Quantum temporal imaging: application of a time lens to quantum optics

We consider application of a temporal imaging system, based on the sum-frequency generation, to a nonclassical, in particular, squeezed optical temporal waveform. We analyze the restrictions on the pump and the phase matching condition in the summing crystal, necessary for preserving the quantum features of the initial waveform. We show that modification of the notion of the field of view in the quantum case is necessary, and that the quantum field of view is much narrower than the classical one for the same temporal imaging system. These results are important for temporal stretching and compressing of squeezed fields, used in quantum-enhanced metrology and quantum communications.Comment: 9 pages, 3 figure

Six types of E−E-functions of the Lie groups O(5) and G(2)

New families of $E$-functions are described in the context of the compact simple Lie groups O(5) and G(2). These functions of two real variables generalize the common exponential functions and for each group, only one family is currently found in the literature. All the families are fully characterized, their most important properties are described, namely their continuous and discrete orthogonalities and decompositions of their products.Comment: 25 pages, 13 figure

On E-functions of Semisimple Lie Groups

We develop and describe continuous and discrete transforms of class functions on a compact semisimple, but not simple, Lie group $G$ as their expansions into series of special functions that are invariant under the action of the even subgroup of the Weyl group of $G$. We distinguish two cases of even Weyl groups -- one is the direct product of even Weyl groups of simple components of $G$, the second is the full even Weyl group of $G$. The problem is rather simple in two dimensions. It is much richer in dimensions greater than two -- we describe in detail $E-$transforms of semisimple Lie groups of rank 3.Comment: 17 pages, 2 figure

Three dimensional C-, S- and E-transforms

Three dimensional continuous and discrete Fourier-like transforms, based on the three simple and four semisimple compact Lie groups of rank 3, are presented. For each simple Lie group, there are three families of special functions ($C$-, $S$-, and $E$-functions) on which the transforms are built. Pertinent properties of the functions are described in detail, such as their orthogonality within each family, when integrated over a finite region $F$ of the 3-dimensional Euclidean space (continuous orthogonality), as well as when summed up over a lattice grid $F_M\subset F$ (discrete orthogonality). The positive integer $M$ sets up the density of the lattice containing $F_M$. The expansion of functions given either on $F$ or on $F_M$ is the paper's main focus.Comment: 24 pages, 13 figure

The rings of n-dimensional polytopes

Points of an orbit of a finite Coxeter group G, generated by n reflections starting from a single seed point, are considered as vertices of a polytope (G-polytope) centered at the origin of a real n-dimensional Euclidean space. A general efficient method is recalled for the geometric description of G- polytopes, their faces of all dimensions and their adjacencies. Products and symmetrized powers of G-polytopes are introduced and their decomposition into the sums of G-polytopes is described. Several invariants of G-polytopes are found, namely the analogs of Dynkin indices of degrees 2 and 4, anomaly numbers and congruence classes of the polytopes. The definitions apply to crystallographic and non-crystallographic Coxeter groups. Examples and applications are shown.Comment: 24 page

Measuring nonclassicality of bosonic field quantum states via operator ordering sensitivity

We introduce a new distance-based measure for the nonclassicality of the states of a bosonic field, which outperforms the existing such measures in several ways. We define for that purpose the operator ordering sensitivity of the state which evaluates the sensitivity to operator ordering of the Renyi entropy of its quasi-probabilities and which measures the oscillations in its Wigner function. Through a sharp control on the operator ordering sensitivity of classical states we obtain a precise geometric image of their location in the density matrix space allowing us to introduce a distance-based measure of nonclassicality. We analyse the link between this nonclassicality measure and a recently introduced quantum macroscopicity measure, showing how the two notions are distinct

Orthogonal Decomposition of Some Affine Lie Algebras in Terms of their Heisenberg Subalgebras

In the present note we suggest an affinization of a theorem by Kostrikin et.al. about the decomposition of some complex simple Lie algebras ${\cal G}$ into the algebraic sum of pairwise orthogonal Cartan subalgebras. We point out that the untwisted affine Kac-Moody algebras of types $A_{p^m-1}$ ($p$ prime, $m\geq 1$), $B_r, \, C_{2^m}, D_r,\, G_2,\, E_7,\, E_8$ can be decomposed into the algebraic sum of pairwise or\-tho\-go\-nal Heisenberg subalgebras. The $A_{p^m-1}$ and $G_2$ cases are discussed in great detail. Some possible applications of such decompositions are also discussed.Comment: 16 pages, LaTeX, no figure

Recursion relations and branching rules for simple Lie algebras

The branching rules between simple Lie algebras and its regular (maximal) simple subalgebras are studied. Two types of recursion relations for anomalous relative multiplicities are obtained. One of them is proved to be the factorized version of the other. The factorization property is based on the existence of the set of weights $\Gamma$ specific for each injection. The structure of $\Gamma$ is easily deduced from the correspondence between the root systems of algebra and subalgebra. The recursion relations thus obtained give rise to simple and effective algorithm for branching rules. The details are exposed by performing the explicit decomposition procedure for $A_{3} \oplus u(1) \to B_{4}$ injection.Comment: 15p.,LaTe