Geometric Phases for Mixed States in Interferometry

Abstract

We provide a physical prescription based on interferometry for introducing the total phase of a mixed state undergoing unitary evolution, which has been an elusive concept in the past. We define the parallel transport condition that provides a connection form for obtaining the geometric phase for mixed states. The expression for the geometric phase for mixed state reduces to well known formulas in the pure state case when a system undergoes noncyclic and unitary quantum evolution.

Experimental measurement of mixed state geometric phase by quantum interferometry using NMR

“…Therefore, the interference pattern given by Eq.7 takes the following form for a mixed input spin state [11],…”

“…As the phase shifter and the unitary operator 'U' are path specific, they are represented by two controlled operations, together given by [11],…”

“…The oscillation pattern of the probability resembles the well known optical interference pattern. According to Sjöqvist et al the shift of interference pattern is a function of the geometric phase acquired by the quantum system during the unitary evolutions, as well as the purity of the initial internal state (such as, polarization of a photon) of the quantum systems involved in the interferometric operation [11]. The geometric phase can be directly measured from the shift of the interference pattern.…”

“…Recently Sjöqvist et al have provided a new description for mixed state geometric phase in terms of quantum interferometry [11]. In a quantum interferometer a quantum system undergoes a series of unitary evolutions, after which the probability of finding the system in one of its eigenstates becomes an oscillatory function of some control parameter.…”

A classical optical approach to the ‘non-local Pancharatnam-like phases’ in Hanbury-Brown–Twiss correlations

“…It has also brought out the relevance of the Bargmann invariants for geometric phase theory [11]. While some attempts have been made to define geometric phases for mixed state evolution [12][13][14][15], the emphasis at the basic level has been on pure states.…”