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Physical Review A

Analytical theory for optimal control of multiphoton transitions

Ilya Grigorenko, Martı́n E. Garcia

Published Jul 17, 2006• 3 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.74.013404

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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.
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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.
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Using optimal control, we establish and link the ultimate bounds in time (referred to as the quantum speed limit) and energy of two- and three-level quantum nonlinear systems which feature 1:2 resonance. Despite the unreachable complete inversion, by using the Pontryagin maximum principle, we determine the optimal time, pulse area, or energy for a given arbitrary accuracy. We show that the third-order Kerr terms can be absorbed in the detuning in order to lock the dynamics to the resonance. In the two-level problem, we determine the nonlinear counterpart of the optimal $\ensuremath{\pi}$-pulse inversion for a given accuracy. In the three-level problem, we obtain an intuitive pulse sequence similar to the linear counterpart but with different shapes. We prove the (slow) logarithmic increasing of the optimal time as a function of the accuracy.