Gonzalo G.E. Ordóñez

Explicit construction of a time superoperator for quantum unstable systems

Abstract A time superoperator T conjugate to the Liouville superoperator LH=[H,] is constructed for a quantum system with one excited state or unstable particle. While there is no time operator conjugate to the Hamiltonian in the wave function space due to the positivity of energy, T may exist in the density matrix space as the spectrum of LH covers all the real axis. This is the first example of an observable that can only be formulated in the Liouville–von Neumann space of density matrices. In our example the expectation value of T gives the lifetime of the unstable particle. Once the time superoperator is obtained it is easy to define an entropy superoperator.

Complex collective states in a one-dimensional two-atom system

We consider a pair of identical two-level atoms interacting with a scalar field in one dimension, separated by a distance x{sub 21}. We obtain collective decaying states, belonging to a complex spectral representation of the Hamiltonian. The imaginary parts of the eigenvalues give the decay rates, and the real parts give the average energy of the collective states. In one dimension there is strong interference between the fields emitted by the atoms, producing cooperative effects even when the atoms are far apart. The decay rates and the energy oscillate with the distance x{sub 21}. Depending on x{sub 21}, the decay rates will either decrease, vanish, or increase as compared with the one-atom decay rate. We have sub- and superradiance at periodic intervals. Our model may be used to study two-cavity electron waveguides. The vanishing of the collective decay rates then suggests the possibility of obtaining stable configurations, where an electron is trapped inside the two cavities.

Using symmetries of the Frobenius-Perron operator to determine spectral decompositions

Abstract The spectral decomposition of the Frobenius-Perron operator for a class of piecewise-linear maps is determined from symmetry transformations of the dyadic Bernoulli map. This approach enables one to construct explicit compact expressions for the eigenstates.

Optically tunable bound states in the continuum

Yingyue Boretz, Gonzalo Ordonez, Satoshi Tanaka, and Tomio Petrosky Center for Studies in Statistical Mechanics and Complex Systems, The University of Texas at Austin, Austin, TX 78712 USA Physics and Astronomy Department, Butler University, 4600 Sunset Ave., Indianapolis, IN 46208 USA Department of Physical Science, Osaka Prefecture University, Gakuen-cho 1-1, Sakai 599-8531, Japan (Dated: November 12, 2013)

MICROSCOPIC ENTROPY FLOW AND ENTROPY PRODUCTION IN RESONANCE SCATTERING

A microscopic dynamical entropy (or ‘H function) for a two-level atom interacting with a field is introduced. The excitation process of the atom due to the resonance scattering of a wave packet is discussed. Three stages of scattering process (before, during and after the collision) are described in terms of entropy production and entropy flow. The excitation of the atom may be considered as the construction of a non-equilibrium structure due to entropy flow. The emission of photons distributes the energy of the unstable state among the field modes, leading to an increase of microscopic entropy. In this process, instability in dynamics associated with resonances plays a central role. The ‘H function is constructed outside the Hilbert space, which allows strictly irreversible time evolution, avoiding probabilistic arguments associated with ignorance. .

Spectral decomposition of tent maps using symmetry considerations

The spectral decomposition of the Frobenius-Perron operator of maps composed of many tents is determined from symmetry considerations. The eigenstates involve Euler as well as Bernoulli polynomials.

Irreversibility, Probabilities and Dressed Unstable States in Quantum Mechanics

The problem of the meaning of quantum unstable states including their dressing is considered. The formulation given in terms of density matrices outside the Hilbert space is used. A dressed unstable state for the Friedrichs model, which is the simplest model that incorporates both bare and dressed quantum states is obtained. Due to resonance singularities that appear in the frequency denominators, quantum unstable systems are categorized as non-integrable systems in the sense of Poincare. The excited unstable state is derived from the stable states through a suitable analytic continuation of the denominators. It is given by an irreducible density matrix with broken time-symmetry. Our state decays following a Markovian equation. There are no deviations from exponential decay neither for short nor for long times, as is the case for the bare state. The dressed state satisfies an uncertainty relation between energy and lifetime. There are experiments that could verify our proposal. A typical one would be the ...

Dynamical formulation of Gaussian white noise

We study the connection between Hamiltonian dynamics and irreversible, stochastic equations, such as the Langevin equation. We consider a simple model of a harmonic oscillator (Brownian particle) coupled to a field (heat bath). We introduce an invertible transformation operator Λ that brings us to a new representation where dynamics is decomposed into independent Markovian components, including Brownian motion. The effects of Gaussian white noise are obtained by the non-distributive property of Λ with respect to products of dynamical variables. In this way we obtain an exact formulation of white noise effects. Our method leads to a direct link between dynamics of Poincaré nonintegrable systems, probability and stochasticity.

Derivation of 1/f noise from resonances

We study non-interacting particles in a small subsystem which is weakly coupled to a reservoir. We show that this class of systems can be mapped into an extended form of the Friedrichs model. We derive from the Hamiltonian dynamics that the number fluctuation in a subsystem is 1/f or 1/fβ noise. We show that this effect comes from the sum of resonances.

POLYNOMIAL SHIFT STATES OF A CHAOTIC MAP

We present a piecewise- linear map of the unit interval in which the resolvent of the Frobenius- Perron operator, considered in a polynomial basis, has an essen-tial singularity at the origin. Associated with the essential singularity are polynomial shift states, which are obtained from creation and annihilation operators in non- self- dual function spaces. Correlation functions of general polynomial observables have decay components that vanish in a finite time.