# Density Matrix via Few Dominant Observables for the Ultrafast Non-Radiative Decay in Pyrazine

## TL;DR

This paper introduces a method to efficiently describe quantum dynamics using a small set of observables, applied to the ultrafast decay in pyrazine.

## Contribution

A novel approach to compactly represent quantum dynamics using dominant observables in the surprisal of the density matrix.

## Key findings

- The ultrafast population decay in pyrazine is accurately captured with six dominant observables.
- Extending the model to 4D vibrational space requires only 11 time-dependent coefficients.
- The method efficiently handles non-adiabatic effects and anharmonicity in quantum dynamics.

## Abstract

Unraveling the density matrix of a non-stationary quantum
state
as an explicit function of a few observables provides a complementary
view of quantum dynamics. We have recently developed a practical way
to identify the minimal set of the dominant observables that govern
the quantal dynamics even in the case of strong non-adiabatic effects
and large anharmonicity [Komarova et al., J. Chem. Phys. 155, 204110 (2021)]. Fast convergence in the number of the
dominant contributions is achieved when instead of the density matrix
we describe the time-evolution of the surprisal, the logarithm of
the density operator. In the present work, we illustrate the efficiency
of the proposed approach using an example of the early time dynamics
in pyrazine in a Hilbert space accounting for up to four vibrational
normal modes, {Q10a, Q6a, Q1, and Q9a}, and two coupled electronic states, the optically
dark and the bright states. Dynamics in four-dimensional (4D)
configurational space involve 19,600 vibronic eigenstates. Our results
reveal that the rate of the ultrafast population decay as well as
the shape of the nuclear wave packets in 2D, accounting only for {Q10a,Q6a} normal
modes, are accurately captured with only six dominant time-independent
observables in the surprisal. Extension of the dynamics to 3D and
4D vibrational subspace requires only five additional constraints.
The time-evolution of a quantum state in 4D vibrational space on two
electronic states is thus compacted to only 11 time-dependent coefficients
of these observables.