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Physical Review A

Geometric phase of entangled spin pairs in a magnetic field

Xiangyu Ge, Miki Wadati

Published Nov 1, 2005• 15 citations
Publisher

Abstract

The entanglement dependence of the geometric phase is studied in the case of two interacting spins under a time-independent magnetic field. A general entanglement-dependence geometric phase is formulated. This unification scheme covers the analyses by Sj\"oqvist for precessing spins and for isotropically interacting spins. From the general formula, the geometric phase is determined explicitly for an isotropic interaction and nonzero magnetic field.

DOI
10.1103/physreva.72.052101

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The entanglement dependence of the geometric phase is studied in the case of two interacting spins under a time-independent magnetic field. A general entanglement-dependence geometric phase is formulated. This unification scheme covers the analyses by Sj"oqvist for precessing spins and for isotropically interacting spins. From the general formula, the geometric phase is determined explicitly for an isotropic interaction and nonzero magnetic field.
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 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.
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 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.
We introduce a connection between entanglement induced by interaction and geometric phases acquired by a composite quantum spin system. We begin by analyzing the evaluation of cyclic (Aharonov-Anandan) and noncyclic (Mukunda-Simon) geometric phases for general spin chains evolving in the presence of time-independent magnetic fields. Then, by considering Heisenberg chains, we show that the interaction geometric phase, namely, the total geometric phase with subtraction of free spin contributions, is directly related to the global (Meyer-Wallach) entanglement exhibited by an initially separable state during its evolution in Hilbert space. This is analytically shown for N=2N=2N=2 spins and numerically illustrated for larger chains. This relationship promotes the interaction geometric phase to an indicator of global entanglement in the system, which may constitute a useful tool for quantum tasks based on entanglement as a resource to their performance.