M. A. Lohe

An Analysis of the Quantum Penny Flip Game using Geometric Algebra

We analyze the quantum penny flip game using geometric algebra and so determine all possible unitary transformations which enable the player Q to implement a winning strategy. Geometric algebra provides a clear visual picture of the quantum game and its strategies, as well as providing a simple and direct derivation of the winning transformation, which we demonstrate can be parametrized by two angles �;� . For comparison we derive the same general winning strategy by conventional means using density matrices.

Efficient third and one-third harmonic generation in nonlinear waveguides.

We investigate the interaction between fundamental and third harmonic fields in a nonlinear waveguide. We develop a method for evaluating the maximum efficiency of third harmonic (upconversion) and one-third harmonic (downconversion) generation by considering the solitonic behavior of the interaction. This method can be used to engineer waveguide parameters and identify the input power that enables maximum conversion efficiency to be achieved.

Full vectorial analysis of polarization effects in optical nanowires

We develop a full theoretical analysis of the nonlinear interactions of the two polarizations of a waveguide by means of a vectorial model of pulse propagation which applies to high index subwavelength waveguides. In such waveguides there is an anisotropy in the nonlinear behavior of the two polarizations that originates entirely from the waveguide structure, and leads to switching properties. We determine the stability properties of the steady state solutions by means of a Lagrangian formulation. We find all static solutions of the nonlinear system, including those that are periodic with respect to the optical fiber length as well as nonperiodic soliton solutions, and analyze these solutions by means of a Hamiltonian formulation. We discuss in particular the switching solutions which lie near the unstable steady states, since they lead to self-polarization flipping which can in principle be employed to construct fast optical switches and optical logic gates.

Nonlinear polarization bistability in optical nanowires.

Using the full vectorial nonlinear Schrödinger equations that describe nonlinear processes in isotropic optical nanowires, we show that there exist structural anisotropic nonlinearities that lead to unstable polarization states that exhibit periodic bistable behavior. We analyze and solve the nonlinear equations for continuous waves by means of a Lagrangian formulation and show that the system has bistable states and also kink solitons that are limiting forms of the bistable states.

Nonlinear Self-Flipping of Polarization States in Asymmetric Waveguides

Waveguides of subwavelength dimensions with asymmetric geometries, such as rib waveguides, can display nonlinear polarization effects in which the nonlinear phase difference dominates the linear contribution, provided the birefringence is sufficiently small. We predict that self-flipping polarization states can appear in such rib waveguides at subwatt power levels. We describe an optical power-limiting device with optimized rib waveguide parameters that can operate at low power with switching properties.

An improved formalism for quantum computation based on geometric algebra--case study: Grover's search algorithm

The Grover search algorithm is one of the two key algorithms in the field of quantum computing, and hence it is desirable to represent it in the simplest and most intuitive formalism possible. We show firstly, that Clifford’s geometric algebra, provides a significantly simpler representation than the conventional bra-ket notation, and secondly, that the basis defined by the states of maximum and minimum weight in the Grover search space, allows a simple visualization of the Grover search analogous to the precession of a spin-

Steady-state and travelling wave solutions with nonlinear polarization attraction

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Nonlinear polarization self-flipping and optical switching

particle. Using this formalism we efficiently solve the exact search problem, as well as easily representing more general search situations. We do not claim the development of an improved algorithm, but show in a tutorial paper that geometric algebra provides extremely compact and elegant expressions with improved clarity for the Grover search algorithm. Being a key algorithm in quantum computing and one of the most studied, it forms an ideal basis for a tutorial on how to elucidate quantum operations in terms of geometric algebra—this is then of interest in extending the applicability of geometric algebra to more complicated problems in fields of quantum computing, quantum decision theory, and quantum information.