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\noindent
{\Large \bf A quantum-kinetic equation for the description of
            internal~ dynamics~ of~ multilevel atomic systems \\
moving through a target matter }

\vspace{.5cm}

\hspace{1.4cm}
O Voskresenskaya\footnote{Current address: Institut f\"ur Theoretische
Physik, Universit\"at Heidelberg, MPI f\"ur Kernphysik, D -
69029 Heidelberg, Deutschland; e-mail:
Olga.Voskresenskaja@mpi-hd.mpg.de}

\vspace{.5cm}


\hspace{1.5cm}\parbox{13cm}{
{\small Joint
Institute for Nuclear Research, Dubna, Moscow Region, 141980 Russia}
\par\bigskip

{\small {\bf Abstract.}
The quantum mechanical consideration of a passage of relativistic
elementary atoms (EA) through a target matter is given.  A
quantum-kinetic equation for the density matrix describing their
internal state evolution at EA rest frame is derived.\/}

%\par\bigskip {\small{\bf PACS.} 36.10.Gv, 36.10.-k, 11.80.Fv, 11.80.La,
%34.50.-s, 02.60.Cb\/}
}

\rm
\section{Introduction}

 For the interpretation of the data of DIRAC experiment
 \cite{dirac,nemen} which aims to measure the lifetime of hydrogenlike
EA consisting of $\pi^+$ and $\pi^-$ mesons ($A_{2\pi}$ atoms) one
needs to have the accurate theory for the description of internal
dynamics of the  $A_{2\pi}$ atoms moving through a target matter.

 During their passage through the target $A_{2\pi}$ (pionium atoms)
 interacts with target atoms that causes the excitation,
 deexcitation or ionization of the $A_{2\pi}$. To describe
 these variations of $A_{2\pi}$ internal states the authors of
 \cite{AT} proposed a set of kinetic equations for the probabilities to
 find the pionium atom in the definite quantum state at some distance
 from the point of $A_{2\pi}$ production.

 It is clear that such ``classical''  description is approximate
 because does not take into account the possible interference (quantum)
 effects. These last can be included in consideration only
 in the framework of density matrix formalism.


\section{Derivation of a density matrix kinetic equation}

The derivation of a kinetic equation for the density
matrix of fast atomic systems passing through a target
matter can be given at target rest frame \cite{voskr},
but more simple these equation can be obtained at
rest frame of the EA.

At this frame the target moves with the velocity $\vec v_0$ and the
electromagnetic field produced by target atoms is described by 4-vector
potential $A_{\mu}=(\Phi,\vec A)$, $\vec A=(\vec v_0/c)\Phi$.

The scalar potential $\Phi$ interacts with the charges of mesons and
the vector potential $A_{\mu}$ with their currents.  Because
the typical velocities of the particles forming EA are of
order $\alpha c\ll c$  ($\alpha$ is the fine structure constant),
we will neglect the term proportional to the current in the
Hamiltonian (see \cite{rel}).
%\newpage

Then internal dynamics of relativistic EA (later, for definiteness,
``of pionium atoms'') is described by the Schr\"odinger
equation
\begin{equation}
\label{eq:d1} i\frac{\partial\psi(\vec
r,t)}{\partial t}=H\psi(\vec r,t)
\end{equation} with the Hamiltonian of the form
\begin{equation} \label{eq:d2} H=H_{0}+V(\vec r,t),\quad
H_0=T+ V_{0}(\vec r)
\end{equation}
and
\begin{equation}
\label{eq:d3}
T=-\Delta/2\mu=-(d/d\vec r)^2/2\mu\,.
\end{equation}
Here, $V_{0}(\vec r)$  are the potential energy of pion-pion interaction and
$V(\vec r,t)$ is the potential energy of interaction between the
pionium and the target atom.

We will suppose that the positions of atoms inside the target are not
varied during the interaction of target with the pionium atom
(the so-called ``frozen'' target approximation).
Then
\begin{equation}
\label{eq:d4}
V(\vec r)=e\sum_i[\Phi\left(\vec r_i(t)-\vec r/2\right)-
\Phi\left(\vec r_i(t)+\vec r/2\right)]\,,
\end{equation}
\begin{equation}
\label{eq:d5}
\vec r_i(t)=\vec r_i(t_0)+\vec v_0(t-t_0)\,,
\end{equation}
\begin{equation}
\label{eq:d6}
\Phi(\vec R)=\gamma\Phi_0\sqrt{\vec
R^2+\gamma^2(\vec v_0\vec R)^2}\,,
\end{equation}
\begin{equation}
\label{eq:d7}
\vec R=\vec r_i(t)\mp\vec r/2,\quad
\gamma=1/\sqrt{1-v_0^2/c^2}\,.
\end{equation}
Here, $\Phi_0$ is the potential of the target atom at it's rest frame,
and we have put the origin of the coordinate system to the
center-of-mass of pionium.

Thus, the solution of the  Schr\"odinger equation  (\ref{eq:d1})
depends on ``frozen'' positions $\vec r_i(t_0)$ of the target atoms
$$\psi(\vec r,t)=\psi\Bigl(\vec r,t;\{\vec r_i(t_0)\}\Bigl)\,.$$

The density matrix of pionium is defined as follows:
\begin{eqnarray}
\rho(\vec r,\vec r^{\,\prime};t)&=&\Bigl\langle
\psi\Bigl(\vec r,t;\{\vec r_i(t_0)\}\Bigl)
\cdot \psi(\vec r^{\,\prime},t;\{\vec r_i(t_0)\})
\Bigl\rangle_{\{\vec r_i(t_0)\}}\,,
\label{eq:d7.5}
\end{eqnarray}
where $\langle\rangle_{\{\vec r_i(t_0)\}}$ means averaging over
all possible positions of target atoms.

Let $t_0$ be the point of time when moving target meet the pionium
atom and $\psi(\vec r, t_0)$ is the value of pionium wave function at
this time. Then at $t>t_0$
\begin{equation}
\psi\Bigl(\vec r,t;\{\vec r_i(t_0)\}\Bigl)=
\int  G\Bigl(\vec r,\vec r_0;t,t_0;\{\vec r_i(t_0)\}\Bigl)
\psi_i(\vec r_0,t_0)
d\vec r_0\,,
\label{eq:d8}
\end{equation}
where $G$ is the Green function of Eq. (\ref{eq:d1}).

According to \cite{fein}, it can be expressed in terms of the path
integral \begin{equation} G(\vec r,\vec r_0;t,t_0;\{\vec r_i(t_0)\}) =
\int D\vec r(t)\exp(iS)\,,
\label{eq:d9}
\end{equation}
with
\begin{equation}
S = S_0+S_1\,,
\label{eq:d10}
\end{equation}
\newpage
\begin{equation}
S_0=\int\limits_{t_0}^{t}dt^{\prime}L_0
\bigl(\vec v(t ^{\prime}),\vec r(t ^{\prime})\bigl)\,,\quad
S_1=-\int\limits_{t_0}^{t}dt^{\prime}V
\bigl(\vec r(t ^{\prime}),t ^{\prime}\bigl)\,,
\label{eq:d11}
\end{equation}

\begin{eqnarray}
L_0\bigl(\vec v(t ^{\prime}),\vec r(t ^{\prime})\bigl)&=&
\mu\vec v^{\,2}(t ^{\prime})/2-
V_0\bigl(\vec r(t ^{\prime})\bigl)\,,
\label{eq:d12}
\end{eqnarray}

$$\vec v(t ^{\prime})=d\vec r(t ^{\prime})/dt ^{\prime}\,.$$

It can be shown (see  \cite{VOSKR99}) that
\begin{eqnarray}
S_1&=&-\sum_i\left\{\chi\Bigl(\vec b_i+\vec s(t_i)/2\Bigl)-
\chi\Bigl(\vec b_i-\vec s(t_i)/2\Bigl)\right\}\vartheta(t-t_i)\,,
\label{eq:d15}
\end{eqnarray}
where
\begin{eqnarray}
\chi(\vec b_{\pm})=\frac{e}{v_0}\int\limits_{-\infty}^{\infty}\Phi
\left(\sqrt{\vec b_{\pm}^2+z^2}\right)dz\,,
\label{eq:d17}
\end{eqnarray}
\begin{equation}
\vec b_{\pm}=\vec b_{i}\pm\frac{\vec s(t_i)}{2},\quad
t_i=t_0+\frac{\vec v_0\cdot \vec r_i(t_0)}{v_0^2}\,,
\label{eq:d16}
\end{equation}
\begin{eqnarray}
\vec b_i&=&\vec r_i(t_0)_{\bot}=\vec r_i(t_0)-
\frac{\vec v_0\cdot \vec r_i(t_0)}{v_0^2}
\cdot \vec v_0\,,
\label{eq:d19}
\end{eqnarray}
\begin{eqnarray}
\vec s(t_i)&=&\vec r(t_i)_{\bot}=\vec r(t_i)-
\frac{\vec v_0\cdot \vec r_i(t_0)}{v_0^2}
\cdot \vec v_0\,,
\label{eq:d20}
\end{eqnarray}
the Heavyside step function $\vartheta(t)$ is 0 for $t<0$ and 1 for
$t>0$.

Substituting (\ref{eq:d9})-(\ref{eq:d20}) into (\ref{eq:d7.5})
and performing the averaging over the ``frozen'' positions of target
atoms with the help of the prescription of \cite{LP81,AS}, one can get
the following representation for the density matrix:  \begin{equation}
\rho(\vec r,\vec r^{\,\prime};t)= \int\widetilde{G}(\vec r,\vec
r^{\,\prime};\vec r_0,\vec r_0^{\,\prime}; t,t_0)\psi_i(\vec
r_0,t_0)\psi^{\ast}_i(\vec r_0^{\,\prime},t_0) d\vec r_0d\vec
r_0^{\,\prime}\,, \label{eq:d21}
\end{equation}
with
\begin{equation}
\widetilde{G}(\vec r,\vec r^{\,\prime};\vec r_0,\vec r_0^{\,\prime};t,t_0)
= \int D\vec r(t)D\vec r^{\,\prime}(t)\exp(i\widetilde{S}_0-W)\,,
\label{eq:d22}
\end{equation}
\begin{eqnarray}
\widetilde{S}_0&=&\int\limits_{t_0}^{t}dt^{\prime}\left\{
L_0\bigl(\vec v(t ^{\prime}),\vec r(t ^{\prime})\bigl)
-L_0\bigl(\vec v^{\,\prime}(t^{\prime}),\vec r^{\,\prime}
(t^{\prime})\bigl)\right\}\,,
\label{eq:d23}
\end{eqnarray}
\begin{eqnarray}
W&=&v_0\gamma n_0\int\limits_{t_0}^{t}dt^{\,\prime}
\Omega\bigl(\vec s(t^{\prime}),\vec s^{\,\prime}(t^{\prime})\bigl)\,,
\label{eq:d24}
\end{eqnarray}
\begin{eqnarray}
\Omega\bigl(\vec s(t^{\prime}),\vec s^{\,\prime}(t^{\prime})\bigl)&=&
\int d^2b\left\{1-\exp\left(i
\Phi\bigl(\vec b,\vec s(t^{\prime}),\vec
s^{\,\prime}(t^{\prime})\bigl) \right)\right\}\,,
\label{eq:d25}
\end{eqnarray}
\begin{eqnarray}
\Phi\bigl(\vec b,\vec s(t^{\prime}),\vec
s^{\,\prime}(t^{\prime})\bigl)&=& \chi\bigl(\vec b+\vec
s(t^{\prime})/2\bigl)-\chi\bigl(\vec b-\vec
s(t^{\prime})/2\bigl)\nonumber\\ &-&\chi\bigl(\vec b+\vec
s^{\,\prime}(t^{\prime})/2\bigl)+\bigl(\vec b- \vec
s^{\,\prime}(t^{\prime})/2\bigl)\,.
\label{eq:d26}
\end{eqnarray}
Here, $n_0$ is the number of atoms in the unite volume of target at
it's rest frame,
$\vec s$ and $\vec s^{\,\prime}$ are the transverse parts of the
vectors  $\vec r$ and $\vec r^{\,\prime}$.

From Eqs. (\ref{eq:d21})-(\ref{eq:d24}) it easily derive (see
\cite{fein}) the following equation for the density matrix:
\begin{eqnarray}
i\frac{\partial\rho(\vec r,\vec r^{\,\prime};t)}{\partial t}&=&
H_{0}(\vec r)\rho(\vec r,\vec r^{\,\prime};t)-
H_{0}(\vec r^{\,\prime})\rho(\vec r,\vec r^{\,\prime};t) \nonumber\\
&&-iv_0\gamma n_0 \Omega(\vec s,\vec s^{\,\prime})
\rho(\vec r,\vec r^{\,\prime};t)\,,
\label{eq:d27}
\end{eqnarray}
where the last operator term describes the Coulomb interaction
between EA and the target atoms with account of all multiphoton
exchanges.   Using a generalized optical potential
of the form
$V_{opt}(\vec s,\vec s^{\,\prime}) =k\Omega(\vec s,\vec s^{\,\prime})$,
where $k=-iv_0\gamma n_0$, we can represent this term as
$V_{opt}\rho(t)$.

The form of Eq.~(\ref{eq:d27}) is similar to the form of
Eq. (116) in Ref.~\cite{chang} describing the internal
dynamics of multilevel atoms in laser fields, where the last term
$\Gamma\rho$ describes the contribution of the spontaneous
relaxation.

The equations of motion for the density matrix elements
\begin{equation}
\label{eq:4.41}
\rho_{ik}=\int\psi_i^{\ast}(\vec r)\psi_k(\vec r^{\,\prime})
\rho(\vec r,\vec r^{\,\prime})d\vec rd\vec r^{\,\prime}
\end{equation}
looks like as follows:
\begin{equation}
\label{eq:4.43}
\frac{\partial\rho_{ik}(t)}{\partial t}=i\Delta_{ik}\rho_{ik}(t)
-v_0\gamma n_0\sum_{l,m}\Omega_{ik,lm}\rho_{lm}(t) \,,
\end{equation}
where
$$\Delta_{ik}=\varepsilon_k-\varepsilon_i\,,$$
\begin{equation}
\label{eq:4.44}
\Omega_{ik,lm}=\int\psi_i^{\ast}(\vec r)\psi_l(\vec r)
\psi_k(\vec r^{\,\prime})\psi^{\ast}_m(\vec r^{\,\prime})
\Omega(\vec s,\vec s^{\,\prime})d\vec rd\vec r^{\,\prime}\,,
\end{equation}
the EA wave functions $\psi_{i(k)}$ and binding energies
$\varepsilon_{i(k)}$ obey the Schr\"odinger equation
\begin{equation}
\label{eq:4.42}
H_0\psi_{i(k)}=\varepsilon_{i(k)}\psi_{i(k)}\,.
\end{equation}

Taking into account the lifetime $\tau_{i}$  of the EA,
we can obtain
\begin{eqnarray} \label{eq:4.45.5}
\frac{\partial\rho_{ik}(t)}{\partial t}&=&
\left[i(\varepsilon_{k}-\varepsilon_{i})-\frac{1}{2}
(\Gamma_i+\Gamma_k)\right]
\rho_{ik}(t)-v_0\gamma n_0\sum_{l,m}\Omega_{ik,lm}\rho_{lm}(t)\,,
\end{eqnarray}
where  $\Gamma_{i(k)}=1/\tau_{i(k)}$  is the EA levels width
(for details see \cite{voskr}).

Properties of the solution of (\ref{eq:d27})
will be discussed in the following paper.
%\newpage

\section*{Acknowledgments}

The author is grateful to Leonid Nemenov, Leonid Afanasyev
and J\"org Raufeisen for stimulating interest to the work and useful
comments.  I would like also to thank Alexander Tarasov for fruitful
discussions.  Also, it is a pleasure for me to thank J\"org H\"ufner
and Boris Kopeliovich for hospitality at the Institute for Theoretical
Physics, Heidelberg University, and the MPI f\"ur Kernphysik,
Heidelberg, where some part of this work was done.

%\newpage
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\bibitem{chang} Chang S and Minogin V 2002 {\it Physics Reports}
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\end{thebibliography}

\end{document}

%The derivation of a analogous equation
%with consideration for lifetime of EA
%is given in the paper \cite{voskr}.



