94 research outputs found
Global Hamilton principal functions of the eikonal equations on and
No full textThis monograph grew out of a series of lectures given at the XXVI Summer School of Mathematical Physics, Ravello, September 2001, organized by G.N.F.M. (gruppo Nazionale di Fisica Matematica) of I.N.d.A.M. (Istituto Nazionale di Alta Matematica, Roma), at the Department of Mathematics of the University of Torino in the academic years 2000/2001 and 2001/2002, and at the Departmento of Physical Sciences of the University of Napoli, May 2003.
The elements of Symplectic Geometry and Analytical Mechanics on which these lectures are based can be found in the literature of the seventies and eighties of the last century. The bibliography is of course far from complete and refers the reader to some of the important contributions. Here, we introduce only the essential notions of symplectic geometry needed for application to the geometrical theory of the Hamilton-Jacobi equation and to the control theory of static systems. Most of these notions are well known, but the way they are assembled and used is new in many respects.
A fundamental role in the present approach is played by the notion of generating family and by two operations: the composition of generating families of symplectic relations and the canonical lift from objects on manifolds (submanifolds, relations, mappings, vector fields, etc.) to symplectic objects on the corresponding cotangent bundles. Generating families describe special subsets of cotangent bundles which we call Lagrangian sets. A Lagrangian set is a Lagrangian submanifold (which may be immersed) if the generating family is a Morse family. However, there are physically interesting examples of Lagrangian sets which are not Lagrangian submanifolds. An advantage of considering generating families as fundamental objects is that, while the composition of two symplectic relations may not be a smooth relation, the composition of two generating families is always a smooth function. In other words, the symplectic creed as formulated by A. Weinstein in his article Symplectic geometry (1981) in the form everything is a Lagrangian submanifold, which means that one should try to express objects in symplectic geometry and mechanics in terms of Lagrangian submanifolds, is here replaced by everything has a generating family.
The geometrical theory of the Hamilton-Jacobi equation is closely related to Geometrical Optics. The symplectic formulation of Hamiltonian Optics presented here differs from other formulations illustrated in papers and well known reference books cited in the Bibliography and it is, in my opinion, very close to the original ideas of Hamilton. From a geometrical view-point a Hamilton-Jacobi equation is a coisotropic submanifold of a cotangent bundle. A geometrical solution is a Lagrangian set described by a generating family and contained in the coisotropic submanifold. There are two fundametal symplectic relations associated with a Hamilton-Jacobi equation, the characteristic relation and the characteristic reduction. The two corresponding generating families are the Hamilton principal function and the complete solution of the Hamilton-Jacobi equation, respectively. By composing the latter with its transpose we get the former. Since the characteristic relation is a singular Lagrangian submanifold, the Hamilton principal function is necessarily a generating family and not a two-point function as in the classical theory. Cauchy data (or sources of systems of rays), mirror and lenses are represented by symplectic relations thus, by generating families. Then the Cauchy problem and the actions of a lens or of a mirror on a system of rays are translated into the composition of generating families.
What is presented here is only a first approach to Geometrical Optics based on the notions of symplectic relation and generating family. We do not cover many important examples of optical phenomena, which can be found in standard reference books (e.g. Synge, Luneburg, Buchdahl) and which probably can be treated within this framework.
Perhaps, the use of generating families and symplectic relations does not yield a revolutionary progress in Hamiltonian Optics, but we are obliged to introduce these concepts if, for example, we want to give a global meaning to the Hamilton characteristic function, as shown in Chapters 3 and 4.
Symplectic relations and generating families can play an interesting role also in the control theory of static systems, including thermostatic systems. Chapter 5 is devoted to this matter. Our approach is based on the notion of control relation and on an extended version of the virtual work principle for constrained systems with non-controlled degrees of freedom (hidden variables). Several examples of singular phenomena concerning static systems and thermostatics are illustrated. In particular, it is shown how the Maxwell rule follows as a theorem from the extended virtual work principle. Thermostatics of simple and composite systems is here described in the four-dimensional state space, with global coordinates (S, V, P, T), entropy, volume, pressure, absolute temperature, endowed with the natural symplectic structure induced by the first principle of thermodynamics.
An outline of the basic tools of calculus on manifolds needed in our discussion is given in Appendix A. A supplementary note (Appendix B) written in collaboration with Franco Cardin (Dipartimento di Matematica Pura e Applicata, Università di Padova), is devoted to the calculus of global principal Hamilton functions for the eikonal equations on the two-dimensional sphere S2 and pseudo-sphere H2
Separability in Riemannian Manifolds
No full textAn outline of the basic Riemannian structures underlying the separation of variables in the Hamilton-Jacobi equation of natural Hamiltonian systems.This paper is a contribution to the Special Issue on Analytical Mechanics and Dif ferential Geometry in honour
of Sergio Benenti. The full collection is available at http://www.emis.de/journals/SIGMA/Benenti.html
Stability of quantum computing in the presence of imperfections
No full textWe model an isolated quantum computer as a two-dimensional lattice of qubits (spin halves) with fluctuations in individual qubit energies and residual short-range interqubit couplings. We show that above a critical inter-qubit coupling strength, quantum chaos sets in and this results in the interaction induced dynamical thermalization and occupation numbers well described by the Fermi-Dirac distribution. This thermalization destroys the noninteracting qubit structure and sets serious requirements for the quantum computer operability. We then construct a quantum algorithm which uses qubits in an optimal way and efficiently simulates a physical model with rich and complex dynamics. The numerical study of the effect of static imperfections in the quantum computer hardware shows that the main elements of the phase space structures are accurately reproduced up to a time scale which is polynomial in the number of qubits. The errors generated by these imperfections are more significant than t..
Dynamics of entanglement in quantum computers with imperfections
No full textThe dynamics of the pairwise entanglement in a qubit lattice in the presence of static imperfections exhibits different regimes. We show that there is a transition from a perturbative region, where the entanglement is stable against imperfections, to the ergodic regime, in which a pair of qubits becomes entangled with the rest of the lattice and the pairwise entanglement drops to zero. The transition is almost independent of the size of the quantum computer. We consider both the case of an initial maximally entangled and separable state. In this last case there is a broad crossover region in which the computer imperfections can be used to create a significant amount of pairwise entanglement. © 2003 The American Physical Society
Effects of single-qubit quantum noise on entanglement purification
No full textWe study the stability under quantum noise effects of the quantum privacy amplification protocol for the purification of entanglement in quantum cryptography. We assume that the E91 protocol is used by two communicating parties (Alice and Bob) and that the eavesdropper Eve uses the isotropic Bu (z) over bar ek-Hillery quantum copying machine to extract information. Entanglement purification is then operated by Alice and Bob by means of the quantum privacy amplification protocol and we present a systematic numerical study of the impact of all possible single-qubit noise channels on this protocol. We find that both the qualitative behavior of the fidelity of the purified state as a function of the number of purification steps and the maximum level of noise that can be tolerated by the protocol strongly depend on the specific noise channel. These results provide valuable information for experimental implementations of the quantum privacy amplification protocol
Chaotic dynamics in superconducting nanocircuits
No full textThe quantum kicked rotator can be realized in a periodically driven superconducting nanocircuit. A study of the fidelity allows the experimental investigation of exponential instability of quantum motion inside the Ehrenfest time scale, chaotic diffusion and quantum dynamical localization. The role of noise and the experimental setup to measure the fidelity is discussed as well
Anomalous Heat Transport in Classical Many-Body Systems: Overview and Perspectives
No full textIn this review paper we survey recent achievements in anomalous heat diffusion, while highlighting open problems and research perspectives. First, we briefly recall the main features of the phenomenon in low-dimensional classical anharmonic chains and outline some recent developments in the study of perturbed integrable systems and the effect of long-range forces and magnetic fields. Selected applications to heat transfer in material science at the nanoscale are described. In the second part, we discuss of the role of anomalous conduction in coupled transport and describe how systems with anomalous (thermal) diffusion allow a much better power-efficiency trade-off for the conversion of thermal to particle current
Eigenstates of an operating quantum computer: hypersensitivity to static imperfections
No full textWe study the properties of eigenstates of an
operating quantum computer
which simulates the dynamical evolution in the regime of quantum chaos.
Even if the quantum algorithm is polynomial in number of qubits nq,
it is shown that the ideal eigenstates become mixed
and strongly modified by static imperfections above a certain
threshold which drops exponentially with nq.
Above this threshold the quantum eigenstate entropy grows linearly with nq
but the computation remains reliable
during a time scale which is polynomial
in the imperfection strength and in nq
Exotic states in the dynamical Casimir effect
No full textWe consider the interaction of a qubit with a single mode of the quantized electromagnetic field and show that, in the ultrastrong coupling regime and when the qubit-field interaction is switched on abruptly, the dynamical Casimir effect leads to the generation of a variety of exotic states of the field, which cannot be simply described as squeezed states. Such effect is a consequence of the intrinsic nonlinearity of the qubit and also appears when initially both the qubit and the field are in their ground state. The non-classicality of the obtained exotic states is characterized by means of a parameter based on the volume of the negative part of the Wigner function. A transition to non-classical states is observed by changing either the interaction strength or the interaction time. The observed phenomena appear as a general feature of nonadiabatic quantum gates, so that the dynamical Casimir effect can be the origin of a fundamental upper limit to the maximum speed of quantum computation and communication protocol
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