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Physical Review A

Exploring the level sets of quantum control landscapes

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

Published May 1, 2006• 64 citations
Publisher

Abstract

A quantum control landscape is defined by the value of a physical observable as a functional of the time-dependent control field $E(t)$ 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 $T$, regardless of the intervening dynamics. We present a technique for exploring a landscape level set, where a scalar variable $s$ is introduced to characterize trajectories along these level sets. The control fields $E(s,t)$ 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 $s$ 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 $f(s,t)$ that appears in the differential equation for the control field. The continuous function $f(s,t)$ 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.

DOI
10.1103/physreva.73.053401

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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.
A controlled quantum system possesses a search landscape defined by the target physical objective as a function of the controls. This paper focuses on the landscape for the transition probability P(i → f) between the states of a finite level quantum system. Traditionally, the controls are applied fields; here, we extend the notion of control to also include the Hamiltonian structure, in the form of time independent matrix elements. Level sets of controls that produce the same transition probability value are shown to exist at the bottom P(i → f)=0.0 and top P(i → f)=1.0 of the landscape with the field and/or Hamiltonian structure as controls. We present an algorithm to continuously explore these level sets starting from an initial point residing at either extreme value of P(i → f). The technique can also identify control solutions that exhibit the desirable properties of (a) robustness at the top and (b) the ability to rapidly rise towards an optimal control from the bottom. Numerical simulations are presented to illustrate the varied control behavior at the top and bottom of the landscape for several simple model systems.
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.
Manipulation of a quantum system can be viewed in the framework of a control landscape defined as the physical objective as a functional of the control. Control landscape analyses have thus far considered linear quantum dynamics. This paper extends the analysis of control landscape topology to nonlinear quantum dynamics with the objective of steering a finite-level quantum system from an initial state to a final target state. The analysis rests on the assumptions that (i) the final state is reachable from the initial state, (ii) the differential mapping from the control to the state is surjective, and (iii) the control resources are unconstrained. Under these assumptions, landscape critical points (i.e., where the slope vanishes) for nonlinear quantum dynamics only appear as the global maximum and minimum; thus, the landscape is free of traps. Moreover, the landscape Hessian (i.e., the second derivative with respect to the control) at the global maximum has finite rank, indicating the presence of a large level set of optimal controls that preserve the value of the maximum. Extensive numerical simulations on finite-level models of the Gross-Pitaevskii equation confirm the trap-free nature of the landscape as well as the Hessian rank analysis, using either an applied electric field or a tunable condensate two-body interaction strength as the control. In addition, the control mechanisms arising in the numerical simulations are qualitatively assessed. These results are a generalization of previous findings for the linear Schr\"odinger equation, and show promise for successful control in a wide range of nonlinear quantum dynamics applications.
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 controlled quantum system possesses a search landscape defined by the observable value as a functional of the control field. Within the search landscape, there exist level sets of controls giving the same observable value. This paper focuses on level sets of the transition probability Pi¿f. For transition probabilities 0i¿fi¿f= 1, a second order D-MORPH algorithm is presented that can explore the perfect control level set. D-MORPH is utilized to identify level set members that exhibit certain desirable secondary characteristics, e.g., minimal pulse fluence. Numerical simulations for finite level systems are presented to illustrate the variety of control behavior found across level set members.
A search for the variables that control a quantum system's dynamics occurs over a landscape, defined as the target objective as a function of the variables. Prior studies show that upon satisfaction of three specific assumptions, the topology of the landscape is free of suboptimal traps that could prematurely halt the search for an optimal control. One key assumption is free access to all necessary control variables; however, in practice, the controls are always limited in some fashion which may result in constraint-induced traps on the landscape. This paper aims to introduce the means to systematically explore the nature of constrained controls that yield suboptimal outcomes. The procedure utilizes kinematic controls, which comprise a simple set of time-independent variables, and then performs a landscape topology-preserving transformation into corresponding dynamic controls. The equivalent landscape topology of these two formulations permits the study of a family of dynamic controls that reflect constrained control landscape behavior. In particular, constrained dynamic controls are identified as isolated points on the landscape or as suboptimal level sets. The wide range of such dynamic controls indicates the richness and complexity of constraint-induced features on the landscape.