Fundamental Cycle of a Periodic Box-Ball System

We investigate a soliton cellular automaton (Box-Ball system) with periodic boundary conditions. Since the cellular automaton is a deterministic dynamical system that takes only a finite number of states, it will exhibit periodic motion. We determine its fundamental cycle for a given initial state.Comment: 28 pages, 6 figure

Ultradiscretization of the solution of periodic Toda equation

A periodic box-ball system (pBBS) is obtained by ultradiscretizing the periodic discrete Toda equation (pd Toda eq.). We show the relation between a Young diagram of the pBBS and a spectral curve of the pd Toda eq.. The formula for the fundamental cycle of the pBBS is obtained as a colloraly.Comment: 41 pages; 7 figure

On the initial value problem of a periodic box-ball system

We show that the initial value problem of a periodic box-ball system can be solved in an elementary way using simple combinatorial methods.Comment: 9 pages, 2 figure

Correlation function for a periodic box-ball system

We investigate correlation functions in a periodic box-ball system. For the two point functions of short distance, we give explicit formulae obtained by combinatorial methods. We give expressions for general N-point functions in terms of ultradiscrete theta functions.Comment: 13 pages, 2 figures, submitted to J. Phys. A: Math. Theo

A crystal theoretic method for finding rigged configurations from paths

The Kerov--Kirillov--Reshetikhin (KKR) bijection gives one to one correspondences between the set of highest paths and the set of rigged configurations. In this paper, we give a crystal theoretic reformulation of the KKR map from the paths to rigged configurations, using the combinatorial R and energy functions. This formalism provides tool for analysis of the periodic box-ball systems.Comment: 24 pages, version for publicatio

Box-ball system: soliton and tree decomposition of excursions

We review combinatorial properties of solitons of the Box-Ball system introduced by Takahashi and Satsuma in 1990. Starting with several definitions of the system, we describe ways to identify solitons and review a proof of the conservation of the solitons under the dynamics. Ferrari, Nguyen, Rolla and Wang 2018 proposed a soliton decomposition of a configuration into a family of vectors, one for each soliton size. Based on this decompositions, the authors have proposed a family of measures on the set of excursions which induces invariant distributions for the Box-Ball System. In this paper, we propose a new soliton decomposition which is equivalent to a branch decomposition of the tree associated to the excursion, see Le Gall 2005. A ball configuration distributed as independent Bernoulli variables of parameter $\lambda<1/2$ is in correspondence with a simple random walk with negative drift $2\lambda-1$ and infinitely many excursions over the local minima. In this case the authors have proven that the soliton decomposition of the walk consists on independent double-infinite vectors of iid geometric random variables. We show that this property is shared by the branch decomposition of the excursion trees of the random walk and discuss a corresponding construction of a Geometric branching process with independent but not identically distributed Geometric random variables.Comment: 47 pages, 33 figures. This is the revised version after addressing referee reports. This version will be published in the special volume of the XIII Simposio de Probabilidad y Procesos Estoc\'asticos, UNAM Mexico, by Birkhause

Noise correlations in a flux qubit with tunable tunnel coupling

We have measured flux-noise correlations in a tunable superconducting flux qubit. The device consists of two loops that independently control the qubit's energy splitting and tunnel coupling. Low frequency flux noise in the loops causes fluctuations of the qubit frequency and leads to dephasing. Since the noises in the two loops couple to different terms of the qubit Hamiltonian, a measurement of the dephasing rate at different bias points provides a way to extract both the amplitude and the sign of the noise correlations. We find that the flux fluctuations in the two loops are anti-correlated, consistent with a model where the flux noise is generated by randomly oriented unpaired spins on the metal surface.Comment: 7 pages, including supplementary materia

Dynamical decoupling and dephasing in interacting two-level systems

We implement dynamical decoupling techniques to mitigate noise and enhance the lifetime of an entangled state that is formed in a superconducting flux qubit coupled to a microscopic two-level system. By rapidly changing the qubit's transition frequency relative to the two-level system, we realize a refocusing pulse that reduces dephasing due to fluctuations in the transition frequencies, thereby improving the coherence time of the entangled state. The coupling coherence is further enhanced when applying multiple refocusing pulses, in agreement with our $1/f$ noise model. The results are applicable to any two-qubit system with transverse coupling, and they highlight the potential of decoupling techniques for improving two-qubit gate fidelities, an essential prerequisite for implementing fault-tolerant quantum computing