Supercanonical transformations and time-dependent Hamiltonians

Starting from general self-adjoint linear combinations of generators of the superalgebra $\frak{osp}(m'/n')$ a time-dependent Hamiltonian of a supersymmetric quantum mechanical system is defined by computing the supercommutator of the linear forms. The resulting Hamiltonian is given by the sum of two quadratic forms in odd, respectively even, generators of the superalgebra describing a class of systems containing bosons and fermions. Linear supercanonical transformations of wave vectors leave invariant a Heisenberg superalgebra and belong to the supergroup OSp$(m'/n')$ . The equations of motion for the supercanonical transformations in the Heisenberg picture are shown to be systems of ordinary differential equations. The unitary time evolution operator is constructed using the adjoint map. For periodic Hamiltonians, it is shown that this is a procedure to obtain effective Floquet Hamiltonians. The examples show that the known superalgebras-based approaches for the nuclear shell model and the Jaynes-Cummings model are incorporated in the lowest-dimensional cases. The presented approach opens the possibility to study quantum control problems defined by linear combinations of superalgebra generators with Grassmannian coefficients

Superlattice model for quantum gates

Probability electronic current conservation in superlattices leads to a natural connection with the qubit and quantum gate concepts of quantum information and the introduction of Bloch spheres and Hopf bundles. Superlattices or multilayer devices with coupled channels are described by sectionally constant potentials in the longitudinal direction and arbitrary lateral dependency, which allow the explicit analytic calculation of the scattering amplitudes connecting ingoing with outgoing wave functions. A superlattice with n open channels and two terminals can be seen as a quantum gate, since both can be viewed as quantum systems with n components, each of which can have two states given by input and output. Taking into account the dimensionality of the respective Hilbert spaces, it results that the one-channel or mode superlattice corresponds to a single qubit; the two-channels case to two qubits, and the three- and four-channels cases to three qubits. The coupling of modes or channels corresponds to the entanglement of qubits. As shown in this work, superlattices with interacting energy modes constitute physical systems which allow to understand and evaluate explicitly the main features used to describe entangled qubits through the study of Hopf fiber bundles on spheres. Explicit examples for superlattice gate models with coupled channels are provided

Superlattices with coupled degenerated spectrum

Two new analytical results are given which are of great help to understand superlattices with coupled modes. The first is an explicit relation for the transfer matrices in terms of Schur functions and Chebyshev polynomials. The second is a condition which generalizes the old well-known Floquet-Bloch trace condition to determine the spectrum. These improvements allow a fast computation of scattering amplitudes without obscuring the calculation with complicated numerical methods. As the energy grows, the eigenvalues degeneracy determines two types of transmission gaps. It is shown that these results could make it possible to design in greater detail the energy spectrum, for the very interesting case including modes coupling and degeneracy. They keep the understanding on the same footing as that of the traditional basic uncoupled problem considered at the beginning of the study of superlattices, like the Kroning-Penning model, or by Floquet-Bloch's theorem, or later by Esaki for heterostructures