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A perturbative approach to the critical time durations in quantum optimal control for multivariate harmonic oscillator

Esma Meral, Metin Demіralp

Published May 27, 2008• 0 citations
OpenAlex

Abstract

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.

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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.
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)
In this work we have investigated the stability and robustness of the optimal control solutions to a quantum system when the control duration goes to zero. The solutions at this limit are called “Instantaneous Solutions”. These investigations are based on the second variation of the cost functional evaluated at control solution values when the first variations of wave and costate functions are related to the first variation of external field amplitude via control equations. This form of cost functional's second variation is purely quadratic in the first variation of the external field amplitude. Investigations are conducted for an illustrative model system, one dimensional quantum harmonic oscillator under linear dipole interaction, purely quadratic objective operator in position, and purely quadratic penalty operator in momentum. We have not found the stability operator's spectrum explicitly. Instead we have employed a bound analysis to understand the system's stability and robustness. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)
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, 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.
This work considers optimal control of quantum-mechanical systems within the framework of perturbation theory with respect to the controlling optical electric field. The control problem is expressed in terms of a cost functional including the physical objective, the penalties, and constraints. The resultant nonlinear variational equations are linearized by considering the lowest-order term in an expansion in powers of the optical-field strength. The optical field is found to satisfy a linear integral equation, and the solution may be expressed in terms of a generalized eigenvalue problem associated with the corresponding kernel. A full determination of the field is specified through the solution to the integral equation and the roots of an accompanying linearized spectral equation for a characteristic multiplier parameter. Each discrete value of the latter parameter corresponds to a particular solution to the variational equations. As a result, it is argued that under very general conditions there will be a denumerably infinite number of solutions to well-posed quantum-mechanical optimal-control problems.