Norio Inui

One-dimensional three-state quantum walk

We study a generalized Hadamard walk in one dimension with three inner states. The particle governed by the three-state quantum walk moves, in superposition, both to the left and to the right according to the inner state. In addition to these two degrees of freedom, it is allowed to stay at the same position. We calculate rigorously the wave function of the particle starting from the origin for any initial qubit state and show the spatial distribution of probability of finding the particle. In contrast with the Hadamard walk with two inner states on a line, the probability of finding the particle at the origin does not converge to zero even after infinite time steps except special initial states. This implies that the particle is trapped near the origin after a long time with high probability.

Localization of two-dimensional quantum walks

The Grover walk, which is related to Grovers search algorithm on a quantum computer, is one of the typical discrete time quantum walks. However, a localization of the two-dimensional Grover walk starting from a fixed point is strikingly different from other types of quantum walks. The present paper explains the reason why the walker who moves according to the degree-four Grover operator can remain at the starting point with a high probability. It is shown that the key factor for the localization is due to the degeneration of eigenvalues of the time evolution operator. In fact, the global time evolution of the quantum walk on a large lattice is mainly determined by the degree of degeneration. The dependence of the localization on the initial state is also considered by calculating the wave function analytically.

Localization of Multi-State Quantum Walk in One Dimension

Particle trapping in multi-state quantum walk on a circle is studied. The time-averaged probability distribution of a particle which moves four different lattice sites according to four internal states is calculated exactly. In contrast with “Hadamard walk” with only two internal states, the particle remains at the initial position with high probability. The time-averaged probability of finding the particle decreases exponentially as distance from a center of a spike. This implies that the particle is trapped in a narrow region. This striking difference is minutely explained from difference between degeneracy of eigenvalues of the time-evolution matrices. The dependence of the particle distribution on initial conditions is also considered.

Soft‐sputtering of insulin films in argon‐cluster secondary ion mass spectrometry

In secondary ion mass spectrometry (SIMS) of organic substances, the dissociation of the sample molecules is crucial. We have developed SIMS equipment capable of bombardment, where the primary ions are argon cluster ions with kinetic energy per atom controllable down to 1  eV. We previously reported the detection of intact ions of insulin and cytochrome C using this equipment. In this paper, we present a detailed characterization of the emission of secondary ions from insulin, focusing on the difference in secondary ion yield between intact ions and fragment ions by varying the incident angle of the cluster ions. The emission intensity of the intact ions was changed drastically due to the exposed dosage and incident angle of the cluster ions in contrast to the fragment ions. We discuss these results based on the manner in which the argon-cluster ions collide with the organic solid.

Numerical Study of Enhancement of the Casimir Force between Silicon Membranes by Irradiation with UV Laser

Irradiation generates free carriers inside silicon and it can cause enhancement of the Casimir force between silicon membranes. We study numerically the temporal behavior of the Casimir force between two parallel silicon membranes after irradiating the surface with UV pulse laser. In the calculation, the free carrier density is obtained by solving the diffusion equation with the generation and recombination terms, and the dielectric function of the illuminated silicon is calculated from them by employing the Drude model. Based on the Lifshitz theory accounting for thickness of the slabs, we calculate the Casimir force as a function of time and consider the finite size effect of the thickness.

Enhanced surface sensitivity in secondary ion mass spectrometric analysis of organic thin films using size‐selected Ar gas‐cluster ion projectiles

A size-selected argon (Ar) gas-cluster ion beam (GCIB) was applied to the secondary ion mass spectrometry (SIMS) of a 1,4-didodecylbenzene (DDB) thin film. The samples were also analyzed by SIMS using an atomic Ar(+) ion projectile and X-ray photoelectron spectroscopy (XPS). Compared with those in the atomic-Ar(+) SIMS spectrum, the fragment species, including siloxane contaminants present on the sample surface, were enhanced several hundred times in the Ar gas-cluster SIMS spectrum. XPS spectra during beam irradiation indicate that the Ar GCIB sputters contaminants on the surface more effectively than the atomic Ar(+) ion beam. These results indicate that a large gas-cluster projectile can sputter a much shallower volume of organic material than small projectiles, resulting in an extremely surface-sensitive analysis of organic thin films.

A Generalized Mode Summation Formula of the Zero-Point Energy in a Cavity

The summation of zero-point energies defined by \(\sum_{m,n}(\pi/2)\sqrt{(n+\beta)^{2}/a^{2}+m^{2}/b^{2}}\) is studied. The parameter β is a positive real number and the zero-point energy of a massless scalar field is given by setting β=0 as a special case. The Casimir energy is obtained by the generalized Abel–Plana formula as a function of β and it is found that the Casimir energy density in the limit b →∞ changes from negative to positive as increasing β. The Casimir effect in the three dimensional case is also considered.

Catalan numbers in a series expansion of the directed percolation probability on a square lattice

We regard the bond directed percolation on a square lattice as a discrete-time Markov process of a one-dimensional interacting particle system. The coefficients in series expansion of the probability of having m particles at time n - 1 are studied. We derive the difference equations for the first and the second series of coefficients and prove that these coefficients are expressed using the ballot numbers, whose special cases are known as the Catalan numbers. As a corollary of our results, we prove a part of the conjecture by Baxter and Guttmann that the correction terms are expressed as rational functions of the Catalan numbers. We also give approximations for the percolation probability using the present formula.

FLUCTUATIONS OF QUANTUM RANDOM WALKS ON CIRCLES

Temporal fluctuations in the Hadamard walk on circles are studied. A temporal standard deviation of probability that a quantum random walker is positive at a given site is introduced to manifest striking differences between quantum and classical random walks. An analytical expression of the temporal standard deviation on a circle with odd sites is shown and its asymptotic behavior is considered for large system size. In contrast with classical random walks, the temporal fluctuation of quantum random walks depends on the position and initial conditions, since temporal standard deviation of the classical case is zero for any site. It indicates that the temporal fluctuation of the Hadamard walk can be controlled.

Fermi Partition Functions of Friendly Walkers and Pair Connectedness of Directed Percolation

Non-crossing random walkers with attractive interactions called friendly walkers (FWs) are studied. A restriction on trajectories, which is analogous to Paulis exclusion principle, is imposed and the Fermi partition functions are defined. We prove a theorem that the pair connectedness of the bond directed percolation (DP) with bond concentration p is related to the Fermi grand partition function of FW if we set the temperature T =-1/( k B ln p ) and the chemical potential µ=- i π/ ln p , where k B is the Boltzmann constant and \({\rm i}=\sqrt{-1}\). The pure imaginary chemical potential means that the DP transition can be regarded as a symmetry breaking of parity in the number of FWs. As a corollary of the theorem, a new method is proposed for calculating the series expansion of the pair connectedness and percolation probability of DP using the low-temperature expansion data of FW.