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arXiv:quant-ph/0309197 · Journal of Physics B Atomic Molecular and Optical Physics

Analytical solution of the optimal laser control problem in two-level systems

Martı́n E. Garcia, Ilia Grigorenko

Published Jun 8, 2004• 13 citations
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Abstract

The optimal control of two-level systems by time-dependent laser fields is studied using a variational theory. We reformulate the optimal control problem in terms of differential equations, which have to be fulfilled by the optimal fields. We obtain general analytical expressions for the optimal pulse shapes leading to global maximization or minimization of different physical quantities in two-level systems.

DOI
10.1088/0953-4075/37/12/011

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The optimal control of two-level systems by time-dependent laser fields is studied using a variational theory. We reformulate the optimal control problem in terms of differential equations, which have to be fulfilled by the optimal fields. We obtain general analytical expressions for the optimal pulse shapes leading to global maximization or minimization of different physical quantities in two-level systems.
We present a variational theory for the optimal control of quantum systems with relaxation over a finite time interval. In our approach, which is a nontrivial generalization of previous formulations and which contains them as limiting cases, the optimal control field fulfills a high-order Euler-Lagrange differential equation, which guarantees the uniqueness of the solution. We solve this equation numerically and also analytically for some limiting cases. The theory is applied to two-level quantum systems with relaxation, for which we determine quantitatively how relaxation effects limit the control of the system.
In this work, we investigate how and to which extent a quantum system can be driven along a prescribed path in Hilbert space by a suitably shaped laser pulse. To calculate the optimal, i.e., the variationally best pulse, a properly defined functional is maximized. This leads to a monotonically convergent algorithm which is computationally not more expensive than the standard optimal-control techniques to push a system, without specifying the path, from a given initial to a given final state. The method is successfully applied to drive the time-dependent density along a given trajectory in real space and to control the time-dependent occupation numbers of a two-level system and of a one-dimensional model for the hydrogen atom.
We study the potential for optimal control of a symmetric double quantum-dot structure interacting with a single pulsed electromagnetic field. We first use the rotating wave and resonant approximations and reduce the dynamics of the system to that of a degenerate three-level $\ensuremath{\Lambda}$-type system. We also formulate the optimal control problem in terms of differential equations that have to be fulfilled by the optimal electromagnetic fields. We then obtain general analytical expressions for the optimal pulse shapes that lead to global maximization of the final population of the target state and of the time-averaged population of the target state in the quantum-dot structure.
Using the Floquet formalism and the adiabatic approximation we develop a theory for optimal control of a multilevel quantum system interacting with an external electric field under multiphoton resonance conditions. Optimal solutions for the case of control over a time interval are presented. We derive a differential equation to determine the shape of the optimal pulse, and show how the order of the multiphoton resonance affects the solution. In particular, we demonstrate that optimal control in the case of two-photon resonance and using only adiabatically shaped control pulses represents a special degenerate case, for which the shape of the pulse cannot be optimized. Thus, for this particular case the final state of the controlled system is completely determined by the pulse energy.
A previously reported method for shaping electromagnetic field pulses to achieve chemical selectivity is extended and applied to a simple multiple level model system. The pulse shaping approach is based on optimal control theory, where both the time-dependent Schrödinger equation and the constant pulse energy are used as constraints on the variational scheme. A conjugate gradient direction method is used to direct the convergence of the iterative process used to calculate the optimum pulse shape. The method is applied to a five-level system interacting with an optical (laser) field. Results demonstrating selectivity and stability are compared to those of other recent related investigations.
Using results from quantum filtering theory and methods from classical control theory, we derive an optimal control strategy for an open two-level system (a qubit in interaction with the electromagnetic field) controlled by a laser. The aim is to optimally choose the laser's amplitude and phase in order to drive the system into a desired state. The Bellman equations are obtained for the case of diffusive and counting measurements for vacuum field states. A full exact solution of the optimal control problem is given for a system with simpler, linear, dynamics. These linear dynamics can be obtained physically by considering a two-level atom in a strongly driven, heavily damped, optical cavity.