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Physical Review A

Entropy and negativity of Fermi-resonance coupling vibrations in a spectroscopic Hamiltonian

Xi-Wen Hou, Ming-Fang Wan, Zhong-Qi Ma

Published Feb 9, 2009• 11 citations
Publisher

Abstract

The entanglement dynamics of Fermi resonance coupling vibrations in a spectroscopic Hamiltonian for molecules $\mathrm{C}\mathrm{H}{\mathrm{D}}_{3}$, $\mathrm{C}\mathrm{H}{\mathrm{Cl}}_{3}$, and $\mathrm{C}\mathrm{H}{\mathrm{F}}_{3}$ is studied in terms of the von Neumann entropy and negativity with three kinds of initial states that are direct products of coherent states, thermal states, and squeezed states of each mode. It is demonstrated that the increasing rate of entanglement in the early-time evolution is the largest for the squeezed states, while that is the smallest for the coherent states. For small parameters in the coherent and thermal states, entanglement displays a quasiperiodicity with the quasiperiod in $\mathrm{C}\mathrm{H}{\mathrm{F}}_{3}$ being the longest. Furthermore, the dynamical correlation between the von Neumann entropy and negativity shows that positively correlated or anticorrelated behaviors of both measures strongly depend on initial states and molecules. For those states with small parameters, both measures exhibit dominantly positive correlation. These are useful for molecular quantum computing and information processing.

DOI
10.1103/physreva.79.022308

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Several measures of entanglement have attracted considerable interest in the relationship of a measure of entanglement with other quantities. The dynamics of entropy, energy and negativity is studied for Fermi-resonance coupled vibrations in substituted methanes with three kinds of initial mixed states, which are the mixed density matrices of binomial states, thermal states and squeezed states on two vibrational modes, respectively. It is demonstrated that for mixed binomial states and mixed thermal states with small magnitudes the entropies of the stretch and the bend are anti-correlated in the same oscillatory frequency, so do the energies for each kind of state with small magnitudes, whereas the entropies exhibit positive correlations with the corresponding energies. Furthermore, for small magnitudes quantum mutual entropy is positively correlated with the interacting energy. Analytic forms of entropies and energies are provided with initial conditions in which they are stationary, and the agreement between analytic and numerical simulations is satisfactory. The dynamical entanglement measured by negativity is examined for those states and conditions. It is shown that negativity displays a sudden death for mixed binomial states and mixed thermal states with small magnitudes, and the time-averaged negativity has the minimal value under the conditions of stationary entropies and energies. Moreover, negativity is positively correlated with the mutual entropy and the interacting energy just for mixed squeezed states with small magnitudes. Those are useful for molecular quantum information processing and dynamical entanglement.
The dynamical entanglement for Fermi coupled C-H stretch and bend vibrations in molecule CHD 3 is studied in terms of two negativities and the reduced von Neumann entropy, where initial states are taken to be direct products of photon-added coherent states on each mode. It is demonstrated that the negativity defined by the sum of negative eigenvalues of the partial transpose of density matrices is positively correlated with the von Neumann entropy. The entanglement difference between photon-added coherent states and usual coherent states is discussed as well.
As an alternative with potential connections to actual experiments, other than the systems more usually used in the field of entanglement, the dynamics of entropy correlation and entanglement between two anharmonic vibrations in a well-established algebraic model, with parameters extracted from fitting to highly excited spectral experimental results for molecules H 2 O and SO 2 , is studied in terms of the linear entropy and two negativities for various initial states that are respectively taken to be the mixed density matrices of thermal states and squeezed states on each mode. For a suitable parameter in initial states the entropies in two stretches can show positive correlation or anti-correlation. And the linear entropy of each mode is positively correlated with the negativities just for the mixed-squeezed states with small parameters in H 2 O while they do not display any correlation in other cases. For the mixed-squeezed states the negativities exhibit dominantly positive correlations with an effective mutual entropy. The differences in the linear entropy and the negativities between H 2 O and SO 2 are discussed as well. Those are useful for molecular quantum computing and quantum information processing.
The dynamics of quantum entanglement described by the von Neumann entropy is studied for the localized states of Fermi-resonance coupling vibrations in molecule CS2, where the interacting energy between the stretching and the bending modes is considered to establish a connection between entanglement and energy. It is shown that entanglement reveals dominant anti-correlation with the interacting energy for the stretch-localized state, while that exhibits dominantly positive correlation for the bend-localized state. The entanglement and the energy for the dislocalized states are discussed as well. Those are useful for molecular quantum computing and quantum information in high dimensional states.
By means of the reduced-density linear entropy, we investigate the properties of dynamical entanglement of vibrations in integrable dimer and realistic small molecules which are initially in the two-mode squeezed vacuum state. It is found that the entropy of the integrable dimer is periodic for weak coupling strength c 1 and small squeezing parameter r, and there exists a beat phenomenon for strong c 1 and large r. Moreover, the entropy of the small molecules is quasi-periodic for small r, and the beat phenomenon occurs in the entropy evolution of the two molecules C 2 D 2 and SO 2 for large r. Our results might be used for molecular quantum computing based on vibrational states.