Minimum-time generation of a uniform superposition in a qubit with only transverse field control

We consider a two-level system with a fixed energy spacing (detuning) between the two levels and a single transverse control field which can take values between zero and a maximum amplitude. Using Pontryagin's maximum principle, we completely solve the problem of generating in minimum time a uniform superposition of the two quantum states when starting from one of them, for all the values of the ratio between the maximum control amplitude and the detuning. For each value of this ratio we find the optimal pulse sequence to have the bang-bang form and calculate the durations of the pulses composing it. The suggested framework is not only restricted to the problem at hand, but it can be also exploited in the problem of fast charging a quantum battery based on a two-level system, as well as for the optimization of pulse sequences used for the controlled preparation of the excited state in a quantum emitter, which is a prerequisite for its usage as a single-photon source.

Figure 1

Duration of the optimal single On pulse versus $r={\mathrm{\Omega }}_0/\mathrm{\Delta }$, for $r\ge 1$.

Figure 2

Trajectory of $\vec{\varphi }$ in the cases where the first and the last On pulses are (a) equal and (b) complementary.

Figure 3

Geometric interpretation of condition (81).

Figure 4

(a) Normalized durations of candidate optimal pulse-sequences when the dimensionless parameter $r={\mathrm{\Omega }}_0/\mathrm{\Delta }$ is in the range $r_{\text{min}}^n\le r<r_{\text{max}}^n$: symmetric pulse-sequence with $n$ Off pulses (green solid line), symmetric pulse-sequence with $n+1$ Off pulses (green dashed line), complementary pulse-sequence with $n$ Off pulses (red solid line). (b) Normalized duration of the optimal pulse sequence versus $r$, forming a “stairway to heaven” as $r\to 0$.

Figure 5

Final Bloch vector on the equator $\vec{r}\left(t_f\right)$ (black arrow), as well as rotation angle $\beta$ and axis $\widehat{\mu }$ (red arrow) corresponding to the total propagator of the optimal sequence, for various values of parameter $r={\mathrm{\Omega }}_0/\mathrm{\Delta }$ in the interval $r_{\text{max}}^{n+1}\le r<r_{\text{max}}^n$: (a) $r=r_{\text{max}}^{n+1}$, (b) $r_{\text{max}}^{n+1}<r<r_{\text{min}}^n$, (c) $r=r_{\text{min}}^n$, (d) $r_{\text{min}}^n<r<r_{\text{max}}^n$.

Figure 6

Percentage deviation between the durations of the intuitive suboptimal and optimal pulse sequences.

Figure 7

Optimal pulse sequences containing a single On pulse. (a) Optimal pulse for $r=1.1$. (b) Corresponding optimal trajectory. (c) Optimal pulse for $r=10$. (d) Corresponding optimal trajectory.

Figure 8

Examples of optimal pulse sequences containing a single Off pulse. (a) Optimal complementary pulse sequence for $r=0.85$. (b) Corresponding optimal trajectory, where the red segments correspond to On pulses while the blue segment to the Off pulse. (c) Optimal symmetric pulse-sequence for $r=0.55$. (d) Corresponding optimal trajectory, where the green segments correspond to On pulses while the blue segment to the Off pulse.

Figure 9

Examples of optimal pulse-sequences containing two Off pulses. (a) Optimal complementary pulse sequence for $r=0.4$. (b) Corresponding optimal trajectory, where the red segments correspond to On pulses while the blue segments to Off pulses. (c) Optimal symmetric pulse sequence for $r=0.35$. (d) Corresponding optimal trajectory, where the green segments correspond to On pulses while the blue segments to Off pulses.

Figure 10

Examples of optimal pulse sequences containing three Off pulses. (a) Optimal complementary pulse sequence for $r=0.26$. (b) Corresponding optimal trajectory, where the red segments correspond to On pulses while the blue segments to Off pulses. (c) Optimal symmetric pulse-sequence for $r=0.2$. (d) Corresponding optimal trajectory, where the green segments correspond to On pulses while the blue segments to Off pulses.