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.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.

Physical Review AMay 1, 200664 citations

Experimental exploration over a quantum control landscape through nuclear magnetic resonance

Qiuyang Sun, István Pelczer, Gregory Riviello +2 more

The growing successes in performing quantum control experiments motivated the development of control landscape analysis as a basis to explain these findings. When a quantum system is controlled by an electromagnetic field, the observable as a functional of the control field forms a landscape. Theoretical analyses have revealed many properties of control landscapes, especially regarding their slopes, curvatures, and topologies. A full experimental assessment of the landscape predictions is important for future consideration of controlling quantum phenomena. Nuclear magnetic resonance (NMR) is exploited here as an ideal laboratory setting for quantitative testing of the landscape principles. The experiments are performed on a simple two-level proton system in a H ${}_{2}$ O-D ${}_{2}$ O sample. We report a variety of NMR experiments roving over the control landscape based on estimation of the gradient and Hessian, including ascent or descent of the landscape, level set exploration, and an assessment of the theoretical predictions on the structure of the Hessian. The experimental results are fully consistent with the theoretical predictions. The procedures employed in this study provide the basis for future multispin control landscape exploration where additional features are predicted to exist.The growing successes in performing quantum control experiments motivated the development of control landscape analysis as a basis to explain these findings. When a quantum system is controlled by an electromagnetic field, the observable as a functional of the control field forms a landscape. Theoretical analyses have revealed many properties of control landscapes, especially regarding their slopes, curvatures, and topologies. A full experimental assessment of the landscape predictions is important for future consideration of controlling quantum phenomena. Nuclear magnetic resonance (NMR) is exploited here as an ideal laboratory setting for quantitative testing of the landscape principles. The experiments are performed on a simple two-level proton system in a H${}_{2}$O-D${}_{2}$O sample. We report a variety of NMR experiments roving over the control landscape based on estimation of the gradient and Hessian, including ascent or descent of the landscape, level set exploration, and an assessment of the theoretical predictions on the structure of the Hessian. The experimental results are fully consistent with the theoretical predictions. The procedures employed in this study provide the basis for future multispin control landscape exploration where additional features are predicted to exist.

arXiv:1502.04045Physical Review AMar 12, 201422 citations

A theoretical analysis of quantum control landscapes

Michael Hsieh

Jan 1, 20080 citations

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.

{}_{2}

O-D

{}_{2}

O sample. We report a variety of NMR experiments roving over the control landscape based on estimation of the gradient and Hessian, including ascent or descent of the landscape, level set exploration, and an assessment of the theoretical predictions on the structure of the Hessian. The experimental results are fully consistent with the theoretical predictions. The procedures employed in this study provide the basis for future multispin control landscape exploration where additional features are predicted to exist.