Figure 1

Illustration of the different angular parameters used in Proposition 1 to characterize a projected trajectory $\gamma \left(t\right)$.Reuse & Permissions

Figure 2

On the left, the projected trajectories for the optimal synthesis of $U^{*}=e^{i\lambda \frac{{\sigma }_z}{2}}$ with $\lambda =\frac{\pi }{2},\pi ,\frac{3\pi }{2},2\pi$. On the right, the projection of the optimal trajectories of $U^{*}=e^{-ia\frac{{\sigma }_z}{2}}{e}^{ib\frac{{\sigma }_y}{2}}{e}^{ia\frac{{\sigma }_z}{2}}$ with $(a,b)=(0,\frac{k\pi }{4})$, where $k=1,2,3,4$. The dots indicate the position of the target states.Reuse & Permissions

Figure 3

On both figures, the two full lines correspond to the trajectories for $U^{*}$ and $-U^{*}$, where the dots correspond to the end points of the trajectories and the arrows to the travel direction. On the left, we see the projected trajectory of a $z$ rotation where $U^{*}=e^{i\frac{\pi }{4}\frac{{\sigma }_z}{2}}$. On the right, the figure displays the projected trajectory of a rotation on the $(x,y)$ plane where $U^{*}=e^{i\frac{\pi }{5}\frac{{\sigma }_y}{2}}$. The dashed lines are the trajectories for the limiting case $U$ such that $t^{*}\left(U\right)=t^{*}(-U)$.Reuse & Permissions

Figure 4

Plot of the minimum time $t^{*}$ to generate the rotations around the ${\vec{n}}_y,\frac{1}{\sqrt{2}}({\vec{n}}_y+{\vec{n}}_z)$, and ${\vec{n}}_z$ axes as a function of the rotation angle $\alpha$. The blue (solid) and red (dashed) curves represent the time to generate $U$ and $-U$, respectively. The black dots and vertical dotted lines underline the angles $\alpha$ for which $t^{*}=\widetilde{t^{*}}$. These angles are $\alpha =\pi ,3\pi$, as expected [see Eq. (27)].Reuse & Permissions

Figure 5

Projected trajectories for the optimal synthesis of $U^{*}\in \mathrm{SU}\left(2\right)$ subjected to a detuning field of intensity $\Delta =0,\frac{1}{2},\frac{3}{2},\frac{5}{2}$ (trajectories $a,b,c,d,$ respectively). On the left, $U^{*}=e^{i\frac{\pi }{2}\frac{{\sigma }_z}{2}}$ corresponds to a rotation about the $z$ axis. On the right, $U^{*}=e^{i\frac{\pi }{4}\frac{{\sigma }_y}{2}}$ corresponds to a rotation about the $y$ axis.Reuse & Permissions

Figure 6

(a) Time-optimal durations $T(U,\Delta )$ and $T(-U,\Delta )$ to generate $U=(0,2.2689,0)$ (blue/solid line) and $-U$ (red/dashed line) as a function of the detuning $\Delta$. (b) Time-optimal controls $u_{\psi }$ of $U$ (blue/solid line) and $-U$ (red/dashed line), respectively, as a function of the detuning $\Delta$. The region defined by the two black curves represents the time-optimal domain $\Omega {}_{\Delta }$ for each value of $\Delta$. The difference of time function ${\mathrm{T}}_{\mathrm{diff}}$ stops being continuous when both $u_{{\psi }_{+}}$ and $u_{{\psi }_{-}}$ do not belong to $\Omega {}_{\Delta }$ (domain $X$ defined by the two vertical lines). The black dots and the white squares represent the values of $\Delta$ for which ${\mathrm{T}}_{\mathrm{diff}}$ changes sign for $\Delta \in X$ and $\Delta \in \mathbb{R}/X$, respectively.Reuse & Permissions

Figure 7

Evolution of the minimum time $t^{*}$ as a function of $\Delta$ for rotations around the three axes ${\vec{n}}_y,\frac{1}{\sqrt{2}}({\vec{n}}_y+{\vec{n}}_z)$, and ${\vec{n}}_z$. The rotation angle is fixed to $\alpha =\pi /2$.Reuse & Permissions

Figure 8

(a) Plot of the time duration $T\left(u_{\psi }\right)$ of the controls $u_{\psi }$ for $\psi \in [-{\varphi }^{*}-2\pi ,-{\varphi }^{*}+2\pi ]$. (b, c) Plot of the restriction $f{}_{\Delta }$ of the end-point mapping for detuning values $\Delta =\frac{1}{2}$ and $\frac{3}{2}$, respectively. The panel (b) illustrates the case $\Delta \le \left|\tan \right(\frac{{\theta }^{*}}{2}\left)\right|$, where the optimal domain $\Omega {}_{\Delta }$ consists of the full domain $[u_{-{\varphi }^{*}-2\pi },u_{-{\varphi }^{*}+2\pi }]$. The panel (c) depicts the case $\Delta >\left|\tan \right(\frac{{\theta }^{*}}{2}\left)\right|$, where the optimal domain $\Omega {}_{\Delta }\subset [u_{-{\varphi }^{*}-2\pi },u_{-{\varphi }^{*}+2\pi }]$. In this example, the angular parameters ${\theta }^{*}=2.2689$ and ${\varphi }^{*}=0$ are fixed and taken as in Fig. 6.Reuse & Permissions