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Acta Physica Sinica

Time evolution of entanglement for the two spin-1/2 particles system

Lu Peng, Shun-Jin Wang, 四川大学物理科学与技术学院理论物理中心,成都 610064

Published Jan 1, 2009• 14 citations
Publisher

Abstract

The Heisenberg spin cluster with two particles controlled by a time-dependent magnetic field is investigated. Since the system possesses an SU（2） algebraic structure, the exact analytical solution to time evolution operator can be obtained by using algebraic dynamical method. Based on the analytical solution, the time evolution of wave function of the two-particle system can be obtained, then the entanglement can be calculated easily. The time evolution of the entanglement for different initial states is studied, and the effect of external field on the entanglement is discussed as well.

DOI
10.7498/aps.58.5955

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The Heisenberg spin cluster with two particles controlled by a time-dependent magnetic field is investigated. Since the system possesses an SU（2） algebraic structure, the exact analytical solution to time evolution operator can be obtained by using algebraic dynamical method. Based on the analytical solution, the time evolution of wave function of the two-particle system can be obtained, then the entanglement can be calculated easily. The time evolution of the entanglement for different initial states is studied, and the effect of external field on the entanglement is discussed as well.
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We study the dynamics evolution of a two-quhit Heisenherg XXX spin chain under a time-dependent rotating magnetic field. Based on the algebraic structure of the non-autonomous system, the exact solution of the Schrödinger equation is obtained by using the method of algebraic dynamics. Based on the time-dependent analytical solution, we further study the entanglement evolution between the two coupled spins for different initial states, and find that the entanglement is determined by the coefficients of the initial state and the coupling constant J of the system.
Introduction: In this study, we investigate the dynamics of the Entanglement in a two-spin system with long-range interaction.Method: For this purpose, we use negativity as the entanglement measurement. Using the time evolution operator, we obtain entanglement dynamics of the system at time t. We consider two different initial states and investigate the dynamical behavior of the system for all of them separately. Finally, we consider both J and D as a function of R and then study the time evolution of the entanglement in different R.Results: We find that in the long-range interaction, the R dependence of the dynamical behavior in the two systems is different. Depending on the initial states, the DM interaction and the magnetic field have no effect on the entanglement dynamics.
We consider the time evolution of entanglement in a finite two-dimensional transverse Ising model. The model consists of a set of seven localized spin-12\frac{1}{2}21​ particles in a two-dimensional triangular lattice coupled through nearest-neighbor exchange interaction in the presence of an external time-dependent magnetic field. The magnetic field is applied in different function forms: step, exponential, hyperbolic, and periodic. We found that the time evolution of the entanglement shows an ergodic behavior under the effect of the time-dependent magnetic fields. Also, we found that while the step magnetic field causes great disturbance to the system, creating rapid oscillations, the system shows great controllability under the effects of the other magnetic fields where the entanglement profile follows closely the shape of the applied field even with the same frequency for periodic fields. This follow-up trend breaks down as the strength of the field, the transition constant for the exponential and hyperbolic forms, or the frequency for periodic field increase leading to rapid oscillations. We observed that the entanglement is very sensitive to the initial value of the applied periodic field: the smaller the initial value is, the less distorted the entanglement profile is. Furthermore, the effect of thermal fluctuations is very devastating to the entanglement, which decays very rapidly as the temperature increases. Interestingly, although a large value of the magnetic field strength may yield a small entanglement, the magnetic field strength was found to be more persistent against thermal fluctuations than the small field strengths.
The geometric phase, in particular the Berry phase, in an entangled state of five spin-1/2 particles is studied. A time-dependent magnetic field is applied to control the time evolution of the cluster. Using the method of algebraic dynamics, we calculate the non-adiabatic geometric phase or Berry phase and the degeneracy energy levels when the magnetic rotates around Z axis. Based on the exact analytical solutions, we show how the Berry phase of the entangled state of this cluster depends on the external magnetic field parameters ω (the angular velocity of the rotating magnetic field) and θ (the angle between the magnetic field and Z axis).