Shuichi Tasaki

Quantum theory of non integrable systems

In 1889 H. Poincare introduced a basic distinction between integrable and non-integrable dynamical systems. This distinction refers to the role of resonances which may lead to divergences. A specially important class of non-integrable systems are «large» Poincare systems (LPS) which have a continuous spectrum and present continuous sets of resonances. As has been shown earlier, LPS play an essential role both in classical and quantum mechanics. Essentially all nontrivial problems of field theory as well as kinetic theory belong this class. We consider here the well known Friedrichs model in which an unstable discrete level is coupled to a continuum. Poincares theorem prevents the existence of solutions of the eigenvalue problem associated to the Hamiltonian which would be analytic in the coupling constant

Control of decoherence: Analysis and comparison of three different strategies

We analyze and compare three different strategies, all aimed at controlling and eventually halting decoherence. The first strategy hinges upon the quantum Zeno effect, the second makes use of frequent unitary interruptions s“bang-bang” pulses and their generalization, quantum dynamical decoupling d, and the third uses a strong, continuous coupling. Decoherence is shown to be suppressed only if the frequency N of the measurements or pulses is large enough or if the coupling K is sufficiently strong. Otherwise, if N or K is large, but not extremely large, all these control procedures accelerate decoherence. We investigate the problem in a general setting and then consider some practical examples, relevant for quantum computation.

The fluctuation theorem for currents in open quantum systems

A quantum-mechanical framework is set up to describe the full counting statistics of particles flowing between reservoirs in an open system under time-dependent driving. A symmetry relation is obtained, which is the consequence of microreversibility for the probability of the nonequilibrium work and the transfer of particles and energy between the reservoirs. In some appropriate long-time limit, the symmetry relation leads to a steady-state quantum fluctuation theorem for the currents between the reservoirs. On this basis, relationships are deduced which extend the Onsager–Casimir reciprocity relations to the nonlinear response coefficients.

Quantum zeno effect

Misra and Sudarshan pointed out, based on the quantum measurement theory, that repeated measurements lead to a slowing down of the transition, which they called the quantum Zeno effect. Recently, Itano, Heinzen, Bollinger and Wineland have reported that they succeeded in observing that effect. We show that the results of Itano et al. can be recovered through conventional quantum mechanics and do not involve a repeated reduction of the wave function

Fick's law and fractality of nonequilibrium stationary states in a reversible multibaker map

Nonequilibrium stationary states are studied for a multibaker map, a simple reversible chaotic dynamical system. The probabilistic description is extended by representing a dynamical state in terms of a measure instead of a density function. The equation of motion for the cumulative function of this measure is derived and stationary solutions are constructed with the aid of deRham-type functional equations. To select the physical states, the time evolution of the distribution under a fixed boundary condition is investigated for an open multibaker chain of scattering type. This system corresponds to a diffusive flow experiment through a slab of material. For long times, any initial distribution approaches the stationary one obeying Ficks law. At stationarity, the intracell distribution is singular in the stable direction and expressed by the Takagi function, which is continuous but has no finite derivatives. The result suggests that singular measures play an important role in the dynamical description of non-equilibrium states.

Generalized spectral decomposition of the β-adic baker's transformation and intrinsic irreversibility

Abstract We construct a generalized spectral decomposition of the Frobenius-Perron operator of the β-adic bakers transformation using a general iterative operator method applicable in principle for any mixing dynamical system. The eigenvalues in the decomposition are related to the decay rates of the autocorrelation functions and have magnitudes less than one. We explicitly define appropriate generalized function spaces, which provide mathematical meaning to the formally obtained spectral decomposition. The unitary Frobenius-Perron evolution of densities, when extended to the generalized function spaces, splits into two semigroups, one decaying in the future and the other in the past. This split, which reflects the asymptotic evolution of the forward and backward K-partitions, shows the instrinsic irreversibility of the bakers transformation.

Intrinsic irreversibility of quantum systems with diagonal singularity

The work of the Brussels-Austin group on irreversibility over the last years has shown that quantum large Poincare systems with diagonal singularity lead to an extension of quantum theory beyond the conventional Hilbert space framework and logic. We characterize the algebra of observables, the states and the logic of the extended quantum theory of intrinsically irreversible systems with diagonal singularity. We illustrate the general ideas for the Friedrichs model.

Quantum Zeno effect

Abstract In 1977, Misra and Sudarshan showed, based on the quantum measurement theory, that an unstable particle will never be found to decay when it is continuously observed. They called it the quantum Zeno effect (or paradox). More generally the quantum Zeno effect is associated to the inhibition of transitions by frequent measurements. This possibility has attracted much interest over the last years. Recently, Itano, Heinzen, Bollinger and Wineland have reported that they succeeded in observing the quantum Zeno effect. This would indeed be an important step towards the understanding of the role of the observer in quantum mechanics. However, in the present paper, we will show that their results can be recovered through conventional quantum mechanics and do not involve a repeated reduction (or collapse) of the wave function.

An analytical construction of the SRB measures for Baker-type maps

For a class of dynamical systems, called the axiom-A systems, Sinai, Ruelle and Bowen showed the existence of an invariant measure (SRB measure) weakly attracting the temporal average of any initial distribution that is absolutely continuous with respect to the Lebesgue measure. Recently, the SRB measures were found to be related to the nonequilibrium stationary state distribution functions for thermostated or open systems. Inspite of the importance of these SRB measures, it is difficult to handle them analytically because they are often singular functions. In this article, for three kinds of Baker-type maps, the SRB measures are analytically constructed with the aid of a functional equation, which was proposed by de Rham in order to deal with a class of singular functions. We first briefly review the properties of singular functions including those of de Rham. Then, the Baker-type maps are described, one of which is nonconservative but time reversible, the second has a Cantor-like invariant set, and the third is a model of a simple chemical reaction R<-->I<-->P. For the second example, the cases with and without escape are considered. For the last example, we consider the reaction processes in a closed system and in an open system under a flux boundary condition. In all cases, we show that the evolution equation of the distribution functions partially integrated over the unstable direction is very similar to de Rhams functional equation and, employing this analogy, we explicitly construct the SRB measures. (c) 1998 American Institute of Physics.

The neutron reflectometer (C3-1-2) at the JRR-3M reactor at JAERI

Abstract A neutron reflectometer has been installed at the cold neutron guide tube (C3-1-2) of the JRR-3M reactor at JAERI. Incident neutrons for the reflectometer have a long wavelength of 12.6 A, with a wavelength resolution of 3.2%. The reflectometer has the advantage of a large reflection angle which is appropriate for studies of low- q and off-specular phenomena. The neutron intensity reduction due to the long wavelength is substantially compensated by the relatively coarse beam divergence and wavelength resolution. The reflectometer is mounted in vertical geometry to yield a beam of 3 × 40 mm 2 and is applicable to measurements of mirror systems formed on flat substrates. Applications of the reflectometer to neutron optics and polymer studies are demonstrated and discussed.