Optical field-strength generalized polarization of multimode single photon states in integrated directional couplers

Multimode polarization degree of mixture optical states in integrated directional couplers

“….E N . We take advantage to underline that this optical field space is a natural one for representing both classical as well as quantum generalized (or standard) polarization [12]. The above Jones vector defines a multimode polarization state where the modes e can be interpreted as spatial polarization modes such as was commented on above.…”

“…, where D is the density matrix of the multimode state. As a direct consequence from Equations (9) and (10) we have that a pure state, that is, a state whose amplitudes jc j and phases are constants on time, the rank[D] ¼ 1 [12] and thus, by taking into account the subadditivity property of the rank of a symmetric matrix, the rank[M] 2 and consequently the maximum number of eigenvalues different to zero are only two. In general, the particular eigenvalues must be obtained by diagonalization of M, which corresponds physically to an orthogonal transformation, and therefore we have only two (or one) real eigenvectors of interest, that is, fũ 1 ,ũ 2 g corresponding to two supermodesẽ 1 ,ẽ 2 .…”

“…This important result has a deep effect on the concept of multimode polarization of pure states, namely, any pure multimode state can be rewritten as a bimode state on a two-dimensional subspaceẼ 1Ẽ2 and therefore we can apply all the well-known results of standard polarization to characterize these states, that is, the curves followed by the Jones vector on that subspace. In particular the case rank[M] ¼ 1 corresponds to a generalized linear polarization [12], that is, the Jones vector would describe a straight line on the…”

“…These multimode optical states can be represented in a multidimensional space where a type of multi-dimensional polarization or multimode polarization associated to such multimode states appears. This kind of generalized classical polarization in N Â N directional couplers has been recently presented [12] and although such a polarization will be tackled in this work, it is worth making it clear from a classical point of view.…”

Nonclassical polarization dynamics in classical-like states

“…Hence partial polarization can then be understood as the rapid and random succession of more or less different polarization states. In the quantum realm, we find that the electric field can never display a well-defined ellipse, in just the same way that particles cannot follow definite trajectories [2][3][4][5][6]. This is because the (field) quadratures satisfy the same commutation relations of position and linear momentum, and it brings about several remarkable consequences: (i) there is no room for the classic, textbook definition of polarization, (ii) the simple and elegant picture of partial polarization as a random succession of definite ellipses gets lost, and (iii) any quantum light state is partially polarized because of unavoidable (quantum) fluctuations.…”