Coherent control of the vibrational dynamics of aligned heteronuclear diatomic molecules

We present an analytical pulse design protocol for controlling the vibrational dynamics of polar diatomic molecules within a given electronic state. Altering the potential energy function via the position-dependent electric permanent dipole moment, the vibrational state population dynamics is directly controlled using appropriately shaped laser pulses in the midinfrared regime. The optimal pulse shapes—that are expected to drive the molecule along user-defined quantum pathways—are obtained by reverse engineering, that is, solving the Schrödinger equation of the nuclei inversely in a relevant subspace. The proposed control scheme is validated by accurately solving the full time-dependent Schrödinger equation of the ${\mathrm{HeH}}^{+}$ molecular ion with two completely different methods: (1) propagating the complex population amplitudes of many field-free eigenstates or (2) propagating directly the nuclear wave packet on a grid. We find that besides smooth transitions, arbitrary Rabi oscillations as well as vibrational ladder climbing can be efficiently controlled with the present scheme. As a result, the molecule is successively excited beyond the potential barrier, leading to enhanced dissociation in the ground electronic state. Rotational effects and possible extensions of the presented control are also briefly discussed.

Figure 1

Schematic representation of the control scenario discussed in this work. The shaped $E\left(t\right)$ laser pulse drives the populations of the controlled vibrational states according to a prescribed quantum path. The amplitude and phase of $E\left(t\right)$ are engineered such that the dynamical energy level shifts are compensated and the molecule is driven to the desired final quantum state superposition. Controlled Rabi oscillations can be also induced between the states of interest; furthermore an appropriate sequence of different $E\left(t\right)$ can induce a vibrational ladder climbing of the molecule. ${\omega }_{\mathrm{res}}$ is the resonance angular frequency, and $\mu$ is the nuclear transition dipole of the controlled pair of states.

Figure 2

Application of the $E\left(t\right)$ engineered laser pulse [Eq. (19)] to control the population dynamics of the $|1\rangle \to |2\rangle$ transition in ${\mathrm{HeH}}^{+}$. A wide range of the $\alpha$ transition parameter is considered (a) to achieve a complete control over the rate of transition and the final target $|1\rangle$ and $|2\rangle$ populations (dashed-dotted horizontal lines). The initial population ratio of $|1\rangle$ and $|2\rangle$ is 0.3:0.7 ($p^i=0.3$), and the final target population distribution of these states is 0.8:0.2 ($p^f=0.8$) [for the control function, see Eq. (18)]. Upon fast transitions, the two-level approximation is violated as other states get populated, and we only approximately reach the target superposition (f). On the other hand, applying sufficiently slow transitions, any desired population dynamics is achieved with $E\left(t\right)$ (see the perfect matching of the control functions and the numerically obtained populations in e). Very fast transitions violate not only the TLA but the RWA (g). The full results are calculated in the space of 10 vibrational states with the TDEC method [Eq. (4)], while Eq. (5) is solved to obtain the TLA populations without the RWA.

Figure 3

Application of the $E\left(t\right)$ engineered laser pulse [Eq. (17)] to control the population dynamics of the $|1\rangle \leftrightarrow |2\rangle$ Rabi oscillations in ${\mathrm{HeH}}^{+}$. A wide range of the $\tau$ transition time parameter is considered (a) to achieve a complete control over the rate of transition and the final target $|1\rangle$ and $|2\rangle$ populations (dashed-dotted horizontal lines). The initial population ratio of $|1\rangle$ and $|2\rangle$ is 0.3:0.7 ($p^i=0.3$), the amplitude of population oscillation is $A=0.5$, and the desired number of Rabi cycles is $n=3$ [for the control function, see Eq. (20)]. Upon fast transitions, the two-level approximation is violated as other states get populated, and we only approximately reach the target path (f). On the other hand, applying sufficiently slow transitions, any desired population dynamics is achieved with $E\left(t\right)$ (see the perfect matching of the control functions and the numerically obtained populations in g). Very fast transitions violate not only the TLA but also the RWA (e). The full results are calculated in the space of 10 vibrational states with the TDEC method [Eq. (4)], while Eq. (5) is solved to obtain the TLA populations without the RWA.

Figure 4

Control of the $|0\rangle \to |2\rangle$ transition of ${\mathrm{HeH}}^{+}$, using $E\left(t\right)$ in Eq. (19) either with engineered $\gamma \left(t\right)$ or with $\gamma \left(t\right)=0$. Due to the multicycle character of the control field (a), the system is driven along the user-defined quantum path (open symbols in b) not only when $\gamma \left(t\right)$ is engineered in an iterative manner (and hence the time-dependent resonance condition is fulfilled at each moment of time), but also when $\gamma \left(t\right)=0$ is applied. As the dynamically shifted molecular levels follow the rapidly oscillating electric field [Eq. (8)], the impact of the nuclear permanent dipoles ${\mu }_{kk}$ is washed out when the laser pulse supports many cycles, and as a consequence, population is transferred efficiently even when the time-dependent resonance condition is not fulfilled accurately ($\gamma \left(t\right)=0$). The applied control parameters are $p^i=1.0$, $p^f=0.4$, and $\alpha =0.0004$ a.u. The presented results are calculated in the space of 10 vibrational states with the TDEC method [Eq. (4)].

Figure 5

Final population of the first excited vibrational state of ${\mathrm{HeH}}^{+}$ as a function of the laser parameter imperfections defined in Eqs. (21a) and (21b). Deviations from the optimal values of the Rabi frequency and the resonance laser frequency lead to a deterioration of the final $|1\rangle$ state population from its target value of 0.8. A proper modification of the carrier laser frequency and the Rabi frequency can, however, still maintain the final target population (see the dashed line). The remaining fixed values of the applied laser pulses are $\alpha =0.0004$ a.u., $p^i=0.3$, and $p^f=0.8$ [see also Fig. 2]. The presented results are obtained by the TDEC method solving Eq. (4) in the space of 10 vibrational states, with $E\left(t\right)$ in Eq. (19).

Figure 6

Vibrational ladder climbing of the ${\mathrm{HeH}}^{+}$ molecule, applying either a single (a, b) or a double (c, d) step size. The applied laser pulse sequences are engineered such that a complete population inversion is induced between the successively coupled pairs of states $|j\rangle \to |k\rangle$ [for the individual $E\left(t\right)$ pulses of the pulse sequences, see Eq. (19)]. For each pulse, $p^i=1$ and $p^f=0$ are applied and the $\alpha$ transition rate parameter is chosen such that the $\mathrm{TLA}+\mathrm{RWA}$, and thus the control scheme, remain valid ($3\times {10}^{-4}\text{a.u.}<\alpha <7\times {10}^{-4}\text{a.u.}$). The presented results are obtained by the TDWP method, setting the molecule initially in its vibrational ground state $|0\rangle$.

Figure 7

Ground-state dissociation of the ${\mathrm{HeH}}^{+}$ molecule induced by strong laser pulses of fwhm = 20 fs duration and ${\omega }_L=1\mathrm{eV}$ carrier frequency. The molecule is prepared initially in a high-lying vibrational state ($\nu =8$) to allow for enhanced excitation beyond the potential barrier of the ground electronic state. (a) The KER spectra obtained by solving the TDSE for increasing laser intensities ($I_k={10}^{k/8}\times {10}^{15}{\text{W/cm}}^2$, $k=0,1,2...,8$) exhibit a pronounced multipeak pattern when the depletion of the initial state is large. The vertical dashed lines indicate the relevant vibrational eigenenergies shifted by the photon energy, ${\omega }_i+{\omega }_L$ ($i=7,8,9$). (b) Time-dependent populations of the bound and unbound states for $I={10}^{7/8}\times {10}^{15}{\text{W/cm}}^2$. Owing to the Rabi oscillations between the initial and continuum states in the strong field, the molecule gradually dissociates. (c) Besides the initially populated $\nu =8$ state (blue line), the neighboring $\nu =7$ (magenta line) and $\nu =9$ (light green line) states also participate in the Rabi dynamics and subsequent dissociation, giving rise to the multipeak pattern of the spectrum found in (a). The presented results are obtained by the TDWP method.