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AIP conference proceedings

Optimal control equations for the one dimensional quantum harmonic oscillator under the influence of external dipole effects

Muzaffer Ayvaz, Metin Demіralp

Published Jan 1, 2012• 0 citations
Publisher

Abstract

This study focuses on the construction of the optimal control equations for one dimensional quantum harmonic oscillator under the influence of external dipol effects and the solution of these equations by using Fluctuationlessness Theorem and a recently developed scheme called Characteristic Evolutions Method. The dipole function of the system has been taken as odd cubic spatial polynomial. Optimal control equations of the system under consideration are constructed by using expectation values of the position and the momentum operators instead of the wave and costate evolutions. It is shown that, the resulting equations are systems of ordinary differential equations and there are infinitely many ODEs. The solution strategy is based on the approximation of the expectation values for the operator products in the sense of Fluctuationlessness Theorem.

DOI
10.1063/1.4771816

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This study focuses on the construction of the optimal control equations for one dimensional quantum harmonic oscillator under the influence of external dipol effects and the solution of these equations by using Fluctuationlessness Theorem and a recently developed scheme called Characteristic Evolutions Method. The dipole function of the system has been taken as odd cubic spatial polynomial. Optimal control equations of the system under consideration are constructed by using expectation values of the position and the momentum operators instead of the wave and costate evolutions. It is shown that, the resulting equations are systems of ordinary differential equations and there are infinitely many ODEs. The solution strategy is based on the approximation of the expectation values for the operator products in the sense of Fluctuationlessness Theorem.
In this study, expectation value dynamics of the quantum harmonic oscillator under the influence of the external dipol effects is discussed in some details. The dipole function of the system has been taken as an odd cubic spatial polynomial and the external field amplitude has been assumed to be known. Instead of the wave function of the system, expectation values of the position and momentum operators are considered as unknowns and certain matrix ordinary differential equations are obtained by using the fluctuation free matrix representation concept. The resulting equations are in the category of Quantum Mechanical Matrix Ordinary Differential Equations (QMMODE) and can be solved by a recently developed scheme called Characteristic Evolutions Method (CEM). We mention also how to implement this approach. The purpose of this work can be considered as a preparation to quantum optimal control problem solutions via expectation value dynamics.
This work considers the quantum optimal control of one dimensional harmonic oscillator under linear control agents. The system’s potential energy function contains a perturbation term which is bounded everywhere in the space variable’s domain. We use the most preferred cost functional to construct the necessary equations. Equations are converted to a boundary value problem for a set of ordinary differential equations containing the expectation values of certain operators and the terms corresponding to the transitions between the states described by the wave and costate functions. Resulting equations involve certain undesired expectation values and transition terms. We use a recently developed scheme called fluctuation expansion to approximate these terms at the sharply localized wave and costate function limits. This enables us to construct n infinite number of ordinary differential equations and accompanying boundary conditions whose both halves are given at the beginning and end of the control. These equations are truncated and then the resulting boundary value problem is solved iteratively.
Abstract This work deals with the optimal control of one dimensional quantum harmonic oscillator under an external field characterized by a linear dipole function. The penalty term is taken as kinetic energy. The objective operator whose expectation value is desired to get a prescribed target value is taken as the square of the position operator. The dipole function hypothesis in the external field is valid only for weak fields otherwise hyperpolarizability terms which contain powers of the field amplitude higher than 1 should be considered. The weak field assumption enables us to develop a first order perturbation approach to get approximate solutions to the wave and costate equations. These solutions contain the field amplitude and another unknown, so‐called deviation constant, through some certain integrals. By inserting these expressions into the connection equation which functionally relates the field amplitude to the wave and costate function it is possible to produce an integral equation. Same manipulations on the objective equation results in an algebraic equation to determine the deviation parameter. The algebraic deviation equation produces incompatibility which can be relaxed by including second order perturbative term of wave function. The integral and algebraic equations mentioned above are asymptotically solved. Their global solutions are left to future works since the main purpose of this work is to obtain the so‐called field and deviation equations. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)
This work is devoted to find the critical time instants for a multivariate quantum harmonic oscillator system under optimal control. All quantum optimal control agents are taken linear in their arguments. The objective operator and the operator whose expectation value is to be suppressed are expressed as linear combinations of the position and momentum operators. The external field influence on the system is approximated only by the dipole polarizability term and the dipole function is taken as a linear combination of the positions. These linearities together with the elastic force structure of the mutual interactions amongst the oscillators brings the whole linearity to the system. We do not concern with the wave and costate functions. Instead, the expectation values of the position and momenta are taken as unknowns together with the two way transitions between the states described by the wave and costate functions, through positions and momenta. The resulting control problem appears to be a boundary value problem in time through a vector valued ODE. The critical time durations of this equation are sought by expanding the deviation parameter with respect to a perturbation parameter. Perturbation terms for these critical time instants can be evaluated analytically.
In this study, the optimal control equations for one dimensional quantum harmonic oscillator under the quadratic control operators together with linear dipole polarizability effects are constructed in the sense of Heisenberg equation of motion. A numerical technique based on the approximation to the non‐commuting quantum mechanical operators from the fluctuation free expectation value dynamics perspective in the classical limit is also proposed for the solution of optimal control equations which are ODEs with accompanying boundary conditions. The dipole interaction of the system is considered to be linear, and the observable whose expectation value will be suppressed during the control process is considered to be quadratic in terms of position operator x. The objective term operator is also assumed to be quadratic.
Recent years brought a new angle of view to the problems involving matrix representations of, especially, functions. It is the utilization of the fuctuation free matrix representation of those functions, where the matrix representation of the function is taken as the image of the matrix representation of the independent variable under the concerned function as a first approximation. The quality of the approximation becomes better as the fluctuations are suppressed by changing the subspace dimension under consideration. In this study, we discuss the quantum optimal control problem of simple harmonic oscillator as a first step towards the construction of possibly an efficient new tool. Here, the expectation value based optimal control equations are formulated and the all agents of the problem are taken linear for simplicity. The resultant linear system of ordinary differantial equations have exact solutions. By using these solutions and the fluctuationlessness approximation, it is possible to evaluate the matrix representation of an analytic function of the position and momentum operators.