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Journal of Experimental and Theoretical Physics

Biphotons as three-level systems: Transformation and measurement

L. A. Krivitskiı̆, S. P. Kulik, Alexander N. Penin, Maria V. Chekhova

Published Oct 1, 2003• 11 citations
Publisher

Abstract

Two algorithms for measuring the polarization state of the biphoton field prepared in the form of a three-level system (qutrit) are considered. On the basis of the general approach developed by Klyshko [21] for describing the polarization properties of single-mode electromagnetic fields in the fourth order in the field, a procedure for measuring the polarization density matrix of qutrits is proposed and implemented.

DOI
10.1134/1.1625075

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Two algorithms for measuring the polarization state of the biphoton field prepared in the form of a three-level system (qutrit) are considered. On the basis of the general approach developed by Klyshko [21] for describing the polarization properties of single-mode electromagnetic fields in the fourth order in the field, a procedure for measuring the polarization density matrix of qutrits is proposed and implemented.
Most protocols of quantum information transmission are based on qubits, or two-level quantum systems. However, in certain cases, it is reasonable to use quantum systems of higher dimensionality, such as, for instance, qutrits - three-level systems. In the present paper, it is shown that single-mode entangled pairs of polarized photons (biphotons) can be used for this purpose. Polarization state of a single-mode biphoton is considered and its representation on the Poincare sphere is suggested. An orthogonality criterion for two biphotons in arbitrary polarization states is formulated. Experiments on polarization transformations of biphotons are described. Finally, propagation of biphotons through fibres is considered.
The procedure of measurement followed by the reconstruction of the quantum state of a three-level optical system is implemented for a frequency-and spatially degenerate two-photon field. The method of statistical estimation of the quantum state from a solution to the likelihood equation and the analysis of the statistical properties of the obtained estimators is developed. Using the root method of estimating quantum states, the initial two-photon (qutrit) wave function is reconstructed from the measured fourth-order field moments.
We investigate, in a general form, entanglement of biphoton qutrits and ququarts, i.e. states formed in the processes of, correspondingly, degenerate and non-degenerate spontaneous parametric down-conversion. Indistinguishability of photons and, for ququarts, joint presence of the frequency and polarization entanglement are fully taken into account. In the case of qutrits, the most general three-parametric families of maximally entangled and non-entangled states are found, and anticorrelation of the degree of entanglement and polarization is shown to occur and to be characterized by a rather simple formula. Biphoton ququarts are shown to be two-qudits with the single-photon Hilbert space dimensionality d =4, which differentiates them significantly from the often used two-qubit model ( d =2). New expressions for entanglement quantifiers of biphoton ququarts are derived and discussed. Rather simple procedures for a direct measurement of the degree of entanglement are described for both qutrits and ququarts.