2 research outputs found
Loading a Bose-Einstein Condensate onto an Optical Lattice: an Application of Optimal Control Theory to The Non Linear Schr\"odinger Equation
Using a set of general methods developed by Krotov [A. I. Konnov and V. A.
Krotov, Automation and Remote Control, {\bf 60}, 1427 (1999)], we extend the
capabilities of Optimal Control Theory to the Nonlinear Schr\"odinger Equation
(NLSE). The paper begins with a general review of the Krotov approach to
optimization. Although the linearized version of the method is sufficient for
the linear Schr\"odinger equation, the full flexibility of the general method
is required for treatment of the nonlinear Schr\"odinger equation. Formal
equations for the optimization of the NLSE, as well as a concrete algorithm are
presented. As an illustration, we consider a Bose-Einstein condensate initially
at rest in a harmonic trap. A phase develops across the BEC when an optical
lattice potential is turned on. The goal is to counter this effect and keep the
phase flat by adjusting the trap strength. The problem is formulated in the
language of Optimal Control Theory (OCT) and solved using the above
methodology. To our knowledge, this is the first rigorous application of OCT to
the Nonlinear Schr\"odinger equation, a capability that is bound to have
numerous other applications.Comment: 11 pages, 4 figures, A reference added, Some typos correcte
Optimal Control of Quantum Dissipative Dynamics: Analytic solution for cooling the three level system
We study the problem of optimal control of dissipative quantum dynamics.
Although under most circumstances dissipation leads to an increase in entropy
(or a decrease in purity) of the system, there is an important class of
problems for which dissipation with external control can decrease the entropy
(or increase the purity) of the system. An important example is laser cooling.
In such systems, there is an interplay of the Hamiltonian part of the dynamics,
which is controllable and the dissipative part of the dynamics, which is
uncontrollable. The strategy is to control the Hamiltonian portion of the
evolution in such a way that the dissipation causes the purity of the system to
increase rather than decrease. The goal of this paper is to find the strategy
that leads to maximal purity at the final time. Under the assumption that
Hamiltonian control is complete and arbitrarily fast, we provide a general
framework by which to calculate optimal cooling strategies. These assumptions
lead to a great simplification, in which the control problem can be
reformulated in terms of the spectrum of eigenvalues of , rather than
itself. By combining this formulation with the Hamilton-Jacobi-Bellman
theorem we are able to obtain an equation for the globaly optimal cooling
strategy in terms of the spectrum of the density matrix. For the three-level
system, we provide a complete analytic solution for the optimal
cooling strategy. For this system it is found that the optimal strategy does
not exploit system coherences and is a 'greedy' strategy, in which the purity
is increased maximally at each instant.Comment: 9 pages, 3 fig