Viswanath Ramakrishna

Competitive tracking of molecular objectives described by quantum mechanics

The control of molecular events by optical fields is sought with the methods of asymptotic inverse tracking, local track generation (model matching), and competitive tracking which are extensions of exact inverse tracking. The methodology is applied to infrared dissociation of a diatomic molecule and selective dissociation of the stronger bond in a highly coupled linear triatomic system. The major appeal of these methods is that they do not require costly iterations unlike other control studies in which optimization techniques are used to design fields to achieve desired molecular objectives. It is found that in exact inverse tracking where a requisite external field is obtained to exactly track a prescribed objective expectation value as a function of time, a high degree of intuition is required to find an a priori objective track such that the required fields are reasonable in terms of intensity and bandwidth. Furthermore, exact inverse tracking does not allow for tracking of multiple observables. The extensions of the inverse tracking method presented in this work help to alleviate these drawbacks. In all of these extensions the requisite field is computed locally in time through minimization of a cost functional which contains terms designed to minimize the error between the objective and actual tracks and also minimize the field fluence. The objective tracks can be prescribed a priori as in exact inverse tracking or from the present evolving system state (local track generation). Competitive tracking allows for the following of multiple observables although none will be tracked exactly. Locally generated tracks (model matching) require less physical intuition because it is easier to specify an objective track with current knowledge of the state of the system. However, the tradeoff with this method is that prediction of the behavior of the tracked observables may be elusive.

Control of molecular motion

In this paper we present a short introduction to the new field of control of molecular motion. Our intention is to outline how the methods and techniques of control theory play a crucial role in the development of this emerging field, and reciprocally how fundamental new problems are motivated by this interaction.

Constructive control of quantum systems using factorization of unitary operators

We demonstrate how structured decompositions of unitary operators can be employed to derive control schemes for finite-level quantum systems that require only sequences of simple control pulses such as square wave pulses with finite rise and decay times or Gaussian wavepackets. To illustrate the technique, it is applied to find control schemes to achieve population transfers for pure-state systems, complete inversions of the ensemble populations for mixed-state systems, create arbitrary superposition states and optimize the ensemble average of dynamic observables.

Brief Parameter variations, relative degree, and stable inversion

Motivated by our prior research on automatic aircraft guidance, we address stable inversion for output tracking and prove that this process is continuous with respect to parameter variations, even when these variations cause a change in relative degree. Our earlier simulations indicated that the stable inversion process is extremely accurate and that only a linear regulator about the desired trajectory is required in the face of reasonable modeling error. The principal novelty in our technique is that a differential equations point of view is taken as opposed to a state-space approach on the (driven) zero dynamics of the system. This is the situation that arises in many applications and it also enables handling the question of changes in relative degree, without having to be encumbered by the change in state space dimension as the parameters change. Linear systems are first studied, since the corresponding result is unknown. Next, a corresponding theorem for nonlinear systems proved by using the Picard process in conjunction with the result for the linear case. The principal contribution of this paper is a result concerning the continuous dependence of a generalized steady state solution of nonlinear driven differential equations, with respect to parameter variations which cause the order of the differential equation to change. Since the notion of steady state is of paramount importance to innumerable engineering situations, the contents of this paper have wider scope.

Stable inversion and parameter variations

Abstract As part of the process of automatically guiding an aircraft, we have been successful in using stable inversion to compute a desired bounded state trajectory and corresponding bounded control. In addition to this feedforward control, we must also construct a regulator to address modeling errors and disturbances. With respect to modeling errors we find that the stable inversion procedures used are so accurate that the regulator can assume a simple form, say a linear regulator about the desired trajectory. We show that under the appropriate assumptions, the bounded state trajectory and bounded control computed through stable inversion depend continuously on the parameters of the system. This is a consequence of a mathematical result that we prove about the continuous dependence of the “particular solution” of a time varying nonlinear system driven by a bounded input. This is distinct from the usual continuous dependence of the initial value problem for systems.

On the exponential of matrices in su(4)

This paper provides explicit techniques to compute the exponentials of a variety of anti-Hermitian matrices in dimension 4. Many of these formulae can be written down directly from the entries of the matrix. Whenever any spectral calculations are required, these can be done in closed form. In many instances only 2 × 2 spectral calculations are required. These formulae cover a wide variety of applications. Conditions on the matrix which render it to admit one of three minimal polynomials are also given. Matrices with these minimal polynomials admit simple and tractable representations for their exponentials. One of these is the Euler–Rodrigues formula. The key technique is the relation between real 4 × 4 matrices and the quaternions.

Active optical lattice filters

Highly integrated, novel architectures for optical filtering can leverage structure, gain and variable delays to provide a multi-use photonic platform. Hardware and software results are presented.

Parametrizing Quantum States and Channels

AbstractThis work describes one parametrization of quantum states and channels and several of its possible applications. This parametrization works in any dimension and there is an explicit algorithm which produces it. Included in the list of applications are a simple characterization of pure states, an explicit formula for one additive entropic quantity which does not require knowledge of eigenvalues, and an algorithm which finds one Kraus operator representation for a quantum operation without recourse to eigenvalue and eigenvector calculations. PACS: 03.67a, 03.67-Hk, 03.67-Lx

Control of classical regime molecular objectives—applications of tracking and variations on the theme

We examine the applicability of variations of exact tracking, including asymptotic tracking, state-dependent trajectory tracking, and competitive tracking, for the problem of generating feasible external laser fields for control of classically modeled molecular motion via external laser fields. Applications are considered to the selective unimolecular dissociation of the stronger bond of highly coupled linear triatomic molecules. Specifically, we explore two variations of exact tracking, namely, asymptotic energy tracking and competitive acceleration tracking. It is found that, in the former case, a high degree of physical intuition is needed to specify a desirable energy track a priori. In the latter case, less physical intuition is necessary since we only need to specify the desired accelerations for given displacements, momenta and potential at each time instead of following a prescribed track. However, a trade-off with this method arises due to the difficulty in predicting the relevant observables such as the dissociation probability before actually integrating the equations of motions.

Composition methods for four-port couplers in photonic integrated circuitry

Planar photonic integrated circuits based on four-port couplers offer enhanced sophistication and functionality. Each four-port coupler is characterized by sixteen signal coupling coefficients governed by ten energy constraints. The ability to generate the constrained sixteen coupling coefficients is needed in the analysis of the four-port coupler. However, the energy constraint equations are nonlinear and cumbersome to solve directly. We introduce two techniques to reduce these signal coupling coefficients to a set of six free parameters. Hence we can characterize all possible couplers in terms of their sixteen constrained coupling coefficients, or either of two sets of six free parameters. This reduction in parameters has significant ramifications for the design, specification, and empirical characterization of these useful building blocks.