Entanglement between electronic and vibrational degrees of freedom in a laser-driven molecular system

We investigate the entanglement between electronic and vibrational degrees of freedom produced by a vibronic coupling in a molecular system described in the Born-Oppenheimer approximation. Entanglement in a pure state of the Hilbert space $\cal{H}$$=$$\cal{H}$$_{el}$$\bigotimes$$\cal{H}$$_{vib}$ is quantified using the von Neumann entropy of the reduced density matrix and the reduced linear entropy. Expressions for these entanglement measures are derived for the $2 \times N_v$ and $3 \times N_v$ cases of the bipartite entanglement, where 2 and 3 are the dimensions of the electronic Hilbert space $\cal{H}$$_{el}$, and $N_v$ is the dimension of $\cal{H}$$_{vib}$. We study the entanglement dynamics for two electronic states coupled by a laser pulse (a $2 \times N_v$ case), taking as an example a coupling between the $a^3\Sigma_{u}^{+} (6s,6s)$ and $1_g(6s,6p_{3/2})$ states of the Cs$_2$ molecule. The reduced linear entropy expression obtained for the $3 \times N_v$ case is used to follow the entanglement evolution in a scheme proposed for the control of the vibronic dynamics in a Cs$_2$ cold molecule, implying the $a^3\Sigma_{u}^{+}(6s,6s)$, $0_g^-(6s,6p_{3/2})$, and $0_g^-(6s,5d)$ electronic states, which are coupled by a non-adiabatic radial coupling and a sequence of chirped laser pulses.