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arXiv:quant-ph/0108058 · Physical Review Letters

Singularities of the Mixed State Phase

Rajendra Bhandari

Published Dec 12, 2002• 52 citations
arXivHTMLPDF

Abstract

A recent proposal of Sjoqvist et.al. to extend Pancharatnam's criterion for phase difference between two different pure states to the case of mixed states in quantum mechanics is analyzed and the existence of phase singularities in the parameter space of an interference experiment with particles in mixed states pointed out. In the vicinity of such singular points the phase changes sharply and precisely at these points it becomes undefined. A closed circuit in the parameter space around such points results in a measurable phase shift equal to 2nπ, where n is an integer. Such effects have earlier been observed in interference experiments with pure polarization states of light, a system isomorphic to the spin-1/2 system in quantum mechanics. Implications of phase singularities for the interpretation of experiments with partially polarized and unpolarized neutrons are discussed. New kinds of topological phases involving variables representing decoherence (depolarization) of pure states are predicted and experiments to verify them suggested.

DOI
10.1103/physrevlett.89.268901

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A recent proposal of Sjoqvist et.al. to extend Pancharatnam's criterion for phase difference between two different pure states to the case of mixed states in quantum mechanics is analyzed and the existence of phase singularities in the parameter space of an interference experiment with particles in mixed states pointed out. In the vicinity of such singular points the phase changes sharply and precisely at these points it becomes undefined. A closed circuit in the parameter space around such points results in a measurable phase shift equal to 2nπ, where n is an integer. Such effects have earlier been observed in interference experiments with pure polarization states of light, a system isomorphic to the spin-1/2 system in quantum mechanics. Implications of phase singularities for the interpretation of experiments with partially polarized and unpolarized neutrons are discussed. New kinds of topological phases involving variables representing decoherence (depolarization) of pure states are predicted and experiments to verify them suggested.
A critique of a recent experiment [Wagh et.al., Phys.Rev.Lett.81, 1992 (7 Sep 1998)] to measure the noncyclic phase associated with a precessing neutron spin in a neutron interferometer, as given by the Pancharatnam criterion, is presented. It is pointed out that since the experiment measures, not the noncyclic phase itself, but a quantity derived from it, it misses the most interesting feature of such a phase, namely the different sign associated with states lying in the upper and the lower hemispheres, a feature originating in the existence of a phase singularity. Such effects have earlier been predicted and seen in optical interference experiments using polarization of light as the spinor [Bhandari, Phys.Rep.281, 1 (Mar 1997)].
Single-particle interference phenomena can be observed with neutrons, and the "entanglement of degrees of freedom", i.e. the contextuality, can be verified and used in further experiments. Entanglement of two photons or atoms is a complementary situation to the double-slit diffraction of a single photon, neutron, or atom. In this respect, neutrons are proper tools for testing quantum mechanics, because they are massive, they couple to electromagnetic fields due to their magnetic moment, and they are subject to all basic interactions, and they are sensitive to topological effects as well. The4π-symmetry of spinor wave functions, the spin-superposition law, and many topological phenomena can be made visible, which shows interesting intrinsic features of quantum physics. Related experiments will be discussed. Deterministic and stochastic partial absorption experiments can be described by Bell-type inequalities. Recent neutron interferometry experiments based on post-selection methods have renewed the discussion about quantum non-locality and the quantum measuring process. It has been shown that interference phenomena can be revived even when the overall interference pattern has lost its contrast. This indicates a persisting coupling in the phase space even in cases of spatially separated Schrödinger cat-like situations. These states are extremely fragile and sensitive against any kind of fluctuations and other de-coherence processes. More complete quantum experiments also show that a complete retrieval of quantum states behind an interaction volume becomes impossible in principle.
A three-dimensional generalization of the sign-change ( pi phase shift) rule for adiabatic cycles of spin-1/2 or two-state wave functions encircling a degeneracy in the parameter space of the Hamiltonian yields a Dirac-type singularity wherein any closed circuit of the adiabatic cycle in which the degeneracy is "looped" results in an observable +/-2pi phase shift. It is concluded that an interferometer loop similarly taken around a magnetic monopole of strength n/2 yields an observable +/-2npi phase shift, n being an integer.
Neutron interferometry provides a powerful tool to investigate particle and wave features in quantum physics. Single particle interference phenomena can be observed with neutrons and the entanglement of degrees of freedom, i.e., contextuality can be verified and used in further experiments. Entanglement of two photons, or atoms, is analogous to a double slit diffraction of a single photon, neutron or atom. Neutrons are proper tools for testing quantum mechanics because they are massive, they couple to electromagnetic fields due to their magnetic moment, they are subject to all basic interactions, and they are sensitive to topological effects, as well. The 4π-symmetry of spinor wave functions, the spin-superposition law and many topological phenomena can be made visible, thus showing interesting intrinsic features of quantum physics. Related experiments will be discussed. Deterministic and stochastic partial absorption experiments can be described by Bell-type inequalities. Neutron interferometry experiments based on post-selection methods renewed the discussion about quantum non-locality and the quantum measuring process. It has been shown that interference phenomena can be revived even when the overall interference pattern has lost its contrast. This indicates a persisting coupling in phase space even in cases of spatially separated Schrödinger cat-like situations. These states are extremely fragile and sensitive against any kind of fluctuations and other decoherence processes. More complete quantum experiments also show that a complete retrieval of quantum states behind an interaction volume becomes impossible in principle, but where and when a collapse of the wave-field occurs depends on the level of experiment.