三能级混合态的量子几何相位

把三能级开放系统的密度矩阵按照Gellmann矩阵展开,然后将展开系数和Bloch球中的方位角对应, 从而获得了Poincaré球内部点和复三维Hilbert空间的非单位矢量即波函数的映射.进一步建议用该非单位矢量来定义混合态的量子几何相位.结果显示该几何相位仅仅与复Hilbert投影空间的几何结构有关, 与开放系统具体的演化路径无关;并且该混合态的几何相位依赖于开放系统的反转粒子数,也是描述开放系统混合度的单值光滑曲线,这个结果意味着混合态的演化的确按照几何相位保持其运动记忆.此外,在纯态的限制下,Berry相位是本文定义的几何相位极限情况.

Geometric quantum phase for three-level mixed state

By expanding the density matrix of the open system in terms of Gell-mann matrix in a three-level system, we parameterize coefficients of expansion by some azimuthal angles and find an identity mapping of the density matrices onto interior points of the unit Poincaré sphere. Thus, the relations between the points on the unit Poincaré sphere and wave functions are extended to connect the interior points in the sphere with the nonunit vector rays corresponding to an open system in complex Hilbert space. Thus,the geometric phases for the open system are proposed to be observed by the nonunit vector rays,where the geometric phase of the pure state is the limiting case of our definition. The results show that this geometric phase merely with duplicate three-dimensional Hilbert projection space geometry structure related, has nothing to do with the open system concrete evolution way; and it depends on population inversion and is a slippy and single-value curve of Bloch radius. Therefore, the mixed state of open system retains indeed a memory of its motion in the form of a geometric phase factor.