On atomic state purity operator, degree of state purity and concurrence in the JC and anti-JC models

Abstract

The state of an atom in a bipartite qubit, Jaynes-Cummings (JC) or anti-Jaynes-Cummings (aJC) interaction is described by a reduced density operator. The purity of the state has been measured by taking the trace of the square of the reduced density operator. In this article, we define the square of the reduced density operator as the state purity operator, composed of a completely pure state part and a completely mixed state part. The coefficient of the completely mixed state part is the mixed state measure, formally obtained as the determinant of the reduced density operator and it is therefore directly related to tangle, the square of concurrence of the bipartite system. Expressed in various equivalent forms, the mixed state measure provides all the characteristic elements of state purity or entanglement, such as eigenvalues of the reduced density operator, nonclassicality measures and a state purity complex amplitude. The argument of the state purity complex amplitude in polar form is the phase of the state purity measure, which defines the degree of purity of the state. We find that the degree of purity and concurrence are complementary quantifiers satisfying a complementarity relation. The general form of the mixed state measure provides an interpretation that concurrence is fully defined by the Bloch radius four-vector in a spacetime frame. Plots of the degree of purity, concurrence and spin excitation number at resonance reveal that the atomic state collapses rapidly to a momentary totally mixed or maximally entangled state over a very short time after the beginning of the interaction, then evolves gently to a pure disentangled state in the middle of the collapse region of the spin excitation number. In off-resonance dynamics with large values of the frequency detuning parameters, the degree of purity, concurrence and spin excitation number develop periodic evolution at the respective maximum (1) or minimum (0) values, simultaneously signifying that at very large detuning, the atom evolves to a perfectly pure disentangled state where the spin excitation number is effectively zero.