It is well established that unpolarized light is invariant with respect to any SU(2) polarization transformation. This requirement fully characterizes the set of density matrices representing unpolarized states. We introduce the degree of polarization of a quantum state as its distance to the set of unpolarized states. We use two different candidates of distance, namely the Hilbert-Schmidt and the Bures metric, showing that they induce fundamentally different degrees of polarization. We apply these notions to relevant field states and we demonstrate that they avoid some of the problems arising with the classical definition.

We discuss a general and systematic method for obtaining effective Hamiltonians that describe different nonlinear optical processes. The method exploits the existence of a nonlinear deformation of the usual su(2) algebra that arises as the dynamical symmetry of the original model. When some physical parameter, dictated by the process under consideration, becomes small, we immediately get a diagonal effective Hamiltonian that correctly represents the dynamics for arbitrary states and long times. We extend the technique to su(3) and su(N), finding the corresponding effective Hamiltonians when some resonance conditions are fulfilled.Comment: 13 Pages, no figures, submitted for publicatio

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We explore the role played by the quantum relative phase in a well-known model of atom-field interaction, namely, the Dicke model. We introduce an appropriate polar decomposition of the atom-field relative amplitudes that leads to a truly Hermitian relative-phase operator, whose eigenstates correctly describe the phase properties, as we demonstrate by studying the positive operator-valued measure derived from it. We find the probability distribution for this relative phase and, by resorting to a numerical procedure, we study its time evolution.Comment: 20 pages, 4 figures, submitted to Phys. Rev.

degree of polarization of a quantum state can be defined as its Hilbert-Schmidt distance to the set of unpolarized states. We demonstrate that the states optimizing this degree for a fixed average number of photonsN present a fairly symmetric, parabolic photon statistics, with a variance scaling asN 2 . Although no standard optical process yields such a statistics, we show that, to an excellent approximation, a highly squeezed vacuum can be considered as maximally polarized. [5], to cite only a few relevant examples, have been systematically formulated within this framework. The rationale behind this is quite clear: once we have identified a convex set with the desired physical properties (classicality, separability, etc.), the distance determines the distinguishability of a state with respect to that set [6]. Apart from its conceptual simplicity, this procedure avoids many undesired problems that can arise in more standard approaches.Irrespective of our particular choice for the distance, a natural question emerges: what states maximize the corresponding measure. A good deal of effort has been devoted to characterize maximally nonclassical or entangled states. However, as far as we know, maximally polarized states have been not considered thus far, except for some trivial cases. It is precisely the purpose of this Letter to fill this gap, providing a complete description of such states, as well as feasible experimental schemes for their generation.Let us start by briefly recalling some basic concepts about quantum polarization. We assume a monochromatic plane wave propagating in the z direction, whose electric field lies in the xy plane. Under these conditions, the field can be fully represented by two complex amplitude operators, denoted byâ H andâ V when using the basis of linear (horizontal and vertical) polarizations. They obey the standard bosonic commutation relations [â j ,â † k ] = δ jk , with j, k ∈ {H, V }. The Stokes operators are then introduced as the quantum counterparts of the classical variables [7], namelŷand their mean values are precisely the Stokes parameters ( Ŝ 0 , Ŝ ), whereŜ T = (Ŝ 1 ,Ŝ 2 ,Ŝ 3 ) and the superscript These states span an invariant subspace of dimension N + 1, and the operatorsŜ act therein as an angular momentum N/2. Any (linear) polarization transformation is generated by the Stokes operators (1). However,Ŝ 0 induces only a common phase shift that does not change the polarization state and can thus be omitted. Therefore, we restrict ourselves to the SU(2) transformations, generated by S. SinceŜ 1 is related toŜ 2 andŜ 3 by the commutation relations, only these two generators suffice. It is well known thatŜ 2 generates rotations around the direction of propagation, whereasŜ 3 represents differential phase shifts between the modes. It follows then that any polarization transformation can be realized with linear optics: phase plates and rotators.Although a precise defintion of polarized light at the quantum level may be controversial [8], there is a wide consensus [9] in vi...