Heat Transport in Harmonic Lattices

We work out the non-equilibrium steady state properties of a harmonic lattice which is connected to heat reservoirs at different temperatures. The heat reservoirs are themselves modeled as harmonic systems. Our approach is to write quantum Langevin equations for the system and solve these to obtain steady state properties such as currents and other second moments involving the position and momentum operators. The resulting expressions will be seen to be similar in form to results obtained for electronic transport using the non-equilibrium Green’s function formalism. As an application of the formalism we discuss heat conduction in a harmonic chain connected to self-consistent reservoirs. We obtain a temperature dependent thermal conductivity which, in the high-temperature classical limit, reproduces the exact result on this model obtained recently by Bonetto, Lebowitz and Lukkarinen.     

Lattices of Quantum Automata

We defined and studied three different types of lattice-valued finite state quantum automata (LQA) and four different kinds of LQA operations, discussed their advantages, disadvantages, and various properties. There are four major results obtained in this paper. First, no one of the above mentioned LQA follows the law of lattice value conservation. Second, the theorem of classical automata theory, that each nondeterministic finite state automaton has an equivalent deterministic one, is not necessarily valid for finite state quantum automata. Third, we proved the existence of semilattices and also lattices formed by different types of LQA. Fourth, there are tight relations between properties of the original lattice l and those of the l-valued lattice formed by LQA.     

Heat Transport in Ordered Harmonic Lattices

We consider heat conduction across an ordered oscillator chain with harmonic interparticle interactions and also onsite harmonic potentials. The onsite spring constant is the same for all sites excepting the boundary sites. The chain is connected to Ohmic heat reservoirs at different temperatures. We use an approach following from a direct solution of the Langevin equations of motion. This works both in the classical and quantum regimes. In the classical case we obtain an exact formula for the heat current in the limit of system size N→∞. In special cases this reduces to earlier results obtained by Rieder, Lebowitz and Lieb and by Nakazawa. We also obtain results for the quantum mechanical case where we study the temperature dependence of the heat current. We briefly discuss results in higher dimensions.     

Harmonic Analysis on Quantum Tori

This paper is devoted to the study of harmonic analysis on quantum tori. We consider several summation methods on these tori, including the square Fejér means, square and circular Poisson means, and Bochner-Riesz means. We first establish the maximal inequalities for these means, then obtain the corresponding pointwise convergence theorems. In particular, we prove the noncommutative analogue of the classical Stein theorem on Bochner-Riesz means. The second part of the paper deals with Fourier multipliers on quantum tori. We prove that the completely bounded L _p Fourier multipliers on a quantum torus are exactly those on the classical torus of the same dimension. Finally, we present the Littlewood-Paley theory associated with the circular Poisson semigroup on quantum tori. We show that the Hardy spaces in this setting possess the usual properties of Hardy spaces, as one can expect. These include the quantum torus analogue of Fefferman’s H₁-BMO duality theorem and interpolation theorems. Our analysis is based on the recent developments of noncommutative martingale/ergodic inequalities and Littlewood-Paley-Stein theory.     

Orthomodular Lattices and a Quantum Algebra

We show that one can formulate an algebra with lattice ordering so as to contain one quantum and five classical operations as opposed to the standard formulation of the Hilbert space subspace algebra. The standard orthomodular lattice is embeddable into the algebra. To obtain this result we devised algorithms and computer programs for obtaining expressions of all quantum and classical operations within an orthomodular lattice in terms of each other, many of which are presented in the paper. For quantum disjunction and conjunction we prove their associativity in an orthomodular lattice for any triple in which one of the elements commutes with the other two and their distributivity for any triple in which a particular element commutes with the other two. We also prove that the distributivity of symmetric identity holds in Hilbert space, although whether or not it holds in all orthomodular lattices remains an open problem, as it does not fail in any of over 50 million Greechie diagrams we tested.     

Energy Propagation in Classical Harmonic Lattices with Diluted Disorder

We consider the problem of a harmonic lattice in which masses’ distribution is a superposition of a random and a periodic distribution. Classical equations for the mass displacement and velocities are solved using a second-order Euler formalism. Energy flow was investigated on two distinct experiments: (i) We injected an initial wave-packet with energy E ₀ and analyzed the dynamics of the resulting energy pulse; (ii) we pumped one of the sides of the lattice with a external signal and then we observed the propagation of the pulse until the other side of chain. Our calculations suggest that the diluted disordered mass distribution promotes energy dynamics at high frequency limit.     

Anomalous Dissipative Quantum Harmonic Oscillator

We investigate the low-temperature statistical properties of a harmonic oscillator coupled to a heat bath, where the low-frequency spectrum vanishes. We obtain the exact result of the zero point energy. Due to the low frequency shortage of environmental oscillators＇ spectral density, the coordinate and momentum correlation functions decay as T^-4 arid T^-6 respectively at zero temperature, where T is the correlation time. The low-temperature behavior of the mean energy does not violate the third law of thermodynamics, but differs largely from the Ohmic spectrum case.     

Bound States and the Third Harmonic Generation in an Electric Field Biased Semi-parabolic Quantum Well

Within the framework of the compact density matrix approach, the third-harmonic generation (THG) in an electric-field-biased semi-parabolic quantum well (QW) has been deduced and investigated. Via variant of displacement harmonic oscillation, the exact electronic states in the semi-parabolic QW with an applied electric field have also been obtained and discussed. Numerical results on typical GaAs material reveal that, electric fields and confined potential frequency of semi-parabolic Q W have obvious influences on the energy levels of electronic states and the THG in the semi-parabolic Q W systems.     

Bound States and the Third Harmonic Generation in an Electric Field Biased Semi-parabolic Quantum Well

Within the framework of the compact density matrix approach, the third-harmonic generation (THG) in an electric-field-biased semi-parabolic quantum well (QW) has been deduced and investigated. Via variant of displacement harmonic oscillation, the exact electronic states in the semi-parabolic QW with an applied electric field have also been obtained and discussed. Numerical results on typical GaAs material reveal that, electric fields and confined potential frequency of semi-parabolic QW have obvious influences on the energy levels of electronic states and the THG in the semi-parabolic QW systems.     

Multiparty Quantum Secret Sharing of Quantum States Using Entanglement States

A multi-partite-controlled quantum secret sharing scheme using several non-orthogonal entanglement states is presented with unconditional security. In this scheme, the participants share the secret quantum state by exchanging the secret polarization angles of the disordered travel particles. The security of the secret quantum state is also guaranteed by the non-orthogonal multi-partite-controlled entanglement states, the participants＇ secret polarizations, and the disorder of the travelling particles. Moreover, the present scheme is secure against the particle-number splitting attack and the intercept-and-resend attack. It may be still secure even if the distributed quantum state is embedded in a not-so-weak coherent-state pulse.     

Entangled Quantum States

Entangled quantum states are an important component of quantum computingtechniques such as quantum error correction, dense coding, and quantumteleportation. We determine the requirements for a state in the Hilbert space C ⁹to be entangled and a solution to the corresponding factorization problem if thisis not the case.     

Quantum Brain States

If conscious observers are to be included in the quantum mechanical universe, we need to find the rules that engage observers with quantum mechanical systems. The author has proposed five rules that are discovered by insisting on empirical completeness; that is, by requiring the rules to draw empirical information from Schrödinger's solutions that is more complete than is currently possible with the (Born) probability interpretation. I discard Born's interpretation, introducing probability solely through probability current. These rules tell us something about brains. They require the existence of observer brain states that are neither conscious nor unconscious. I call them ready brain states because they are on stand-by, ready to become conscious the moment they are stochastically chosen. Two of the rules are selection rules involving ready brain states. The place of these rules in a wider theoretical context is discussed.     

Asymptotic Description of Fast Thermal Processes in Scalar Harmonic Lattices

Physics of the Solid State - Thermal vibrations in a d-dimensional (d = 1, 2) scalar harmonic lattice of a simple structure are under consideration. Redistribution of the averaged kinetic and...     

Lattices and Quantum Logics with Separated Intervals, Atomicity

It is well known that a Boolean algebra B isatomic (atomistic) iff the interval topology on B isHausdorff. But this no longer holds for orthomodularlattices (quantum logics). There exist (even complete) atomic orthomodular lattices the intervaltopology of which is not Hausdorff. We show that anothercharacterization of atomicity for Boolean algebras isthe following: A Boolean algebra B is atomic iff B has separated intervals. Furthermore, we showthat the interval topology on a complete orthomodularlattice L is Hausdorff iff L has separated intervals iffL is atomic and it has separated intervals. An orthomodular lattice L with orthomodularMacNeille completion has separatedintervals iff L is atomic and it has separated intervalsiff the interval topology on isHausdorff.     

Quantum Waveguide Properties of Bethe Lattices with a Ring

Based on waveguide theory we investigate electronic transport properties of Bethe lattices with a mesoscopic ring threaded by a magnetic flux. The generalized eigen-function method (GEM) is used to calculate the transmission and reflection coefficients up to the fifth generation of Bethe lattices. The relationships among the transmission coefficient T, magnetic flux Φ and wave vector kl are investigated in detail. The numerical results are shown by the three-dimensional plots and contour maps. Some resonant-transmission features and the symmetry of the transmission coefficient T to flux Φ are observed and discussed.     

Extended and Periodic States in One-Dimensional Aperiodic Lattices

The eigenstates of an electron in two cases of one-dimensional (ID) aperiodic lattices are investigated analytically in the tight-binding approximation. The two cases of aperiodic lattices investigated are 1D Thue-Morse (TM) lattice and ID hierarchical system, both of which are non-quasiperiodic. It is shown that both the aperiodic models possess extended states. In particular, in 1D hierarchical system, there exists a finite fraction of periodic states independent of the hierarchical parameter R.     

KAM for the Quantum Harmonic Oscillator

In this paper we prove an abstract KAM theorem for infinite dimensional Hamiltonians systems. This result extends previous works of S.B. Kuksin and J. P?schel and uses recent techniques of H. Eliasson and S.B. Kuksin. As an application we show that some 1D nonlinear Schr?dinger equations with harmonic potential admits many quasi-periodic solutions. In a second application we prove the reducibility of the 1D Schr?dinger equations with the harmonic potential and a quasi periodic in time potential.