[en] We review the properties of the Kronecker (direct, or tensor) product of square matrices A⊗B⊗C⋯ in terms of Hubbard operators. In its simplest form, a Hubbard operator X_n^i,j can be expressed as the n-square matrix which has entry 1 in position (i,j) and zero in all other entries. The algebra and group properties of the observables that define a multipartite quantum system are notably straightforward in such a framework. In particular, we use the Kronecker product in Hubbard notation to get the Clebsch–Gordan decomposition of the product group SU(2)×SU(2). Finally, the n-dimensional irreducible representations so obtained are used to derive closed forms of the Clebsch–Gordan coefficients that rule the addition of angular momenta. Our results can be further developed in many different directions. -- Highlights: •The Kronecker product is studied in terms of Hubbard operators. •Complicated calculations involving large matrices are reduced to simple relations of subscripts. •The algebraic properties of the quantum observables of multipartite systems are studied. •The Clebsch–Gordan coefficients are given in terms of hypergeometric ₃F₂ functions. •The results can be further developed in many different directions

[en] The progress of the factorization method since the 1935 work of Dirac is briefly reviewed. Though linked with older mathematical theories the factorization seems an autonomous 'driving force', offering substantial support to the present day Darboux and Baecklund approaches

[en] It is shown that the radial part of the hydrogen Hamiltonian factorizes as the product of two not mutually adjoint first-order differential operators plus a complex constant ε. The 1-SUSY approach is used to construct non-Hermitian operators with hydrogen spectra. Other non-Hermitian Hamiltonians are shown to admit an extra 'complex energy' at ε. New self-adjoint hydrogen-like Hamiltonians are also derived by using a 2-SUSY transformation with complex conjugate pairs ε, ε-bar

[en] The solution of the Schroedinger equation for a position-dependent mass quantum system is studied in two ways. First, the interaction is found which must be applied to a mass m(x) in order to supply it with a particular spectrum of energies. Second, given a specific potential V(x) acting on the mass m(x), the related spectrum is found. The method of solution is applied to a wide class of position-dependent mass oscillators and the corresponding coherent states are constructed. The analytical expressions of such position-dependent mass coherent states preserve the functional structure of the Glauber states

[en] A set of Hamiltonians that are not self-adjoint but have the spectrum of the harmonic oscillator is studied. The eigenvectors of these operators and those of their Hermitian conjugates form a bi-orthogonal system that provides a mathematical procedure to satisfy the superposition principle. In this form the non-Hermitian oscillators can be studied in much the same way as in the Hermitian approaches. Two different nonlinear algebras generated by properly constructed ladder operators are found and the corresponding generalized coherent states are obtained. The non-Hermitian oscillators can be steered to the conventional one by the appropriate selection of parameters. In such limit, the generators of the nonlinear algebras converge to generalized ladder operators that would represent either intensity-dependent interactions or multi-photon processes if the oscillator is associated with single mode photon fields in nonlinear media.

[en] The time-evolution of the quantum correlations between two qubits that are coupled to a pair of photon baths is studied. We show that conditioned transitions occurring in the entire system have influence on the time-evolution of the subsystems. Then, we show that the study of the population inversion of each of the qubits is a measure of the correlations between them that is in agreement with the notion of concurrence. (paper)

[en] We discuss some of the geometric properties of the state spaces in Quantum Mechanics with emphasis in the two-level systems case. Then the dynamics of a two-level atom affected by radiation fields is revisited in both the semiclassical and quantum approaches. In the calculations we use Hubbard operators to reduce the matrix operations into simple relations of subscripts. The time-evolution of the energy states of the atom is associated to trajectories either on the 2-sphere or in the three-dimensional unit ball according to the classical or quantum approach we are using.

[en] We analyze the time-evolution of a pair of atoms that are initially entangled and are each of them in an independent isolated QED cavity. The cavities are assumed to contain a radiation bath of a given number of photons. The sudden death and recovering of entanglement are studied in terms of the concurrence introduced by Hill and Wootters. This last gives the value 1 for maximal entanglement and 0 for unentangled systems. We find that the entanglement of the bipartite atomic systems depends on the number of photons in the cavities when the radiation bath is assumed to be in a Fock state.

[en] The properties of the Kronecker product are revisited in terms of Hubbard operators. The simplest representation of a Hubbard operator X^i,j_n is a square matrix of size n with an entry equal to 1 and zero elsewhere. This framework simplifies the calculation of the Kronecker product of arbitrary matrices no matter the size or the number of the involved factors. Some applications are presented, these include the algebra of permutation matrices, the Hadamard matrix, the XXX Heisenberg model and the interaction of an atom with radiation fields

[en] We consider the relations between nonstationary quantum oscillators and their stationary counterpart in view of their applicability to study particles in electromagnetic traps. We develop a consistent model of quantum oscillators with time-dependent frequencies that are subjected to the action of a time-dependent driving force, and have a time-dependent zero point energy. Our approach uses the method of point transformations to construct the physical solutions of the parametric oscillator as mere deformations of the well known solutions of the stationary oscillator. In this form, the determination of the quantum integrals of motion is automatically achieved as a natural consequence of the transformation, without necessity of any ansätz. It yields the mechanism to construct an orthonormal basis for the nonstationary oscillators, so arbitrary superpositions of orthogonal states are available to obtain the corresponding coherent states. We also show that the dynamical algebra of the parametric oscillator is immediately obtained as a deformation of the algebra generated by the conventional boson ladder operators. A number of explicit examples is provided to show the applicability of our approach. (paper)