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\null
\vspace{4cm}\noindent
{\bf QUANTUM ADIABATIC EVOLUTION\footnotemark}
\footnotetext{Lecture delivered by C.-E.Pfister}
\\ \\ \\
Alain Joye\footnotemark and Charles-Edouard Pfister\footnotemark
\\ \\
\hspace*{2.6cm}\setcounter{footnote}{1}
\footnotemark Centre de Physique Th\'eorique\newline\hspace*{2.6cm}
CNRS, Luminy Case 907\newline\hspace*{2.6cm}
F-13288 Marseille France\newline\hspace*{2.6cm}
\footnotemark Ecole Polytechnique F\'ed\'erale de Lausanne
\newline\hspace*{2.6cm}
D\'epartement de Math\'ematiques\newline\hspace*{2.6cm}
CH-1015 Lausanne Switzerland
\renewcommand{\thefootnote}{\arabic{footnote}}
\setcounter{footnote}{0}
\\ \\ \\
{\bf ADIABATIC THEOREM OF QUANTUM MECHANICS}
\\
The notion of adiabatic evolution or adiabatic process is an important
theoretical concept,
which occurs at several places in Physics. The main feature of this concept
is that although the
process is very slow, global changes can take place without local changes.
Adiabaticity is at the border between dynamics and statics. This concept was
introduced by Boltzmann
in Classical Mechanics through the notion of
adiabatic invariants. In Thermodynamics
adiabatic processes play an important role. In Quantum Mechanics, if
the state of the system
is an eigenfunction $\psi(t_0)$ for the eigenvalue
$e(t_0)$ at $t=t_0$, then in the adiabatic
limit the
state of the system at time $t=t_1$ is an eigenfunction
$\psi(t_1)$ for the eigenvalue $e(t_1)$,
provided the energy-level $e(t)$ remains
isolated during the time-interval $[t_0,t_1]$. Even if
$H(t_0)= H(t_1)$, the
eigenfunction $\psi(t_1)$ is generally
different from $\psi(t_0)$ by a phase which can be decomposed into
a dynamical phase related to the energy-level $e(t)$ and a geometric phase
related to the
spectral subspaces visited during the
adiabatic process. This is the fundamental
observation of Berry \footnotemark[1] which
gave rise to extensive developments during the last
ten years \footnotemark[2],\footnotemark[3],\footnotemark[4].
The adiabatic theorem in Quantum Mechanics has
been proven very early by Born and
Fock \footnotemark[5]. It was refined at several
occasions. We mention in particular
the papers of Kato \footnotemark[6] and Garrido
\footnotemark[7] which brought new
ideas into the subject. In this
short paper we describe recent mathematical results obtained
during the last four
years, mainly in the case of analytic time-dependent
quantum systems, where few
rigorous results were known before. Let $H(t)$ be
the Hamiltonian of a time-dependent
system. We suppose that $H(t)$ is a self-adjoint operator,
uniformly bounded from below,
defined on a dense domain $D$, independent of $t$,
of a Hilbert space ${\cal H}$. We suppose
that $H(t)$ is at least of class $C^2$. The time
evolution of the system is governed by
the evolution operator $U(s,s')$, solution of the
Schr\"odinger equation $(\hbar =1)$
\be
i\frac{d}{d s}U(s,s')=H(\eps s)U(s,s'), \,\,\, U(s',s')=\un
\ee
where $1/\eps$ is the typical time-scale over which the Hamiltonian changes
significantly. This equation is conveniently studied
using a rescaled time, $t=\eps s$,
which leads to
\be
i\eps\frac{d}{d t}U(t,t')=H(t)U(t,t'), \,\,\, U(t',t')=\un
\ee
where, with an abuse of notations, we have again denoted the evolution by $U$.
It should be stressed, although we do not write
it explicitly, that $U(t,t')$ is a
function of $\eps$. The adiabatic limit corresponds
to the limit $\eps$ tending to zero which
is a singular limit.
The adiabatic theorem states that there exists an
approximate solution $V(t,t')$ of equation (2),
called adiabatic evolution, such that
\be
\sup_{t\in[t_0,t_1]}\| U(t,t_0)-V(t,t_0)\| ={\cal O} (\eps )
\ee
The main assumptions needed to prove this result
are the smoothness of the Hamiltonian and
the \newline
{\bf Gap Condition:} {\it there exists $g>0$ so
that for each $t$ the spectrum $\sigma (t)$ of $H(t)$
can be separated into two closed
subsets $\sigma_1(t)$, $\sigma_2(t)$ and }
$$ \mbox{dist}(\sigma_1(t),\sigma_2(t))\geq g$$
By convention $\sigma_1(t)$ is always a bounded subset
of $\sigma(t)$. We denote the spectral
projector associated with $\sigma_1(t)$ by $Q(t)$.
The main feature of $V(t,t')$ is that it is {\it compatible with the
decomposition, or follows the decomposition, of the Hilbert space}
${\cal H}$ into
\be
{\cal H}=Q(t){\cal H}\oplus (\un -Q(t)){\cal H}
\ee
which is expressed by the intertwining property
\be
V(t,t')Q(t')=Q(t)V(t,t')
\ee
Let us consider
now the consequences of (3) and (5).
Let $\psi_0$ be an initial wave-function at $t=t_0$,
such that $Q(t_0)\psi_0=\psi_0$. At time $t>t_0$
the wave-function is $\psi(t)=U(t,t_0)\psi_0$, and
$\phi(t)=V(t,t_0)\psi_0$ is an approximate
solution. Because of property (5)
$Q(t)\phi(t)=\phi(t)$,
i.e. $\phi(t)$ belongs to the spectral subspace $Q(t){\cal H}$ of $H(t)$.
This information, combined
with (3),
implies that the probability that the system makes
a transition from the subspace $Q(t_0){\cal H}$
to the subspace $(Q(t){\cal H})^\perp$ is of order $\eps^2$.
In the special case
$\mbox{dim}Q(t)\equiv 1$, $\psi_0$ is an eigenfunction of
$H(t_0)$ for the eigenvalue
$e(t_0)=\mbox{tr}Q(t_0)H(t_0)$
and $\phi(t)$ an
eigenfunction of $H(t)$ for the eigenvalue
$e(t)=\mbox{tr}Q(t)H(t)$. We can give a more explicit
expression for $\phi(t)$,
\be
\phi(t)=V(t,t_0)\psi_0=\e{-i/\eps\int_{t_0}^{t}e(s)ds}\varphi(t)
\ee
where $\varphi(t)$ is the unique eigenfunction of
eigenvalue $e(t)$ which is determined by the
conditions $\bra\frac{d}{ds}\varphi(s)|\varphi(s)\ket=0$,
$t_0\leq s\leq t$, and $\varphi(t_0)=
\psi_0$.
The adiabatic evolution $V(t,t')$ is solution of the equation
\be
i\eps\frac{d}{d t}V(t,t')=
\left(H(t)+i\eps \left[\frac{d }{d t}Q(t),Q(t)\right]
\right)V(t,t')\;, \,\,\, V(t',t')=\un
\ee
The result (3) can be improved when
$H(t)$ is of class $C^{q+2}$, $q>0$. For
$j=1,2,\ldots ,q$ we can construct
decompositions of the Hilbert space ${\cal H}$ into
\be
{\cal H}=Q_{j}(t){\cal H}\oplus (\un -Q_{j}(t)){\cal H}
\ee
and evolutions compatible with these decompositions,
which are better approximate solutions of (2).
Before doing this
let us recall the main result about smooth evolutions
(not necessarily unitary), which are
compatible with a given decomposition of the Hilbert
space (4) \footnotemark[8], \footnotemark[9].
There is a particular evolution
$W(t,t')$ satisfying (5), which is a solution of
\be
i\frac{d }{d t}W(t,t')=i\left[\frac{d }{d t}Q(t),Q(t)\right]W(t,t')\;,
\,\,\, W(t',t')=\un
\ee
This evolution has a geometric interpretation
given below, and it depends only on $Q(t)$.
All other smooth evolutions $\widehat{V}$ satisfying
the intertwining property (5)
are solutions of equations
\be
i\frac{d }{dt}\widehat{V}(t,t')=
\left(B(t)+i\left[\frac{d }{d t}Q(t),Q(t)\right]\right)\widehat{V}(t,t')\,,
\,\,\, \widehat{V}(t',t')=\un
\ee
where $B(t)$ commutes with $Q(t)$. The structure
of equation (7) is now clear, and the best
choice \footnotemark[10] of $B(t)$ with respect
to (3) is $B(t)=\eps^{-1} H(t)$. It is not difficult
to show that the eigenfunction $\varphi(t)$ in (6) is simply $W(t,t')\psi_0$.
We come to the geometric interpretation of $W$,
and give at the same time an illustration of
the adiabatic theorem. Let $H(m)$ be a family of Hamiltonians $H(m)$,
satisfying the gap condition for each value of the parameter $m$
belonging to some manifold $M$, and let $\sigma_1(m)$ consist of a single
eigenvalue $e(m)$ with $\mbox{dim}Q(m)\equiv p$. We also suppose that
the family of all subspaces $Q(m){\cal H}$,
$m\in M$, forms a vector bundle $F$ of base $M$,
whose points are all $(m,\phi)$ with $m\in M$
and $\phi\in Q(m){\cal H}$, and whose fiber above
$m'$ is the set of all points $(m',\phi)$,
$\phi\in Q(m'){\cal H}$. The wave-function $\psi_0$ is an
eigenfunction of $H(m_0)$ for the eigenvalue
$e(m_0)$. We decide to change the value of the parameter
$m_0$ into $m_1$ in an adiabatic way, and ask what
is the wave-function $\psi_1$ of the system after
this change. To do that we choose a path
$\gamma$ in $M$, parametrized by $t\in[0,1]$,
with $\gamma(0)=m_0$, and $\gamma(1)=m_1$.
The speed of the change of the parameter $m$
is controlled by the parameter $\eps>0$. Using
the {\it rescaled }time
$\psi_1=\psi(1)$ is the solution of
\be
i\eps \frac{d }{d s}\psi(s)=H(\gamma(s))\psi(s)\,, \,\,\, \psi(0)=\psi_0
\ee
The adiabatic approximation of $\psi_1$ is
$V(1,0)\psi_0$ and has an expression like (6),
\be
V(1,0)\psi_0=\e{-i/\eps\int_{0}^{1}e(\gamma(s))ds}W(1,0)\psi_0
\ee
The phase $\eps^{-1}\int_{0}^{1}e(\gamma(s))ds$ is
the {\it dynamical phase}. It depends on the
parametrization of the path $\gamma$, and it records how long is
the process for changing the value of the
parameter. In this approximation $W(s,0)\psi_0$ is an eigenfunction
of $H(\gamma(s))$ for all $s\in[0,1]$.
The operator $W(1,0)$ depends only on the path $\gamma$,
and not on its parametrization.
Since the bundle $F$ is
constructed with the $Q(m)$, we automatically get
a natural connection, because we have
a natural decomposition of the tangent spaces into
a vertical part and an horizontal part.
In order to simplify the notations let us suppose
that $M$ is embedded in $\R^n$,
so that $F$ is embedded in $\R^n\times{\cal H}$. Let
$f=(m,\phi)\in F$. Any tangent vector
$v_f$ at
$f$ can be viewed as a velocity vector of
a curve $c(t)=(c_1(t),c_2(t))$ in
$\R^n\times{\cal H}$ with $c(t)\in F$, and
$c(0)=f$, i.e. $v_f=(\dot{c_1}(0),\dot{c_2}(0))_f$.
The vertical vectors at $f$ are velocity vectors of
curves $c(t)$ with $c_1(t)\equiv m$. They
are of the form $(0,\dot{c_2}(0))_f$ with
$\dot{c_2}(0)=Q(m)\dot{c_2}(0)$. Using the projection
$Q(m)$ we have a decomposition of $v_f$ into a vertical vector
$(0,Q(m)\dot{c_2}(0))_f$ and an {\it horizontal} vector
$(\dot{c_1}(0),(\un-Q(m))\dot{c_2}(0))_f$, and thus
we have a connection. The operator
$W(1,0)$ in (12) is {\it the holonomy operator describing
the parallel transport
corresponding to the above connection along the path $\gamma$.}
To show this we must verify that
the velocity vector to the curve $(\gamma(s),W(s,0)\psi_0)$,
$s\in[0,1]$, is horizontal
for any $s$ and any $\psi_0\in Q(m_0)$. This is immediate since
\be
Q(\gamma(s))\left(\frac{d}{ds}Q(\gamma(s))\right)Q(\gamma(s))=0
\ee
as a consequence of $Q(\gamma(s))^2=Q(\gamma(s))$.
Let us now construct new decompositions (8) of ${\cal H}$.
We define the projections $Q_j$
as approximate solutions of the Heisenberg equation
\be
i\eps\frac{d }{dt}A(t)=[H(t),A(t)]
\ee
They are obtained by an iteration process \footnotemark[11],\footnotemark[12].
We set $Q_0:=Q$, $K_{-1}=0$ and
$K_j:=i\left[\frac{d }{d t}Q_j(t),Q_j(t)\right]$, $j=0,\ldots $.
For each $t$ let
$\Gamma$ be a simple closed path in the complex plane,
counterclockwise oriented, and
encircling $\sigma_1(t)$. We suppose that the distance from
$\Gamma$ to the spectrum
$\sigma(t)$ is at least $g/2$. Clearly $Q_0(t)$ is an
approximate solution of (14) up
to an error term ${\cal O}(\eps)$. Let us suppose that
we have constructed $Q_{k}(t)$,
$k=0,\ldots ,j-1$. Then $Q_{j}(t)$ is the spectral
projection of the operator
$H_{j}(t):=H(t)-\eps K_{j-1}(t)$, defined by the Riesz formula
\be
Q_{j}(t)=-\frac{1}{2\pi i}\oint_{\Gamma}(H_j(t)-\lambda)^{-1}d\lambda
\ee
Provided $\eps$ is small enough $Q_j$ is well defined
up to $j=q$ if $H$ is $C^{q+2}$.
Using regular perturbation theory it is easy to show that
$Q_j(t)$ is an approximate solution of (14) up to an error term
${\cal O}(\eps^{j+1})$.
If it happens that all
derivatives of the resolvent of $H(t_0)$ are zero up to order $p$, then
$Q_j(t_0)=Q(t_0)$, $j\leq p$.
We know how to find evolutions $V_j$ compatible with (8)
by solving equation (10). The best
choice, if we want $V_j$ to give the best approximate solution
of (2), which is compatible
with the decomposition (8) of ${\cal H}$, is
\be
B(t)=\eps^{-1} H_j(t)
+Q_j(t)K_{j-1}(t)Q_j(t)+(\un-Q_j(t))K_{j-1}(t)(\un-Q_j(t))
\ee
Notice that $V_0=V$.
\\
\\
\noindent
{ \bf Adiabatic Theorem}
\\
{\bf A)} {\it Let $H(t)$ be of class $C^{q+2}$,
$q\geq 0$, and let the gap condition be satisfied for all
$t\in [t_0,t_1]$.
Then there exists $\eps_j >0$ so that for all $\eps <\eps_j$,
the evolution $V_j$,
$0\leq j\leq q$ is defined in $[t_0,t_1]$, and}
$$\sup_{t\in [t_0,t_1]}\| U(t,t_0)-V_j(t,t_0)\|=
{\cal O} \left( \eps^{j+1}
\right)$$
{\bf B)} {\it Let $H(t)$ have an analytic continuation in the strip
$S_{\alpha}=\{t+is\in\C : |s|<\alpha\}$.
Let the gap condition be satisfied for all
$t\in\R$, and let $c(t)$ be an integrable function, with
$\lim_{t\ra\pm\infty}c(t)=0$, and $H^+$,
$H^-$ be two self-adjoint operators such that for all
$\varphi \in D$}
$$ \sup_{|s|<\alpha}\|(H(t+is)-H^{\pm})\varphi\|
\leq c(t)(\|\varphi\|+\|H^{\pm}\varphi\|)\;\;,
\;\;\pm t>>1$$
{\it Then there exist $\eps^* >0$ and a positive constant $\kappa$,
such that for all $\eps<\eps^*$ and
for all $t,t'\in\R$ we have a decomposition of $\cal H$
into }
${\cal H}=Q_{*}(t){\cal H}\oplus (\un -Q_{*}(t)){\cal H}$
{\it and a compatible evolution $V_*(t,t')$ such that,}
$$
\| U(t,t')-V_*(t,t')\|={\cal O} \left(\e{-\kappa /\eps}\right)
$$
{\it In both cases}
$$\| Q(t)-Q_j(t)\|={\cal O} (\eps)\;\;,\;\;\| Q(t)-Q_*(t)\|=
{\cal O} (\eps)$$
{\it Moreover $\lim_{t\ra\pm\infty}\| Q(t)-Q_*(t)\|=0$.}\\
\noindent
{\bf Comments:}
1) The proof \footnotemark[12] of the theorem is to
give an explicit lower bound for
$\eps_j$ or $\eps^{*}$ by proving the following estimate.
By hypothesis we can find
$a$, $b$ and $c$ such that for all $\lambda\in \Gamma$
\be
\|\frac{d^p}{dt^p}(\lambda-H(t))^{-1}\|\leq ac^p\frac{p!}{(1+p^2)}\;\;,
\;\;\|\frac{d^p}{dt^p}K_0(t)\|\leq bc^p\frac{p!}{(1+p^2)}
\ee
with $p=0,\ldots,q+1$ in case A), all $p$ in case B),
and in that case $b=b(t)$ is a bounded
integrable function. Then there exist
$\eps(a,\|b\|_{\infty})$, a constant
$d=d(a,\|b\|_{\infty})$ such that for all $j$,
$j=0,\ldots,q$ in case A) or $j\leq q^*=[1/ecd\eps]$
in case B), all $\eps\leq\eps(a,\|b\|_{\infty})$,
\be
\|K_j(t)-K_{j-1}(t)\|\leq b\eps^jd^jc^j\frac{j!}{(1+j^2)}
\ee
In case A), using this information we can repeat the
proof of Theorem 3.1 in \footnotemark[14].
In case B) we set $Q_*(t):=Q_{q^*}(t)$ and choose
as evolution $V_*$ the solution of (10) with
$B(t)=\eps^{-1}H_{q^*}$. By writing the differential equation
for $V_*^{-1}(t,t')U(t,t')$
the proof is immediate. Notice that we have an explicit simple
upper bound for the error term in that case
\be
\| U(t,t')-V_*(t,t')\|\leq
\left |\int_{t'}^{t}b(s)ds\right|2(\pi[1/ecd\eps])^{1/2}\exp(-2[1/ecd\eps])
\ee
The condition on the asymptotic behaviour
of $H(t)$ is not needed to establish (19), but is
used in the application of remark 2) below. Of course, if it is
not verified, then $b(t)$ is in general not integrable.
2) In case B)
if $\psi(t)$ is a normalized solution of the Schr\"odinger
equation such that $\lim_{t\ra -\infty}\|(\un -Q(t))\psi(t)\|=0$,
then the transition probability to
the spectral subspace $(\un -Q(+\infty)){\cal H}$ at
time $t=+\infty$ is exponentially small
\be
\lim_{t\ra +\infty}\|(\un -Q(t))\psi(t)\|^2={\cal O}
\left(\e{-2\kappa /\eps}\right)
\ee
This result was first obtained in \footnotemark[13] in
the case of matrices, and in the general
case in \footnotemark[14]. The method of the proof is
quite different than the above mentioned method.
The evolution is studied along special complex paths in the complex time-plane.
The method gives good estimations of $\kappa$ in the case
of matrices and even optimal
values of $\kappa$ in the case of $2\times 2$ matrices.
Recently Martinez \footnotemark[15] has rederived result
(20) by a microlocal analysis.
We also mention \footnotemark[16], however the results are weaker.
3) Nenciu \footnotemark[17] gave a closely related proof of B), which precedes
the above sketched proof \footnotemark[12],
and Sj\"ostrand \footnotemark[18] obtained similar results by a
microlocal analysis.
4) The evolution $V_*$ is called {\it superadiabatic} because of the
exponential estimate.
\newpage
\noindent
{ \bf TWO-LEVEL SYSTEMS}
\\
We consider here only the case of analytic systems
with two energy levels $e_1(t)$ and $e_2(t)$,
which have the same permanent degeneracy. As above
$Q(t)$ is the spectral projection
associated with $e_1(t)$. The paradigm model when
$\mbox{dim}Q(t)\equiv1$ is a spin $1/2$ in an
external time-dependent magnetic field \footnotemark[13],
and when $\mbox{dim}Q(t)\equiv2$ it
is a spin $3/2$ in a time-dependent quadrupole electric
field \footnotemark[19].
We suppose that $\mbox{tr}H(t)\equiv 0$, so
that $H(t)^2=\rho(t)\un$ with
$\rho(t)\geq g^2/4$ in order that the gap condition
be satisfied. We have
\be
e_1(t)=-\sqrt{\rho(t)}, \,\,\, \sqrt{1}=1\;\;\;\;{\rm and}\;\;\;\;
Q(t)=\frac{1}{2}\left(\un +\frac{H(t)}{e_1(t)}\right)
\ee
We suppose that the conditions of part B) of
the adiabatic theorem are satisfied.
For these simple systems the adiabatic evolution
can be written explicitly as
\bea
V(t,t')&=&\exp\left(-i/\eps\int_{t'}^te_1(s)ds\right)W(t,t')Q(t') \nonumber\\
&+&\exp\left(-i/\eps\int_{t'}^te_2(s)ds\right)W(t,t')(\un -Q(t'))
\eea
where $W(t,t')$ is given by (9).
Let $\chi_-\in Q(-\infty){\cal H}$ and
$\chi_+\in (\un -Q(+\infty)){\cal H}$ be two normalized
eigenvectors of $H^-$ and $H^+$. The transition
probability from $\chi_-$ to $\chi_+$ is
\be
{\cal P}(\eps,\chi_-,\chi_+)=\lim_{{\scriptstyle t_0\ra -\infty}
\atop {\scriptstyle t_1\ra +\infty}}
\left|\bra\chi_+ | U(t_1,t_0)\chi_-\ket\right|^2
\ee
In order to derive an
asymptotic formula for ${\cal P}(\eps,\chi_-,\chi_+)$,
we observe that the evolution $U$ is
analytic in the strip $S_{\alpha}$, whereas $V$ is
in general multivalued and has
singularities at the zeros of $\rho$ in $S_{\alpha}$. Indeed,
let us assume that there exists a complex eigenvalue
crossing at $z_j$ (and by symmetry at
$\overline{z_j}$),
i.e. $\rho(z_j)=0$. The eigenvalues and
eigenprojections have a singularity at $z_j$ and $\overline{z_j}$
(see (21)).
We further suppose that $z_j$
is a zero of order one for $\rho(z)$ with $\mbox{Im}z_j>0$.
We can continue analytically
$e_i(t)$, $i=1,2$, and $Q(t)$ in the
punctured strip $S_{\alpha}$, which is defined by removing the zeros of
$\rho$ from $S_{\alpha}$, but the analytic continuation
is not unique. We fix a point $t$
of the real axis
and make the analytic continuation of $e_1$ and $Q$
along a clockwise oriented
simple loop $\gamma_j$ based at $t$,
encircling $z_j$ and no other zero of $\rho$. After
analytic continuation we obtain
the values $e_1(t|\gamma_j)=e_2(t)$, and $Q(t|\gamma_j)=(\un -Q(t))$.
A simple computation shows that
\be
i\left[\frac{d }{d t}Q(t),Q(t)\right]=
\frac{i}{4\rho(t)}\left[\frac{d }{d t}H(t),H(t)\right]
\ee
and therefore the generator of $W$ is a single-valued analytic
function in
the punctured strip. $W$ itself is multi-valued and
$W(t|\gamma_j)$, the analytic continuation of
$W(t,t)$ along $\gamma_j$, is such that
\be
W(t|\gamma_j)Q(t)=(\un -Q(t))W(t|\gamma_j)
\ee
The operator $W(t|\gamma_j)$, which describes the
transport of vectors along $\gamma_j$,
is not unitary anymore since $\gamma_j$ is a path in the
complex plane and the projection $Q(z)$ is not
self-adjoint when $z\not \in \R$.
If we start with a vector in $Q(t)$,
then, at the end of the loop $\gamma_j$, we
get a vector in the orthogonal subspace. When the
eigenvalues are not degenerate,
we can thus define a {\em complex} phase $\theta_j(t)$ by the relation
\be
W(t|\gamma_j)\phi_1(t)=\exp(-i\theta_j(t))\phi_2(t)
\ee
where $\phi_i(t)$ is a (normalized) eigenvector of
$H(t)$ for the eigenvalue $e_i(t)$, $i=1,2$.
The important point is that $W(t|\gamma_j)$ and
$\theta_j(t)$ depend only on the homotopy
class of the closed path $\gamma_j$ based at $t$ in the punctured strip.
Moreover, $\mbox{Im}\theta_j(t)$ is
independent of the base point $t$ of $\gamma_j$ since $t$ is {\em real}.
It is natural to write the evolution $U(t,0)$ as
$U(t,0):=V(t,0)A(t)$. Using the fact that the
evolution $U$ is analytic in the strip, i.e. path
independent, whereas $V$ is path dependent, we can
derive a useful identity for $A$ by considering the Schr\"odinger
equation along two different paths,
the real axis and a path $\eta_j$ in the complex plane going
above $z_j$ \footnotemark[19]. This identity allows
us to express the transition probability (23)
in terms of the analytic continuations of the
eigenvalues and eigenprojections along the
path $\gamma_j$ and of $A$ along the path $\eta_j$ \footnotemark[19].
We can perform the asymptotic
analysis of the transition probability (23) as $\eps\ra 0$
whenever we have a good control
of the analytic continuation of $A$ along $\eta_j$. At
this step it is necessary to impose
further conditions on the behaviour of the multi-valued function
$\Phi(z):=-2\int_t^z\sqrt{\rho}(z')dz'$.
These conditions specify the global behaviour in the strip
of the so-called Stokes
lines, which are the level-lines $\mbox{ Im}\Phi(\cdot)=\mbox{ Im}\Phi(z_j)$,
where $z_j$ is a zero
of ${\rho}$
\footnotemark[13],\footnotemark[19]. The results of the
analysis are given in Table 1
when the above mentioned conditions are satisfied
for one (and then necessarily unique)
simple zero $z_j$ of $\rho$. This case is generic. In
the table we have dropped the
index $j$.
\\ \\ \\
\hspace*{.5cm}{\bf Table 1.} Asymptotic transition probability
in the generic case.
\\ \\ \hspace*{.5cm}
\small
\begin{tabular}{ll} \hline
\multicolumn{1}{c}{\bf Characteristics of $H(t)$} & \multicolumn{1}{c}
{{\bf Transition probability} ${\cal P}(\eps,\chi_-,\chi_+)$}
\\ \hline
& \\ General formula\footnotemark[19] & \\
\begin{tabular}{l}
Complex hermitian\\dim$Q(t){\cal H}>1$
\end{tabular}
&$\begin{array}{l}
\biggl(\left|\bra\chi_+|W(+\infty,t)W(t|\gamma)W(t,-\infty)\chi_-\ket\right|^2
+{\cal O} (\eps)\biggr)\times\\
\exp\left(\frac{2}{\eps}\mbox{ Im}\int_{\gamma}e_1(z)dz\right),
\,\,\,\hspace*{3.4cm} \eps \ra 0\end{array}$\\
& \\ Dykhne formula\footnotemark[20] & \\
\begin{tabular}{l}
Real symmetric\\dim$Q(t){\cal H}=1$
\end{tabular}
&$\;\;\left(1+{\cal O}
(\eps)\right)\exp\left(\frac{2}{\eps}\mbox{ Im}\int_{\gamma}e_1(z)dz\right)
,\;\; \eps \ra 0$\\
& \\ Geometrical prefactor\footnotemark[21],
\footnotemark[13],\footnotemark[22] & \\
\begin{tabular}{l}
Complex hermitian\\dim$Q(t){\cal H}=1$
\end{tabular}
&$\;\;\left(1+{\cal O} (\eps)\right)
\exp\left(\frac{2}{\eps}\mbox{ Im}\int_{\gamma}e_1(z)dz+
2\mbox{Im}\theta\right)
,\;\; \eps \ra 0$\\
& \\ Landau-Zener formula\footnotemark[23],\footnotemark[24] & \\
\begin{tabular}{l}
Complex hermitian\\dim$Q(t){\cal H}=1$\\
Avoided crossing:\\$e_2(t)-e_1(t)\stackrel{\delta\ra 0}{\simeq}$ \\
$\sqrt{a^2t^2+2ct\delta+b^2\delta^2}$
\end{tabular}
&\begin{tabular}{l}$\left(1+{\cal O} (\delta)+{\cal O}
(\eps)\right)\exp\left(-\frac{\pi\delta^2}{2\eps}
\left(\frac{b^2}{a}-\frac{c^2}{a^3}\right)
(1+{\cal O} (\delta))\right)$,\\
$\hspace*{7.3cm}\eps,\delta\ra 0$\end{tabular}\\
& \\ Accurate formula\footnotemark[25] & \\
\begin{tabular}{l}
Complex hermitian\\dim$Q(t){\cal H}=1$
\end{tabular}
&\begin{tabular}{l}$\left(1+{\cal O} \left(\e{-\kappa /\eps}\right)\right)
\exp\left(\frac{2}{\eps}\mbox{Im}\int_{\gamma}e_1^*(z,\eps)dz+
2\mbox{ Im}\theta^*(\eps)\right)$,\\
$\hspace*{6.4cm}\;\;\kappa>0,\;\; \eps \ra 0$
\end{tabular}\\
\hline
\end{tabular}
\normalsize
\\ \\ \\
\noindent
{\bf Remarks:}
We can express the decay rate as
\be
2\mbox{ Im}\int_{\gamma_j}e_1(z)dz=
2\mbox{Im}\int_{t}^{z_j}(e_1(z)-e_2(z))dz=2\mbox{ Im}
\Phi(z_j)<0
\ee
with the integration path from $t$ to $z$
inside $\gamma_j$. There is a geometric interpretation
of the Stokes lines as particular geodesics in a
metric naturally associated with the
quadratic differential $\rho(z)d^2z$ \footnotemark[13]. In this metric
$|\mbox{ Im}\Phi(z_j)|$ is the distance of
$z_j$ to the real axis, and always in this metric
it is the nearest complex-eigenvalue crossing in
the upper half-plane to the real axis.
Except for the real case and the
Landau-Zener formula, we have a nontrivial prefactor
of geometric nature which is
\be
0\leq \left|\bra\chi_+|W(+\infty,t)
W(t|\gamma)W(t,-\infty)\chi_-\ket\right|^2
\leq \|W(t|\gamma)\|^2
\ee
and which reduces to $\exp\left(2\mbox{Im}\theta\right)$
when dim$Q(t){\cal H}=1$.
The geometric nature of the prefactor is evident in (28):
we transport the vector $\chi_-$
along the real axis from $-\infty$ to $t$,
then along the path $\gamma$ around the complex-eigenvalue crossing, and
finally again along the real axis up to $+\infty$. We get
a vector in $(\un -Q(+\infty)){\cal H}$ and we
compare this transported vector with $\chi_+$.
The geometric prefactor was measured recently in a spin
experiment\footnotemark[22].
The case of the avoided crossing leading to the
Landau-Zener formula is important in Physics.
The last row of the Table 1 yields an asymptotic formula accurate
up to exponentially small
{\it relative}
errors. The expressions $e_1^*$ and $\theta^*$ are
defined as $e_1$ and $\theta$ with $H_{q^*}$ in
place of $H$ (see comment 1 on the adiabatic theorem).
\\ \\ \\
\hspace*{.5cm}{\bf Table 2.} Asymptotic transition
probability in non generic cases.
\\ \\ \hspace*{.5cm}
\small
\begin{tabular}{ll} \hline
\multicolumn{1}{c}{\bf Characteristics of $H(t)$} & \multicolumn{1}{c}
{{\bf Transition probability} ${\cal P}(\eps,\chi_-,\chi_+)$}
\\ \hline
& \\ High order degeneracy\footnotemark[26],\footnotemark[27] & \\
\begin{tabular}{l}
Real symmetric\\dim$Q(t){\cal H}=
1$\\$\rho(z)\simeq (z-z_1)^n$\\$H(z)\simeq (z-z_1)^m$, $2m\leq n$
\end{tabular}
&$\begin{array}{l}
\left(4\sin^2\left(\frac{\pi (n-2m)}{2(n+2)}\right)+{\cal O}
\left( \eps^p \right)\right)\times\\
\exp\left(\frac{2}{\eps}\mbox{Im}\int_{t}^{z_1}(e_1(z)-e_2(z))dz\right),
\;\;p>0,\;\;\eps \ra 0 \end{array}$\\
& \\ St\"uckelberg oscillations\footnotemark[28],\footnotemark[25] & \\
\begin{tabular}{l}
Complex hermitian\\dim$Q(t){\cal H}=1$\\ $q$ simple zeros $z_j$\\
of $\rho(z)$ contribute
\end{tabular}
&$\begin{array}{l}
\left|\sum_{j=1}^{q}
\exp\left(-i\theta_j-\frac{i}{\eps}
\int_{\gamma_j}e_1(z)dz\right)\right|^{2}\;+\\
{\cal O} (\eps)\exp\left(-\frac{2}{\eps}|\mbox{ Im}
\int_{\gamma_1}e_1(z)dz|\right),\;\;
\eps \ra 0\end{array}$\\
& \\ Conjugated effects\footnotemark[26] & \\
\begin{tabular}{l}
Real symmetric\\dim$Q(t){\cal H}=1$\\$q$ zeros $z_j$ characterized\\
by $(n_j,m_j)$\\ as above contribute
\end{tabular}
&$\begin{array}{l}
\left|\sum_{j=1}^{q}A_j(n_j,m_j)
\exp\left(-\frac{i}{\eps}
\int_{t}^{z_j}(e_1(z)-e_2(z))dz\right)\right|^{2}\;+\\
{\cal O} (\eps^{p})
\exp\left(-\frac{2}{\eps}|\mbox{Im}\int_{t}^{z_1}(e_1(z)-e_2(z))dz|\right),\\
A_j(n_j,m_j)=2\sigma_j
\sin\left(\frac{\pi (n_j-2m_j)}{2(n_j+2)}\right),\;\;
\sigma_j=\pm 1,\;\; \eps \ra 0 \end{array}$\\
& \\ Accurate formula\footnotemark[25] & \\
\begin{tabular}{l}
Complex hermitian\\dim$Q(t){\cal H}=
1$\\ $q$ simple zeros $z_j$\\of $\rho(z)$ contribute
\end{tabular}
&$\begin{array}{l}
\left|\sum_{j=1}^{q}
\exp\left(-i\theta_j^{*}(\eps)-\frac{i}{\eps}
\int_{\gamma_j}e_1^{*}(z,\eps)dz\right)
\right|^{2}\;+\\
{\cal O} \left(\e{-\kappa /\eps}\right)
\exp\left(-\frac{2}
{\eps}|\mbox{ Im}\int_{\gamma_1}e_1(z)dz|\right),\;\;
\kappa > 0,\;\;\eps \ra 0 \end{array}$\\
\hline
\end{tabular}
\normalsize
\\ \\
In Table 2 we have given the results in the non
generic case, when there is an eigenvalue crossing,
which is a zero of
$\rho(z)$ of higher order and/or when several eigenvalue
crossings (say q) determine the asymptotics
of the transition probability.
These cases give rise to different prefactors and/or
interference phenomena in the transition
probability. The above outlined method must be slightly
modified. The conditions on the Stokes
lines, which are needed to prove the results below
are given in the quoted references.
\noindent
{\bf Remarks:}
For the precise definition of $\sigma_j$, as well as
the paths of integration in
the first and third line see \footnotemark[26]. Notice
also that in the second and third lines
\be
\mbox{Im}\int_{\gamma_j}e_1(z)dz <0
\;\;,\;\;\mbox{Im}\int_{t}^{z_j}(e_1(z)-e_2(z))dz<0
\ee
are independent of $j=1,\ldots,q$. In the
last line the energy $e_1^{*}$, or the phases
$\theta_j^{*}$ are defined using $H_{q^{*}}$, and we have
\be
\mbox{Im}\int_{\gamma_j}e_1^{*}(z,\eps)dz=
\mbox{Im}\int_{\gamma_j}e_1(z)dz +
{\cal O}(\eps^{2})
\ee
In the discussion the eigenvalue-crossings in the upper half-plane only
play a role, because we consider the transition from the lower energy-level
to the higher energy-level.
Finally we mention that these methods, the iteration
scheme and the analysis in the complex plane, can be applied to
study similar problems,
like the semi-classical limit of the above barrier reflection
coefficient for
the stationary Schr\"odinger equation, or the adiabatic
invariant of a time-dependent
classical oscillator \footnotemark[25].
\\ \\
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\normalsize
\end{document}