deletions | additions       diff --git a/Optimal control.tex b/Optimal control.tex  index 2b1e931..05c6db0 100644  --- a/Optimal control.tex  +++ b/Optimal control.tex 
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status of its implementation, and describe some of the results that have been  obtained with it until now.  ``Quantum control'' \texit{Quantum control}  can be loosely defined as the manipulation of physical processes at the quantum level. We are concerned here with the theoretical  branch of this discipline, whose most general formulation is precisely  QOCT. This is, in fact, a particular case of the general mathematical 
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answer the following question: given a quantum process governed by a  Hamiltonian that depends on a set of parameters, what are the values of those  parameters that maximize a given observable that depends on the behaviour of  the system? In mathematical terms: let a set of parameters $u_1,\dots,u_M \equiv u$ determine the Hamiltonian of a system $\hat{H}[u,t]$, so that the evolution of the system also depends on the value taken by those parameters:  \begin{eqnarray} \begin{align}  {\rm i}\frac{{\rm d}}{{\rm d}t}\vert\psi(t)\rangle & =&  \hat{H}[u,t]\vert\psi(t)\rangle\,,  \\  \vert\psi(0)\rangle & =&  \vert\psi_0\rangle\,, \end{eqnarray} \end{align}  i.e. the solution of Schr{\"{o}}dinger's equation determines a map $u  \longrightarrow \psi[u]$ (QOCT can also be formulated for the density matrix, in terms of von Neumann's or Lindblad's equation, for example). Suppose we wish to optimize a  functional of the system $F=F[\psi]$. QOCT is about finding the extrema of 
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The first step is the selection of the parameters $u$, that constitute the \emph{search space}. Frequently, these parameters are simply the values that the \emph{control function} (typically, the electric field amplitude) takes at the time intervals that are used to discretize the propagation interval, i.e. it is a ``real time parametrization''. However, more sopnisticated parametrizations allow to fine-tune the search space, introducing constraints and penalties into the formulation.  Then, one must choose an algorithm for maximizing multi-dimensional  functions such as $G$. One possibility is the family of \emph{gradient-less} algorithms, that only require a procedure to compute the value of the function, and do not need the gradient. In this case, the previous equations are obviously not needed. One only has to propagate the system forwards in time, which is what octopus can do best. The value of the function $G$ can then be computed from the evolution of $\psi$ obtained with this propagation, and feed it into the optimization procedure. Octopus implements a few gradient-less algorithms. However, the most efficient optimizations can be obtained if the gradient information can be employed. In that case, we can use standard schemes, such as for example the family of conjugate-gradient algorithms, or the Broyden-Fletcher-Goldfarb-Shanno (BFGS) quasi-Newton scheme -- we use the implementation of these algorithms included in the GSL mathematical library~\cite{Galassi2009}. Some ad-hoc algorithms, developed explicitly for QOCT, exist. These may in some circumstances be faster than the general purpose ones. Some of those have also been implemented in Octopus (see, e.g. Refs~\cite{Zhu1998,Zhu1998a,Ohtsuki1999}).