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Quantum Electronics

Polarisation properties of single-mode biphotons

L. A. Krivitskiı̆, S. P. Kulik, Gleb Maslennikov, Maria V. Chekhova

Published Jan 31, 2005• 3 citations
Publisher

Abstract

The paper reviews recent research by the authors on the polarisation properties of correlated photon pairs, also called biphotons, generated via spontaneous parametric down-conversion in the collinear frequency—degenerate regime. From the viewpoint of their polarisation properties, they represent qutrits, i.e., three-dimensional quantum systems, and are quite promising for quantum information applications.

DOI
10.1070/qe2005v035n01abeh002891

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The paper reviews recent research by the authors on the polarisation properties of correlated photon pairs, also called biphotons, generated via spontaneous parametric down-conversion in the collinear frequency—degenerate regime. From the viewpoint of their polarisation properties, they represent qutrits, i.e., three-dimensional quantum systems, and are quite promising for quantum information applications.
We investigate in a general form entanglement of biphoton qutrits and ququarts, i.e. states formed in the processes of collinear and, correspondingly, degenerate and non-degenerate Spontaneous Parametric Down-Conversion. Indistinguishability of photons and, for ququarts, joint presence of the frequency and and polarization entanglement are fully taken into account. In the case of qutrits the most general 3-parametric families of maximally entangled and non-entangled states are found, and anti-correlation 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=4d=4d=4, which differs them significantly from the often used two-qubit model (d=2d=2d=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.
The works devoted to studying the polarization properties of a two-photon light generated upon spontaneous parametric down-conversion in the collinear frequency-degenerate regime are briefly overviewed, with emphasis on the studies carried out by us over a period from 1999 to 2001 within the framework of the project “Polarization Optics of Biphotons” of the Russian Foundation for Basic Research. In particular, the polarization state of a two-photon light was analyzed and its pictorial mapping onto the Poincaré sphere was proposed. The experiments on polarization transformations of a two-photon light were performed; based on these transformations, a method was suggested for ternary quantum information coding. A two-photon state with the orthogonal photon polarizations was synthesized experimentally from the two beams of identically polarized correlated photons, and the spectral properties of this state were investigated. Finally, a method was suggested for measuring the polarization state of a two-photon light in the collinear frequency-degenerate case (“tomography”).
Summary form only given. Consider a pair of collinear frequency-degenerate photons-a biphoton, which can be obtained by spontaneous parametric down-conversion (SPDC). Its state vector/sup 1/ is |/spl Psi/>=C/sub 1/|2,0>+C/sub 2/|0,2>+C/sub 3/|1,1>, (1) with |n,m>, n+m=2 denoting n H-polarized photons and m V-polarized photons. The first and the second terms describe type-I biphotons (correlated photons have similar H or V polarizations), and the third one describes type-II biphotons (correlated photons have orthogonal polarizations). In the present work we study an anticorrelation properties of the artificially synthesized type-II biphotons.
Summary form only given. Consider a pair of collinear frequency-degenerate photons-a biphoton, which can be obtained by spontaneous parametric down-conversion (SPDC). Being a three-state quantum system, it is a ternary analog of a qubit. Its state vector is |/spl psi/>=c/sub 1/|2,0>+c/sub 2/|0,2>+c/sub 3/|1,1>, with |n,m>,n+m=2 denoting n X-polarized photons and m Y-polarized photons. The first two terms describe type-I biphotons (correlated photons have similar polarizations), and the third one type-II biphotons (correlated photons have orthogonal polarizations). An arbitrary biphoton state is given by four parameters.
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.