Optimization of short coherent control pulses

The coherent control of a small quantum system is considered. For a two-level system coupled to an arbitrary bath we consider a pulse of finite duration. We derive the leading and the next-leading order corrections to the evolution operator due to the noncommutation of the pulse and the bath Hamiltonian. The conditions are computed that make the leading corrections vanish. The pulse shapes optimized in this way are given for $\pi$ and $\frac{\pi }{2}$ pulses.

Figure 1

(a) Plot of a generic pulse $v\left(t\right)$. The $x$- and $y$-axis values are expressed in units of ${\tau }_p$ and $1/{\tau }_p$, respectively. The shaded region represents an example for $\phi \left(t\right)$, that is the integrated pulse amplitude between the instant ${\tau }_s$ and a generic instant $t$ (see text). (b) Sketch of the approximately equivalent idealized instantaneous pulse.

Figure 2

Composite $\pi$ and $\pi /2$ pulses with piecewise constant amplitude in the symmetric case. In the $\pi$ pulse, the sign changes occur at ${\tau }_1={\tau }_p/7$ ($6{\tau }_p/7$, respectively) and the amplitude is $a_{\text{max}}=7\pi /6\left[1/{\tau }_p\right]$. In the $\pi /2$ pulse, the sign changes occur at ${\tau }_1=0.13155{\tau }_p$ (${\tau }_p-{\tau }_1$, respectively) and the amplitude is $a_{\text{max}}=1.65765\left[1/{\tau }_p\right]$.

Figure 3

Composite $\pi$ and $\pi /2$ pulses with piecewise constant amplitude made from two elementary pulses in the asymmetric case. For the $\pi$ pulse, the sign change occurs at ${\tau }_1=3{\tau }_p/4$, the amplitude is $a_{\text{max}}=\pi \left[1/{\tau }_p\right]$, and the instant of the equivalent ideal pulse is ${\tau }_s=\left({\tau }_p/2\right)\left(1-1/\pi \right)$. For the $\pi /2$ pulse, ${\tau }_1=0.78220{\tau }_p$, the amplitude is $a_{\text{max}}=1.39156\left[1/{\tau }_p\right]$ and the instant of the equivalent ideal pulse is ${\tau }_s=0.23128{\tau }_p$.

Figure 4

Composite $\pi$ and $\pi /2$ pulses with piecewise constant amplitude made from three elementary pulses in the asymmetric case. For the $\pi$ pulse, the sign changes occur at ${\tau }_1={\tau }_p/13$ and at ${\tau }_2=6{\tau }_p/13$, the amplitude is $a_{\text{max}}=13\pi /6\left[1/{\tau }_p\right]$, and the instant of the equivalent ideal pulse is ${\tau }_s={\tau }_p/2$. For the $\pi /2$ pulse, the sign changes occur at ${\tau }_1=0.44834{\tau }_p$ and at ${\tau }_2=0.80813{\tau }_p$, the amplitude is $a_{\text{max}}=2.80074\left[1/{\tau }_p\right]$, and the instant of the equivalent ideal pulse is ${\tau }_s=0.219{\tau }_p$.

Figure 5

Continuous $\pi$ and $\pi /2$ pulses in the symmetric case. The parameter $a$ as defined in the ansatz (42) takes the value $-3.73002/{\tau }_p$ for the $\pi$ pulse and the value $-1.78259/{\tau }_p$ for the $\pi /2$ pulse. The horizontal lines indicate the amplitude for a totally constant, uncorrected $\pi$ pulse (solid line) and $\pi /2$ pulse (dashed-dotted line), respectively.

Figure 6

Continuous $\pi$ and $\pi /2$ pulses in the asymmetric case. For the $\pi$ pulse, the parameter $a$ as defined in the ansatz (43) takes the value $3.39308/{\tau }_p$ and the position ${\tau }_s$ of the equivalent ideal pulse takes the value $0.33806{\tau }_p$. For the $\pi /2$ pulse, the parameter $a$ is $1.54714/{\tau }_p$ and the position ${\tau }_s$ of the equivalent ideal pulse is $0.28093{\tau }_p$. The horizontal lines indicate the amplitude for a totally constant, uncorrected $\pi$ pulse (solid line) and $\pi /2$ pulse (dashed-dotted line), respectively.