Electronic decoherence following photoionization: Full quantum-dynamical treatment of the influence of nuclear motion

Photoionization using attosecond pulses can lead to the formation of coherent superpositions of the electronic states of the parent ion. However, ultrafast electron ejection triggers not only electronic but also nuclear dynamics—leading to electronic decoherence, which is typically neglected on time scales up to tens of femtoseconds. We propose a full quantum-dynamical treatment of nuclear motion in an adiabatic framework, where nuclear wave packets move on adiabatic potential energy surfaces expanded up to second order at the Franck-Condon point. We show that electronic decoherence is caused by the interplay of a large number of nuclear degrees of freedom and by the relative topology of the potential energy surfaces. Application to ${\mathrm{H}}_2\mathrm{O}$, paraxylene, and phenylalanine shows that an initially coherent state evolves to an electronically mixed state within just a few femtoseconds. In these examples the fast vibrations involving hydrogen atoms do not affect electronic coherence at short times. Conversely, vibrational modes involving the whole molecular skeleton, which are slow in the ground electronic state, quickly destroy it upon photoionization.

Figure 1

The spatial overlap of vibrational wave packets, and thus the electronic coherence, depends on the topology of the potential energy surfaces relative to each other. This is illustrated here for the motion along one normal-mode coordinate $Q_i$. After excitation from the ground state (gray), the nuclear wave packets evolve on the cationic potential energy surface (orange). Left side: Different gradients and curvatures lead to diminishing spatial overlap and decoherence. Right side: The potential energy surfaces are displaced only vertically. The nuclear wave packets move synchronously; spatial overlap is high throughout the propagation.

Figure 2

Evolution of electronic purity in ${\mathrm{H}}_2\mathrm{O}$ with four cationic states. The propagation using one mode only (blue dashed lines) is compared to the propagation of the full-dimensional system (orange solid line). The system, initially prepared in an equally weighted superposition of all cationic states, evolves to a mixed state within $1\mathrm{fs}$. This is attributed to the loss of coherence along the bending mode as well as the interplay of the three vibrational modes.

Figure 3

Evolution of electronic purity in paraxylene with two cationic states. One-dimensional simulations (blue dotted lines) are compared to the propagation of the full-dimensional system (orange solid line). The interplay of the large number of modes leads to a mixed state on a time scale of $2\text{–}3\mathrm{fs}$. Recurrence can be seen in the one-dimensional simulations, but is suppressed when all vibrational modes are taken into account.

Figure 4

Electronic coherence is reduced by the interplay of all vibrational modes. The ten fastest modes, corresponding to the C-H vibrations, maintain coherence. It is the slow modes, corresponding to vibrations of the whole molecule and the benzene-ring carbon atoms, that account for the fast decoherence.

Figure 5

Electronic decoherence in paraxylene for an initially pure state $\mathrm{\Psi }(\mathbf{Q},0)={\chi }_0(\mathbf{Q},0)\left(\sqrt{c_1}|1\rangle +\sqrt{c_2}|2\rangle \right)$. The time scale where electronic decoherence is destroyed is independent of the choice of weights $c_1,c_2$. A mixed state is reached within few femtoseconds, where $\mathrm{Tr}\left({\rho }^2\right)=c_1^4+c_2^4$.

Figure 6

In paraxylene, a subset of 12 modes that are coupled to each other $\left(\right|{\gamma }_{ij}^{\left(\mu \right)}|>{10}^{-3})$ is given a second single-particle function (SPF) per mode (orange dotted line), compared to one SPF per mode (blue solid line). This suppresses the small recurrence at $8\text{–}10\mathrm{fs}$.

Figure 7

Electronic decoherence in paraxylene including (blue solid line) and neglecting (orange dashed line) the kinetic energy operator in the quantum-dynamical calculation.

Figure 8

Evolution of electronic purity in phenylalanine with two cationic states. One-dimensional simulations for each of the 63 normal modes (blue dotted lines) are compared to the propagation including all modes (orange solid line) for an initially pure state. Electronic decoherence occurs within $1\mathrm{fs}$.