Figure 1

Illustration of switching strategies. (a) corresponds to Eq. (1) and shows the time variation of the switching function. The strict adiabatic limit corresponds to $\tau \to \infty$. However, approximate adiabaticity can be obtained for finite $\tau$. (b) shows the time dependence of two eigenvalues that undergo an avoided crossing. The orange (light gray) curve indicates the eigenvalue whose associated eigenstate contributes to the instantaneous time-evolved state when the evolution is adiabatic. The speed change of $S\left(t\right)$ needed to achieve adiabatic evolution is related to the size of the avoided crossing. When the evolution is not adiabatic, the wave function, if the crossing is traversed fully diabatically, is described by the red (dark gray) curve. (c) shows a stepwise variation of $\Delta \omega$ (see Appendix ) and the electric field for a pulse train. The adiabatic limit for Floquet eigenstates corresponds to $n\to \infty$, $[\Delta {\omega }_n-\Delta {\omega }_{n-1}]\to 0$, $\forall n$. (d) gives a pictorial representation of the two kinds of avoided crossings that can arise for quasienergy curves (see text).Reuse & Permissions

Figure 2

The top row shows quasienergies $\varepsilon$, in units of $B/\hslash$, within the first Brillouin zone (see Appendix ) for quasienergy states with initial quantum number $M\le 3$ and even $J$ as a function of $\Delta \omega$ (dimensionless) for a train with pulse parameters $\sigma =0.003\hslash /B$ and $T=0.02\hslash /B$. The middle row shows the same for odd $J$. The bottom row shows the evolution of the average instantaneous alignment $\langle {\cos }^2\theta \rangle$ as a function of time $t$, in units of $(\hslash /B)$, for ten initial states, as indicated in the plots. The color scale (gray scale) in the curves for the quasienergy plots indicates the quasienergy state associated with each alignment curve. A fixed field step such that the increment in $\Delta \omega$ is 1 has been chosen for all the calculations.Reuse & Permissions

Figure 3

Instantaneous average alignment during a sequence of laser pulses with $\sigma =0.003\hslash /B$, $T=0.02\hslash /B$, and $\Delta \omega =135.25$. Time $t$ is in units of $\hslash /B$. The initial state is a single Floquet eigenstate (solid line) or a linear combination of two Floquet states (dashed line). These two initial rotational wave packets were obtained by increasing the field strength during a pulse sequence. The single Floquet state was obtained using $\Delta \omega$ steps of size 0.25. The mixed state was obtained when $\Delta \omega$ changed by steps of 2.5. For the mixed state, alignment oscillates during the subsequent pulse train. However, for the single Floquet eigenstate the alignment is quasiconserved.Reuse & Permissions

Figure 4

Sensitivity of avoided crossings to the period of the laser pulses. Top and middle rows show quasienergies $\varepsilon$, in units of $B/\hslash$, as a function of $\Delta \omega$ (dimensionless) within the first Brillouin zone and average alignment ${\langle \langle {\cos }^2\theta \rangle \rangle }_T$ during each pulse for two pulse sequences, with $\sigma =T/10$. Plots in the left column are for $T=0.16\hslash /B$, and plots in the right column are for $T=0.18\hslash /B$. Plots in the top row show quasienergy curves for even $J$. Plots in the middle row correspond to odd $J$. Alignment as a function of pulse number $n$ is shown in the bottom row. Alignment of the initial state $|J=0,M=0\rangle$ is similar for both cases. However, the state $|J=1,M=0\rangle$ loses the alignment for the longer pulse due to the existence of a strong avoided crossing near $\Delta \omega =135$ with a state whose progenitor state is $|J=5,M=0\rangle$ (dashed curve in the corresponding quasienergy plot). For a step of 0.5 in $\Delta \omega$, the crossing is adiabatically traversed. Diabatic traversing of the crossing would avoid the misalignment. However, due to the accidental degeneracy between the two states, larger field steps do not lead to pure diabatic crossing but to a mixture of the two states involved in the crossings, with the subsequent loss of alignment. For the shorter pulse, the initial state $|J=1,M=0\rangle$ does not show any crossings, although it is approaching one with the highly excited state $|J=17,M=0\rangle$ (dashed curve in the corresponding quasienergy plot).Reuse & Permissions

Figure 5

Quasienergies $\varepsilon$, in units of $B/\hslash$, as a function of dimensionless $\Delta \omega$ (top and middle panels) and average alignment ${\langle \langle {\cos }^2\theta \rangle \rangle }_T$ (bottom panel) during a laser period as a function of pulse number $n$ for a pulse train with $\sigma =0.03\hslash /B$ and $T=0.3\hslash /B$. The rotational ground state $|J=0,M=0\rangle$ ends up less aligned than the initial state $|J=1,M=0\rangle$. The ground state is nearly degenerate with $|J=6,M=0\rangle$ (dashed curve in the quasienergy plot shown in the top panel) for a large range of $\Delta \omega$ values. Thus, diabatic crossing is not possible. Adiabatic following is theoretically possible but requires a very small field step. A wave packet formed by a linear combination of both states develops, which gives rise to oscillations in the alignment. The state $|J=1,M=0\rangle$ is nearly degenerate, near $\Delta \omega =0$, with the excited state $|J=17,M=0\rangle$ (dashed curve in the quasienergy plot shown in the middle panel). The interaction between both states is negligible, and $|J=1,M=0\rangle$ gets aligned when the field increases. The time evolution has been calculated using $\Delta \omega$ steps of 0.1.Reuse & Permissions