Evolution of Level Sets in Hamilton Parameter Space

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 availab

Evolution of Level Sets in Hamiltonian Parameter Space

Fariel Shafee

Department of Physics

Princeton University

Princeton, NJ 08540

Abstract:

We develop a formalism to study the use of the Level Set Method in the investigation of the evolution of observables in terms of parameters of the Hamiltonian, both of the system itself and of a 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 preliminary scrutiny of the possibility of using temperature as a relevant parameter in ensembles of quantum systems.

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 availab

1. INTRODUCTION

We have previously reported [1,2] on the applicability of the Level Set Method (LSM) [3-5] to quantum control problems. In that work we concentrated on the level sets in the state function space. The expectation value of an observable for a given system with fixed parameters of course depends on the superposition of the eigenstates of the operator corresponding to the observable and we studied how level sets with constant values of the observables could be defined in the Hilbert space and how control operators created from a control Hamiltonian could move the system in the Hilbert space in such a way as to preserve the expectation value of the observable (EVO). The path from a given state to another with a different EVO is of course not unique, and we commented, if laser pulses were chosen for control, as is possible in many current chemical contexts, how simple combinations of operators could attain the target.

In a different scenario the space to be addressed may not be the Hilbert space of state vectors, but that of the parameters determining the potential and hence the Hamiltonian. For example, if we are interested in the ground state energy, or the energy of the first excited state, or in a specific spectral line, then we are not concerned with the mixture of states in determining the observable in issue, and therefore in the orientation of the state vector in Hilbert space. On the other hand virtually all observables, even those belonging to pure states, or to specific combinations of eigenstates will in general depend on the parameters of the Hamiltonian and when more than one parameter is involved, we expect to get contours in the parameter space which may be attacked by techniques developed by the LSM in a classical context.

In the next section we shall first try to make these notions mathematically more precise in developing a framework for the LSM for the space of Hamiltonian parameters. In the following section we shall study a few simple examples with analytic solutions. In section 4 we shall be concerned with numerical solutions in more complicated cases where this method may actually find appropriate use. In the last section we present our conclusions regarding possibilities and shortcomings of this approach.

2. LEVEL SETS IN HAMILTONIAN SPACE Let us consider the system Hamiltonian

2

p 2(r,p,)(,r)S I i m H a V a =+ (1)where the a i are parameters of the potential.

Obviously all eigenstates of the Hamiltonian will in general be functions of the ai, and hence any expectation value

()||()i i a a θψθψ<>=<>

(2)will also be a function of the a i .

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 availab

Hence in the a i space we expect to have the equal EVO contours

()i a c

θ= (3)

If now we also introduce a control Hamiltonian interacting with the system ()c c j H H b = (4)with the parameters b j , then the equal EVO contours will be represented in the product space of the two sets of parameters, because the eigenstates will be perturbed to new functional forms involving the b j parameters:

(,)i j a b c θ= (5)It is possible that one or more of the parameters are environmental ones, affecting both H s and H c .Let us call this common set e k . This would reduce the dimensionality of the parameter space, but not change anything else. Let us choose one of these common parameters and call it s. Then (,,)a b s c θ= (6)where we have dropped the indices from the noncommon parameters a and b . If we have only one a and one b , then in three dimensions Eq. 6 corresponds to a surface, and for a particular s, we get the intersection of the surface with the relevant plane giving us a contour. As s changes the contour also changes, and we have a “motion” of the level set, such that each point (a,b) on the contour moves to a new point (a’,b’) on a different contour in the parameter space with changing s (Fig. 1, 2).

Hence we can talk about the points on the level set having a velocity in the parameter space with s representing time.

da a ds

db b ds u u =

= (7)

Since for any point the co-ordin …… 此处隐藏：20342字，全部文档内容请下载后查看。喜欢就下载吧 ……