Time Evolution of Entanglement of Electrons and Nuclei and Partial Traces in Ultrafast Photochemistry

Broad in energy optical pulses induce ultrafast molecular dynamics where nuclear degrees of freedom are entangled with electronic ones. We discuss a matrix representation of wave functions of such entangled systems. Singular Value Decomposition, (SVD) of this matrix provides a representation as a sum of separable terms. Their weights can be arranged in decreasing order. The representation provided by the SVD is equivalent to a Schmidt decomposition. If there is only one term or if one term is already a good approximation, the system is not entangled. The SVD also provides either an exact or a few term approximation for the partial traces. A simple example, the dynamics of LiH upon ultrafast excitation to several non-adiabatically coupled electronic states is provided. The major contribution to the entanglement is created during the exit from the Franck Condon region. An additional contribution is the entanglement due to the nuclear motion induced non-adiabatic transitions.