Singular Extremals for the Time-Optimal Control of Dissipative Spin $\frac{1}{2}$ Particles

We consider the time-optimal control by magnetic fields of a spin $\frac{1}{2}$ particle in a dissipative environment. This system is used as an illustrative example to show the role of singular extremals in the control of quantum systems. We analyze a simple case where the control law is explicitly determined. We experimentally implement the optimal control using techniques of nuclear magnetic resonance. To our knowledge, this is the first experimental demonstration of singular extremals in quantum systems with bounded control amplitudes.

Figure 1

Plot of the optimal trajectories (blue or dark gray curve) and of the inversion recovery sequence (green or gray curve) in the plane ($y$, $z$) for $T_1=740\mathrm{ms}$, $T_2=60\mathrm{ms}$ and ${\omega }_{\max }/(2\pi )=32.3\mathrm{Hz}$. The experimentally measured trajectories are represented by filled squares and open diamonds, respectively. The corresponding control laws are represented in the lower panel. In the upper panel, the small insert represents a zoom of the optimal trajectory near the origin. The dotted line is the switching curve originating from the horizontal singular line. The vertical green line corresponds to the intuitive solution. The solid blue curve is the optimal trajectory near the origin.

Figure 2

(top) Ratio $T_{\mathrm{opt}}/T_{\mathrm{IR}}$ as a function of ${\omega }_{\max }/(2\pi )$. The horizontal line indicates the position of the limit ratio when ${\omega }_{\max }\to +\infty$. (bottom) Optimal trajectories (left part) and the inversion recovery sequences (right part) for four values of ${\omega }_{\max }/(2\pi )$, 2.7, 7, 32.3, and 500 Hz. The vertical lines of the top panel correspond to the four solutions of the bottom panel. The solid curve is the case considered in Fig. 1. The black dot and the cross represent, respectively, the positions of the initial and final points. The small arrows indicate the way the trajectories are followed.

Figure 3

Contour plot of the function $d\dot{r}/d\theta$ as a function of $y$ and $z$. The solid lines represent the set of zeros of $d\dot{r}/d\theta$ or the singular set $S$ (see the text). The time-minimum and time-maximum singular lines are plotted, respectively, in black and red (light gray). The circle is the projection of the Bloch sphere in the plane ($y$, $z$).