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AIP conference proceedings

A Fluctuation Analysis at the Classical Limit for the Expectation Dynamics of a Single Quartic Quantum Anharmonic Oscillator

Muzaffer Ayvaz, Metin Demіralp, Theodore E. Simos, George Psihoyios, Ch. Tsitouras

Published Jan 1, 2010• 2 citations
Publisher

Abstract

The definition of the fluctuation terms in the matrix representations of operators, especially function operators, brings a lot of facilitations to the numerical analysis of matrix representation issues. This work considers a one‐dimensional quantum oscillator with purely quartic anharmonicity and focuses on its position and momentum expectations. By using certain level commutator operation it is possible to get two ODEs as equations of motion, which are in classical nature when the fluctuation terms are omitted. To understand how fluctuation terms affect expectation dynamical analysis we add extra temporal unknowns and construct extra ODEs over them. There seem to be an infinite ODEs chain for fluctuation altough their truncations seem to be utilizable as approximations. Here basic theory is summarized. Further details will be prasented at the conference presentation.

DOI
10.1063/1.3498311

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The definition of the fluctuation terms in the matrix representations of operators, especially function operators, brings a lot of facilitations to the numerical analysis of matrix representation issues. This work considers a one‐dimensional quantum oscillator with purely quartic anharmonicity and focuses on its position and momentum expectations. By using certain level commutator operation it is possible to get two ODEs as equations of motion, which are in classical nature when the fluctuation terms are omitted. To understand how fluctuation terms affect expectation dynamical analysis we add extra temporal unknowns and construct extra ODEs over them. There seem to be an infinite ODEs chain for fluctuation altough their truncations seem to be utilizable as approximations. Here basic theory is summarized. Further details will be prasented at the conference presentation.
The variation of the expectation matrix of position and momentum operator in time can serve us to investigate the evolution of a quantum system in time. This brings the utilization of the ODEs instead of Schrödinger's equation at the expense of incapability for the calculation of the wave function. As long as we deal with the observables which can be expressed in terms of position and momentum operators this may be quite practical to know about the quantum dynamics of the system under consideration. Expectation matrix of an operator becomes a function of the expectation matrices of the position and momentum operator when the fluctuations diminish to zero. At this limit, the time‐variant ODEs for the expectation matrices of the position and momentum operator can be handled by using the matrix algebraic tools even in the case of nonlinearities in the potential function. This work presents certain details about these points.
A new method called “Fluctuation Expansion” has been recently proposed for solving quantum dynamical problems . This method is succesfully applied to the dynamics of expectation values such that not the wave function itself but the expectation values of the position and momentum operator are taken into consideration and then the expectation value of certain entities depending on position and momentum operators are expressed via fluctuation expansion. This brings an infinite number of new unknown, fundamental and associate fluctuation terms. The derivatives of expectation values of the position and momentum operators and fluctuation operators are expressed in terms of these terms again. The resulting infinite set of differential equations and accompanying initial conditions are truncated at certain fluctuation order and then tried to be solved. Here we take a one dimensional time independent Hamiltonian and consider the zeroth and first order truncations. Equations obtained in this way are investigated in the sense of phase space which is defined by the expextation values of the position and momentum operator and the fluctuations entering the truncated equations. We find certain subspaces of the phase space such that there, certain expressions remain constant . This helps us to reduce the dimension of the problem.
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.
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.
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.
The time evolution of a quantum single-quartic anharmonic oscillator is considered. The study is carried on in operational form by use of the raising and lowering operators of the oscillator. The equation of motion is solved by application of a new integration method based on iteration techniques, and the rigorous solutions that describe the time development of the displacement and momentum operators of the oscillator are obtained. These operators are presented as a Laplace transform and a subsequent inverse Laplace transform of suitable functionals. Finally, the results are employed to describe the time evolution of a quasiclassical anharmonic oscillator.