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\markboth{\small \em \quad Bilha Segev \hfill}{\hfill \small \em 
Evolution kernels for phase-space distributions \quad}

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\title{EVOLUTION KERNELS FOR PHASE-SPACE DISTRIBUTIONS}

\author{BILHA SEGEV}

\address{ Department of Chemistry,
        Ben-Gurion University of the Negev
        POB 653\\ Beer-Sheva 84105, Israel} 

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\maketitle
\abstracts{Different aspects of
tunneling in the Wigner representation are studied.
The evolution kernels for the Wigner function and 
for other phase-space distributions are considered and
multidimensional tunneling is studied in the context of Fermi golden rule for 
transitions between potential surfaces. Marinov's path integral for the 
evolution kernel for the Wigner function is considered.}

%\vspace{0.4cm}

\tableofcontents

\newpage

\section*{To the memory of Michael Marinov}
Just a few years ago Prof.\ Michael Marinov was my teacher and D.Sc.\
advisor at the Technion, Israel Institute of Technology. We were a group
of students, trying to understand the secrets of the universe, or at least
of quantization, through many lectures, stimulating group meetings,
seminars and --- yes, hard work. Michael was a superb teacher much admired
by all. He gave wonderful lectures on field theory, scattering theory, and
quantum mechanics. I particularly remember his lectures on Grassmann
algebra and on supersymmetry.  This was a wonderland of truth and beauty. 
I shall always carry with me Michael's deep respect and admiration to
nature and its laws which we try to understand through the adventure of
science. 

\section{Introduction}
In my work with Marinov we studied tunneling.\cite{8}$^{\mbox -}$\cite{5d}
One object of our research was the 
evolution kernel of the Wigner function in 
one-dimensional potential scattering.\cite{5e}
In my
recent work I extended the treatment to other phase-space representations and to
other physical conditions.\cite{SH00,Segev01}
The Wigner function was used for various applications and many
papers and reviews were written about it.\cite{1a}$^{\mbox -}$\cite{more3} 
Traditionally,
the similarity to classical distributions encouraged applications in
semiclassical theories. Here, following my work with Marinov, 
I take a complementary view point 
emphasizing application of the Wigner
representation to purely quantum effects with no classical counterpart. This
work includes application to scattering, to wave-packet propagation, 
and to energy transfer processes within a single molecule
including in particular nonclassical Frank-Condon factors and radiationless
transitions in polyatomic molecules. What these different phenomena have in
common is their nonclassical nature which is treated here within a
phase-space Wigner function approach to tunneling.

Quantum propagators can be quite different from classical propagators. The
classical equations of motion define classical trajectories in phase space
for the coordinates, $q(t)$, and momenta, $p(t)$, as a function of the time $t$. 
The classical propagators for Liouville 
distributions in phase space are therefore $\delta $
functions over the classical trajectories defining a one-to-one mapping
between single points of the initial and final distributions. In quantum
mechanics, on the other hand, a single point of the phase-space distribution
at one time can in principle be causally connected to many points of this
phase-space distribution at another time. 
Furthermore, quantum propagators may include quantum jumps,
sudden changes in the momentum or quick changes in the coordinate 
of the phase-space quasidistributions.
Two points of the phase-space distribution at two different 
times can be causally connected even when no classical trajectory connects them.
These essentially quantum features are
the subject of this work.

\section{Evolution kernel for the Wigner function}
\subsection{Propagators}
A quantum state can be described by a wave function $\psi(q)$ or more
generally by a density matrix $\hat{\rho}$ and its Weyl symbol ---
the Wigner function
$\rho(q,p)$,
\begin{eqnarray}
\rho(q,p)=
\frac{1}{2\pi}\int
\left\langle q+\frac{\eta}{2}\big|\,\hat{\rho}\,\big|q-\frac{\eta}{2}\right\rangle
e^{-ip\eta}d\eta  \ .                                     
\end{eqnarray}
Here and in the following $\hbar=1$ 
and all integrals run from ${-\infty}$ to ${\infty}$ 
unless otherwise specified.
The evolution operator $ \hat{U}(t)=\exp({-i\hat{H}t}) $
gives the time evolution of the wave function 
$ |\psi_t\rangle = \hat{U}(t)|\psi_0\rangle   $
as well as that of the density operator $\hat{\rho}$:
 $\hat{\rho}_t      = \hat{U}(t)\hat{\rho}_0\hat{U}^+(t) $.
Propagators are defined to give this time evolution in some 
representation of the quantum state.

Feynman propagator, $ \langle q|\hat{U}(t)|q_0\rangle $,
gives the time evolution of the wave function in the coordinate
representation
\begin{eqnarray}
\psi_t(q) = \langle q|\psi_t\rangle = \int d q_0\,
\langle q|\hat{U}(t)|q_0\rangle \psi_0(q_0) \ ,              
\end{eqnarray}
and a similar propagator, $
\langle p|\hat{U}(t)|p_0\rangle$, is defined in the momentum
representation.

In an analogous way, the phase-space propagator $L_t(q,p;q_0,p_0)$ is
defined to give the time evolution of the Wigner function, 
\begin{eqnarray}
\rho_t(x) = \int dx L_t(x,x_0)\rho_0(x_0) \ ,
\label{r=Lr}
\end{eqnarray}
where $x\equiv(q,p)$.

The phase space propagator in terms of 
the propagators in the  coordinate
and 
in the momentum representations, respectively, is given by
\begin{eqnarray}
&&\!\!\!\!\!\!\!  L_t(q,p;q_0,p_0)  \label{L=pup}\\[1mm]
&&\!\!\!\!\!\!\!  =\frac{1}{2\pi}\int d\eta d\eta_0
e^{i(p\eta-p_0\eta_0)}
\left\langle q-\frac{\eta}{2} |\hat{U}|q_0-\frac{\eta_0}{2} \right\rangle
\left[\left\langle q+\frac{\eta}{2}
|\hat{U}|q_0+\frac{\eta_0}{2}\right\rangle\right]^* 
 \nonumber \\[1mm]
&&\!\!\!\!\!\!\!  =\frac{1}{2\pi}\int d\sigma d\sigma_0
e^{i(q\sigma-q_0\sigma_0)}
\left\langle p + \frac{\sigma}{2}|\hat{U}|p_0+\frac{\sigma_0}{2}\right\rangle
\left[\left\langle p-\frac{\sigma}{2}
|\hat{U}|p_0-\frac{\sigma_0}{2}\right\rangle\right]^*  . %2.5
\nonumber
\end{eqnarray}
$L$ is real and satisfies
\begin{eqnarray}
&&\int dx L_t(x,x_0)= 1 =
\int dx_0 L_t(x,x_0)  \ , \nonumber \\
&&L_{t_1+t_2}(x,x_0)=
\int dx^\prime L_{t_2}(x,x^\prime)
L_{t_1}(x^\prime,x_0)  \ .  
\end{eqnarray}

Two examples are: a free particle, for which
\begin{eqnarray}
&&\langle q|\hat{U}|q_0\rangle =
\sqrt\frac{m}{2\pi it}
\exp\left[\frac{im}{2t}(q-q_0)^2\right]  \ ,\nonumber \\[1mm] 
&&\langle p|\hat{U}|p_0\rangle =
\exp\left({-i t {p^2}/{2m}}\right) \delta(p-p_0)  \ , \nonumber \\[2mm] 
&&L_t(q,p;q_0,p_0) =
\delta(p-p_0)\delta\left(q-q_0 - t {p_0}/{m} \right)  \ ,
\end{eqnarray}
and a harmonic oscillator (with $m = 1$, $\omega = 1$),
for which
\begin{eqnarray}
&&\langle q|\hat{U}|q_0\rangle =
(2\pi i \sin t)^{-\frac{1}{2}}
\exp \left[i\, \frac{(q^2+q^2_0)\cos t - 2q   q_0}{2\sin t} \right]
\ , \nonumber \\ [1mm]
&&\langle p|\hat{U}|p_0\rangle =
(2\pi i \sin t)^{-\frac{1}{2}}
\exp \left[i\, \frac{(p^2+p^2_0)\cos t - 2p   p_0}{2\sin t} \right] 
\ , \nonumber \\ [1mm]
&&L_t(q,p;q_0,p_0) =
\delta(p-p_0\cos t + q_0 \sin t) 
\delta(q-q_0\cos t - p_0 \sin t) \ .
\end{eqnarray}

The phase-space propagator provides an explicit distinction between
classical and quantum dynamics. 
For any Hamiltonian
$ H = p^2/2m+V(q) $
with a potential $V(q)$ that is, at most, quadratic in the coordinate
$q$, the phase-space propagator is a $\delta$ function on the classical
paths. It is well known that in this case
the classical and quantum dynamics are basically the
same, as   Wigner's 
integro-differential equation for quadratic potentials is equivalent to
Liouville's classical equation. On the other hand, 
any deviation of $L$ from a $\delta$ function is a
manifestation of the difference 
between classical and quantum dynamics. 

\subsection{Phase-space propagator for barrier penetration}
The phase-space propagator for 
barrier penetration is obtained by substituting in Eq.\,(\ref{L=pup})
the momentum propagator for one dimensional scattering, giving:
\begin{eqnarray}
L_t(q,p;q_0,p_0) &=&
\delta(p-p_0) \frac{1}{2\pi}\int d\sigma
e^{i\sigma(q_0+t \frac{p_0}{m}-q)}\, 
a(p_0-{\sigma}/{2})\,a^*(p_0+{\sigma}/{2})
  \nonumber \\[1mm]
&+& \delta(p+p_0)  \frac{1}{2\pi}\int d\sigma
e^{i\sigma(q_0+t\frac{p_0}{m}+q)}\,
b(p_0-{\sigma}/{2})\, b^*(p_0+{\sigma}/{2}) \nonumber \\
&+& \left[\frac{1}{\pi}\, e^{2iqp_0-2i(q_0+t \frac{p_0}{m} )p} \;
b^*(p_0-p)\, a(p+p_0)+c.c.\right] \ ,
\end{eqnarray}
where $a(k)$ and $b(k)$ are the transmission and reflection amplitudes, respectively,
for the time-independent Schr\"odinger equation. 
The integrals over $\sigma$ converge since for real $k$ 
$a(k)$ and $b(k)$ are physical
amplitudes, regular and bounded:
\begin{eqnarray}
 0<|a(k)|<1~, & |a(0)|=0~, & |a(k\rightarrow \infty)|=1~, \nonumber \\[1mm]
 0<|b(k)|<1~, & |b(0)|=1~, & |b(k\rightarrow \infty)|=0~.
\end{eqnarray}   

One can get a better understanding of the phase-space propagator
by integrating in order to obtain either the coordinate  
or the momentum probability distributions. The probability of
having a final momentum $p$ is: 
\begin{eqnarray}
{\cal P}_t(p) &=& \int_{-\infty}^{\infty} d q \rho_t(q,p) \nonumber \\
&=& |a(p)|^2 \int_{-\infty}^{\infty} d q_0 \rho_0(q_0,p) 
+|b(-p)|^2 \int_{-\infty}^{\infty} d q_0 \rho_0(q_0,-p) \nonumber \\
&&+[b^*(-p)a(p)+b(-p)a^*(p)] \int_{-\infty}^{\infty} d q_0 \rho_0(q_0,0) \ .
\end{eqnarray}   
If the initial Wigner function has a well defined momentum, then 
$\rho_0(q_0,0)=0$ and the last term vanishes, giving:
\begin{eqnarray}
{\cal P}_t(p) =|a(p)|^2 {\cal P}_0(p)  +|b(-p)|^2 {\cal P}_0(-p) \ ,
\end{eqnarray}   
where for a pure state ${\cal P}_t(p)=|\Psi_t(p)|^2 $.
In the same way, the probability of 
finding the particle in the final coordinate $q$ is given by:
\begin{eqnarray}
{\cal P}_t(q) &=& \int d p\ \rho_t(q,p) =\int {dq_0dp_0} \rho_0(q_0,p_0)  
 \nonumber \\
&&\times\left\{
\frac{1}{2\pi} \int d\sigma
\exp\left[{i\sigma(q_f -q)} \right]\
a(p_0-{\sigma}/{2})\,a^*(p_0+{\sigma}/{2}) \right. \nonumber
\\[1mm] 
&&+ 
\frac{1}{2\pi}  
\int d\sigma
\exp\left[{i\sigma(q_f +q)}\right]\
b(p_0-{\sigma}/{2})\,b^*(p_0+{\sigma}/{2}) 
  \nonumber \\[1mm]
&&+ \left. \left[
\frac{\exp\left(i 2qp_0\right)}{4 \pi^2} 
\int d\sigma
\exp\left({i\sigma q_f }\right)\
b(p_0-{\sigma}/{2})\, a^*(p_0+{\sigma}/{2}) 
+c.c.  \right] \right\}\ , \nonumber \\[2mm]
q_f &\equiv & q_0+\frac{p_0}{m}t \ .
\end{eqnarray}   
If the time is large enough, so that
$q_f \rightarrow \infty $ 
for every $(q_0,p_0)$ for which
$\rho_0(q_0,p_0)\neq0$,  
the integrands oscillate rapidly, giving
${\cal P}_t(q) \rightarrow 0$ except for $q\sim q_f$ or $q\sim - q_f$.
Here too, the last term does not contribute.

For any initial Wigner function that is prepared outside the potential 
range with a well defined momentum, and after a sufficiently long time,
only the first two terms in the propagator contribute to
$\rho_t(q,p)$. The phase-space propagator for Wigner functions
with a well defined momentum, incident on a local potential 
barrier is therefore given by:
\begin{eqnarray}
\!\!\!\!\! L_t(q,p;q_0,p_0) &= &\delta(p-p_0) T(q_f-q,p_0)
+ \delta(p+p_0) R(q_f+q,p_0)\ ,
\end{eqnarray}   
where
\begin{eqnarray}
T(q_f-q,p_0)\equiv \!\int\!\! d\sigma \ t(\sigma) = \frac{1}{2\pi}\!\int\!\!
 d\sigma\,
e^{i\sigma(q_f-q)}\, 
a(p_0-{\sigma}/{2})\, a^*(p_0+{\sigma}/{2})
 \ ,
  \nonumber \\[1mm]
R(q_f+q,p_0)\equiv \!\int\!\! d\sigma \ r(\sigma) =\frac{1}{2\pi} \!\int\!\! d\sigma\,
e^{i\sigma(q_f+q)}\,
b(p_0-{\sigma}/{2})\, b^*(p_0+{\sigma}/{2}) \ .
\label{thet}
\end{eqnarray}
The propagator has two distinct parts: a tunneling propagator in which
the final momentum equals the initial one and a reflection propagator in
which the momentum changes sign.  Both are consistent with energy
conservation. 
When any initial Wigner function is incident on the barrier the result
after a long time will be two
Wigner functions: one that has penetrated the
barrier and moves in the original direction and one that has been reflected
and moves in the opposite direction. 

The tunneling propagator depends only on the initial 
momentum $p_0$, on the
coordinate difference between free and actual motion $q_f-q$, and on
the forward scattering amplitude $a(k)$. 
In the same way, the reflection propagator depends only on the initial 
momentum $p_0$, on the
difference between the absolute value of the final
coordinate and its value for free motion $q_f+q$, and on
the backward scattering amplitude $b(k)$. 

For a few barriers, $a(k)$ and $b(k)$, the transmission and reflection
amplitudes respectively, are known from the exact solution of the 
time-independent
Schr\"odinger equation. For these barriers one can substitute
them  and explicitly perform the integrations. 

For a general potential barrier $a(k)$ and $b(k)$ are unknown.
However, their
analytic properties in the complex $k$ plane can be deduced from
the theory of differential equations, and used in order 
to obtain a universal
form for $T$ and $R$.
For real $k$, $a(k)$ and $b(k)$ are regular and bounded.
We have shown that they are also regular in the immediate
neighborhood of the real $k$ axis. The singularities
of $a(k)$ and $b(k)$ are poles, which are all at
finite distances from the real $k$ axis, 
and an essential singularity at
infinity. Furthermore, $a(k)$ is
analytic in the upper half of the complex $k$ plane.
The fact that $a(k)$ has no singularities for $ \Im k >0$ is related
to causality.\cite{8} 

The transmission $T \neq 0$ only for $q \sim q_f \gg 0$, 
and the reflection  $R \neq 0$ only for $q \sim - q_f \ll 0$.
The coefficient of $\sigma$ in the exponent, $ q_f-|q| $,
has a clear physical meaning,
it is the path delay: the difference between the final coordinate $q$
and the coordinate a free particle would assume if it has passed
through the barrier without noticing it, $q_f$, 
or if it has been reflected from 
an infinite wall at the middle of the barrier, $-q_f$. 
For $ q_f>|q| $, or for $ q_f<|q| $,
one can close the contour in the upper half, or the lower half, of
the complex $\sigma$ plane, respectively,
and use  Caushy's theorem in order to perform the integrations.
$T$ and $R$ can be given, in this way, by a sum over the residues at
the poles of the integrands in the complex 
$\sigma$ plane. The singularities of the integrands $t(\sigma)$, $r(\sigma)$
are induced by the singularities of $a(k)$ and $b(k)$. 
Each singular point in the complex $k$ plane induces
two singular points in the complex $\sigma$ plane, located symmetrically
on both sides of the imaginary 
$\sigma$ axis, with complex conjugates 
contributions to the integrals, with opposite signs.

If, for example, $b(k)$ has a  singularity in the complex $k$ plane at
$k=\kappa$, then $r(\sigma)$ will have two induced singularities of
the same kind  in the complex $\sigma$ plane, at
$\sigma_1 = 2(p_0-\kappa)$ and at  $\sigma_2 = -2(p_0-\kappa^*)$.
If, the singularity is a simple pole with the
residue $K$, 
the residue of  $r(\sigma)$ at $\sigma_1 $
will be 
$-2 K\,b^*(2 p_0-\kappa^*)\,\exp{[i 2(p_0-\kappa)(q_f+q)]}$
while the residue of  $r(\sigma)$ at $\sigma_2 $
will be
$2 K^* \,b(2 p_0-\kappa^*)\,\exp{[-i 2(p_0-\kappa^*)(q_f+q)]}$.
Using the notation:
\begin{eqnarray}
B e^{-i \beta} \equiv 4 K^* ~b(2 p_0-\kappa^*)\ ,
\end{eqnarray}   
this two residues, multiplied by $( 2 \pi i)$, add up to
\begin{eqnarray}
2 \pi B e^{2 \Im \kappa (q_f+q)} \sin \left[ 2 (p_0- \Re  \kappa)(q_f+q)+
\beta \right]
\ .
\end{eqnarray}   

In the same way, if $a(k)$ has a simple pole at $k=\kappa$ with the
residue $K$, then $t(\sigma)$ will have two induced poles  
at $\sigma_1 = 2(p_0-\kappa)$ and at  $\sigma_2 = -2(p_0-\kappa^*)$
whose two residues, multiplied by $( 2 \pi i)$, add up to
\begin{eqnarray}
2 \pi A e^{2 \Im \kappa (q_f-q)} \sin  \left[ 2 (p_0- \Re  \kappa)(q_f-q)
+\alpha\right]
\ , 
\end{eqnarray}  
where
\vspace*{-3mm}
\begin{eqnarray}
A e^{-i \alpha} \equiv  -4 K^* ~a(2 p_0-\kappa^*)\ .
\end{eqnarray}   

We have deduced from causality that  
$a(k)$ has no singularities for $ \Im k >0$.
This implies that the $t$ has no singularities in the lower
half of the complex $\sigma$ plane and
\vspace*{-3mm}
\begin{eqnarray}
T\left(q_0+t {p_0}/{m}-q <0\right)=0 \ .
\end{eqnarray}   
Given a point in the initial Wigner function,
with the initial coordinate $q_0$ and the initial
momentum $p_0$, the final coordinate, as obtained by the asymptotic
phase-space propagator we have found,
can not exceed its value for free motion, or 
causality will be violated.

Finally, the universal form for the transmission and reflection 
propagators for barrier penetration in phase space is given by:
\begin{eqnarray}
&&T(q_f-q,p_0) = \delta (q_f-q) \nonumber \\[1mm]
&&\ \  -  \Theta (q_f-q) \sum_{n}
A_n e^{2 \Im \kappa_n (q_f-q)} \sin [ 2 (p_0- \Re  \kappa_n)(q_f-q)+\alpha_n]\ ,
\nonumber \\[2mm]
&&R(q_f+q,p_0) = \nonumber \\[1mm]
&&\ \  \ \Theta (q_f+q) \sum_{m} 
B_m e^{2 \Im \kappa_m (q_f+q)} \sin [ 2 (p_0- \Re  \kappa_m)(q_f+q)+\beta_m]
\nonumber \\[1mm]
&&\ \  + \Theta [-(q_f+q)] \sum_{l} 
C_l e^{2 \Im \kappa_l (q_f+q)} \sin [ 2 (p_0- \Re  \kappa_l)(q_f+q)+\gamma_l]\ ,
\label{sumpoles}
\end{eqnarray}   
where $n$ numbers
the singularities of $a(k)$ with $\Im k <0$, $m$ numbers
the singularities of $b(k)$ with $\Im k <0$, and $l$ numbers
the singularities of $b(k)$ with $\Im k >0$. Furthermore, 
$\kappa_n$, $\kappa_m$, and
$\kappa_l$ are the positions of these singularities in the complex $k$
plane. For a given potential barrier they are constant parameters, while
$A_n $, $\alpha_n$, $B_m $, $\beta_m$, $C_l$ and $\gamma_l$ are functions
of $p_0$, whose exact forms are determined by 
the details of the barrier's shape.

The coordinate dependence of the propagators has the same functional form 
for every local potential. The propagators are functions of the path-delay,
$q_0+t {p_0}/{m}-|q|$, which is the coordinate difference between free 
and actual 
motion. The probability for a large  path-delay decreases exponentially,
but the
most probable path-delay is not zero, since
the exponential decline is modulated by an oscillatory term. 
The rate of the exponential decline, and the frequency of the oscillations,
are given by the distance of the singularities of the transition
and reflection amplitudes, in the complex momentum plane, from the 
imaginary and real axis, respectively. The coefficients 
and the phase-shifts depend on the details of the 
barrier shape and on the initial momentum. 

It is important to note that the phase-space
propagator is not a $\delta$ function. Each point in the 
initial Wigner function is propagated, 
not into a single point, but into a region of 
points in the final Wigner function. 
This is a manifestation
of the difference between classical and quantum dynamics
during barrier penetration. Notice that, 
asymptotically, after the particle had passed the barrier
or has been reflected from it,
the propagation in time is a simple coordinate
translation, where each point in 
the final Wigner functions is 
translated with a different velocity, given classically
by its momentum. 

\subsection{Two limits:
semiclassical versus sudden approximation}
In this section two complementary limits 
for our propagator will be considered: the semiclassical limit, and
the limit of a $\delta$ function
potential barrier, where the semiclassical
approximation fails. The phase-space propagator was expressed above
via sums over
the singularities of the amplitudes. For a general potential barrier
these are usually infinite sums. We will show that 
in the limit of a sharp barrier only the first terms in these sums
contribute, while
in the semiclassical approximation the various terms
acquire large phases and are equally important. 

In the limit where the area under the barrier is kept constant while its
width tends to zero and its height to infinity, the potential approaches
a $\delta$ function:
\vspace*{-3mm}
\begin{eqnarray}
V^{\delta}(q)= \Omega \delta(q) \ , \label{Vdelta}
\end{eqnarray} 
(taking, for simplicity, $m=1$).
The amplitudes for this potential are:
\begin{eqnarray}
a^{\delta}(k)=\frac{k}{k+ i \Omega} \ ,\qquad
b^{\delta}(k)=\frac{ i \Omega}{k+ i \Omega} \ . \label{abdelta}
\end{eqnarray} 
Both have only one simple pole in the complex $k$ plane at
$\kappa = - i \Omega$.
Applying the formalism described above, the phase-space propagator
for this barrier is:
\begin{eqnarray}
&&L_t^{\delta}(q,p;q_0,p_0) = \delta(p-p_0) T^{\delta}(q_f-q,p_0)
+ \delta(p+p_0) R^{\delta}(q_f+q,p_0) \ , \nonumber \\[2mm]
&&T^{\delta}(q_f-q,p_0) =  \delta (q_f-q)  \nonumber \\[2mm]
&&~~~~~~~~~~~-   ~ \Theta (q_f-q) 
A^{\delta}(p_0)\exp \left[{- 2 \Omega (q_f-q)}\right]  ~ 
\sin \left[ 2 p_0(q_f-q) + \alpha^{\delta} \right] \ ,
\nonumber \\[2mm]
&&R^{\delta}(q_f+q,p_0)= \Theta (q_f+q)~
B^{\delta}(p_0) \sin \left[ 2 p_0(q_f+q)\right] \ ,
\nonumber \\[2mm]
&& A^{\delta}(p_0) =
{2 \Omega}\sqrt{(2 p_0)^2+\Omega^2}/{p_0}  \ ,
%\nonumber \\[2mm]
\quad B^{\delta}(p_0)= {2 \Omega^2}/{p_0}  \ ,
\nonumber \\[1mm]
&& \alpha^{\delta}= - \arctan \left( {2 p_0}/{\Omega} \right) \ .
\label{deltaprop}
\end{eqnarray}   

In the opposite limit of a wide and smooth barrier and when the initial
momentum is small compared to the barrier's height, the semiclassical
approximation to barrier penetration, WKB, applies. 
To leading order in this approximation the transition
amplitude is given by:
\begin{eqnarray}
&&a^{sc}(k)= \exp \left[\varphi (k)\right] \ ,\nonumber \\[1mm]
&& \varphi (k) = i \int^{\infty}_{-\infty} \left[ 
\sqrt{k^2 - 2 m V(q)} - k \right] d q \ , \nonumber \\[1mm]
&&|a^{sc}(k)|^2=  \exp  \left[ - 2 \int^{q_{>}}_{q_{<}}
\sqrt{2 m V(q) - k^2 }  d q \right] \ ,
\end{eqnarray} 
where $q_{>}(k)$ and $q_{<}(k)$ are the turning points.
One can see that  in the semiclassical approximation, 
the integrand of the transmission propagator,
$a^{sc}(p_0-\sigma/2)\, a^{sc}{}^*(p_0+\sigma/2)$, analytically
continued off the real $\sigma$ axis
is an entire function of $\sigma$ 
whose only singularity is an essential
singularity at infinity. 
The result of Eq.\,(\ref{sumpoles}) does not apply 
since the residue theorem can not be used 
in order to calculate the phase-space propagator. 
One should go back and substitute 
$a^{sc}(k)$ in Eq.\,(\ref{thet}), obtaining:
\begin{eqnarray}
T^{sc} = \frac{1}{2\pi}\int d \sigma \exp \left[ i \sigma
(q_f-q) + \varphi ( p_0 - {\sigma}/{2} ) +
\varphi^* ( p_0 + {\sigma}/{2} ) \right] \ .
\end{eqnarray} 
For potential barriers, $V(q)$, which are wide and smooth enough so
that $\varphi$ can be expanded as a converging power series of $\sigma$
around $p_0$, one has:
\begin{eqnarray}
&&\varphi ( p_0 - {\sigma}/{2} ) +
\varphi^* ( p_0 + {\sigma}/{2} ) = 
\varphi_0 - i \varphi_1 \sigma + \varphi_2 \sigma^2
 - i \varphi_3 \sigma^3 + \ldots~~~~~ \ , \nonumber \\[1mm]
&&\varphi_0 = 2 \Re \varphi(p_0)                   
 \ , \quad
\varphi_1 = \Im \varphi^{'}(p_0) 
 \ , \quad
\varphi_2 = \frac{\Re \varphi^{''}(p_0)}{4}
 \ , \quad
\varphi_3 = \frac{\Im \varphi^{'''}(p_0)}{24}\ ,
  \nonumber \\[1mm]      
&&\varphi_{2 n} =\frac{\Re \varphi^{(2 n)}(p_0)}{ 2^{2 n-1} (2 n)!} 
\ , \quad
\varphi_{2 n+1} = \frac{\Im \varphi^{(2 n+1)}(p_0)}{ 2^{2 n} (2 n+1)!}
 \ .
\end{eqnarray} 

Neglecting all orders of $\sigma$ but one, a $\delta$ function is 
obtained:
\begin{eqnarray}
&& T^{sc} = |a^{sc}(p_0)|^2 \int^{\infty}_{-\infty} d \sigma
\exp \left[ i \sigma ( q_f -q - \varphi_1 ) \right] \\[1mm]
&&\ =  \exp \left({\!- 2 \int^{q_{>}}_{q_{<}}\!\!\!
\sqrt{2 m V(q) - p_0^2} ~ d q }\right) \delta \left[ q - \left(
q_0 + \frac{p_0}{m} t - \Im \varphi^{'}(p_0) \right) \right] \ .
\nonumber
\end{eqnarray} 
In this limit the propagation is similar to classical propagation with 
a time delay given by the first derivative
of the phase shift $\varphi_1 = \Im \varphi^{'}(p_0)$.

Keeping terms up to $\sigma^3$ gives,
\begin{eqnarray}
T^{sc}\simeq |a^{sc}(p_0)|^2 \frac{1}{\pi} \int^{\infty}_{0} d \sigma
e^{\varphi_2 \sigma^2} \cos \left[ \sigma (q_f-q-\varphi_1 )-
\sigma^3 \varphi_3 \right] \ .
\end{eqnarray} 
Application of this semiclassical approximation
to a general barrier may be subtle as 
this integral converges only for $\varphi_2 \leq 0$. 
In fact, $\varphi$
acquire additional phase shifts at the turning points
whenever the barrier is not  smooth enough.
However, for ``well-behaved" barriers,  for which $\varphi_2 \leq 0$,
an Airy function, sometimes modified
by an exponential decline, is obtained, which is typical
to semiclassical Wigner functions. This was demonstrated for the
Poschl-Teller barrier.\cite{5b} 

The semiclassical 
propagator obtained in this way does not display the properties 
of the exact propagators. These deviations are artifacts of the approximation,
not real properties of the quantum propagation.

\section{Propagation of other phase-space distributions}
\subsection{Phase-space representations}
Different representations of quantum mechanics in phase space were
introduced over the years. 
A simple definition for most 
of these representations can be given in the following way:
\begin{eqnarray}
\varrho _{t}^{\zeta }(q,p) &=&\frac{1}{4\pi ^{2}}\int d\xi \int d\eta \int
dq^{\prime }\left\langle q^{\prime }+\frac{\hbar }{2}\eta \big|\, \hat{\rho}%
_{t}\,\big| q^{\prime }-\frac{\hbar }{2}\eta \right\rangle  \nonumber \\[2mm]
&&\times \zeta (\xi ,\eta )\exp [i\xi (q^{\prime }-q)-i\eta p]\,,
\end{eqnarray}
where $\hat{\rho}$ is the density matrix.\cite{1l} 
In this section we do not set $\hbar=1$.  
The function $\zeta (\xi ,\eta )$ defines
the representation. Not every $\zeta (\xi ,\eta )$ is allowed.\cite{1h,4muga} We focus on the
best known and most used phase-space representations: the Wigner function,
the standard-ordered function, the Kirkwood (antistandard-ordered) function,
the $P$ function, and the Husimi function, for which $\zeta (\xi ,\eta )$ is
respectively given by: 
\begin{eqnarray}
&&\zeta ^{{\rm W}}(\xi ,\eta )=1\;, \\[1mm]
&&\zeta ^{{\rm S}}(\xi ,\eta )=\exp \left[ -i\frac{\hbar }{2}\ \xi \eta %
\right] \;, \\[1mm]
&&\zeta ^{{\rm K}}(\xi ,\eta )=\exp \left[ i\frac{\hbar }{2}\ \xi \eta %
\right] \;, \\[1mm]
&&\zeta ^{{\rm P}}(\xi ,\eta )=\exp \left[ \frac{\hbar }{4m\kappa }\xi ^{2}%
\right] \exp \left[ \frac{\hbar m\kappa }{4}\eta ^{2}\right] \;, \\[1mm]
&&\zeta ^{{\rm H}}(\xi ,\eta )=\exp \left[ -\frac{\hbar }{4m\kappa }\xi ^{2}%
\right] \exp \left[ -\frac{\hbar m\kappa }{4}\eta ^{2}\right] \;,
\end{eqnarray}
where $\kappa $ and $m$ are constants defining the representation. The
Glauber $Q$ function is a special case of the Husimi function. Other
phase-space distribution, e.g.\ with singular kernels are not considered
here but can be considered as well.\cite{4muga}

Time evolution can be considered in phase space in terms of evolution kernels
for each one of these phase-space quasi-distributions. For each 
phase-space distributions a phase-space propagator is defined as
the evolution kernel in the following way: 
\begin{equation}
\varrho _{t}^{\zeta }(q,p)=\int dq_{0}\int dp_{0}\ {\cal L}_{t}^{\zeta
}(q,p;q_{0},p_{0})\varrho _{0}^{\zeta }(q_{0},p_{0})\,.
\end{equation}

\subsection{Free propagation}
The free propagator depends on the representation. 
In the
classical limit of $\hbar \rightarrow 0$ we recover for all the
representations the simple propagator: 
\begin{equation}
\lim_{\hbar \rightarrow 0}{\cal L}_{t}^{\zeta }(q,p;q_{0},p_{0})=\delta
(p-p_{0})\delta (q_{0}+\frac{p}{m}\,t-q)\;,
\end{equation}
but for finite $\hbar $ the propagators are not the simple delta function of
free classical propagation. Unique in this respect is the free propagator
for the Wigner function whose simple form is exact in the quantum regime as
well as in the classical one: 
\begin{equation}
{\cal L}_{t}^{{\rm W}}(q,p;q_{0},p_{0})=\delta (p-p_{0})\delta
(q_{0}+\frac{p}{m}\,t-q)\;.
\end{equation}
In all the other representation the integrals defining the free propagator
diverge for finite $\hbar $. Nevertheless, the
propagators defined by these divergent integrals are well defined
distributions. In particular, they are normalized and integrable.
In all the representations, excluding Wigner, there is a competition between
the limits of large time and small $\hbar $ and deviations of the free
propagators from their classical limit grow with time.

The free propagators for the standard-ordered function and the Kirkwood
(antistandard-ordered) functions are, respectively: 
\begin{eqnarray}
{\cal L}_{t}^{{\rm S/K}}(q,p;q_{0},p_{0})&\!\! =&\!\!
\frac{\delta (p_{0}-p)}{2\pi }\!%
\int\!\! d\xi \exp \left[ i\xi \left( q_{0}+\frac{p}{m}\,t-q\right) \pm i\frac{%
\hbar t}{2m}\xi ^{2}\right]  \label{freeSKint} \\[1mm]
&\!\! =&\!\!\delta (p_{0}-p)\sqrt{\frac{\pm im}{2\pi t\hbar }}\ \exp \left[ \frac{\mp
im}{2t\hbar }(q_{0}+\frac{p}{m}\,t-q)^{2}\right] \;,  \label{freeSK}
\end{eqnarray}
where the integral representation of Eq.\,(\ref{freeSKint}) is
exact while the expression in Eq.\,(\ref{freeSK}) was obtained
after a regularization. The free propagators for the $P$ function and the
Husimi distribution are: 
\begin{eqnarray}
{\cal L}_{t}^{{\rm H}}(q,p;q_{0},p_{0}) &=&-\frac{1}{2\pi }\sqrt{\frac{1}{%
\pi m\hbar \kappa }}\exp \left[ -\frac{m}{\hbar \kappa t^{2}}\left(
q_{0}+\frac{p}{m}\,t-q\right) ^{2}\right] \nonumber \\[1mm]
&&\times\int d\eta \exp \left[ -i\eta \left( q_{0}+\frac{p}{m}\,t-q\right) 
+\frac{%
\hbar \kappa t^{2}}{4m}\eta ^{2}\right] \\[1mm]
&=&-{\cal L}_{t}^{{\rm P}}(q,p_{0};q_{0},p) \ .
\end{eqnarray}

The difference between the Wigner representation and other representations
here is striking. Free propagation in all other representations is based on
interference. The value of each initial point contributes to the values of
many final points and the value of each final point is determined by the
values of many initial points. The Wigner function is the only phase-space
function freely propagating in a completely classical way. 

Furthermore, while the propagators 
for the Wigner, standard-ordered, and Kirkwood functions
explicitly display momentum conservation with $\delta (p-p_{0})$ and a
symmetry between initial and final coordinates and momenta, the propagators
for the $P$ function and the Husimi function do not have these properties.
In fact the propagation of the Husimi function seems to be carried out with
a Gaussian distribution of momenta around the final momentum, while the
relation to the initial momentum is ill defined, and the reverse applies to
the propagation of the $P$ function. Clearly momentum is conserved
regardless of the representation chosen to describe the process. Integration
of the time evolution in all these representations would give the same
result for any observable consequences of momentum conservation. In some
representations, however, momentum is conserved explicitly in the evolution
kernel: the value of these quasidistributions at a phase-space point
propagates only into phase-space points with the same momentum. In other
representations the intrinsic property of momentum conservation is obscured.
The point-to-point influence functional does not display it and it is
recovered only after integration.

\subsection{Scattering}
If the propagation is asymptotically free yet includes a small region of
interaction, a scattering matrix can be defined. For single channel
scattering, as well as for the elastic channel of multichannel scattering, a
single amplitude $a(p)$ defines the
propagation.
The propagators in the different representations are: 
\begin{eqnarray}
&&{\cal L}_{t}^{{\rm W}}(q,p;q_{0},p_{0}) =\delta (p-p_{0})\frac{1}{2\pi }
\int d\xi \,\,a(p+{\hbar }\xi/{2} )a^{\ast }(p-{\hbar }\xi/{2} ) 
\nonumber \\[1mm]
&&\ \ \ \ \ \ \ \ \ \times \exp \left [i\xi\left  (q-q_{0}-tu^{{\rm W}}(p,\xi ) \right) \right]\ , \\[1mm]
&&{\cal L}_{t}^{{\rm S}}(q,p;q_{0},p_{0}) =\delta (p-p_{0})\frac{1}{2\pi }
\int d\xi \,a(p)a^{\ast }(p-\hbar \xi )  \nonumber \\[1mm]
&&\ \ \ \ \ \ \ \ \ \times \exp\left  [i\xi\left  (q-q_{0}-tu^{{\rm S}}(p,\xi ) \right) \right]\ , \label{aS}
\\[1mm]
&&{\cal L}_{t}^{{\rm K}}(q,p;q_{0},p_{0}) =\delta (p-p_{0})\frac{1}{2\pi }
\int d\xi \,a(p+\hbar \xi )a^{\ast }(p)  \nonumber \\[1mm]
&&\ \ \ \ \ \ \ \ \ \times \exp \left [i\xi \left (q-q_{0}-tu^{{\rm K}}(p,\xi ) \right) \right]\ ,  
\label{aK} \\[1mm]
&&{\cal L}_{t}^{{\rm P}}(q,p;q_{0},p_{0}) =\int dp_{1}\,\frac{1}{\sqrt{\pi
m\kappa \hbar }}\exp \left[ -\frac{1}{m\kappa \hbar }(p_{0}-p_{1})^{2}\right]
\label{aP}\\[1mm]
&&\times \frac{1}{2\pi }\int d\eta \,\exp \left[ i\ \eta (p_{1}-p)\right]
\;\exp \left[ \frac{\hbar m\kappa }{4}\eta ^{2}\right]  \nonumber \\[1mm]
&&\times \frac{1}{2\pi }\int d\xi \,a(p_{1}+{\hbar }\xi/{2} )\,a^{\ast
}(p_{1}-{\hbar }\xi/{2} )  
\exp \left[ i\ \xi \left(q-q_{0}-tu^{{\rm W}}(p_{1},\xi )\right)\right] \,\,
\nonumber  \\[1mm]
&&={\cal L}_{t}^{{\rm H}}(q,p_{0};q_{0},p)\,.  \label{aH}
\end{eqnarray}
where we have introduced the auxiliary velocity functions: 
\begin{eqnarray}
u^{{\rm W}}(p,\xi ) &=&\frac{p}{m}\,, \\[1mm]
u^{{\rm K}}(p,\xi ) &=&\frac{p}{m}+\frac{\hbar \xi }{2m}\,, \\[1mm]
u^{{\rm S}}(p,\xi ) &=&\frac{p}{m}-\frac{\hbar \xi }{2m}\,.
\end{eqnarray}
Note that while $u^{{\rm W}}(p,\xi )$ does not depend on $\xi $, $u^{{\rm K/S
}}(p,\xi )$ do.
The propagators for single channel scattering in the different phase-space
representations are given by bilinear integral transforms of the scattering
amplitude. They provide different mappings from the energy domain where the
amplitudes are defined to the time domain where the propagators act.
Mappings between the energy and time domains are particularly interesting
for the discussion of causality.

The scattering amplitude $a(p)\,$and its properties depend on the specific
problem considered. 
It often has the following properties when
analytically continued into the complex momentum plane:
\begin{itemize}
\item  $a^{\ast }(p) = a(-p)$, or $a^{\ast }(p) = - a(-p)$\,.
\item  $\,a(p)$ is analytic in the upper half of the complex $p$ plane\,.
\item  $a(p)\rightarrow 1$ as $|p|\rightarrow \infty $.
\end{itemize}
These properties were proven for the
Schr\"{o}dinger equation with a positive potential,\cite{8} but are more general.
Whenever the transition amplitude has these properties, the propagators
can be written with $a(-p+{\hbar }\xi)$ instead of $a^{\ast}(p-{\hbar }\xi)$ etc.
The integral over $\xi $ can then be considered as a contour integral in the
complex $\xi $ plane. Because $a(-p\pm \hbar \xi )$ and $a(-p\pm \hbar \xi /2)$
are analytic in the upper $\xi $ plane it is possible to deform the contour
of $\int_{-\infty }^{\infty }d\xi $ to the arc at infinity of the upper $\xi $
plane, where $|a|\rightarrow 1$. In general, this will not give any simple
result or insight. The integral over the arc at infinity can assume very
complicated forms and would often diverge for an arbitrary dependence of the
auxiliary $u$ functions on $\xi $. Only for cases in which $u(p,\xi )=v(p)$
does not depend on $\xi $, a simple result follows. In these cases the
integral over $\xi $ would vanish identically for $q>q_{0}+tv(p)$. For $%
q<q_{0}+tv(p)$ different velocity functions, different scattering amplitudes, and
different representations would all give different propagators but under the
assumptions specified above all these different propagators will vanish for
coordinates too far apart at times that are too short. The limit on the
propagation is then locally set by the velocity $v(p)$.

The manifestation in phase space of the analytical properties of the scattering
matrix then depends on the representation. In previous section we have shown
that in the Wigner representation: 
\begin{equation}
{\cal L}_{t}^{{\rm W}}(q>q_{0}+tp/m)=0\,.
\end{equation}
This property of the Wigner function tells us that no information can be
transferred faster than free motion as long as the assumptions regarding the
analytical properties of the scattering amplitude hold.\cite{5b,5e} 
In this section we
have checked for this property in the other phase-space representations 
and found that no such simple restriction applies to the
other phase-space propagators. In them, the velocity functions $u$ depend on 
$\xi $, and the contributions from the arcs at infinity diverge. We note
however that the reason for the lost simplicity is not in the scattering
process but rather in the free propagation which is so simple for the Wigner
function yet so complicated for the other phase-space quasi-distributions.
Note that the Husimi function propagator as was noticed before does not
explicitly display momentum conservation. As a result, while it is confined
by classical free propagation, this free propagation can be with any
momentum, hence the propagation velocity is not really confined.

\section{Fermi golden rule in phase space}
In the previous sections propagation in phase space was considered in 
one dimension and 
with a given Hamiltonian. The advantage of the Wigner 
representation was demonstrated. In this section we a priori restrict 
the discussion to the
Wigner representation, yet include the more complicated dynamics of transitions
between multidimensional potential energy surfaces.

Radiationless transitions in
molecules,\cite{jortner} and output
coupling
from a Bose-Einstein Condensate to form an atom
laser,\cite{Munich99,yoni} are very different
physical
processes. Yet they share essential features: both involve an irreversible 
transition
between distinct energy surfaces
in multidimensional complex systems. We consider these processes in the Wigner
phase-space representation. 
Fermi Golden Rule in phase space gives: 
\begin{equation}
\frac{2\pi }{\hbar }\kappa ^{2}{\rm Tr}\left[ \hat{\varrho _{i}}^{\prime }%
\hat{\varrho _{f}}\right] =\frac{2\pi }{\hbar }\kappa ^{2}{\int }dqdp\rho
_{f}(q,p)\rho _{i}^{\prime }(q,p) \label{overlap}
\end{equation}
for the transition rate. Here $\kappa ^{2}$ is an integral over the
electronic degrees of freedom, $\rho _{i}^{\prime }(q,p)$ is the Wigner
transform of the incoming state density matrix, $\hat{\varrho _{i}}^{\prime }
$, and $\rho _{f}(q,p)$ is the Wigner transform of the outgoing state
density matrix, $\hat{\varrho _{f}}$. 
The incoming density matrix is 
$\hat{\varrho_{i}}^{\prime }=\hat{\lambda}\hat{\varrho_{i}}\hat{\lambda
}^{\dagger }$
where $\hat{\varrho_{i}}$ is the initial density matrix and 
$\hat{\lambda}$ is the coupling operator. 
For a pure initial state 
$\hat{\varrho_{i}}=|\psi _{i}\rangle \langle \psi _{i}|$ 
where $\psi _{i}$ is an eigenfunction of $\hat{H}_{i}$.
The manifold of final states $
\{\varphi _{k}\}$ includes all eigenfunctions of $\hat{H}_{f}$ with a given
energy: 
\begin{eqnarray}
\hat{\varrho _{f}}=\Sigma _{E_{k}=E}\ |\varphi _{k}\rangle \langle
\varphi _{k}|=\delta \left( E-\hat{H}_{f}\right) \, , \nonumber
\end{eqnarray}
where $E=E_{i}$ for
radiationless transitions and $E=E_{i}\pm \hbar \omega $ for radiative
transitions. The initial and final hamiltonians $\hat{H}_{i}$ and $
\hat{H}_{f}$ differ because of an electronic transition in molecules and a
change of hyperfine state of an output coupled atom in the BEC.

The phase-space overlap integral of Eq.\,(\ref{overlap}) is a subject of our
study\,\cite{SH00} as well as of previous
works.\cite{heller,bruno}
The
integral yields an approximation for the rate, while a study of the
integrand shows where in phase space the transition occurs. For the cases in
which the surfaces do not cross in classically allowed points, surface
jumping to the point or region in phase space that dominate the overlap
integral, constitute the dominant relaxation channel in multi-channel and
multidimensional relaxation processes.\cite{SH00}

\subsection{Radiationless transitions in polyatomic molecules}
Relaxation of electronically excited molecules plays a part in many chemical
processes. In a radiative transition the population on the upper surface
undergoes a { vertical } transition according to the Condon principle. A
photon is emitted with a frequency that overcomes the energy gap between the
surface potentials. The initial conditions for the nuclear dynamics on the
ground surface potential are given by the position and momentum of the wave
packet on the upper surface. The Condon principle applies as well to
radiationless transition in the case where the surfaces cross in a
classically allowed position. In this case, the population transfer between
the surfaces is in the region where the electronic surfaces cross and the
electronic energy of the excited molecule is transferred to the molecular
internal vibrational degrees of freedom smoothly, without the nuclei jumping
in either position or momentum discontinuously.

We consider a radiationless transition between nested surface potentials
where the two surface potentials do not cross in a classically allowed
region. If the radiative transition channel is not available a vertical
transition is impossible and the molecule must decay by a sudden change of
its nuclei position or momentum. We call this process {\em surface jumping }
where the jumping we refer to is a quantum jump and can be related to the
general non-classical phenomena of tunneling. 

One of the crucial steps in the formal description of the transition between
the surfaces involve the calculation of the Frank-Condon (FC) overlap
integrals between the initial and final vibronic states. The value of the FC 
{ integral} serves as a measure for the probability of the transition and
when several modes of vibration are competing for the electronic energy the
relative FC factors can be a defining parameter for the division of the
energy. For a radiationless transition it can be shown, using the reflection
principle, or by using the stationary phase approximation, that for the case
of crossing surfaces the contribution to the FC integral is maximal in the
region where the electronic states cross. The integral defining the FC
factor is controlled by the vicinity of this leakage point. {\em Surface
jumping }occurs for the cases in which the electronic surfaces do not cross
in classically allowed points, the FC factor is exponentially small, the
reflection principle does not apply, and there is no straightforward way to
predict which region in space would dominates the FC factor.

We suggest a simple method for
calculating such weak Franck-Condon factors. The Wigner transform of the
wavefunction on an initial Born-Oppenheimer state is calculated for the
donor potential surface and projected onto the acceptor energy surface
energy shell. The integrated projection yields an approximation for the
relevant Franck-Condon factors, while the phase-space integrand shows where in
phase space the leakage occurs between the donor and acceptor states. This
in turn determines the initial conditions on the acceptor surface for
subsequent internal vibrational relaxation and energy flow. 

A model of the $S_{2}\rightarrow S_{0}$ vibronic relaxation transition of
the benzene molecule where 30 modes of vibration compete for the
electronic energy was studied within this approach. For this
model, we have shown that almost all the energy must go to a single $C-H $
local stretching. The initial conditions for vibrations of this mode will
be a coordinate jump of the hydrogen atom toward the ring. All the other
modes undergo an almost vertical transition, where the energy that they
take is proportional to their equilibrium displacement between the two
surfaces.

Our analysis shows that, almost by definition,
this is a general phenomena: surface jumping in
radiationless transitions in polyatomic molecules involves an excitation of
one degree of freedom of the system which takes most of the energy while the
rest of the degrees of freedom undergo a vertical or an almost vertical
transition. For larger energy gaps - more energy is transferred to this
dominant accepting mode, which need not be a normal mode of the system.

\subsection{Radiative decay} 
A similar situation occurs in radiative transitions when the energy of the
photon is too big or too small to account for the energy difference between
the surfaces, i.e.\ in the wings of an absorption band.\cite{heller} If we
quantize the
radiation field, it becomes clear that induced radiative transitions are
very similar to radiationless decay, in that a given level (molecule +
photon) is weakly coupled to a dense set of quasidegenerate levels (excited
molecule+ one less photon). In an electronic absorption for example, we can
raise the lower potential energy surface by $\hbar \omega $, where $\omega $
is the frequency of the light; using the radiation coupling as the
perturbation, we can now treat the coupling as if it were radiationless.
Radiative transitions are thus more experimentally flexible than
radiationless ones, in that the photon energy can be used to put the two
surfaces out of contact with each other. This corresponds spectroscopically
to being the red or blue wings of the absorption band. 

\subsection{Transitions from a thermal state} 
A semiclassical approximation for the transfer rate of a thermal
distribution of atoms or molecules between two potential surfaces under the
influence of an external electromagnetic field is obtained by projecting the
initial thermal phase-space distribution into a set of energy surfaces
corresponding to the initial energy distribution shifted by the energy
supplied by the external field. To leading order, the transition
rate from a
thermal state is given by 
\begin{equation}
\Gamma _{T}(\Delta )\approx \frac{2\pi \kappa^{2}}{\hbar }\frac{1}{(2\pi
\hbar )^{d}}\int dq\int dp\, P_{T}(H_{i})\, \delta (H_{f}-H_{i}-\hbar \Delta )\,,
\label{approxfermi}
\end{equation}
which farther simplifies in cases where $H(q,p)=p^{2}/2m+V(q)$ to 
\begin{equation}
\Gamma _{T}(\Delta )\approx \frac{2\pi \kappa ^{2}}{\hbar }\frac{1}{(2\pi
\hbar )^{d}}\int dq\,\delta (V_{f}-V_{i}-\hbar \Delta )\int dp\,P_{T}(H_{i})\,,
\label{approxfermisimple}
\end{equation}
where $\kappa $ is the electronic term, $\Delta $ is the
detuning between
the frequency of the electromagnetic field and the splitting between the
levels, $P_{T}(E)$ is the thermal occupation probability of a state in the
initial potential surface with energy $E$ and $H_{i}(q,p)$ and $H_{f}(q,p)$
are the phase-space representations of the Hamiltonians corresponding to the
two potential surfaces. 

This forms the basis for a 
calculation of the transfer rate 
of a gas
of atoms or
molecules with initial Boltzmann
or Bose-Einstein thermal distributions.
We have applied it to the calculation of finite temperature effects on
weak output coupling of a magnetically trapped atomic bose gas so as to
form an atom laser, where the different surfaces correspond to different 
hyperfine levels of the atoms; as well as to model systems where the
approximate results were successfully compared with exact quantum
calculations.      
  
\section{Path integral for the propagator}
The path integral\,\cite{pi0} was a subject close to Marinov's heart.
His report\,\cite{pim1} is a classic monograph on this subject.
In his recent paper\,\cite{5a} and lecture notes\,\cite{pim3}
Marinov developed a new type of phase-space
path integral for the evolution kernel of the Wigner function.

There were several issues motivating this work.
Mostly, Marinov wanted a path integral that would 
preserve the canonical invariance of the theory and
would not introduce a polarization in phase space.
He was concerned with the fact that Feynman's path integral for 
the evolution kernel in real space introduces such a polarization, and wrote:
\begin{quote}
``The problem here is that the trajectories in 
the phase space must belong to a special class, determined in a 
non-invariant manner"
\end{quote}
and therefore: 
\begin{quote}
``One could not get the standard path integral for 
coordinate spaces having a non-Euclidean topology".\cite{pim3}
\end{quote}
In this section I revisit Marinov's path integral and study some of its properties.

\subsection{General structure}
A path integral for the phase-space propagator of the Wigner function
was developed by Marinov.
He has shown that for $H = p^2 / 2m + V(q)$
the phase-space propagator is
\begin{eqnarray}
&&L_t(q,p;q_0,p_0) = \nonumber  \\
&&\lim_{N\rightarrow\infty}
\prod^{N-1}_{\nu=1}
\left[ \int dq_\nu\int dp_\nu \right]
\prod^N_{\nu=1} 
\delta \left[(q_\nu-q_{\nu-1})-\frac{t}{N}   
\frac{p_\nu+p_{\nu-1}}{2m} \right]\nonumber  \\
&&\times \prod^N_{\nu=1}
\left[ \int \frac{d\eta_\nu}{2\pi} \exp\Big{\{}
-i \Big[ \eta_\nu(p_\nu-p_{\nu-1})\right.% \right. \right.
\nonumber \\
&&+\frac{t}{N} 
\left. 
%\left. \left.
\left(V\left(\frac{q_\nu+q_{\nu-1}+\eta_\nu}{2}\right)-
V\left(\frac{q_\nu+q_{\nu-1}-\eta_\nu}{2}
\right)\right)\Big]\Big{\}}\right]  \label{path}
\end{eqnarray}
where $p_n=p$ and $q_N=q$. 
We will see that
sometimes this path integral reduces to a sum over classical trajectories, 
in other cases it is dominated by classical 
trajectories but allows for some quantum fluctuations around the classical paths, and
finally, it includes the possibility for quantum jumps, which can even dominate
if no classical paths exist.

\subsection{Classical trajectories}
For potentials that  are at most
quadratic in the coordinate, the phase-space propagator is a $\delta$
function on the classical paths. It is easy to verify this property 
starting from the path integral.

For the simplest case of a free particle $V=0$ and
\begin{eqnarray}
&&\!\!\!\!\!\!\! L_t^0(q,p;q_0,p_0) = \lim_{N\rightarrow\infty} 
\prod^{N-1}_{\nu=1}
\left[\int dq_\nu\int dp_\nu\right]
\prod^N_{\nu=1} \Big{\{}\delta\Big[(q_\nu-q_{\nu-1})%
\nonumber\\[1mm] %A3
&&\!\!\!\!\!\!\!- \frac{t}{N}\frac{(p_\nu+p_{\nu-1})}{2m}\,\Big]\,
\delta(p_\nu-p_{\nu-1})\Big{\}} 
= \delta(p-p_0)\delta\left(q-q_0-\frac{p_0}{m}\,t\right). 
\end{eqnarray}
For a potential which is at most quadratic in $q$,
$V(q+\eta/2)-V(q-\eta/2) = \eta\,\partial
V/\partial q  $,
\begin{eqnarray}
&&\!\!\!\!\!\!\! L_t^{\rm quad.}(q,p;q_0,p_0)= \lim_{N\rightarrow\infty} 
\prod^{N-1}_{\nu=1}
\left[\int dq_\nu\int dp_\nu\right] 
\prod^N_{\nu=1} \Big{\{}\delta\Big[(q_\nu-q_{\nu-1})
\nonumber\\[1mm]
&&\!\!\!\!\!\!\! - \frac{t}{N}   %A3
\,\frac{p_\nu+p_{\nu-1}}{2m}\Big]\delta\left(p_\nu-p_{\nu-1}
+\frac{t}{N}\frac{\partial V}{\partial q} \right) \Big{\}} 
= \delta(p-p_{cl})\delta(q-q_{cl}). 
\end{eqnarray}

When the potential is not quadratic, quantum fluctuations start to play a role. 
A semiclassical approximation,
expanding the exponent to second order in the coordinates, shows that 
the propagator is dominated by the classical paths, yet this treatment is restricted
to those cases where such an expansion is justified. Quantum corrections 
can then be analyzed by keeping higher order terms in the expansion, 
giving the propagator a finite width around the classical path. 
What if the expansion is wrong, say because it is mathematically ill defined, 
or when no classical path exists?  An alternative approach to quantum effects can
be developed by resuming the path integral as a sum
over momenta jumps. This approach is introduced in the next subsection.

\subsection{Quantum fluctuations and quantum jumps}
The integral over $\eta$ includes 
the possibility for quantum fluctuations and quantum jumps. 

For simplicity, let us focus on cases where the potential 
is $0$ or $V_0$ on different segments of space 
so that ${\partial V}/{\partial q}$ is either $0$ 
or ill defined. In these cases $V\left[\left({q_\nu+q_{\nu-1}+\eta_\nu}\right)/2\right]
-
V\left[\left({q_\nu+q_{\nu-1}-\eta_\nu}\right)/2\right]$ is $0$, $V_0$ or $-V_0$ 
for different values of $q_\nu+q_{\nu-1}$ and $\eta$. The integral 
over $\eta$
can be calculated exactly to give a sum over such functions as 
$\sin \left[(p_\nu-p_{\nu-1})(q_\nu+q_{\nu-1})\right]/(p_\nu-p_{\nu-1})$ 
with different coefficients depending on the values of 
$\bar{q}_\nu=(q_\nu+q_{\nu-1})/2$, $\tilde{p}_\nu=p_\nu-p_{\nu-1}$
and on the coordinate segments defining the potential. In general:
\begin{eqnarray}
F_\nu\!\equiv\!\!
\int \!\! \frac{d\eta_\nu}{2\pi} \exp\!\left\{\!
-i \left[ \eta_\nu \tilde{p}_\nu
 +\frac{t}{N} 
\left(V\!\left(\bar{q}_\nu + \frac{\eta_\nu}{2}\right)-
V\!\left(\bar{q}_\nu - \frac{\eta_\nu}{2}\right)
\right)\right]\right\} \!
- \! \delta(\tilde{p}_\nu)\ .
\end{eqnarray}
Different potentials would give different functions $F_\nu=F_\nu(\tilde{p}_\nu;\bar{q}_\nu)$. 

With this definition, the trajectories can be redefined and resummed by the number $\sigma$ of 
steps in which the momentum changes or {\it jumps}, the time intervals 
$\nu_1,  \, \ldots, \, \nu_\sigma$ in which the jumps occur and the size of these 
momentum jumps
$\alpha_1, \, \ldots, \, \alpha_\sigma$,
\begin{eqnarray}
&&L_t(q,p;q_0,p_0) =  \lim_{N\rightarrow\infty}
\sum_{\sigma=0}^N \sum_{\nu_1=1}^{N} \int d \alpha_1
\sum_{\nu_2=\nu_1+1}^{N} \int d \alpha_2 ... 
\sum_{\nu_\sigma}^{N} \int d \alpha_\sigma
 \nonumber \\
&&\times\prod^{N-1}_{\nu=1}
\left[ \int dq_\nu\int dp_\nu \right]
\prod^N_{\nu=1} 
\delta \left[(q_\nu-q_{\nu-1})-\frac{t}{N}   
\frac{p_\nu+p_{\nu-1}}{2m} \right]\nonumber  \\
&&\times\prod_{\nu\neq\nu_1,...\nu_\sigma}
\delta(p_\nu-p_{\nu-1}) 
\prod_{\nu=\nu_1,...\nu_\sigma}
\delta(p_\nu-p_{\nu-1}-\alpha_i) F_\nu(\alpha_i;q_\nu+q_{\nu-1}) \nonumber \\
&& =  \lim_{N\rightarrow\infty}
\sum_{\sigma=0}^N \sum_{\nu_1=1}^{N} \int d \alpha_1
F_1(\nu_1,\alpha_1;q_0,p_0) 
\nonumber \\
&&
\times\sum_{\nu_2=\nu_1+1}^{N} \int d \alpha_2 
F_2(\nu_1,\nu_2,\alpha_1,\alpha_2;q_0,p_0)
\nonumber \\
&&\times \, \ldots 
\sum_{\nu_\sigma}^{N} \int d \alpha_\sigma
F_\sigma(\nu_1,\ldots , \nu_\sigma, \alpha_1,\ldots , \alpha_\sigma ;q_0,p_0)
\nonumber \\
&&\times
\delta \left[q-q_0- \frac{t}{m} \left({p_0 + \sum_{i=1}^\sigma
\frac{\nu_i}{N} \alpha_i} \right)\right]
\delta \left(p-p_0- \sum_{i=1}^{\sigma}
 \alpha_i \right) \ . \label{jumpath}
\end{eqnarray}
The functions $F_{2,3, \, \ldots, \, \sigma}$ depend on $\nu_1$ and $\alpha_1$ because $q_{2,3, \, \ldots, \, \sigma}$ do.

This new form for Marinov's 
path integral is conceptually very simple. Each point in the 
Wigner function is propagated by an evolution kernel which is a 
sum over simple trajectories. Each trajectory is characterized 
by the number, times, and strength of sudden changes of its momentum. 
In between the jumps the path is a free particle path, and with each jump comes a
factor $F$ which depends on the position of the jump and on the
potential in an essentially non-local way. 
The difficulty in calculating propagators for specific potentials
is two-fold: first one has to find the factors $F$ and second to sum over the 
infinite number of paths. 

\section{Conclusions}
Quantum mechanics can be
studied in many different representations. The physical results of an
experiment or the theoretical predictions for an observable effect do not
depend on the representation chosen but a clever choice often simplifies the
analysis and sometimes helps our physical intuition. In this work several
cases where considered where fundamental or complicated problems
considerably simplify in the Wigner representation. 
Several questions remain: 
\begin{itemize}
\setlength{\itemsep}{0cm}
\item In studying the propagator of section 2
above we have neglected a third term which appeared 
in addition to tunneling and reflection. 
Preliminary studies indicate  that this third term 
is responsible for transient ringing effects at the barrier region.
Farther work is needed to resolve the properties of these ringing effects.
\item The transition rates according to Fermi Golden Rule was 
applied within a Born-Oppenheimer approximation, yet it may be more general. 
Extending the 
treatment beyond the adiabatic approximation is worth looking into.
\item The path integral for the propagator was written as a sum over momentum jumps. 
Summation of the paths to rederive the closed form expression for the propagator
in the limit of a narrow barrier and large time, remains to be done.
\end{itemize}

\section*{Acknowledgments}
\addcontentsline{toc}{section}{\numberline{}Acknowledgments}
This research was supported by Grant No.\ 2000118 from the
United States - Israel Binational Science Foundation (BSF), Jerusalem,
Israel.
I dedicate this paper to the memory of my teacher and advisor Michael Marinov. 

\section*{References}
\addcontentsline{toc}{section}{\numberline{}References}

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\end{document}

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