Driving the formation of the RbCs dimer by a laser pulse: A nonlinear-dynamics approach

We study the formation of the RbCs molecule by an intense laser pulse using nonlinear dynamics. Under the Born-Oppenheimer approximation, the system is modeled by a two-degree-of-freedom rovibrational Hamiltonian, which includes the ground electronic potential energy curve of the diatomic molecule and the interaction of the molecular polarizability with the electric field of the laser. As the laser intensity increases, we observe that the formation probability first increases and then decreases after reaching a maximum. We show that the analysis can be simplified to the investigation of the long-range interaction between the two atoms. We conclude that the formation is due to a very small change in the radial momentum of the dimer induced by the laser pulse. From this observation, we build a reduced one-dimensional model which allows us to derive an approximate expression of the formation probability as a function of the laser intensity.

Figure 1

(a) Electronic potential energy curve $\varepsilon \left(R\right)$ of the RbCs and (b) parallel ${\alpha }_{\parallel }\left(R\right)$ and perpendicular ${\alpha }_{\perp }\left(R\right)$ components of the molecular polarizability of the RbCs molecule.

Figure 2

Equipotential curves of the potential energy surface $V(R,\theta ,t)$ during the plateau $\left[g\right(t)=1]$ for a laser field strength $F=1.5\times {10}^{-3}$ a.u.

Figure 3

Formation probability as a function of $F$ for an initial energy $E_0=3\times {10}^{-9}$ computed from Hamiltonian (1). The parameters of the pulse are $T_{\mathrm{ru}}=T_{\mathrm{rd}}=5\text{ps}$ and $T_{\mathrm{p}}=70\text{ps}$ (shaded red line), $T_{\mathrm{ru}}=T_{\mathrm{rd}}=15\text{ps}$ and $T_{\mathrm{p}}=70\text{ps}$ (solid green line) and $T_{\mathrm{ru}}=T_{\mathrm{rd}}=15\text{ps}$ and $T_{\mathrm{p}}=140\text{ps}$ (dotted blue line), respectively.

Figure 4

Evolution of the energy of an ensemble of trajectories with initial energy $E_0=3\times {10}^{-9}$ a.u. The amplitude of the laser field is $F=1.5\times {10}^{-3}$ a.u. The parameters of the pulse are $T_{\mathrm{ru}}=T_{\mathrm{rd}}=15\text{ps}$ and $T_{\mathrm{p}}=70$ ps. The solid red and the dashed blue lines indicate the dissociation energy (9) and the zero energy, respectively.

Figure 5

Probability distribution of the energy of an ensemble of trajectories with an initial energy $E_0=3\times {10}^{-9}$ a.u. after a ramp-up of 15 ps (dashed red line). The dotted blue line is the probability energy distribution of an ensemble given by Eq. (11). The dissociation energy $E_d$ for this electric field is denoted with the solid green vertical line. The parameters of the pulse are $T_{\mathrm{ru}}=T_{\mathrm{rd}}=15\text{ps}$ and $T_{\mathrm{p}}=70\text{ps}$ and the amplitude of the electric laser field is $F=1.5\times {10}^{-3}$ a.u.

Figure 6

Poincaré section $\left(P_R=0,{\dot{P}}_R<0\right)$ of Hamiltonian (1) for an energy $E=-3.98\times {10}^{-4}$ a.u. and for an electric field $F=1.5\times {10}^{-3}$ a.u.

Figure 7

Trajectories in the plane $(Rsin\theta ,Rcos\theta )$ of Hamiltonian (1) for $F=1.5\times {10}^{-3}\text{a.u.}$ and energy $E_0=-3.98\times {10}^{-4}$ a.u. (a) Rotational trajectory with initial conditions $\theta =\pi /2,P_{\theta }=50$, and $P_R=0$ (red square in Fig. 6); (b) rotational chaotic trajectory with initial conditions $\theta =1.45,P_{\theta }=0$, and $P_R=0$ (green dot in Fig. 6); (c) vibrational chaotic trajectory with initial conditions $\theta =1.1,P_{\theta }=0$, and $P_R=0$ (blue star in Fig. 6); and (d) vibrational regular trajectory with initial conditions $\theta =0.2,P_{\theta }=0$, and $P_R=0$ (purple diamond in Fig. 6).

Figure 8

First recurrence time (in ps) in the Poincaré section $(P_R=0,\dot{P_R}<0)$ in the plane $(\theta ,P_{\theta })$ for $F=1.5\times {10}^{-3}\text{a.u.}$ and energy $E=-3.98\times {10}^{-4}$ a.u. The color axis has been saturated at 1000 ps for clarity. In the middle region, the recurrence time reaches above 1400 ps. Note that a logarithmic scale is used in the color code.

Figure 9

Poincaré section $\left(P_R=0,{\dot{P}}_R<0\right)$ of Hamiltonian (1) for an energy $E=-3.976\times {10}^{-4}$ a.u. and for an electric field $F=1.5\times {10}^{-3}$ a.u.

Figure 10

Probability distribution of the energy of an ensemble of trajectories with an initial energy $E_0=3\times {10}^{-9}$ a.u. after the a ramp-down of 15 ps. The vertical red line indicates the zero energy value. The amplitude of the laser field is $1.5\times {10}^{-3}$ a.u. and the parameters of the pulse are $T_{\mathrm{ru}}=T_{\mathrm{rd}}=15\text{ps}$ and $T_{\mathrm{p}}=70\text{ps}$.

Figure 11

Histogram of the initial conditions leading to formation (solid red line) and leading to dissociation (shaded blue line). The parameters of the laser are $F=1.5\times {10}^{-3}$ a.u., $T_{\mathrm{ru}}=T_{\mathrm{rd}}=15\text{ps},$ and $T_{\mathrm{p}}=70\text{ps}$. The energy of the trajectories is $E_0=3\times {10}^{-9}$ a.u.

Figure 12

Formation probability as a function of $F$ for an initial energy $E_0=3\times {10}^{-9}$ a.u. computed using Hamiltonian (12). The parameters of the pulse are $T_{\mathrm{ru}}=T_{\mathrm{rd}}=5\text{ps}$ and $T_{\mathrm{p}}=70\text{ps}$ (dashed red line), $T_{\mathrm{ru}}=T_{\mathrm{rd}}=15\text{ps}$ and $T_{\mathrm{p}}=70\text{ps}$ (solid green line), and $T_{\mathrm{ru}}=T_{\mathrm{rd}}=15\text{ps}$ and $T_{\mathrm{p}}=140\text{ps}$ (dotted blue line), respectively.

Figure 13

Probability distributions of an ensemble of trajectories with an initial energy $E_0=3\times {10}^{-9}$ a.u. after a ramp-up of $15\text{ps}$ obtained with formula (16) (solid red line) and with Hamiltonian (12) (dashed blue line). The amplitude of the laser field is $F=1.5\times {10}^{-3}\text{a.u.}$ The dashed green vertical line indicates the dissociation energy $E_d=-F^2{\alpha }_{\parallel }\left(\infty \right)/4$ while the dotted purple vertical line denotes the energy $E_d=-F^2{\alpha }_{\parallel }\left(R_{\max }\right)/4$.

Figure 14

Formation probability for Hamiltonian (12) as a function of $F$ for an initial energy $E_0=3\times {10}^{-9}$ a.u. The parameters of the pulse are $T_{\mathrm{ru}}=15\text{ps},T_{\mathrm{p}}=70\text{ps}$, and no ramp-down.

Figure 15

Periods of our ensemble of trajectories for $F=1.5\times {10}^{-3}$ a.u. using Eq. (17). Note the logarithmic scale in the vertical axis.

Figure 16

Formation probability as a function of $F$ for an initial energy $E_0=3\times {10}^{-9}$ a.u. obtained from the long-range Hamiltonian (20) (dashed red line) and the full Hamiltonian (12) (solid green line). The parameters of the pulse are $T_{\mathrm{ru}}=T_{\mathrm{rd}}=15\text{ps}$ and $T_{\mathrm{p}}=70\text{ps}$.

Figure 17

(a) Evolution as a function of $F$ of the roots $R_1$ and $R_2$ of $E\left(R\right)$ given by Eq. (24). (b) Evolution of $R_2-R_1$ as a function of $F$. The parameters of the pulse are $T_{\mathrm{ru}}=15\text{ps},T_{\mathrm{p}}=70\text{ps}$, and $T_{\mathrm{rd}}=15\text{ps}$.

Figure 18

Formation probability given by Eq. (27) as a function of $F$ (dotted blue line). For completeness, the formation probability as a function of $F$ obtained from the long-range Hamiltonian (20) (dashed red line) and from the full Hamiltonian (12) (solid green line) are also shown. The black vertical dashed arrow is located at the value $F\approx 0.00107$ a.u. given by Eq. (28). For this value of $F$, it is expected to find the maximum of the formation probability. The parameters of the pulse are $T_{\mathrm{ru}}=15\text{ps},T_{\mathrm{p}}=70\text{ps}$, and $T_{\mathrm{rd}}=15\text{ps}$. The initial energy of the trajectories is $E_0=3\times {10}^{-9}$ a.u.