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Physics of Particles and Nuclei Letters

Correlations of polarizations and entangled states in the two-photon system

Vladimir Lvovich Lyuboshitz, Vladimir Lvovich Lyuboshitz

Published Nov 19, 2009• 0 citations
Publisher

Abstract

The correlations of the linear and circular polarizations in the system of two photons have been theoretically investigated. The polarization of a two-photon state is described by the one-photon Stokes parameters and by the components of the correlation “tensor” in the Stokes space. It is shown that in the case of two-photon decays π0 → 2γ, η → 2γ, K 0 → 2γ, K 0 → 2γ and the cascade process |0〉 → |1〉 + γ → |0〉 + 2γ(|0〉 and |1〉 are states with the spin 0 and 1, respectively) the final two-photon state represents a characteristic example of the entangled (nonfactorizable) state, and the correlations between the Stokes parameters in all these decays have the purely quantum character: the incoherence inequalities of the Bell type for the components of the correlation “tensor”, established previously for the case of classical “mixtures”, are violated. The general analysis of the registration procedure for two correlated photons by two one-photon detectors is performed.

DOI
10.1134/s154747710907022x

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The correlations of the linear and circular polarizations in the system of two photons have been theoretically investigated. The polarization of a two-photon state is described by the one-photon Stokes parameters and by the components of the correlation “tensor” in the Stokes space. It is shown that in the case of two-photon decays π0 → 2γ, η → 2γ, K 0 → 2γ, K 0 → 2γ and the cascade process |0〉 → |1〉 + γ → |0〉 + 2γ(|0〉 and |1〉 are states with the spin 0 and 1, respectively) the final two-photon state represents a characteristic example of the entangled (nonfactorizable) state, and the correlations between the Stokes parameters in all these decays have the purely quantum character: the incoherence inequalities of the Bell type for the components of the correlation “tensor”, established previously for the case of classical “mixtures”, are violated. The general analysis of the registration procedure for two correlated photons by two one-photon detectors is performed.
Polarization correlation measurement of two-photon state generated in type-II spontaneous parametric downconversion (SPDC) has been performed in a single input port beam splitter configuration. Two orthogonally polarized entangled photons with degenerate k vectors were directed to the single input port 50%-50% beam splitter. Polarization were measured with linear and circular polarization entangled state by the use of two polarization analyzers followed by two photon counting detectors. A strong correlation of polarization has been observed. The type of two-photon state generated in Type-II SPDC was tested. A strong dependence of the polarization correlation upon the spectral bandwidth of the detector spectral filter was found and a different correlation between linear and circular polarization entangled state were observed. It seems that the standard theoretical models for SPDC need to be improved in order to provide a comprehensive treatment for the result of the experiment.
Spin correlations of e + e − pair productions of two colliding photons are investigated and explicit expressions for their corresponding probabilities are derived and found to be energy (speed) dependent, for initially linearly and circularly polarized photons, different from those obtained by simply combining the spins of the relevant particles, for initially polarized photons. These expressions also depend on the angles of spin of e + (and/or of e −), for initially linearly polarized photons, but not for circularly polarized photons, as a function of the energy. It is remarkable that these explicit results obtained from quantum field theory show a clear violation of Bell's inequality of Local Hidden Variables theories at all energies beyond that of the threshold one for particle production, in support of quantum field theory in the relativistic regime. We hope that our explicit expression will lead to experiments, of the type described in the bulk of this paper, which can monitor energy (and speed) in polarization correlation experiments.
From the work by Perrie et al. [Phys. Rev. Lett. 54, 1790 (1985)], photon pairs from the 2s 1/2 → 1s 1/2 (two-photon) decay of atomic hydrogen are known to be quantum mechanically correlated. In these experiments, the polarization states of the photons emitted in back-to-back geometry were shown to violate the Bell inequality as a qualitative sign of nonlocality and entanglement. In the present contribution, we analyze how these nonlocal quantum correlations, as given by the violation of the Bell inequality, differ from the concurrence as a true entanglement measure. Results are shown for both quantifiers in dependence of the decay geometry and the initial polarization of the atoms for the 2s 1/2 → 1s 1/2 and 3d 5/2 → 1s 1/2 two-photon decay of atomic hydrogen. These results display the difference between nonlocality and entanglement and, hence, may stimulate further experiments on nonlocal quantum correlations in atomic systems.
A theoretical study of the angular correlation and angular variation of entanglement between a photoelectron and a fluorescent photon in photoionization of atoms is provided. The formulation is developed for experiments where spin polarization of a photoelectron and polarization of the subsequent fluorescent photon are detected in a coincidence measurement. The applied formalism of parameterization of all polarization sensitivities of the detectors using Stokes parameters and of the use of statistical and efficiency tensors provides straightforward implementation for any experimental geometry. The framework is demonstrated in the case of calcium photoionization followed by radiative decay.
Experiments testing Bell's inequalities generally uses sources of entangled photons emitted in opposite direction, sent through polarizer of specific different orientation and finally correlated. But the correlation result would be the same if photons was emitted by only one source of 360° rotating polarization, separated in two flux by a semi-transparent mirror, and finally correlated as in the standard experiment. Therefore, it would demonstrate that the result is not due to any entangled state of the photons.