In this paper the time evolution of von Neumann entropy, as a measure of entanglement between V-type three-level atoms and the union of a two-mode field, is studied. The atom–field interaction is assumed to occur in a Kerr-type medium with an intensity-dependent coupling. Introducing a Casmir operator whose eigenvalues, 𝑁, give total excitations in the system and commutes with the governing Hamiltonian, it is concluded that the latter is block-diagonal with ever growing dimensions. As we shall show, however, each block consists of two $2 \times 2$ blocks while all the others, $(N −1)$ in number, are $3 \times 3$. We then proceed to analytically calculate the time-evolution operator which is also block-diagonal, each block with the same properties as that of the Hamiltonian. Our calculations show that, as expected, the atom–field entanglement oscillates which, depending upon the initial state, exhibits the phenomenon of collapse revivals. It is further shown that collapse revivals occur whenever both $2 \times 2$ blocks are involved in the time evolution of the system. Properties of such behaviour in the entanglement are also discussed in detail.

The fact that spin–momentum of massive particles become entangled (disentangled) as seen by moving observers, is used to investigate the properties of von Neumann entropy, as a measure of spin–momentum entanglement. To do so, we partition the total Hilbert space into momentum and spin subspaces so that the entanglement occurs between total spin states and total momenta of two spin-$\dfrac{1}{2}$ particles. Assuming that the occurrence of spin–momentum states is determined by Gaussian probability distributions, we show that the degree of entanglement ascends for small rapidities, reaches a maximum and diminishes at high rapidity. We further report how the characteristics of this behaviour vary as the widths of distributions change. In particular, a separable state, resulting from equal distribution widths, indeed becomes entangled in moving frames.

In the present article, we use negativity to investigate the entanglement between two massive particles in the spin degrees of freedom, as seen by moving observers. Assuming that the occurrence of spin-momentum states is determined by Gaussian probability distributions, we show that the degree of entanglement monotonically descends to a diminishingly small value at high rapidities. We further report, how the characteristics of this behaviour vary as the widths of distributions change. In particular, the degree of maximally entangled spin–spin states, resulting from equal distribution widths, is shown to exhibit extrema, depending on the width, at certain rapidities. The material presented in this paper then supports the idea that, for relativistic particles, a consistent reduced spin density (from which the negativity is derived) is impossible to construct.