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Physical Review A

Observable-preserving control of quantum dynamics over a family of related systems

Adam E. Rothman, Tak-San Ho, HERSCHEL A. RABITZ

Published Aug 25, 2005• 85 citations
Publisher

Abstract

Quantum control aims at the manipulation of atomic- and molecular-scale dynamics phenomena. An important objective in this regard is the understanding of dynamical control within a family of related quantum systems. To explore this issue, diffeomorphic changes in the system Hamiltonian $\mathcal{H}(s,t)$ are introduced by scanning over a homotopy parameter $s$ and then monitoring the control field response needed to maintain the value of a specified target observable. This operation is implemented through a procedure referred to as diffeomorphic modulation under observable-response-preserving homotopy (D-MORPH). The governing D-MORPH differential equation determining the control laser field $E(s,t)$ is shown to explicitly allow for innumerable solutions, with each characterized by the choice of an arbitrary function $f(s,t)$ of $s$ and time $t$. The presence of $f(s,t)$ in the D-MORPH differential equation makes clear the origin of multiple control fields that produce the same observable objective. A stable algorithm is presented for practical execution of D-MORPH with the only criterion that the Hamiltonian $\mathcal{H}(s,t)$ permit reaching the objective over the full domain of $s$ being sampled. Both analytic and numerical examples are presented to illustrate the D-MORPH concept.

DOI
10.1103/physreva.72.023416

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Quantum control aims at the manipulation of atomic- and molecular-scale dynamics phenomena. An important objective in this regard is the understanding of dynamical control within a family of related quantum systems. To explore this issue, diffeomorphic changes in the system Hamiltonian H(s,t)\mathcal{H}(s,t)H(s,t) are introduced by scanning over a homotopy parameter sss and then monitoring the control field response needed to maintain the value of a specified target observable. This operation is implemented through a procedure referred to as diffeomorphic modulation under observable-response-preserving homotopy (D-MORPH). The governing D-MORPH differential equation determining the control laser field E(s,t)E(s,t)E(s,t) is shown to explicitly allow for innumerable solutions, with each characterized by the choice of an arbitrary function f(s,t)f(s,t)f(s,t) of sss and time ttt. The presence of f(s,t)f(s,t)f(s,t) in the D-MORPH differential equation makes clear the origin of multiple control fields that produce the same observable objective. A stable algorithm is presented for practical execution of D-MORPH with the only criterion that the Hamiltonian H(s,t)\mathcal{H}(s,t)H(s,t) permit reaching the objective over the full domain of sss being sampled. Both analytic and numerical examples are presented to illustrate the D-MORPH concept.
A quantum control landscape is defined by the value of a physical observable as a functional of the time-dependent control field E(t)E(t)E(t) for a given quantum-mechanical system. Level sets through this landscape are prescribed by a particular value of the target observable at the final dynamical time TTT, regardless of the intervening dynamics. We present a technique for exploring a landscape level set, where a scalar variable sss is introduced to characterize trajectories along these level sets. The control fields E(s,t)E(s,t)E(s,t) accomplishing this exploration (i.e., that produce the same value of the target observable for a given system) are determined by solving a differential equation over sss in conjunction with the time-dependent Schr\"odinger equation. There is full freedom to traverse a level set, and a particular trajectory is realized by making an a priori choice for a continuous function f(s,t)f(s,t)f(s,t) that appears in the differential equation for the control field. The continuous function f(s,t)f(s,t)f(s,t) can assume an arbitrary form, and thus a level set generally contains a family of controls, where each control takes the quantum system to the same final target value, but produces a distinct control mechanism. In addition, although the observable value remains invariant over the level set, other dynamical properties (e.g., the degree of robustness to control noise) are not specifically preserved and can vary greatly. Examples are presented to illustrate the continuous nature of level-set controls and their associated induced dynamical features, including continuously morphing mechanisms for population control in model quantum systems.
This paper presents a new tracking method where the target observable O(s,T) at the final dynamical time T follows a predefined track P(s) with respect to a homotopy tracking variable s>or=0. The procedure calculates the series of control fields E(s,t) required to accomplish observable homotopy tracking by solving a first-order differential equation in s for the evolution of the control field. Controls produced by this technique render the desired track for all s without encountering field singularities. This paper also extends the technique to the case where the field-free Hamiltonian and dipole moment operator change with s in order to explore the control of new physical systems along the track. Several simulations are presented illustrating the various uses for this quantum tracking control technique.
We introduce a quantum optimal control algorithm for energy minimization that combines the diffeomorphic modulation under observable response preserving homotopy (D-MORPH) gradient and the Broyden Fletcher Goldfarb Shanno (BFGS) iterative scheme for nonlinear optimization. An extended set of controls defining the time-dependent mass, dipole moment, and external perturbational field are optimized to find an effective Hamiltonian that steers the dynamics of the system into the global minimum without getting trapped into local minima. The algorithm is illustrated as applied to energy minimization on rugged surfaces and golf potentials comparable to those previously explored for testing quantum annealing methodologies.
The general objective of quantum control is the manipulation of atomic scale physical and chemical phenomena through the application of external control fields. These tailored fields, or photonic reagents, exhibit systematic properties analogous to those of ordinary laboratory reagents. This analogous behavior is explored further here by considering the controlled response of a family of homologous quantum systems to a single common photonic reagent. A level set of dynamically homologous quantum systems is defined as the family that produces the same value(s) for a target physical observable(s) when controlled by a common photonic reagent. This paper investigates the scope of homologous quantum system control using the level set exploration technique (L-SET). L-SET enables the identification of continuous families of dynamically homologous quantum systems. Each quantum system is specified by a point in a hypercube whose edges are labeled by Hamiltonian matrix elements. Numerical examples are presented with simple finite level systems to illustrate the L-SET concepts. Both connected and disconnected families of dynamically homologous systems are shown to exist.
We derive a simple approximate analytic solution for the driving field that will generate a desired target vibrational wavepacket in a diatomic molecule. This solution is valid for arbitrary population transfer from the initial to the final electronic states. Analytic solutions to the control problem are possible when the control field is of shorter duration than the characteristic time scale of the system being manipulated. For target states that are localized wavepackets, our technique yields multiple solutions for the control field. We describe a procedure to choose from the set of solutions one that achieves both the target state and population transfer with high accuracy, for a large class of target states. The fidelity of each solution is tested by comparing the state obtained from a direct numerical integration of Schrödinger's equation using the analytic solution with the specified target state.
Quantum mechanics, though a probabilistic theory, gives a 'deterministic' answer to the question of how the present determines the future. In essence, in order to predict future probabilities, we need to (numerically) propagate the time-dependent Schrödinger equation from the present to the future. It is interesting to note that classical mechanics of macroscopic bodies, though reputed to be a deterministic theory, does not allow, due to chaos (which unfortunately is more prevalent than integrability), such clear insights into the future. In contrast, small (e.g., atomic, molecular and photonic) systems which are best understood using the tools of quantum mechanics, do not suffer from chaos, rendering the prediction of the probability-distributions of future events possible. The field of quantum control deals with an important modification of this task, namely, it asks: given a wave function in the present, what dynamics, i.e. what Hamiltonian, guarantees a desired outcome or 'objective' in the future? In practice one may achieve this goal of modifying and finding the desired Hamiltonian by introducing external fields, e.g. laser light. It is then possible to reach the objective in a 'trial-and-error' fashion, performed either numerically or in the laboratory. We can guess or build a Hamiltonian, do an experiment, or propagate the initial wave function to the future, compare the result with the desirable objective, and correct the guess for the Hamiltonian until satisfactory agreement with the objective is reached. A systematic way of executing this procedure is the sub-field called 'optimal control'. The trial-and-error method is often very time consuming and rarely provides mechanistic insight. There are situations where analytical solutions exist, rendering the control strategies more transparent. This is especially so when one can identify quantum interferences as the heart of quantum control, the essence of the field called 'coherent control'. The experience accumulated so far by many groups is that quantum interferences are the mechanism underlying most successful control scenarios and control experiments. The identification of the control mechanism is especially facile when the time-dependence of the Hamiltonian merely serves to prepare a state which then evolves in the absence of external fields. Analytical theories also exist when the explicit time dependence of the Hamiltonian can be treated adiabatically. Under these circumstances one can identify the quantum interfering pathways that lead to control, allowing for the simultaneous exploration of entire 'landscapes' (and not just a single 'desirable' objective) resulting from varying all the experimental 'knobs' at our disposal, e.g., relative- and envelope-phases, intensities, polarizations, and pulse-sequencing. Such 'pulse-shaping' knobs are now available down to the sub-femtosecond regime in broad spectral regions ranging from the UV to the IR. We expect that comprehensive investigations of the quantum interference required to control dynamical processes will result in a more detailed understanding of the dynamics themselves. Together with experimental progress---the stage is now set for many exciting applications in fundamental and applied sciences by 'tailoring' light--matter interactions essentially at will.