Exploring the level sets of quantum control landscapes

Adam Rothman, Tak-San Ho, and Herschel Rabitz

Phys. Rev. A 73, 053401 – Published 1 May 2006

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 Schrödinger 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.

Received 8 July 2005

DOI:https://doi.org/10.1103/PhysRevA.73.053401

Authors & Affiliations

Adam Rothman^*, Tak-San Ho^†, and Herschel Rabitz^‡

Department of Chemistry, Princeton University, Princeton, New Jersey 08544-1009, USA

^*Electronic address: arothman@princeton.edu
^†Electronic address: tsho@princeton.edu
^‡Electronic address: hrabitz@princeton.edu

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Vol. 73, Iss. 5 — May 2006

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Images

Figure 1
Starting from an initial control field $E (s = 0, t)$ and using 1024 $s$ steps on the interval $s ∊ [0, 1]$ , the control fields $E_{\pm} (s, t)$ are evolved subject to the functions $f_{\pm} (s, t)$ in Eq. (18). (a) Under fluence maximization with $f_{+} (s, t)$ , the control field grows in amplitude and incorporates complex structure; under fluence minimization with $f_{-} (s, t)$ , the progression of fields decreases in amplitude, slightly in this case. (b) Cross sections of the fields in (a) at $s = 0$ and $s = 1$ are plotted. The field at $s = 0$ is the same for $E_{+} (0, t)$ and $E_{-} (0, t)$ , but the different functions $f_{+} (s, t)$ and $f_{-} (s, t)$ lead the fields to evolve in dramatically different ways for $s > 0$ . In particular, $E_{+} (1, t)$ incorporates new structure while $E_{-} (1, t)$ remains closer in structure to the original field. (c) The population of state ∣8⟩ is displayed at $s = 0$ and also at $s = 1$ for both the fluence minimizing and maximizing cases. It is seen that fluence minimization tends to simplify the intermediate-time dynamics, while fluence maximization tends to complicate the dynamics. In all cases, the population at the final dynamical time $t = T$ is preserved. The differences in the intermediate-time dynamics here are taken as evidence of broader differences in the control control mechanism along each level set trajectory. (d) The field metric $D_{s} [E]$ , defined in Eq. (16), is plotted for fluence minimization and maximization. For fluence maximization, the metric increases rapidly with $s$ ; under fluence minimization, the metric levels off as the field approaches minimal fluence while still accomplishing the requisite population transfer.Reuse & Permissions
Figure 2
The trace of the Hessian in Eq. (22) as a function of $s ∊ [0, 1]$ is shown for the high-yield level set example along both the fluence minimizing $[f_{-} (s, t)]$ and fluence maximizing $[f_{+} (s, t)]$ trajectories. Starting from the central axis $(s = 0)$ , the $s$ parameterization of the two trajectories points in opposite directions along the abscissa. Inset: close-up of the behavior near $s = 0$ . The cusp in the plot is due to a discontinuity in the derivative $\frac{\partial E (s, t)}{\partial s}$ caused by the different constants $c_{+}$ and $c_{-}$ in $f_{\pm} (s, t)$ [cf. Eqs. (5, 18)].Reuse & Permissions
Figure 3
The control fields along the level set described by $f_{1} (s, t)$ are displayed. The control field continually changes with $s$ ; at $s = 1$ , it has a substantially different envelope structure than the field at $s = 0$ , due to the influence of $f_{1} (s, t)$ . Yet fields along the entire $s$ interval render the same value of the target observable population of 0.47.Reuse & Permissions
Figure 4
The intermediate-time dynamics are displayed for states ∣1⟩, ∣4⟩, ∣6⟩, and ∣8⟩ at (a) $s = 0$ , (b) $s = 0.5$ , and (c) $s = 1$ . The populations of the states not pictured show similar oscillatory behavior and are omitted to make the figures more clear. Importantly, despite the differences in the controlled dynamics, the population of state ∣8⟩ at $t = T$ remains fixed at 0.47 for all $s$ . The dynamics appear to gain complexity with $s$ , indicating that distinct control mechanisms must operate at each $s$ value that leave the final population of state ∣8⟩ unchanged.Reuse & Permissions

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