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Journal of Northeast Normal University

Geometric phase and Berry's phase for the entangled system of pure state with two particles

Zheng Hai-xia

Published Jan 1, 2007• 0 citations
OpenAlex

Abstract

Above the work of E.Sjoqvist,the geometric phase and the entanglement of the system composed of a pair of noninteracting spin-1/ 2 particles pressing in an external time-independent magnetic field are considered using a new method.The relation between them is gained newly:only if the entanglement is zero will the geometric phase of the composite systems reduce to a sum of one-particle geometric phase.Further more,the Berry's phase and the entanglement of the system composed of a pair of noninteracting spin-1/2 particles pressing in a rotating magnetic field are studied.The same conclusion is gained.

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Above the work of E.Sjoqvist,the geometric phase and the entanglement of the system composed of a pair of noninteracting spin-1/ 2 particles pressing in an external time-independent magnetic field are considered using a new method.The relation between them is gained newly:only if the entanglement is zero will the geometric phase of the composite systems reduce to a sum of one-particle geometric phase.Further more,the Berry's phase and the entanglement of the system composed of a pair of noninteracting spin-1/2 particles pressing in a rotating magnetic field are studied.The same conclusion is gained.
The entanglement dependence of Berry's geometric phase is studied in the case of two spin-1/2 particles with a spin-spin interaction.Berry's phases for two spin-1/2 entanglement particles in the presence of spin-spin interaction and noninteraction are calculated,respectively.The expression of calculating geometric phase,when entanglement vanishes,is obtained.Moreover,the geometric phase is discussed when only one of the two particles is affected by the external magnetic field.It is shown that its geometric phase is not equal to that of single particle which is acted by the external magnetic field,and the result is related toαthat affect the strength of entanglement. A general entanglement-dependence geometric phase is formulated.
The entanglement dependence of the noncyclic geometric phase is analyzed. A pair of noninteracting spin-12\frac{1}{2}21​ particles prepared in an arbitrarily entangled state and precessing in an external time-independent uniform magnetic field is considered. It is shown that the geometric phase reduces to a sum of one-particle geometric phases for product states and takes on the two values corresponding to the phase factors $\ifmmode\pm\else\textpm\fi{}1$ for maximally entangled states. If only one of the particles is affected by the magnetic field it is demonstrated that the influence of entanglement on the geometric phase may be interpreted as an effective reduction of the degree of polarization of the affected particle. The generalization to more than two precessing spin-12\frac{1}{2}21​ particles, in the particular case where Schmidt decompositions exists, is briefly outlined. The geometric phase for a pair of spin-12\frac{1}{2}21​ particles with a spin-spin interaction is calculated. In this model we show that the noncyclic geometric phase for a certain class of states may be interpreted solely in terms of the solid angle enclosed by the geodesically closed curve on a two-sphere parametrized by the evolving Schmidt coefficients. This suggests a geometric interpretation of Schmidt decompositions for spin-12\frac{1}{2}21​ pairs analogous to that of the Poincar\'e sphere for a single spin 12.\frac{1}{2}.21​.
The geometric phase for states of two spin-1/2 particles in rotating magnetic field is calculated, in particular, the noncyclic and cyclic non-adiabatic phases for the general case are explicitly derived and discussed. We find that the cyclic geometric phase for the entangled state can always be written as a sum of the phases of the two particles respectively; the same cannot be said for the noncyclic phase. We also investigate the geometric phase of mixed state of one particle in a biparticle system, and we find that the geometric phase for one subsystem of an entangled system is always affected by another subsystem of the entangled system.
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).
In this work, we study the geometric phases and quantum entanglement of a composite system consisting of two interacting spin-1/2 particles in the presence of an external magnetic field. The conditions for real and complex eigenvalues are derived and discussed. Based on the non-Hermitian Berry phase, we propose the geometric phase of projective quantum states, discuss their relationship, and analyze their geometric properties. We expound on how the geometric phase of projective quantum states is contingent upon the coupling strength and the nonreciprocity between two subsystems, illustrating the connection between the geometric phase of the composite system and the geometric phases of its subsystems. By studying the quantum entanglement of the subsystems, the role of the correlation between the subsystem and the nonreciprocity is discussed. The results show that the quantum entanglement is influenced and determined by the nonreciprocity of the coupling. These schemes can provide a way to manipulate the nonreciprocal correlation between the subsystems to control the non-Hermiticity, geometric phase, and entanglement of the subsystems.