Elementary Quantum Gates Based on Intrinsic Interaction Hamiltonian

A kind of new operators, the generalized pseudo-spin operators are introduced and a universal intrinsic Hamiltonian of two-qubit interaction is studied in terms of the generalized pseudo-spin operators. A fundamental quantum gate U(θ) is constructed based on the universal Hamiltonian and shown that the roles of the new quantum gate U(θ) is equivalent, functionally, to the joint operation of Hadamard and C-Not gates.     

Symplectic Symmetries of Hamiltonian Systems

Abstract

We prove a generalization to the case of s × s matrix linear differential operators of the classical theorem of E. Cotton giving necessary and sufficient conditions for equivalence of eigenvalue problems for scalar linear differential operators. The conditions for equivalence to a matrix Schrödinger operator are derived and formulated geometrically in terms of vanishing conditions on the curvature of a gl(s, R)-valued connection. These conditions are illustrated on a class of matrix differential operators of physical interest, arising by symmetry reduction from Dirac’s equation for a spinor field minimally coupled with a cylindrically symmetric magnetic field.     

Intrinsic Properties of Quantum Systems

A new realist interpretation of quantum mechanics is introduced. Quantum systems are shown to have two kinds of properties: the usual ones described by values of quantum observables, which are called extrinsic, and those that can be attributed to individual quantum systems without violating standard quantum mechanics, which are called intrinsic. The intrinsic properties are classified into structural and conditional. A systematic and self-consistent account is given. Much more statements become meaningful than any version of Copenhagen interpretation would allow. A new approach to classical properties and measurement problem is suggested. A quantum definition of classical states is proposed.     

Relaxation Times for Hamiltonian Systems

Usually, the relaxation times of a gas are estimated in the frame of the Boltzmann equation. In this paper, instead, we deal with the relaxation problem in the frame of the dynamical theory of Hamiltonian systems, in which the definition itself of a relaxation time is an open question. We introduce a lower bound for the relaxation time, and give a general theorem for estimating it. Then we give an application to a concrete model of an interacting gas, in which the lower bound turns out to be of the order of magnitude of the relaxation times observed in dilute gases.     

Variational Symmetry in Non-integrable Hamiltonian Systems

Abstract

We consider the variational symmetry from the viewpoint of the non-integrability criterion towards dynamical systems. That variational symmetry can reduce complexity in determining non-integrability of general dynamical systems is illustrated here by a new non-integrability result about Hamiltonian systems with many degrees of freedom.     

Two New Classes of Isochronous Hamiltonian Systems

Abstract

An isochronous dynamical system is characterized by the existence of an open domain of initial data such that all motions evolving from it are completely periodic with a fixed period (independent of the initial data). Taking advantage of a recently introduced trick, two new Hamiltonian classes of such systems are identified.     

Anticontrol of Quantum Chaos in Hamiltonian Systems

We present a method of realizing anticontrol chaos in a quantum confined system consisting of N two-levelatoms within a cavity, using a time-delayed optic field. The delay time plays a construction and organization role forproducing temporal chaos, while the interaction between atoms and photons creates spatial chaos. The chaos is quitesensitive to small time delayed. The spectral decomposition of the Hamiltonian obtained by using projection methodologyreveals that evolution of the left eigenvectors shows quite complicated chaotic fashions. The method we proposed maybe easily tested in experiment, and provides a general method using a sort of driving optic field to achieve anticontrol ofchaos for quantum systems.     

Weakly Mixing Invariant Tori of Hamiltonian Systems

We note that every finite or infinite dimensional real-analytic Hamiltonian system with a quasi-periodic invariant KAM torus of finite dimension d≥ 2 can be perturbed in such a way that the new real-analytic Hamiltonian system has a weakly mixing invariant torus of the same dimension. Received: 24 April 1998/ Accepted: 14 January 1999     

Lie Symmetries for Hamiltonian Systems Methodological Approach

This paper proposes an algorithm for the Lie symmetries investigation in the case of a 2D Hamiltonian system. General Lie operators are deduced firstly and, in the the next step, the associated Lie invariants are derived. The 2D Yang-Mills mechanical model is chosen as a test model for this method. PACS: 05.45.-a; 02.30.Ik     

Invariant Tori in Hamiltonian Systems with Impacts

It is shown that a large class of solutions in two-degree-of-freedom Hamiltonian systems of billiard type can be described by slowly varying one-degree-of-freedom Hamiltonian systems. Under some non-degeneracy conditions such systems are found to possess a large set of quasiperiodic solutions filling out two dimensional tori, which correspond to caustics in the classical billiard. This provides a unified proof of existence of quasiperiodic solutions in convex billiards and other systems with impacts including classical billiard in electric and magnetic fields, dual billiard, and Fermi–Ulam systems. Received: 8 September 1999 / Accepted: 16 November 1999     

Variational Formulation for the Multisymplectic Hamiltonian Systems

A variational formulation for the multisymplectic Hamiltonian systems is presented in this Letter. Using this variational formulation, we obtain multisymplectic integrators from a variational perspective. Numerical experiments are also reported.Mathematical Subject Classifications (2000). 70G50, 58Z05.     

An Approximate KAM-Renormalization-Group Scheme for Hamiltonian Systems

We construct an approximate renormalization scheme for Hamiltonian systems with two degrees of freedom. This scheme is a combination of Kolmogorov–Arnold–Moser (KAM) theory and renormalization-group techniques. It makes the connection between the approximate renormalization procedure derived by Escande and Doveil and a systematic expansion of the transformation. In particular, we show that the two main approximations, consisting in keeping only the quadratic terms in the actions and the two main resonances, keep the essential information on the threshold of the breakup of invariant tori.     

A Note on the Integrability of Hamiltonian Systems

We consider a family of Hamiltonian systems

Systems of conservation laws with third-order Hamiltonian structures

We investigate n-component systems of conservation laws that possess third-order Hamiltonian structures of differential-geometric type. The classification of such systems is reduced to the projective classification of linear congruences of lines in \(\mathbb {P}^{n+2}\) satisfying additional geometric constraints. Algebraically, the problem can be reformulated as follows: for a vector space W of dimension \(n+2\), classify n-tuples of skew-symmetric 2-forms \(A^{\alpha } \in \varLambda ^2(W)\) such that

$$\begin{aligned} \phi _{\beta \gamma }A^{\beta }\wedge A^{\gamma }=0, \end{aligned}$$

for some non-degenerate symmetric \(\phi \).

     

Method of Generating N-dimensional Isochronous Nonsingular Hamiltonian Systems

In this paper we develop a straightforward procedure to construct higher dimensional isochronous Hamiltonian systems. We first show that a class of singular Hamiltonian systems obtained through the Ω-modified procedure is equivalent to constrained Newtonian systems. Even though such systems admit isochronous oscillations, they are effectively one degree of freedom systems due to the constraints. Then we generalize the procedure in terms of Ω _i -modified Hamiltonians and identify suitable canonically conjugate coordinates such that the constructed Ω _i -modified Hamiltonian is nonsingular and the corresponding Newton's equation of motion is constraint free. The procedure is first illustrated for two dimensional systems and subsequently extended to N-dimensional systems. The general solution of these systems are obtained by integrating the underlying equations and is shown to admit isochronous as well as amplitude independent quasiperiodic solutions depending on the choice of parameters.