucked away high up on a cliff or perched precariously on the edge of a sandstone precipice. I think it's the imagining of what their life must have been like that piques my curiosity. Trying to picture them building these carefully crafted structures which still bear their fingerprints in many cases; picturing women grinding corn while the kids play nearby; picturing them sitting in the warm winter sun of their south-facing veranda with the fresh fluffy snow of a recent storm blanketing the landscape around them. All of these images fill my mind as I sit and admire the ruins of once spectacular structures. A recent trip to Hovenweep and Canyon of the Ancients was no exception. I admire these places because, as the light seeps through the window in this picture, images of the past seeps through these ruins in my mind. Each ruin is in itself, a window to the past.
Sunrise at Inspiration Point at Bryce Canyon National Park as the beautiful clouds began to break up and scatter over the park.
When the light of the morning hit the rocks on this lovely Sunday morning, we couldn't help but stop to take in the brilliant glow. It was as if the light came from within the towers themselves and the rest of the canyon was glowing from their brilliance. This image was selected by John Fielder as one of 50 out of 450 applicants to be displayed in an exhibition at the Sangre de Cristo Arts Center (Pueblo, CO) entitled "America's Parks: A Centennial Celebration". It will will be displayed in the Hoag Gallery June 7 through August 7, 2016.
When the light of the morning hit the rocks on this lovely Sunday morning, we couldn't help but stop to take in the brilliant glow. It was as if the light came from within the towers themselves and the rest of the canyon was glowing from their brilliance. This image was recognized as one of the top 250 (out of thousands of international entries) in the 2015 Denver Audubon Society's "Share the View" photo contest.


Pen Pals & Picture Books: This is one 'a peeling' activity!
This is one 'a peeling' activity!
I know, stop with the lame titles-- sorry, it really is lame. However, please keep reading because this book is anything but lame! Also, let me apolgize in advance for what I know will be whacked out spacing! Blogger does not play well with others sometimes!
Okay, I am always starting posts with, "This book is_____!" (Insert positive adjective in blank.) But really, this one is so so creative! You. Will. Love. It.
First, you'll need some...let's call them "features." We used apples as bases (and I threw some celery on there too because I thought it it might be cool.) You could really use anything your kids will eat. Also, I went primarily sweet here, but you could do some really cool savory stuff too. Oh! If my kids would only eat potatoes! You could make something really fun with baked potato slices and all their fixins'-- or sweet potatoes with little marshmallow faces! Anything goes, really!
A couple side notes-- interspersed with photos (1)This was too many options, I wanted to put a lot of stuff out to see what they'd do and so you'd get some ideas. But really, you just want a couple different shapes and textures. (2)I put the peanut butter and nutella as "glue" which also wasn't totally necessary, but sticky fun. We have a couple of nutella freaks in this house though, and so the sight of a big glob of chocolate-ee hazelnut-ee goodness was enough to start a feeding frenzy. Second step, start building. You will find that the pieces can be anything. I originally put the fruit strips pieces on to use for hair, but it turns out they make good mouths, legs, headbands, uni brows. And, those little celery pieces, we found they could be pretty much anything.
As you can see, we had two schools of thought. One child....I need to put as much stuff on as I can so I can eat more. (Above is a turtle, like in the book.) The other child... who needs to put all this stuff on when we can just eat it, except the peel, which I ate off for her before hand.
Third: create, munch, and chat. We chatted about what kind of expressions our faces had-- that banana mouth below, is it surprised? scared? We talked about celery and peanut butter and the delicious combination they are, oh and we talked a lot about passing the nutella. Sweet and simple fun!
PS- Saxton and Joost (yes, we're on a first name basis) have a lot of other great books sure to been seen on this very blog in the future, maybe even with more tasty activities. Really, you've got to check them out.
ps- that turtle is so cool! pss oldest son just came in, looked at computer and said "wow! can we do that? I could make a catfish!"
A catfish! ha ha why is it so funny to me that a kids favorite animal is a catfish? maybe because it is hilarious. MORE fun food! I want to come over when you do this! Also nutella...oh wow it is my manna sent from the heavens.


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Drew Pearson, Larry Allen, Charles Haley To Be Inducted Into Cowboys Ring Of Honor
Jerry Jones granted Drew Pearson his overdue place in the Ring Of Honor, announcing that he would be inducted along with Larry Allen and Charles Haley in November.
By Brett Perryman Aug 19, 2011, 4:45pm CDT
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Jerry Jones announced Friday afternoon that Drew Pearson, Larry Allen and Charles Haley would be inducted to the Cowboys ring of honor. The ceremony will be held on November 6, when the Cowboys face the Seattle Seahawks.
Jones last inducted new members in 2005, when he brought in Troy Aikman, Emmitt Smith and Michael Irvin together. The new trio is less connected. Pearson was named to the 1970's all-decade team in an era where receivers didn't pile up the numbers that their counterparts today do. He was an unstoppable force in big games for the Cowboys, though, was first team all-pro three times, and caught the legendary hail mary pass from Roger Staubach.
Haley and Allen were members of Jones' Cowboys, with Haley playing a key role in all three Super Bowl winning teams, and Larry Allen winning one in 1995. Haley owns a total of five Super Bowl rings, most in NFL history, thanks to his three with the 49ers before being dealt to Dallas before the 1992 season. Allen is regarded by many as the best offensive lineman to play for the Cowboys. He was named to six first team all-pro teams and was on the all decade team for both the 1990's and 2000's. Jones said Friday that Allen was universally regarded as the best player on the Cowboys for most of his time with the team, including the years from 1994 to 1999 where the Triplets led the team.


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The Enemy of My Enemy
By W.E.B. Griffin and William E. Butterworth IV
Read by Scott Brick
Part of A Clandestine Operations Novel
Category: Military Fiction | Suspense & Thriller
Category: Military Fiction | Suspense & Thriller | Audiobooks
Nov 26, 2019 | 416 Pages
Dec 11, 2018 | 570 Minutes
About The Enemy of My Enemy
Special agent James Cronley Jr. finds that fighting both ex-Nazis and the Soviet NKGB can lead to strange bedfellows, in the dramatic new Clandestine Operations novel about the birth of the CIA and the Cold War.
A month ago, Cronley managed to capture two notorious Nazi war criminals, but not without leaving some dead bodies and outraged Austrian police in his wake. He's been lying low ever since, but that little vacation is about to end. Somebody–Odessa, the NKGB, the Hungarian Secret Police?–has broken the criminals out of jail, and he must track them down again.
But there's more to it than that. Evidence has surfaced that in the war's last gasps, Heinrich Himmler had stashed away a fortune to build a secret religion, dedicated both to Himmler and to creating the Fourth Reich. That money is still out there in the hands of Odessa, and that infamous organization seems to have acquired a surprising–and troubling–ally.
Cronley is fast finding out that the phrase "the enemy of my enemy is my friend" can mean a lot of different things, and that it is not always clear which people he can trust and which are out to kill him.
Special agent James Cronley, Jr., finds that fighting both ex-Nazis and the Soviet NKGB can lead to strange bedfellows, in the dramatic new Clandestine Operations novel about the birth of the CIA and the Cold War.
A month ago, Cronley managed to capture two notorious Nazi war criminals, but not without leaving some dead bodies and outraged Austrian police in his wake. He's been laying low ever since, but that little vacation is about to end. Somebody–Odessa, the NKGB, the Hungarian Secret Police?–has broken the criminals out of jail, and now he must track them down again.
Cronley is fast finding out that the phrase, "the enemy of my enemy is my friend," can mean a lot of different things, and that it is not always clear which people he can trust, and which are out to kill him.
Also in A Clandestine Operations Novel
Also by W.E.B. Griffin, William E. Butterworth IV
See all books by W.E.B. Griffin, William E. Butterworth IV
About W.E.B. Griffin
W.E.B. Griffin was the author of seven bestselling series: The Corps, Brotherhood of War, Badge of Honor, Men at War, Honor… More about W.E.B. Griffin
About William E. Butterworth IV
William E. Butterworth IV has been an editor and writer for more than 25 years, and has worked closely with his father… More about William E. Butterworth IV
Published by G.P. Putnam's Sons
Nov 26, 2019 | 416 Pages | 4-3/16 x 7-1/2 | ISBN 9780735213074
Published by Random House Large Print
Dec 24, 2018 | 464 Pages | 6-1/8 x 9-1/4 | ISBN 9781984827586
Dec 11, 2018 | 400 Pages | 6 x 9 | ISBN 9780735213067
Dec 11, 2018 | 400 Pages | ISBN 9780735213081
CD | $45.00
Dec 11, 2018 | 570 Minutes | ISBN 9780525640301
People Who Read The Enemy of My Enemy Also Read


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Watch Asura Cryin'
Asura Cryin' is an anime series. It is about a teenage boy who enters high school. This teenage boy is named Tomoharu Natsume. Tomoharu is tormented still by the death of his friend Misao Minakami. He had known Misao sense he was a child. Tomoharu's friend Misao had died three years earlier in a plane crash. Tomoharu barely survived the plane crash. He moves, starts at a new high school, and wants to enjoy life on his own.
He is having fun in his carefree high school life, until a girl shows up. This pretty girl is named Shuri Kurosaki. She has trunk for Tomoharu that she says is from his brother Naotaka Natsume. She says Naotaka asked her to deliver the trunk to Tomoharu. A girl named Kanade Takatsuki is after the trunk that Tomoharu's brother sent to him. Kanade is dressed like a shrine maiden, and the trunk has secrets to the world they live in. Kanade is after the secrets this trunk holds within it. It is Tomoharu's job the protect the trunk, and keep Kanade from getting her hands on it.
Thursday 9:00 AM et/pt on TVTM
2 Seasons, 39 Episodes
Cast: Miyu Irino
Watch Full Episodes of Asura Cryin'
The Gears of Choice ...
Despair at the Diver...
My Feelings for You ...
Remaining Life Scat....
Adjoined Death and P...
Future Without Memor...
Asura Cryin' Full Episode Guide
All Sources Free Sources Custom
Select the sources you want displayed in the episode guide.
26 Crunchyroll Episodes
Season 2 1 full episode
Episode 26 - The Gears of Choice
Tomoharu and the others fight desperately against Tokiya's determination to take back Aki at the cost of all other worlds. However, even with his powered up Kurogane-kai and daughter Persephone, Tomoharu is powerless against Tokiya's four Asura Machinas. The situation turns dire when he loses the igniter to Tokiya.
Episode 25 - E 25
Unable to wait for the central vortex to stabilize, Tokiya uses Hagane, Shirogane, Bismuth, and Rhodonite simultaneously to create a gravitational gate. With the second world already vanishing, Yo, Reishiro, Rikka and the others make preparations for their final battle. Amidst the action, still in shock from losing her sister Yukari, Shuri reveals the truth. She is in fact Yukari and has been fighting to save her sister Shuri since she became a burial doll. Upon seeing Yukari's desperation, Tomoharu promises to save her this time.
Tomoharu returns to the second world as an Asura Cryin' due to his contract with Kanade. However, with the GD's in a state of ruin and the absence of Shuri, Tomoharu realizes he lacks the power to defeat Tokiya, a man determined to create the third world even if it destroys the second. As Tomoharu prepares for the gate to open in 10 days, Reishiro and Yo suddenly appoint him president of the third student council. Still in shock from the surprising turn of events, Tomoharu hardens his resolve when Yo tells him to do what he must.
The vanishing world and the stray daughters turn out to be the product of the gravity reactor experiment's failure. Reunited with Misao in the research facility that triggered Armageddon, Tomoharu descends into the dimensional diving chamber where Naotaka is maintaining Kurogane. Among the physical bodies of burial dolls without souls, Tomoharu spots the first world's Kanade.
In the battle against Takaya, the new Asura Cryin', Tomoharu, Kanade, and Ania are sucked into the whole in space opened by Hagane's powers. On the other side, a solitary Tomoharu wakes up at the Meiotei. His new home is a world much like his own but with subtle differences. Wandering city streets in a confused state, Tomoharu witnesses a daughter attacking a human. The daughter is easily defeated by a mysterious blonde girl who seems to know something about Tomoharu's situation.
Immediately after learning Naotaka is actually Tomoharu from the first world and this world's Tomoharu died in that plane crash three years ago, Tomoharu and the other members of the Science Club find themselves under arrest by the Student Union. During the Student Union's investigation, the mysterious masked woman strikes again and makes off with Ania. Tomoharu and the others seek Toru for help, but their assailants seem to have predicted their actions and pay a visit to the Third Student Council first. With no choice but to turn Tomoharu over to their attackers, the Science Club finds an unexpected ally in the GD assigned to kill Hiwako, Yo.
Rakurowa High School is buzzing with anticipation for the upcoming Sports Festival. Thanks to Higuchi's scheme, Tomoharu and Kanade are paired as a team in the three-legged race, but the two quickly find themselves in trouble. When a coagulant is spilled on the ribbon tying their legs, the two are forced to spend all day and night attached to each other! The next day, Tomoharu accompanies Kanade to her mother's grave on the anniversary of her death, but...
Tomoharu is attacked in the ruins beneath Kuremizaki by GD Kyomu. Unable to call on Kurogane, Tomoharu loses Nia to Kyomu. Tomoharu awakens to find himself in the company of Kagakagari, where he learns the reason Christina and Kotori are helping Kagakagari and the terrible cost the trio is paying. The group sets out for the technology to free burial dolls, but...!
Tomoharu and the others spend their summer vacation working at a seaside pension. Tomoharu's heart is heavy with the secret of the burial doll, but that doesn't stop Higuchi and An from scheming to hook up Kanade and Tomoharu by sending them to legendary love spot Kuremizaki fortress observatory under the guise of a test of courage. Meanwhile, Misao's unhappy that Tomoharu is keeping a secret from her.
Since the disappearance of Aine, Tomoharu's head is full of thoughts of saving Misao. Seeking his brother Naotaka who started everything, Tomoharu goes to Toru to find out how to reach Naotaka. Toru surprises Tomoharu when she asks him to convince Science Club president Tokiya to go to school in exchange for her help.
Tomo's misunderstanding gets the Science Club in trouble once again, and as punishment, the members are made to clean the school pool. However, Tomoharu and the others follow a suspicious koala and find themselves in a labyrinthine underground tunnel system. Led by the koala, Tomoharu and the others find themselves at the Rakuroa ruins...!
Season 1 25 full episodes
Episode 26 - Episode 26 - The Gears of Choice
Episode 25 - Despair at the Diverging Past and Cau...
"Unable to wait for the central vortex to stabilize, Tokiya uses Hagane, Shirogane, Bismuth, and Rhodonite simultaneously to create a gravitational gate. With the second world already vanishing, Yo, Reishiro, Rikka and the others make preparations for their final battle. Amidst the action, still in shock from losing her sister Yukari, Shuri reveals the truth. She is in fact Yukari and has been fighting to save her sister Shuri since she became a burial doll. Upon seeing Yukari's desperation, Tomoharu promises to save her this time."
Episode 24 - My Feelings for You Destroy the World
"Tomoharu returns to the second world as an Asura Cryin' due to his contract with Kanade. However, with the GD's in a state of ruin and the absence of Shuri, Tomoharu realizes he lacks the power to defeat Tokiya, a man determined to create the third world even if it destroys the second. As Tomoharu prepares for the gate to open in 10 days, Reishiro and Yo suddenly appoint him president of the third student council. Still in shock from the surprising turn of events, Tomoharu hardens his resolve when Yo tells him to do what he must."
Episode 23 - Remaining Life, Scattered Life
"The vanishing world and the stray daughters turn out to be the product of the gravity reactor experiment's failure. Reunited with Misao in the research facility that triggered Armageddon, Tomoharu descends into the dimensional diving chamber where Naotaka is maintaining Kurogane. Among the physical bodies of burial dolls without souls, Tomoharu spots the first world's Kanade."
Episode 22 - Adjoined Death and Peace
Episode 21 - Future Without Memories
"In the battle against Takaya, the new Asura Cryin', Tomoharu, Kanade, and Ania are sucked into the whole in space opened by Hagane's powers. On the other side, a solitary Tomoharu wakes up at the Meiotei. His new home is a world much like his own but with subtle differences. Wandering city streets in a confused state, Tomoharu witnesses a daughter attacking a human. The daughter is easily defeated by a mysterious blonde girl who seems to know something about Tomoharu's situation."
Episode 20 - Moment of Annihilation, Darkness of E...
"Immediately after learning Naotaka is actually Tomoharu from the first world and this world's Tomoharu died in that plane crash three years ago, Tomoharu and the other members of the Science Club find themselves under arrest by the Student Union. During the Student Union's investigation, the mysterious masked woman strikes again and makes off with Ania. Tomoharu and the others seek Toru for help, but their assailants seem to have predicted their actions and pay a visit to the Third Student Council first. With no choice but to turn Tomoharu over to their attackers, the Science Club finds an unexpected ally in the GD assigned to kill Hiwako, Yo."
Episode 19 - The Truth Dedicated to You
Episode 18 - The Sad Connection between Magic and ...
"Rakurowa High School is buzzing with anticipation for the upcoming Sports Festival. Thanks to Higuchi's scheme, Tomoharu and Kanade are paired as a team in the three-legged race, but the two quickly find themselves in trouble. When a coagulant is spilled on the ribbon tying their legs, the two are forced to spend all day and night attached to each other! The next day, Tomoharu accompanies Kanade to her mother's grave on the anniversary of her death, but..."
Episode 17 - The Cost of Calling the Sacrifice's N...
"Tomoharu is attacked in the ruins beneath Kuremizaki by GD Kyomu. Unable to call on Kurogane, Tomoharu loses Nia to Kyomu. Tomoharu awakens to find himself in the company of Kagakagari, where he learns the reason Christina and Kotori are helping Kagakagari and the terrible cost the trio is paying. The group sets out for the technology to free burial dolls, but...!"
Episode 16 - Free and Restricted Choices
"Tomoharu and the others spend their summer vacation working at a seaside pension. Tomoharu's heart is heavy with the secret of the burial doll, but that doesn't stop Higuchi and An from scheming to hook up Kanade and Tomoharu by sending them to legendary love spot Kuremizaki fortress observatory under the guise of a test of courage. Meanwhile, Misao's unhappy that Tomoharu is keeping a secret from her."
Episode 15 - The Burial Doll Eaten by the Demon
"Since the disappearance of Aine, Tomoharu's head is full of thoughts of saving Misao. Seeking his brother Naotaka who started everything, Tomoharu goes to Toru to find out how to reach Naotaka. Toru surprises Tomoharu when she asks him to convince Science Club president Tokiya to go to school in exchange for her help."
Episode 14 - The Disappearing World and the Ruins ...
"Tomo's misunderstanding gets the Science Club in trouble once again, and as punishment, the members are made to clean the school pool. However, Tomoharu and the others follow a suspicious koala and find themselves in a labyrinthine underground tunnel system. Led by the koala, Tomoharu and the others find themselves at the Rakuroa ruins...!"
Episode 13 - Mark of the Cursed Criminal
Having failed to take the igniter by force, Kagakagari takes Rakurowa High School's sophomores hostage. Still unsure of the igniter's purpose, Tomoharu slips out to make a deal with Kagakagari while Toru and Rikka quarrel, followed secretly by Kanade. Can they stop the Asura Cryin' out to destroy the alleged "second world?"
Episode 12 - Until the End of the World
Episode 11 - Together in the Ordinary and Extraord...
Successfully stopping Kagakagari's ritual that would have killed the city's demons, Tomoharu returns to regular school life. Aine's bouyant mother Kirie invites Tomoharu to a party to celebrate the anniversary of her daughter's death in cheerful fashion. Much to everyone's surprise, Kirie describes the late Aine as boisterous and mischievous, and Reishiro and Reiko's pasts are revealed. With these thoughts on their minds, the party goes on...
Episode 10 - Connecting the Chains of Time
Having lost Ania to Asura Cryin' Kagakagari, Tomoharu and the others search the room of Kotori, a student teacher working at Rakurowa High School during Christina's stay in Japan. On the other side of the city, Ania is reunited with her sister, her body already vanishing from the strain of powering the Splitter, a machine used to separate burial dolls from their Asura Machina. Requiring the luck eater's abilities to power the Splitter, Kagakagari attempts to use Ania in place of her weakened sister...
Episode 9 - The Impossible Taboo
Episode 8 - The Youngest Daughter of the King of ...
Episode 7 - The Dream that Dispersed in the Heart...
With the installation of the stabilizer, Misao becomes visible to everyone. Seeing the new Misao adapting well to life as an ordinary school girl, Tomoharu reflects on the past. At the same time, Yo and Shuri think back to the events leading up to the accident and the day their peaceful lives came to an end.
Episode 6 - Sacrifice Beyond the Darkness
Shuri's memory is back, but Tomoharu is out with a cold. That doesn't stop former student council president and Kanto Student Union Guardian Dragoon Yo from coming after Shuri's stabilizer. Unable to hand over the stabilizer that will free a burial doll from its Asura Machina, Shuri and Kanade fight back. Armed with Shirogane's ability to cut through space, Yo is a formidable opponent. Trying to buy time until Tomoharu's recovery, Shuri holds Yo off. However, it seems the two share a personal history...
Episode 5 - Intersecting Hearts and Bodies
Episode 4 - Thoughts with Nowhere to Go
Episode 3 - The Shadow of Science
Misao disappears fighting Reishiro's Asura Machina Hisui, and Kanade is held captive. Filled with rage, Tomoharu tries to call on Kurogane again. With the silver trunk in hand, Tomoharu goes to confront the First Student Council. He is stopped by Reishiro's ghost Aine.
Episode 2 - Destruction of the Future
Without a clear understanding of the situation, Tomoharu becomes a handler of an Asura Machina. Taking a break from his life's new problems, Tomoharu goes to visit his landlord Shioizumi, a strange old man living in a room full of spirals. "I hope you can make the decision that's right for you." Confused by the old man's words, Tomoharu bumps into Kanade, who happens to live at the shrine. Tomoharu seeks answers to his questions about Misao, Kanade, and the Asura Machina, but...
Episode 1 - Asura Machina
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Season 6, Episode 10
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Asura Cryin' is currently available to watch free on Crunchyroll and stream on TVTM. You can also stream, download Asura Cryin' on demand at Crunchyroll online.


Joe Blogs: Hold your heads high!!
Tonight was supposed to be a double celebration, it ended up being only for one though but it was a good one. We were out to celebrate our friends birthday and were hoping for England to win the Rugby. It wasn't to be with the Rugby and even though we celebrated the Try that never was we soon turned our attention away from the sport that we don't really understand anyway onto celebrating the birthday. We started in Bromley and after the game moved onto Croydon. I was driving so it was a smooth journey other than trying to find somewhere to park. We spent our night in Walkabout and had it out to the cheese before departing at close on what we believed to be a shortcut. How wrong were we as we walked the long way around, I switched shoes with my girl and then wore my socks through before locating the car again and heading off in a race to beat morning!!


text:
Physicians take critical look at AJCC staging system, melanoma prognostication tool
Ilya Petrou, M.D.
, Cheryl Guttman Krader, BS, Pharm
Cheryl Guttman Krader, BS, Pharm
Cheryl Guttman Krader is a contributor to Dermatology Times, Ophthalmology Times, and Urology Times.
The latest melanoma staging system from the American Joint Committee on Cancer (AJCC) and a tool developed for predicting the clinical outcome of individual patients with localized or regional cutaneous melanoma (www.melanomaprognosis.org) have stirred debate among dermatopathologists.
International report - The latest melanoma staging system from the American Joint Committee on Cancer (AJCC) and a tool developed for predicting the clinical outcome of individual patients with localized or regional cutaneous melanoma (	http://www.melanomaprognosis.org/) have stirred debate among dermatopathologists.
According to Clay J. Cockerell, M.D., the frequency with which new recommendations on melanoma pathology reporting and prognostication have been issued by the AJCC in the past few years has created frustration for clinicians trying to remain current. He also questions the process by which the recommendations are developed, noting that the responsible committee is composed predominantly of surgeons and oncologists, whereas dermatologists and dermatopathologists, who are on the front line of making the diagnosis, were minimally represented.
Furthermore, retrospective studies often are used as an information source for developing the recommendations, and there is no vetting of the recommendations by the specialty societies that are important stakeholders prior to finalization, says Dr. Cockerell, director, division of dermatopathology, University of Texas Southwestern Medical Center, Dallas.
Focusing on the tool for melanoma prognostication, Dr. Cockerell says he questions its heavy weighting on histology.
"The impetus for the most recent changes to the AJCC staging system reflects a desire to provide more accurate prognostication for individual patients diagnosed with melanoma. However, despite what the committee would like, the microscope is simply not a crystal ball," Dr. Cockerell says.
"There are several problems with using melanoma histologic variables for prognostication, and it is also important to keep in mind that disease course and patient outcome are influenced by the individual's immune response plus other factors that are not taken into account by looking under the microscope," he says.
Outlining the shortcomings of the AJCC criteria for prognostication, Dr. Cockerell says that one of the problems is that the histologic variables cannot be applied to partial biopsies in many cases. In addition, several of the variables used, including Breslow thickness, ulceration, regression, and vertical growth phase, are subjective.
The simplified reporting of ulceration as either present or absent is also problematic, as it poses the potential for overcalling ulceration, a feature that places patients in a worse prognostic category, he explains.
"Identifying ulceration as absent or present does not allow differentiation between an advanced lesion that is necrotic and infarcted and a thin lesion that appears ulcerated as a result of external trauma," he says. "At least one study performed by Googe et al. (J Cutan Pathol. 2011;38(1):112-113) has shown that ulceration in thin melanomas is rare, so dermatologists should be wary if ulceration is reported in a thin melanoma, especially those less than 1 mm in thickness."


First up they've done up the interior and the new chairs I found very comfortable. The menu still comes on a blackboard. The music was a bit too much like lounge music but that's just down to personal taste no doubt.
I went with a starter and a main leaving the option open for a dessert.
I started with the Salade de Gesiers Confis et Lardon, oeuf poche et oignons rouges confis. I wasn't disappointed. The confis des oignons was tasty.
Now visually this looks quite nice but I wasn't really too sure what I was eating. The green puree was I thought a green pea puree but it may have been broccoli at least that's what I think we were told it was. It wasn't nice at all and reminded me of the time I sampled a green bean baby pot of food for Charlotte (it was one of the last pots we bought) I've had to look up the white root vegetable as well the crosne which I have to admit I'm not a fan of it didn't really add anything to the meal. The duck was beautiful … the wine sauce went very well with it.
There was a bowl of grey looking mash on the side I forgot to take a picture of. It didn't look too appetising but was amazing once I tucked in. Very heavy on the garlic though if that's an issue for you.
Val's Filet Mignon was Filet Mignon. She wasn't a fan of the veg and ended up leaving most. In all honestly I can't blame her. The Filet Mignon was nice though it's quite difficult to get filet mignon wrong. Again two different meals using pretty much the same base.
I opted to skip dessert and this is where Val made a mistake. She went with dessert and the caramelised apple with vanilla ice cream dessert.
The biscuit layer was WAY too crunchy. The apple .. well you can't really get an apple wrong (it's filled with caramel … maybe a bit overkill) The caramel was full of sugary lumps and it was all just WAY too sweet for my liking.
All in all I'll admit I miss the old restaurant and staff and I have some good memories from there. Val preferred the new decor. I preferred the old even if it was a bit dated. The old place seemed to have simpler dishes and with some of these it feels like they are trying a bit too hard.
We opted for a small bottle of sparkling water and just a jug of water. So I can't comment on their wine selection. This brought the total for lunch to I think 35€ a tad on the pricey side when it's not ticking all the boxes.
They have a small terrace outside which is right beside the tram. On a nice sunny day it's nice to watch the trams go by though as you can see in the first picture above it is rather close.