This paper addresses the nature and multiplicity of an optimally designed electric field for controlling quantum-dynamical processes. A rather general cost functional is considered, with only mild conditions called for amongst the various operators involved. An explicit upper bound on the magnitude of the controlling electric field is attained in terms of the norms of various operators entering into the control cost functional. An earlier work employing first-order perturbation theory arguments showed that, under rather mild assumptions, a denumerably infinite number of control-field solutions exists for the optimal control problem. In the present work, it is shown that through a bound on the remainder of the nonlinear terms in the expansion, this same conclusion concerning the control-field multiplicity continues to hold.
The optimal control of the path to a specified final state of a quantum-mechanical system is investigated. The problem is formulated as a minimization problem over appropriate function spaces, and the well-posedness of this problem is is established by proving the existence of an optimal solution. A Lagrange-multiplier technique is used to reduce the problem to an equivalent optimization problem and to derive necessary conditions for a minimum. These necessary conditions form the basis for a gradient iterative procedure to search for a minimum. A numerical scheme based on finite differences is used to reduce the infinite-dimensional minimization problem to an approximate finite-dimensional problem. Numerical examples are provided for final-state control of a diatomic molecule represented by a Morse potential. Within the context of this optimal control formulation, numerical results are given for the optimal pulsing strategy to demonstrate the feasibility of wave-packet control and finally to achieve a specified dissociative wave packet at a given time. The optimal external optical fields generally have a high degree of structure, including an early time period of wave-packet phase adjustment followed by a period of extensive energy deposition to achieve the imposed objective. Constraints on the form of the molecular dipole (e.g., a linear dipole) are shown to limit the accessibility (i.e., controllability) of certain types of molecular wave-packet objectives. The nontrivial structure of the optimal pulse strategies emphasizes the ultimate usefulness of an optimal-control approach to the steering of quantum systems to desired objectives.
This work presents a general framework for assessing the quality and robustness of control over quantum dynamics induced by an optical field
E(t). The control process is expressed in terms of a cost functional, including the physical objectives, penalties, and constraints. The first variations of such cost functionals have traditionally been utilized to create designs for the controlling electric fields. Here, the second variation of the cost functional is analyzed to explore (i) whether such solutions are locally optimal, and (ii) their degree of robustness. Both issues may be assessed from the eigenvalues of the stability operator S whose kernel $K(t,\ensuremath{\tau})$ is related to $\ensuremath{\delta}\mathcal{E}(t)/\ensuremath{\delta}\mathcal{E}(\ensuremath{\tau}){|}_{c}$ for
0<t, $\ensuremath{\tau}<~T,$ where
T is the target control time. Here
c denotes the constraint that the field satisfies the optimal control dynamical equations. The eigenvalues \ensuremath{\sigma} of S satisfying $\ensuremath{\sigma}<1$ assure local optimality of the control solution, with $\ensuremath{\sigma}=1$ being the critical value separating optimal solutions from false solutions (i.e., those with negative second variational curvature of the cost functional). In turn, the maximally robust control solutions with the least sensitivity to field errors also correspond to $\ensuremath{\sigma}=1.$ Thus, sufficiently high sensitivity of the field at one time
t to the field at another time \ensuremath{\tau} (i.e., $\ensuremath{\sigma}>1$) will lead to a loss of local optimality. An expression is obtained for a bound on the stability operator, and this result is employed to qualitatively analyze control behavior. From this bound, the inclusion of an auxiliary operator (i.e., other than the target operator) is shown to act as a stabilizer of the control process. It is also shown that robust solutions are expected to exist in both the strong- and weak-field regimes.
A new formalism for the optimal control of quantum mechanical physical observables is presented. This approach is based on an analogous classical control technique reported previously. [1] Quantum Lagrange multiplier functions are used to preserve a chosen subset of the observable dynamics of interest. As a result, a corresponding small set of Lagrange multipliers needs to be calculated and they are only a function of time. This is a considerable simplification over traditional quantum optimal control theory. [2] The success of the new approach is based on taking advantage of the multiplicity of solutions to virtually any problem of quantum control to meet a physical objective. A family of such simplified formulations is introduced and numerically tested. Results are presented for these algorithms and compared with previous reported work on a model problem for selective unimolecular reaction induced by an external optical electric field. 1 I.
We investigate the problem of control of quantum systems. Various notions of controllability of quantum systems are discussed and Lie algebraic techniques are employed to derive necessary and sufficient conditions for the resulting degrees of controllability. The results are employed to study the degree of controllability for various model systems. The problem of optimal control is formulated as a problem of maximisation of the expectation value of an observable of the system subject to dynamical constraints and costs. A class of iterative algorithms is discussed, and shown to converge to a solution of the Euler-Lagrange equations, which is a necessary condition for optimality of a solution. A numerical realisation of the algorithm is presented and potential implementation of the resulting control fields in the laboratory is discussed. The algorithm is applied to find optimal control fields for a variety of applications including selective and simultaneous excitation of individual quantum dots in a globally addressed ensemble, creation of superposition states, and selective excitation of certain vibrational modes of a simple molecule such as Hydrogen Fluoride. In each case we systematically explore the parameter space of the algorithm to select solutions with desirable physical features from the multitude of possible choices. As there are many possible optimal control fields which give similar yields the question of whether some are more robust than others is investigated. A number of optimal fields are analysed quantitatively in terms of robustness with regard to temporal and spectral noise affecting the fields, uncertainty in the model parameters and the effect of experimental limitations such as restrictions on the bandwidth of the fields. It is shown that whilst the solutions tend to be sensitive to uncertainty in the energy levels chosen for the model, they appear to be surprisingly tolerant to even severe bandwidth limitations.
An alternative perturbative formulation is introduced to reduce the task of designing optimal controls of molecular motion. The method provides multiple optimal control fields as eigenvectors corresponding to the positive eigenvalues of a matrix {
Amn}. Analytical equations for the matrix elements
Amn are derived. Numerical examples are given for simple three- and four-state molecular systems. The method is shown to be computationally simple and efficient in providing good quality control results. \textcopyright{} 1996 The American Physical Society.