Pulse-driven quantum dynamics beyond the impulsive regime

We review various unitary time-dependent perturbation theories and compare them formally and numerically. We show that the Kolmogorov-Arnold-Moser technique performs better owing to both the superexponential character of correction terms and the possibility to optimize the accuracy of a given level of approximation which is explored in detail here. As an illustration, we consider a two-level system driven by short pulses beyond the sudden limit.

Figure 1

Comparison of the first-order Magnus expansion (dots), the one-iteration type-$B$ KAM expansion (solid line), and the one-iteration type-$C$ KAM expansion (dashed line) for $ϵ=0.5$ and $t_1=t_1^{\prime }=0$: (a) common logarithm of the error ${\Delta }_1$ and (b) error ${\delta }_1$ as a function of $A$. The one-iteration type-$A$ KAM expansion coincides in the case $t_1^{\prime }=0$ with the first-order Magnus expansion. All quantities are dimensionless.

Figure 2

Comparison of the first-order Magnus expansion (dots), the first-order Dyson expansion (circles), the one-iteration type-$B$ KAM expansion with $t_1=t_1^{\prime }=0$ (solid line), and the optimized one-iteration type-$B$ KAM expansion with $t_1=0.5$, $t_1^{\prime }=0.22$ (dash-dot line) for $A=1$: (a) common logarithm of the error ${\Delta }_1$ and (b) error ${\delta }_1$ as a function of $ϵ$. All quantities are dimensionless.

Figure 3

Contours of the eigenvalue $g_2$ for the KAM expansion of type $B$ as a function of $t_1$ and $t_1^{\prime }$, for $A=1$ and $ϵ=0.5$. The values closer to 0 appear darker. All quantities are dimensionless.

Figure 4

Comparison of the one-iteration type-$A$ KAM expansion (dots), the one-iteration type-$B$ KAM expansion with $t_1=0.5$ (solid line), and the one-iteration type-$C$ KAM expansion with $t_1=0.7$ (dash-dot line) for $A=1$ and $ϵ=0.5$: (a) common logarithm of the error ${\Delta }_1$ and (b) eigenvalue $g_2$ as a function of $t_1^{\prime }$. All quantities are dimensionless.

Figure 5

Comparison of the second-order Magnus expansion (dots), the second-order Dyson expansion (circles), the two-iteration type-$B$ KAM expansion $\left(t_1=t_1^{\prime }=t_2=t_2^{\prime }=0\right)$ with the infinite series of commutators (solid line), two commutators (dashed line), four commutators (squares), and the optimized $\left(t_1=0.5,t_1^{\prime }=0.22,t_2=0.66,t_2^{\prime }=0.8\right)$ two-iteration type-$B$ KAM expansion (dash-dot line) for $A=1$: (a) common logarithm of the error ${\Delta }_2$ and (b) error ${\delta }_1$ as a function of $ϵ$. All quantities are dimensionless.