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arXiv:quant-ph/0411155

Evolution of Level Sets in Hamilton Parameter Space

Fariel Shafee

Published Nov 20, 2004• 0 citations
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Abstract

We develop a formalism to study the use of Level Set Method (LSM) in the investigation of evolution of observables in terms of parameters of the Hamiltonian, both of the system itself and the control part. A simple example with an analytic solution available perturbatively is examined. We show that B splines can quite accurately and smoothly interpolate surfaces corresponding to constant expectation value of observables which form the level sets as projections on a finite mesh of data. Lastly we make a brief priliminary scrutiny of teh possibility of using temperature as a relevant parameter in ensembles of quantum systems.

DOI
10.48550/arxiv.quant-ph/0411155

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We develop a formalism to study the use of Level Set Method (LSM) in the investigation of evolution of observables in terms of parameters of the Hamiltonian, both of the system itself and the control part. A simple example with an analytic solution available perturbatively is examined. We show that B splines can quite accurately and smoothly interpolate surfaces corresponding to constant expectation value of observables which form the level sets as projections on a finite mesh of data. Lastly we make a brief priliminary scrutiny of teh possibility of using temperature as a relevant parameter in ensembles of quantum systems.
We investigate the level set method (LSM) in a specific quantum context; namely the dipole transition moment for a system with a nontrivial Morse potential. We draw equal moment sets in the two-dimensional space of two important parameters of the potential, namely the depth of the potential and its width. Another variable is introduced as a scale and we see "motions" of the level sets normal to the contours, as in classical contexts such as fluid dynamics or in epitaxial crystal growth. Presumably interpolating the level sets normally by smooth functions such as splines may give a fairly accurate method of combining the variables to keep the dipole moment invariant.
We examine the relevance of Level Set Methods (LSM)in coherent control quantum systems where the objective is to retain or attain a particular expectation value of a given measurable. The differences with the usual applications of LSM, where continuous closed interfaces are involved, and the quantum case, where we may have a discrete number of points to deal with, are noted. The question of optimization in this new context is also clarified. Simple examples with symmetric and asymmetric multidimensional potentials are briefly considered.
We study the application of a generalized form of the level set method used in classical physical contexts to quantum optimal control situations. The set of OCT equations needed to keep the expectation value of an observable constant is first discussed and the dimensionality of the actual parameter space carefully considered. Then we see how concepts of level set methods emerge that may help solve the inverse problem associated with designing the control Hamiltonian with greater speed. The formal equations and the algorithm are presented.
A quantum spline is a smooth curve parametrized by time in the space of unitary transformations, whose associated orbit on the space of pure states traverses a designated set of quantum states at designated times, such that the trace norm of the time rate of change of the associated Hamiltonian is minimized. The solution to the quantum spline problem is obtained, and is applied in an example that illustrates quantum control of coherent states. An efficient numerical scheme for computing quantum splines is discussed and implemented in the examples.
Quantum splines are curves in a Hilbert space or, equivalently, in the corresponding Hilbert projective space, which generalize the notion of Riemannian cubic splines to the quantum domain. In this paper, we present a generalization of this concept to general density matrices with a Hamiltonian approach and using a geometrical formulation of quantum mechanics. Our main goal is to formulate an optimal control problem for a nonlinear system on [Formula: see text] which corresponds to the variational problem of quantum splines. The corresponding Hamiltonian equations and interpolation conditions are derived. The results are illustrated with some examples and the corresponding quantum splines are computed with the implementation of a suitable iterative algorithm.
In controlled quantum dynamics, a level set is defined as the collection of control fields that produce a specific value for a particular observable. This paper explores the relationship between individual solutions to a control problem, and presents the first experimentally observed quantum control level sets, which are found to be continuous submanifolds. Level sets are observed for two photon transitions where the control is the spectral phase function, which is expressed as a fourth-order polynomial. For the systems studied here, the level sets are shown to be closed surfaces in the spectral phase control space. A perturbation analysis provides insight into the observed topology of the level set, which is shown to be preserved by the low-order polynomial phase representation. Each of the multiple control fields forming a level set preserves the observable value by its own distinct manipulation of constructive and destructive quantum interferences. Thus, the richness of quantum control fields meeting a particular observable value is accompanied by an equally diverse family of control mechanisms.