Inverse problem for the Landau-Zener effect
D. A. Garanin \And R. Schilling
We consider the inverse Landau-Zener problem which consists in finding the energy-sweep functions $W(t)\equiv \varepsilon_1(t)-\varepsilon_2(t)$ resulting in the required time dependences of the level populations for a two-level system crossing the resonance one or more times during the sweep. % We find sweep functions of particular forms that let manipulate the system in a required way, including complete switching from the state 1 to the state 2 and preparing the system at the exact ground and excited states at resonance.
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lzinvepl.tex
%G12957 Submitted 18 February 2002


\documentstyle[psfig]{europhys}
\input euromacr
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\hspace{-1.7ex}\raisebox{-0.65ex}{$\sim$}\, }

\newcommand{\lesssim}{
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\input{tcilatex}

\begin{document}

%\euro{29}{6}{1-$\infty$}{1995}
\euro{XX}{X}{1-$\infty$}{2002} \Date{1 February 2002}
\shorttitle{D. A. GARANIN \And R. SCHILLING: INVERSE PROBLEM FOR
THE LANDAU-ZENER EFFECT}

\bibliographystyle{prsty}

\title{Inverse problem for the Landau-Zener effect}
%
\author{D. A. Garanin \And R. Schilling}
%
\institute{Institut f\"ur Physik, Universit\"at Mainz, D-55099
Mainz, Germany}% (\institute)
%
\pacs{ \Pacs{03}{65.-w}{Quantum mechanics}
 \Pacs{75}{10.Jm}{Quantized spin models}
\Pacs{03}{67.Lx}{Quantum computation} }
%
%
\maketitle
%
\vspace{-1cm}
%
\begin{abstract}
We consider the inverse Landau-Zener problem which consists in
finding the energy-sweep functions $W(t)\equiv
\varepsilon_1(t)-\varepsilon_2(t)$ resulting in the required time
dependences of the level populations for a two-level system
crossing the resonance one or more times during the sweep.
%
We find
sweep functions of particular forms that let manipulate the system
in a required way, including complete switching from the state 1
to the state 2 and preparing the system at the exact ground and
excited states at resonance.
\end{abstract}

\vspace{-1cm}
 The Landau-Zener (LZ) effect
\cite{lan32,zen32,dobzve97}, which consists in the
quantum-mechanical transition between the two time-dependent
levels of a two-level system (states 1 and 2) caused by crossing
the resonance, is a well known phenomenon in many areas of
physics.
%
Well known applications of the LZ effect are those to molecular
collision and  dissociation\cite{lanlif3,crohug77}.
%
Recently, LZ effect has been used to measure the tunnel splitting
in molecular magnets whereby detecting topological interference
effects \cite{werses99} which were predicted theoretically
\cite{losdivigri92,delhen92,garg93}.
%
If the system was in state 1 before crossing the resonance, then,
for a constant sweep rate $v$, the probability to stay in this
state after crossing the resonance is given by
%
\begin{equation}\label{LZProbab}
%
P=\exp \left( -\frac{\pi \Delta ^{2}}{2\hbar v}\right),
%
\end{equation}
%
where $v\equiv  |\dot{W}(t)|$, $W(t)\equiv
\varepsilon_{1}(t)-\varepsilon _{2}(t)$, $\varepsilon
_{1,2}(t)=\langle \psi _{1,2}|\hat{H}|\psi _{1,2}\rangle$, $\Delta
\equiv 2\langle \psi _{1}|\hat{H}|\psi _{2}\rangle$, $W(t)$
satisfies $W(\pm \infty )=\pm \infty $ and $\Delta $ is the tunnel
level splitting.
%
For {\em fast} energy sweep rates $v$ the system spends too little
time in the vicinity of the resonance so that the tunneling matrix
element $\langle \psi _{1}|\hat{H}|\psi_{2}\rangle $ cannot bring
the system into the state 2 (say, onto the other side of a
potential barrier), and the system remains in the initial state 1.
%
For {\em slow} sweep rates the probability to remain in the
initial state becomes exponentially small, and the system travels
into the state 2 remaining on the lower {\em exact} energy term
$\varepsilon _{-}(t)$ of
\begin{equation}
\varepsilon _{\pm }(t)=\frac{1}{2}\left[ \varepsilon
_{2}(t)+\varepsilon _{1}(t)\pm \sqrt{W^{2}(t)+\Delta ^{2}}\right]
. \label{epspmDef}
\end{equation}
%(see Fig.\ \ref{lzi-termosc}$a$)
The corresponding {\em adiabatic} time dependence of the
probability to be in state 1 is given by
\begin{equation}
|c_{1}(t)|^{2}=|\langle \psi _{1}|\psi _{-}(t)\rangle |^{2}=\frac{1}{2}\left( 1-%
\frac{W(t)}{\sqrt{W^{2}(t)+\Delta ^{2}}}\right) .
\label{ProbSlow}
\end{equation}
In the case where $\hbar v\sim \Delta ^{2}$ and $v$ is constant,
the solution of the Schr\"{o}dinger equation for $|c_{1}(t)|^{2}$
 can be found analytically \cite{lan32,zen32,dobzve97} and it is oscillating at $t>0$
 %(see Fig.\ \ref{lzi-termosc}$b)$
 slowly approaching $P$ given by Eq.\
(\ref{LZProbab}) at $t\rightarrow \infty$.

%\begin{figure}[t]
%\unitlength1cm
%\begin{picture}(7,5)
%\psfig{file=lzi-term.eps,angle=-90,width=7.5cm}
%
%\end{picture}
%\begin{picture}(7,5)
%\psfig{file=lzi-osc.eps,angle=-90,width=7.5cm}
%
%\end{picture}
%\caption{\label{lzi-termosc} (a) -- $W$-dependence of exact energy
%eigenvalues of a two-state system;
%
%(b) -- Time dependence of the
%quantum probability $|c_{1}|^{2}$ in course of the linear energy
%sweep ($\hbar=\Delta=v=1$).
%
%The dashed line shows the asymptotic value of $|c_{1}|^{2}$ given
%by $P$ of Eq.\ (\protect\ref{LZProbab}). }
%\end{figure}

The known results above are valid for the linear energy sweep,
$W(t)=vt.$ One can consider, however, other sweep functions $W(t)$
that behave {\em nonlinearly} in the vicinity of the resonance,
$W\sim \Delta $.
%
If the tunneling resonance is not too narrow and its position is
well defined, nonlinear sweeps of a controlled form can be
realized experimentally.
%
Although no general analytical solution of the Schr\"{o}dinger
equation is known for the LZ problem with an arbitrary $W(t),$ one
can look for the functions $W(t)$ that result in a desired time
dependence of the probabillity $|c_{1}(t)|^{2}$, thus solving the
{\em inverse} LZ problem.
%
It is the main motivation of this Letter to study this inverse
problem which, for instance, will allow to determine $W(t)$ needed
to manipulate and to prepare a quantum system in a specific state.

One class of solutions of the time-dependent Schr\"odinger
equation are smooth, non-oscillating functions of time that
describe the complete switching of the system from the state 1 to
the state 2, e.g., Eq.\ (\ref {ProbSlow}).
%
Another class of solutions are those starting in $\psi _{1}$ at
$t=-\infty $ off resonance and ending in a given superposition of
the exact states $\psi_{\pm }$ at resonance [$W(t)=0$ for $t\geq
0]$.
%
The most interesting variants of the latter are the system in the
ground state $\psi _{-}$ at resonance and in the excited state
$\psi_{+}$ at resonance.
%
These possibilities to manipulate the system in a required way
could prove to be useful for the quantum computing (QC).
%
Another interesting possibility is to sweep at a high rate
(nonlinearly in time) far from the resonance thus making the whole
process of crossing the resonance much shorter.
%
Exploring these new types of solution of the LZ problem is the aim
of this Letter.

To this end, we consider the Schr\"{o}dinger equation for the coefficients
in the wave function $\psi (t)=c_{1}(t)\psi _{1}+c_{2}(t)\psi _{2}$ that can
be written in the form
\begin{eqnarray}\label{pqEq}
\hbar \dot p(t) &=&-\frac{i}{2}\Delta q(t),\qquad \qquad \qquad
\quad
p(-\infty )=1  \nonumber \\
\hbar \dot q(t) &=& i W(t)q(t)-\frac{i}{2}\Delta p(t),\qquad
q(-\infty )=0,\qquad W(-\infty )=-\infty
\end{eqnarray}
with $p(t)\equiv e^{i\varepsilon _{1}t}c_{1}(t)$ and $q(t)\equiv
e^{i\varepsilon _{1}t}c_{2}(t)$.
%
We are looking for sweep functions $W(t)$ that result in a
required time dependence of the state populations.
%
 Before solving this inverse LZ problem
analytically, we will illustrate numerically the possibility of
time-symmetric solutions of Eq.\ (\ref{pqEq}) going from $
p(-\infty )=1$ to $p(\infty )=0$ for strongly nonlinear sweep
functions $ W(t)$.
%
 The simplest of these cases is the cubic-parabola sweep with $%
W(t)=-2.5t+1.83t^{3}$ where we have set $\Delta =\hbar =1$ in Eq.\ (\ref
{pqEq}).
%
The time dependence of the probability $|p(t)|^2$ shown in Fig.\
\ref{lzi-cub} can be fitted by $|p|^{2}(t)=[1-\tanh (\sinh
(t))]/2$ which is, however, not the analytical solution of Eq.\
(\ref{pqEq}).

\begin{figure}[t]
\unitlength1cm
\begin{picture}(7,5)
\psfig{file=lzi-cubw.eps,angle=-90,width=7.5cm}
%
\end{picture}
\begin{picture}(7,5)
\psfig{file=lzi-cubp.eps,angle=-90,width=7.5cm}
%
\end{picture}
\caption{The cubic-parabola energy sweep with $W(t)=-2.5t+1.83t^3$
and $\Delta=\hbar=1$.
%
Time dependence of the probability $|c_1(t)|^2=|p(t)|^2$ can be
fitted with $|p|^2=[1-\tanh( \sinh t)]/2$ with graphic accuracy.
%
The function $|p|^2=[1-\tanh t]/2 $ is shown for comparison. }
\label{lzi-cub}
\end{figure}

To find analytical solutions of the inverse LZ problem, one should
rewrite Eq.\ (\ref{pqEq}) in terms of the probability $m\equiv
|p|^{2}$ and some phase variable.
%
We thus define
\begin{equation}
p=\sqrt{m}\exp (i\varphi _{p}),\qquad q=\sqrt{1-m}\exp (i\varphi _{q})
\label{pqviamphi}
\end{equation}
and substitute it in Eq.\ (\ref{pqEq}).
%
Evidently the global phase of the system
$(\varphi_{p}+\varphi_{q})/2$ is irrelevant and the resulting
Schr\"{o}dinger equation can be formulated in terms of $m$ and the
phase difference $\varphi =\varphi _{q}-\varphi _{p}$:
%
\begin{eqnarray}\label{mphiEq}
\hbar \dot m &=&\Delta \sqrt{m(1-m)}\sin \varphi   \nonumber \\
\hbar \dot\varphi &=&W+\frac{\Delta }{2}\frac{1-2m}{\sqrt{m(1-m)}}%
\cos \varphi .
\end{eqnarray}
%
This system of equations can be brought into a more elegant form by
introducing
\begin{equation}
x\equiv 2m-1,\qquad -1<x<1 , \label{xDef}
\end{equation}
%
\begin{equation}
y\equiv \arcsin x,\qquad -\pi /2<y<\pi /2,  \label{yDef}
\end{equation}
and the reduced variables
\begin{equation}
\tau \equiv \left( \Delta /\hbar \right) t,\qquad
\overline{W}\equiv W/\Delta   \label{tauDef}.
\end{equation}
Then Eqs.\ (\ref{mphiEq}) can be put into the reduced form
\begin{eqnarray}
\partial _{\tau }y &=&\sin \varphi   \nonumber \\
\partial _{\tau }\varphi  &=&\overline{W}-\tan y\cos \varphi .
\label{ypsiEqRed}
\end{eqnarray}
The boundary condition for this equation is $y(-\infty )=\pi /2$
whereas the value of $\varphi (-\infty )$ is irrelevant.
%
In general, far from the resonance $\varphi (\tau )$ strongly
oscillates with time and the derivative $\partial _{\tau }y$ is
not small.
%
Below we will consider special kinds of solutions of Eq.\
(\ref{ypsiEqRed}) which are characterized by a smooth,
non-oscillating dependence $\sin\varphi (\tau )$ and $\partial
_\tau{ }y(-\infty)=0$.
%
We will solve the inverse
problem for Eq.\ (\ref{ypsiEqRed}) which consists in finding $\overline{W}%
(\tau )$ resulting in a required function $x(\tau )$.
%
One easily obtains from Eq.\ (\ref{ypsiEqRed}) a general formula
\begin{equation}
\overline{W}(\tau )=\partial _{\tau }\varphi +\tan y\cos \varphi =\partial
_{\tau }^{2}y\left/ \cos \varphi \right. +\tan y\cos \varphi ,\qquad \cos
\varphi =\pm \sqrt{1-(\partial _{\tau }y)^{2}}.  \label{WbarRes}
\end{equation}
The sign in front of $\cos \varphi $ determines the sign of the
function $\overline{W}(\tau )$.
%
Since $|c_{1}(-\infty )|^{2}=1$ and thus $x(-\infty )=1$ and
$y(-\infty )=\pi /2$ one has $\tan y(-\infty )=\infty$.
%
To comply with the condition $W(-\infty )=-\infty $ one has to
choose, in general, the negative sign:
\begin{equation}
\overline{W}(\tau )=-\partial _{\tau }^{2}y\left/ \sqrt{1-(\partial _{\tau
}y)^{2}}\right. -\tan y\sqrt{1-(\partial _{\tau }y)^{2}}.  \label{WbarRes1}
\end{equation}

Before considering particular cases, we will comment on some
general properties of the functions $x(\tau )$ and
$\overline{W}(\tau )$.
%
 If $x(\tau )$ is an odd function of time
then $y(\tau )$ is also odd and $\partial _{\tau }y(\tau )$ is
even, whereas $\partial _{\tau }^{2}y(\tau )$ and $\tan y(\tau )$
are odd.
%
Then from the first of Eqs.\ (\ref{ypsiEqRed}) follows
that $\sin \varphi (\tau )$ is even.
%
If the sign of $\cos \varphi $ does not change, which is the
generic situation, $\cos \varphi (\tau )$ is even.
%
Then from Eq.\ (\ref{WbarRes}) follows that $\overline{W}(\tau )$
is odd.
%
In a similar way one can show that if $x(\tau )$ is even (the
system returns into the initial state $\psi _{1}$ after crossing
the resonance) then $\overline{W }(\tau )$ is even, too.

There is an important special case, however, which is realized if
the derivative $\partial _{\tau }y$ at some point $\tau _{0}$
attains its {\em maximal} value $|\partial _{\tau }y|=1$ which
follows from the first of Eqs.\ (\ref{ypsiEqRed}).
%
At this point $\cos \varphi $ turns to zero and changes its sign
to positive. The latter corresponds to the phase $ \varphi $ going
from the interval $\pi /2<\varphi <\pi $ into the interval $
0<\varphi <\pi /2$.
%
Note that $\partial _{\tau }^{2}y=0$ at $\tau =\tau _{0}$ thus
Eq.\ (\ref{WbarRes1}) does not diverge.
%
If $\tau _{0}=0$ then $\cos\varphi (\tau )$ is an odd function:
$\cos \varphi<0$ for $\tau<0$ and $\cos \varphi>0$ for $\tau>0$.
%
As a result, the parity of $\overline{W}(\tau)$ becomes inverse to
the parity of $x(\tau)$.
%
If $\tau _{0}\neq 0$ then for a symmetric function $x(\tau )$ the
similar happens at $\tau =-\tau_{0}$.
%
That is, $\cos \varphi (\tau)$ changes sign two times and is an
even function, thus the parity of $\overline{W}(\tau )$ coincides
with that of $x(\tau )$.
%
It should be noted that parity considerations for the direct LZ
problem do not apply: Symmetric forms of $\overline{W}(\tau )$ do
not necessarily result in symmetric solutions $x(\tau )$.
%(see Fig.\ \ref{lzi-termosc}$b$)
%
If, however, the solution $x(\tau )$ satisfies $x(\pm \infty)=\mp
1$ or $x(\pm \infty )=\pm 1$ and $\overline{W}(\tau)$ is time
symmetric, then $x(\tau)$ is also symmetric.
%
This follows from the time reversibility of the Schr\"{o}dinger
equation and the symmetry of the boundary values of $x(\tau )$.

Another implication of this analysis is the existence of the {\em
maximal} rate of the probability change.
%
We have seen that $|\partial _{\tau }y|\leq 1$, then from
$\partial _{\tau }x=\sqrt{1-x^{2}}\partial_{\tau }y$ follows
$|\partial _{\tau }x|\leq 1$.
%
This limitation is physically clear: Since tunneling between
$\psi_{1}$ and $\psi_{2}$ is caused by the matrix element $\Delta
\equiv 2\langle \psi_{1}|\hat{H}|\psi_{2}\rangle$, the latter sets
up the natural scale for the tunneling rate, see Eq.\
(\ref{tauDef}).
%
To completely tunnel from $\psi_{1}$ to $\psi_{2}$, the system
should spend some minimal time in the vicinity of the resonance,
that is, the sweep cannot be too fast.
%
Below we will discuss two most important families of the functions
$x(\tau )$ and corresponding $\overline{W}(\tau )$ which depend on
$\tau $ in the combination $u=a\tau$, the parameter $a$ being the
measure of the sweep rate.
%
We will see that there is a maximal value $a=a_{\max }$ for which
$|\partial _{\tau }y|=1$ at some $\tau =\tau _{0}$.
%
We will also see that although the functions $x(\tau )$ do
formally exist for $a>a_{\max }$, one cannot extend
$\overline{W}(\tau )$ into this region, as a result of the
physical limitation on the tunneling rate.
%
Thus solutions $x(\tau)$ with $a>a_{\max} $ are not realizable.

One of possible solutions is of the form
\begin{equation}
x=(-1)^{n}\frac{u^{n}}{\sqrt{1+u^{2n}}},\qquad u\equiv a\tau .  \label{xnSol}
\end{equation}
For $n$ odd it describes switching of the probability from $\psi
_{1}$ to $\psi _{2}$, whereas for $n$ even the system returns into
the initial state.
%
For $n=1$ the form of Eq.\ (\ref{xnSol}) corresponds to Eq.\
(\ref{ProbSlow}) by taking into account Eq.\ (\ref{xDef}) with
$m=|c_{1}|^{2}$.
%
However, the reader should note that Eq.\ (\ref{ProbSlow}) is
realized with the very slow sweep $\overline{W}(\tau )$ of an
arbitrary functional form, whereas Eq.\ (\ref{xnSol}) is valid for
a particular form of $\overline{W}(\tau )$ which is not
necessarily slow.
%
One obtains from Eqs.\ (\ref{yDef}) and (\ref{xnSol})
\begin{eqnarray}
\partial _{\tau }y &=&\frac{1}{\sqrt{1-x^{2}}}\partial _{\tau }x=(-1)^{n}%
\frac{anu^{n-1}}{1+u^{2n}}  \nonumber \\
\partial _{\tau }^{2}y &=&(-1)^{n}\frac{a^{2}nu^{n-2}\left[ n-1-(n+1)u^{2n}%
\right] }{\left( 1+u^{2n}\right) ^{2}}, \label{yders}
\end{eqnarray}
whereas $\tan y =x/\sqrt{1-x^{2}}=(-1)^n u^{n}$. The derivative
$\partial_{\tau }y$ has a chance to attain the value 1 at its
maximum that is determined by $\partial _{\tau }^{2}y=0$, i.e., at
\begin{equation}
u=u_{\max }=\pm \left( \frac{n-1}{n+1}\right) ^{1/(2n)}.  \label{umax}
\end{equation}
At this point one has
\begin{equation}
|(\partial _{\tau }y)_{\max }|=\left\{
\begin{array}{ll}
a, & n=1 \\
a\frac{n+1}{2}\left( \frac{n-1}{n+1}\right)^{(n-1)/(2n)}, & n>1,
\end{array}
\right.
\end{equation}
thus the condition
\begin{equation}
a\leq a_{\max }=\left\{
\begin{array}{ll}
1, & n=1 \\
\frac{2}{n+1}\left( \frac{n-1}{n+1}\right) ^{2n/(n-1)}, & n>1
\end{array}
\right.
\end{equation}
should be fulfilled. For $a<a_{\max }$ Eq.\ (\ref{WbarRes1}) yields
\begin{equation}
\overline{W}(\tau )=\frac{(-1)^{n-1}u^{n}}{\sqrt{1-\left[ anu^{n-1}/\left(
1+u^{2n}\right) \right] ^{2}}}\left[ 1-\frac{a^{2}n\left[
(2n+1)u^{2n-2}-(n-1)u^{-2}\right] }{\left( 1+u^{2n}\right) ^{2}}\right]
,\qquad u=a\tau .  \label{WbarsqrtnRes}
\end{equation}
This function is plotted for $n=1$ and $n=5$ and different values
of the parameter $a$ in Figs.\ \ref{lzi-n1and5}$a$ and
\ref{lzi-n1and5}$b$.
%
For $a\rightarrow a_{\max }$ discontinuities in Eq.\
(\ref{WbarsqrtnRes}) are formed at $u=\pm u_{\max }$.
%
These discontinuities result from the invariably negative choice
for $\cos \varphi $.
%
For $a=a_{\max }$ the sign of $\cos \varphi $ can change at
$u=u_{\max }$ and then these discontinuities
disappear.
%
This form of $\overline{W}(\tau )$ includes the factor ${\rm
sgn}\left[ \left( 1+u^{2n}\right)^{2}-\left( anu^{n-1}\right)
^{2}\right] $, in addition to Eq.\ (\ref{WbarsqrtnRes}).
%
Note that Eq.\ (\ref{WbarsqrtnRes}) cannot be extended into the
region $a>a_{\max }$ preserving $\overline{W}(\tau )$ real, in
contrast to Eq.\ (\ref{xnSol}).
%
The small- and large-argument forms of $\overline{W}(\tau )$ are
\begin{equation}
\overline{W}(\tau )\propto (-1)^{n-1}\left\{
\begin{array}{ll}
u^{n}, & |u|\gg 1 \\
u^{n-2}, & |u|\ll 1,\qquad n>1
\end{array}
\right. .  \label{WnLims}
\end{equation}
In the case $n=1$, Eq.\ (\ref{WbarsqrtnRes}) simplifies to
\begin{equation}
\overline{W}(\tau )=\frac{u}{1+u^{2}}\frac{1-3a^{2}+2u^{2}+u^{4}}{\sqrt{%
1-a^{2}+2u^{2}+u^{4}}}.  \label{WbarsqrtRes}
\end{equation}
One can see that for $u\gg 1$ the function $\overline{W}(\tau )$ describes a
linear energy sweep with the velocity $v=a$.
%
For $u\sim 1$ the form of $\overline{W}(\tau )$ is nonlinear and
for $a<a_{\max }=1$ the resonance is crossed three times,
including $u=0$.
%
For $a=1$ Eq.\ (\ref{WbarsqrtRes}) yields the even function
\begin{equation}
\overline{W}(\tau )=\frac{2-2\tau ^{2}-\tau ^{4}}{(1+\tau ^{2})\sqrt{2+\tau
^{2}}}  \label{Wtaun1a1}
\end{equation}
This sweep crosses the resonance two times, in the forward and
backward directions, which means that the system is in the {\em
excited} state $\psi_2$ at the end of the sweep at $t=\infty$.
%
For $a<1$ and far from the resonance Eq.\ (\ref{WbarsqrtRes})\ \
has the form
\begin{equation}
W(t)=\Delta \overline{W}(\tau )=\Delta a\tau =\left( \Delta
^{2}a/\hbar \right) t,
\end{equation}
thus the corresponding linear sweep rate is $v=\Delta^{2}a/\hbar$.
%
The maximal sweep rate that ensures the complete switching to
$\psi _{2}$ is realized at $a=1$ is $v_{\max }=\Delta ^{2}/\hbar$.
%
For a comparison, in the regular LZ effect with the probability to
stay in the initial state given by Eq.\ (\ref{LZProbab}) the sweep
rate $v\ll \pi \Delta ^{2}/(2\hbar ) $ is required for a complete
switching.
%
For $v=\Delta^{2}/\hbar $ the probability to stay in the initial
state is still $P=0.208$.

\begin{figure}[t]
\unitlength1cm
\begin{picture}(7,5)
%
\psfig{file=lzi-swn1.eps,angle=-90,width=7.5cm}
%
\end{picture}
%
\begin{picture}(7,5)
%
\psfig{file=lzi-swn5.eps,angle=-90,width=7.5cm}
%
\end{picture} \\
%
\begin{picture}(7,5)
%
\psfig{file=lzi-cpln1.eps,angle=-90,width=7.5cm}
%}
%
\end{picture}
%
\begin{picture}(7,5)
%
\psfig{file=lzi-cpln5.eps,angle=-90,width=7.5cm}
%
\end{picture}
%
\caption{ \label{lzi-n1and5}
%
$a$ -- Energy sweep function $\overline{W}(\tau)$ of Eq.\
(\protect\ref{WbarsqrtnRes}) with $n=1$ [or of Eq.\
(\protect\ref{WbarsqrtRes})] for different values of the sweep
rate $a$.
%
$b$ -- The same with $n=5$.
%
$c$ -- The population of the exact excited level, $|c_+|^2$, for
$n=1$ and different values of $a$.
%
$d$ -- The same for $n=5$.
%
Dashed lines in all graphs correspond to $a=a_{\rm max}$ and
$\overline{W}(\tau)$ without jumps [see, e.g., Eq.\
(\protect\ref{Wtaun1a1})].
 }
\end{figure}



Another solution that describes switching from state 1 to state 2
is
\begin{equation}
x=-\tanh u,\qquad u=a\tau .
\end{equation}
Here the sweep function determined from Eq.\ (\ref{WbarRes1}) reads
\begin{equation}
\overline{W}(\tau )=\frac{\sinh u}{\sqrt{1-a^{2}/\cosh ^{2}u}}\left[ 1-\frac{%
2a^{2}}{\cosh ^{2}u}\right] .  \label{Wtaua}
\end{equation}
In the range $1/\sqrt{2}<a<1$ the function $\overline{W}(\tau )$
changes its sign three times and its derivative at $\tau =0$ is
negative.
%
With $a\to 1$ this derivative increases unlimitedly and a jump of $%
\overline{W}(\tau )$ at $\tau =0$ is formed.
%
For $a=a_{\max }=1$ the solution with eliminated jump is an even
function
\begin{equation}
\overline{W}(\tau )=\frac{2}{\cosh \tau }-\cosh \tau .  \label{Wtau}
\end{equation}
This sweep function crosses the resonance two times.
%
 Again, Eq.\ (\ref {Wtaua}) cannot be continued into the region $a>1$.

Now we return to the solution for the probability $x$ described by Eq.\ (\ref
{xnSol}) and analyze its behavior for large $n$.
%
It has a form of three plateaus $x\cong 1$ for $u\lesssim -1$,
$x\cong 0$ for $-1\lesssim u\lesssim 1$, and $x\cong -1$ for
$1\lesssim u$ with two narrow transition regions between them.
%
In the central region $-1\lesssim u\lesssim 1$ one has
$|c_{2}|^{2}\cong |c_{1}|^{2}=(1+x)/2\cong 1/2$, i.e., the system
is in the mixture of the states $\psi _{1}$ and $\psi _{2}$ with
equal weights.
%
The sweep function $\overline{W}(\tau )$ of Eq.\
(\ref{WbarsqrtnRes}) is fast approaching the resonance at $u\cong
-1$ and remaining in the close
vicinity of the latter until $u\cong 1$.
%
Since the probabilities $|c_{1}|^{2}\cong |c_{2}|^{2}\cong 1/2$ do
not oscillate in the range $-1\lesssim u\lesssim 1$ (unlike the
case in which the system is instantaneously brought into
resonance), one is lead to the conclusion that the system is in
one of the exact quantum states $\psi _{-}$ or $\psi _{+}$.
%
It is naturally of interest to question in which of these exact
states does the system come under the influence of the energy
sweep described by Eq.\ (\ref{WbarsqrtnRes}).
%
We will see below that both variants can be realized.
%
This opens a possibility to prepare the system in the ground or
excited state in
resonance by the appropriate choice of $\overline{W}(\tau )$.
%
Indeed, for $n\geq 3$ one has $\overline{W}(\tau )=\partial_{\tau
}y=\partial _{\tau }^{2}y=0$ at $\tau =0, $ thus from Eq.\
(\ref{ypsiEqRed})\ one can see that the sweep can be stopped at
$\tau =0$ and the system will remain in this state.
%
To demonstrate this behavior one should rewrite the wave function
of the system $\psi =c_{1}(t)\psi_{1}+c_{2}(t)\psi_{2}$ determined
above in the basis of the exact states
\begin{equation}
\psi_{\pm }(t)=\frac{1}{\sqrt{2}}\left[ \pm K_{\pm }(t)\psi
_{1}+K_{\mp }(t)\psi _{2} \right] ,\qquad K_{\pm }(t)\equiv
\sqrt{1\pm \frac{\overline{W}(t)}{\sqrt{1+ \overline{W}^{2}(t)}}}
\label{psipm}
\end{equation}
corresponding to the time-dependent ``energy eigenvalues'' of Eq.\
(\ref{epspmDef})\ as $\psi =c_{+}(t)\psi _{+}(t)+c_{-}(t)\psi
_{-}(t)$.
%
The corresponding probabillities are given by
\begin{equation}
|c_{\pm }(t)|^{2}=\frac{1}{2}\left\{ 1\pm
\frac{\overline{W}(t)x(t)+\sqrt{1-x^{2}(t)} \cos \varphi(t)
}{\sqrt{1+\overline{W}^{2}(t)}}\right\} .  \label{cpm2}
\end{equation}
One can see that in resonance at $t=0$ ($\overline{W}=x=0$, $\cos
\varphi =\pm 1$) the situation depends on the sign of $\cos
\varphi$.
%
For sweep functions given by Eq.\ (\ref{WbarsqrtnRes}) with
$a<a_{\max }$ one has $\cos \varphi =-1$ and the system is in the
ground state ($|c_{-}(t)|^{2}=1$, $ |c_{+}(t)|^{2}=0)$.
%
For the special sweep functions with $a=a_{\max }$ one has $\cos
\varphi =1$ and the system is in the excited state
($|c_{-}(t)|^{2}=0,$ $|c_{+}(t)|^{2}=1)$, the transition from the
ground state $\psi_{-}(t )$ to the excited state $\psi _{+}(t) $
occuring for $n\gg 1$ in the vicinity of $u=u_{\max}$ of Eq.\
(\ref{umax}).
%
These features are illustrated for the sweep functions of Eqs.\
(\ref{WbarsqrtnRes}) and (\ref{Wtaun1a1}) in Figs.\
\ref{lzi-n1and5}$c$ and \ref{lzi-n1and5}$d$.
%
Note that for $n=1$ and $a=a_{\max}=1$ the sweep function of Eq.\
(\ref{Wtaun1a1}) returns to $-\infty$ at $\tau=\infty$ and the
system ends up in the excited state $\psi_+=\psi_2$.

To conclude, we have solved the inverse Landau-Zener problem for a
two-level system, i.e., we have established the form of the sweep
function $W(t)\cong \varepsilon_1(t)-\varepsilon_2(t)$ needed to
insure a given probability $|c_1(t)|^2$ to be in the state 1 at
time $t$.
%
We have discussed two examples for which the system performs a
{\em complete} transition from 1 to 2.
%
In contrast to the direct LZ problem with a linear sweep, this can
be accomplished by a {\em finite} sweeping rate.
%
Further we have found such sweep functions that prepare the system
in its exact ground state or in its exact excited state at
resonance.
%
It is not difficult to find $W(t)$ that prepare the system in an
{\em arbitrary} mixture of the exact states at resonance, which
can be potentially used in quantum computing based on tunneling
units.
%



%\bibliography{gar,tunneling}


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\end{thebibliography}



\end{document}

cond-mat0202314.extracted_bib
\begin{thebibliography}{1}

\bibitem{lan32}
{L. D. Landau}, Phys. Z. Sowjetunion {\bf 2},  46  (1932).

\bibitem{zen32}
C. Zener, Proc. R. Soc. London A {\bf 137},  696  (1932).

\bibitem{dobzve97}
{V. V. Dobrovitski and A. K. Zvezdin}, Europhys. Lett. {\bf 38},
377  (1997).

\bibitem{lanlif3}
{L. D. Landau and E. M. Lifshitz}, {\em Quantum {M}echanics}
(Pergamon, London,
  1965).

\bibitem{crohug77}
{D. S. F. Crothers and J. G Huges}, J. Phys. B {\bf 10},  L557
(1977).

\bibitem{werses99}
{W. Wernsdorfer and R. Sessoli}, Science {\bf 284},  133  (1999).

\bibitem{losdivigri92}
{D. Loss, D. P. DiVincenzo, and G. Grinstein}, Phys. Rev. Lett.
{\bf 69},  3232
   (1992).

\bibitem{delhen92}
{J. von Delft and C. L. Henley}, Phys. Rev. Lett. {\bf 69},  3236
(1992).

\bibitem{garg93}
{A. Garg}, Europhys. Lett. {\bf 22},  205  (1993).

\end{thebibliography}