Quantum Optical Random Walk: Quantization Rules and Quantum Simulation of Asymptotics

Rules for quantizing the walker+coin parts of a classical random walk are provided by treating them as interacting quantum systems. A quantum optical random walk (QORW), is introduced by means of a new rule that treats quantum or classical noise affecting the coin's state, as sources of quantization. The long term asymptotic statistics of QORW walker's position that shows enhanced diffusion rates as compared to classical case, is exactly solved. A quantum optical cavity implementation of the walk provides the framework for quantum simulation of its asymptotic statistics. The simulation utilizes interacting two-level atoms and/or laser randomly pulsating fields with fluctuating parameters.Comment: 18 pages, 3 figure

Prime decomposition and correlation measure of finite quantum systems

Under the name prime decomposition (pd), a unique decomposition of an arbitrary $N$-dimensional density matrix $\rho$ into a sum of seperable density matrices with dimensions given by the coprime factors of $N$ is introduced. For a class of density matrices a complete tensor product factorization is achieved. The construction is based on the Chinese Remainder Theorem and the projective unitary representation of $Z_N$ by the discrete Heisenberg group $H_N$. The pd isomorphism is unitarily implemented and it is shown to be coassociative and to act on $H_N$ as comultiplication. Density matrices with complete pd are interpreted as grouplike elements of $H_N$. To quantify the distance of $\rho$ from its pd a trace-norm correlation index $\cal E$ is introduced and its invariance groups are determined.Comment: 9 pages LaTeX. Revised version: changes in the terminology, updates in ref

q-Symmetries in DNLS-AL chains and exact solutions of quantum dimers

Dynamical symmetries of Hamiltonians quantized models of discrete non-linear Schroedinger chain (DNLS) and of Ablowitz-Ladik chain (AL) are studied. It is shown that for $n$-sites the dynamical algebra of DNLS Hamilton operator is given by the $su(n)$ algebra, while the respective symmetry for the AL case is the quantum algebra su_q(n). The q-deformation of the dynamical symmetry in the AL model is due to the non-canonical oscillator-like structure of the raising and lowering operators at each site. Invariants of motions are found in terms of Casimir central elements of su(n) and su_q(n) algebra generators, for the DNLS and QAL cases respectively. Utilizing the representation theory of the symmetry algebras we specialize to the $n=2$ quantum dimer case and formulate the eigenvalue problem of each dimer as a non-linear (q)-spin model. Analytic investigations of the ensuing three-term non-linear recurrence relations are carried out and the respective orthonormal and complete eigenvector bases are determined. The quantum manifestation of the classical self-trapping in the QDNLS-dimer and its absence in the QAL-dimer, is analysed by studying the asymptotic attraction and repulsion respectively, of the energy levels versus the strength of non-linearity. Our treatment predicts for the QDNLS-dimer, a phase-transition like behaviour in the rate of change of the logarithm of eigenenergy differences, for values of the non-linearity parameter near the classical bifurcation point.Comment: Latex, 19pp, 4 figures. Submitted for publicatio

Quantum Walks of SU(2)_k Anyons on a Ladder

We study the effects of braiding interactions on single anyon dynamics using a quantum walk model on a quasi-1-dimensional ladder filled with stationary anyons. The model includes loss of information of the coin and nonlocal fusion degrees of freedom on every second time step, such that the entanglement between the position states and the exponentially growing auxiliary degrees of freedom is lost. The computational complexity of numerical calculations reduces drastically from the fully coherent anyonic quantum walk model, allowing for relatively long simulations for anyons which are spin-1/2 irreps of SU(2)_k Chern-Simons theory. We find that for Abelian anyons, the walk retains the ballistic spreading velocity just like particles with trivial braiding statistics. For non-Abelian anyons, the numerical results indicate that the spreading velocity is linearly dependent on the number of time steps. By approximating the Kraus generators of the time evolution map by circulant matrices, it is shown that the spatial probability distribution for the k=2 walk, corresponding to Ising model anyons, is equal to the classical unbiased random walk distribution.Comment: 12 pages, 4 figure

Non-positivity of the Wigner function and bounds on associated integrals

The Wigner function shares several properties with classical distribution functions on phase space, but is not positive-definite. The integral of the Wigner function over a given region of phase space can therefore lie outside the interval [0,1]. The problem of finding best-possible upper and lower bounds for a given region is the problem of finding the greatest and least eigenvalues of an associated Hermitian operator. Exactly solvable examples are described, and possible extensions are indicated.Comment: 5 pages, Latex2e fil

Superconducting Coherent States for an Extended Hubbard Model

An extended Hubbard model with phonons is considered. q-coherent states relative to the superconducting quantum symmetry of the model are constructed and their properties studied. It is shown that they can have energy expectation lower than eigenstates constructed via conventional processes and that they exhibit ODLRO.Comment: 7 pages, 3 figure

Free Dirac evolution as a quantum random walk

Any positive-energy state of a free Dirac particle that is initially highly-localized, evolves in time by spreading at speeds close to the speed of light. This general phenomenon is explained by the fact that the Dirac evolution can be approximated arbitrarily closely by a quantum random walk, where the roles of coin and walker systems are naturally attributed to the spin and position degrees of freedom of the particle. Initially entangled and spatially localized spin-position states evolve with asymptotic two-horned distributions of the position probability, familiar from earlier studies of quantum walks. For the Dirac particle, the two horns travel apart at close to the speed of light.Comment: 16 pages, 1 figure. Latex2e fil