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Physical Review B

Optimal control of a symmetric double quantum-dot nanostructure: Analytical results

Spyridon G. Kosionis, A. F. Terzis, Emmanuel Paspalakis

Published May 7, 2007• 38 citations
Publisher

Abstract

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.

DOI
10.1103/physrevb.75.193305

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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 \ensuremathΛ\ensuremath{\Lambda}\ensuremathΛ-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.
We theoretically investigate the coherent transfer of one electron between the ground states of a double coupled quantum dot structure. The coherent transfer of the electron is externally controlled by applied electromagnetic fields with on- or close-resonance driving frequencies and various shapes and duration. We derive the analytical expressions for the parameters of the external fields by approximating the quantum dot system as a three-level Λ-type system. The analytical solutions are compared with numerical results and good agreement is found. The control methods developed here are applicable in symmetric and asymmetric quantum dot nanostructures.
The optimal control of the path to a specified final state of a quantum-mechanical system is investigated. The problem is formulated as a minimization problem over appropriate function spaces, and the well-posedness of this problem is is established by proving the existence of an optimal solution. A Lagrange-multiplier technique is used to reduce the problem to an equivalent optimization problem and to derive necessary conditions for a minimum. These necessary conditions form the basis for a gradient iterative procedure to search for a minimum. A numerical scheme based on finite differences is used to reduce the infinite-dimensional minimization problem to an approximate finite-dimensional problem. Numerical examples are provided for final-state control of a diatomic molecule represented by a Morse potential. Within the context of this optimal control formulation, numerical results are given for the optimal pulsing strategy to demonstrate the feasibility of wave-packet control and finally to achieve a specified dissociative wave packet at a given time. The optimal external optical fields generally have a high degree of structure, including an early time period of wave-packet phase adjustment followed by a period of extensive energy deposition to achieve the imposed objective. Constraints on the form of the molecular dipole (e.g., a linear dipole) are shown to limit the accessibility (i.e., controllability) of certain types of molecular wave-packet objectives. The nontrivial structure of the optimal pulse strategies emphasizes the ultimate usefulness of an optimal-control approach to the steering of quantum systems to desired objectives.
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.
We present an optimal and robust quantum control method for efficient population transfer in asymmetric double quantum-dot molecules. We derive a long-duration control scheme that allows for highly efficient population transfer by accurately controlling the amplitude of a narrow-bandwidth pulse. To overcome fluctuations in control field parameters, we employ a frequency-domain quantum optimal control theory method to optimize the spectral phase of a single pulse with broad bandwidth while preserving the spectral amplitude. It is shown that this spectral-phase-only optimization approach can successfully identify robust and optimal control fields, leading to efficient population transfer to the target state while concurrently suppressing population transfer to undesired states. The method demonstrates resilience to fluctuations in control field parameters, making it a promising approach for reliable and efficient population transfer in practical applications.