Star-unitary transformations. From dynamics to irreversibility and stochastic behavior

We consider a simple model of a classical harmonic oscillator coupled to a field. In standard approaches Langevin-type equations for {\it bare} particles are derived from Hamiltonian dynamics. These equations contain memory terms and are time-reversal invariant. In contrast the phenomenological Langevin equations have no memory terms (they are Markovian equations) and give a time evolution split in two branches (semigroups), each of which breaks time symmetry. A standard approach to bridge dynamics with phenomenology is to consider the Markovian approximation of the former. In this paper we present a formulation in terms of {\it dressed} particles, which gives exact Markovian equations. We formulate dressed particles for Poincar\'e nonintegrable systems, through an invertible transformation operator \Lam introduced by Prigogine and collaborators. \Lam is obtained by an extension of the canonical (unitary) transformation operator $U$ that eliminates interactions for integrable systems. Our extension is based on the removal of divergences due to Poincar\'e resonances, which breaks time-symmetry. The unitarity of $U$ is extended to ``star-unitarity'' for \Lam. We show that \Lam-transformed variables have the same time evolution as stochastic variables obeying Langevin equations, and that \Lam-transformed distribution functions satisfy exact Fokker-Planck equations. The effects of Gaussian white noise are obtained by the non-distributive property of \Lam with respect to products of dynamical variables. Therefore our method leads to a direct link between dynamics of Poincar\'e nonintegrable systems, probability and stochasticity.Comment: 24 pages, no figures. Made more connections with other work. Clarified ideas on irreversibilit

Some properties of the resonant state in quantum mechanics and its computation

The resonant state of the open quantum system is studied from the viewpoint of the outgoing momentum flux. We show that the number of particles is conserved for a resonant state, if we use an expanding volume of integration in order to take account of the outgoing momentum flux; the number of particles would decay exponentially in a fixed volume of integration. Moreover, we introduce new numerical methods of treating the resonant state with the use of the effective potential. We first give a numerical method of finding a resonance pole in the complex energy plane. The method seeks an energy eigenvalue iteratively. We found that our method leads to a super-convergence, the convergence exponential with respect to the iteration step. The present method is completely independent of commonly used complex scaling. We also give a numerical trick for computing the time evolution of the resonant state in a limited spatial area. Since the wave function of the resonant state is diverging away from the scattering potential, it has been previously difficult to follow its time evolution numerically in a finite area.Comment: 20 pages, 12 figures embedde

Exact Markovian kinetic equation for a quantum Brownian oscillator

We derive an exact Markovian kinetic equation for an oscillator linearly coupled to a heat bath, describing quantum Brownian motion. Our work is based on the subdynamics formulation developed by Prigogine and collaborators. The space of distribution functions is decomposed into independent subspaces that remain invariant under Liouville dynamics. For integrable systems in Poincar\'e's sense the invariant subspaces follow the dynamics of uncoupled, renormalized particles. In contrast for non-integrable systems, the invariant subspaces follow a dynamics with broken-time symmetry, involving generalized functions. This result indicates that irreversibility and stochasticity are exact properties of dynamics in generalized function spaces. We comment on the relation between our Markovian kinetic equation and the Hu-Paz-Zhang equation.Comment: A few typos in the published version are correcte

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 $x_{21}$. We restrict our attention to states where one atom is excited and the other is in the ground state, in symmetric or anti-symmetric combinations. We obtain exact 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, leading to long-range cooperative effects. The decay rates and the energy oscillate with the distance $x_{21}$. Depending on $x_{21}$, the decay rates will either decrease, vanish or increase as compared with the one-atom decay rate. We have sub- and super-radiance at periodic intervals. Our model may be used to study two-cavity electron wave-guides. The vanishing of the collective decay rates then suggests the possibility of obtaining stable configurations, where an electron is trapped inside the two cavities.Comment: 14 pages, 14 figures, submitted to Phys. Rev.

Electron Trapping in a One-Dimensional Semiconductor Quantum Wire with Multiple Impurities

We demonstrate the trapping of a conduction electron between two identical adatom impurities in a one-dimensional semiconductor quantum-dot array system (quantum wire). Bound steady states arise even when the energy of the adatom impurity is located in the continuous one-dimensional energy miniband. The steady state is a realization of the bound state in continuum (BIC) phenomenon first proposed by von Neuman and Wigner [Phys. Z. 30, 465 (1929)]. We analytically solve the dispersion equation for this localized state, which enables us to reveal the mechanism of the BIC. The appearance of the BIC state is attributed to the quantum interference between the impurities. The Van Hove singularity causes another type of bound state to form above and below the band edges, which may coexist with the BIC

Microscopic nonequilibrium structure in quantum fields and H-functions

AbstractExact dynamical models for entropy production and entropy flow are introduced by means of the complex spectral representation and illustrated by using a simple conservative Hamiltonian system for multileveled atoms coupled to an external time-dependent field. The dynamical form of the H-function is introduced corresponding to the Friedrichs model coupled with the external periodic force. The external force destroys the monotonicity of the H-function evolution and leads to a model for the “entropy flow”. As the result of competition between the dissipation inside the system and the regeneration of the excited states due to the external field, there appears a steady structure of the emitted field around the unstable particle, which corresponds to the “dissipative structure” and is intrinsic to the system as it does not depend on its initial preparation. The behavior of the H-function is investigated for the “velocity inversion experiment”

Causality, delocalization and positivity of energy

In a series of interesting papers G. C. Hegerfeldt has shown that quantum systems with positive energy initially localized in a finite region, immediately develop infinite tails. In our paper Hegerfeldt's theorem is analysed using quantum and classical wave packets. We show that Hegerfeldt's conclusion remains valid in classical physics. No violation of Einstein's causality is ever involved. Using only positive frequencies, complex wave packets are constructed which at $t = 0$ are real and finitely localized and which, furthemore, are superpositions of two nonlocal wave packets. The nonlocality is initially cancelled by destructive interference. However this cancellation becomes incomplete at arbitrary times immediately afterwards. In agreement with relativity the two nonlocal wave packets move with the velocity of light, in opposite directions.Comment: 14 pages, 5 figure

Real measurements and Quantum Zeno effect

In 1977, Mishra and Sudarshan showed that an unstable particle would never be found decayed while it was continuously observed. They called this effect the quantum Zeno effect (or paradox). Later it was realized that the frequent measurements could also accelerate the decay (quantum anti-Zeno effect). In this paper we investigate the quantum Zeno effect using the definite model of the measurement. We take into account the finite duration and the finite accuracy of the measurement. A general equation for the jump probability during the measurement is derived. We find that the measurements can cause inhibition (quantum Zeno effect) or acceleration (quantum anti-Zeno effect) of the evolution, depending on the strength of the interaction with the measuring device and on the properties of the system. However, the evolution cannot be fully stopped.Comment: 3 figure

Complex Energy Spectrum and Time Evolution of QBIC States in a Two-Channel Quantum wire with an Adatom Impurity

We provide detailed analysis of the complex energy eigenvalue spectrum for a two-channel quantum wire with an attached adatom impurity. The study is based on our previous work [Phys. Rev. Lett. 99, 210404 (2007)], in which we presented the quasi-bound states in continuum (or QBIC states). These are resonant states with very long lifetimes that form as a result of two overlapping continuous energy bands one of which, at least, has a divergent van Hove singularity at the band edge. We provide analysis of the full energy spectrum for all solutions, including the QBIC states, and obtain an expansion for the complex eigenvalue of the QBIC state. We show that it has a small decay rate of the order $g^6$, where $g$ is the coupling constant. As a result of this expansion, we find that this state is a non-analytic effect resulting from the van Hove singularity; it cannot be predicted from the ordinary perturbation analysis that relies on Fermi's golden rule. We will also numerically demonstrate the time evolution of the QBIC state using the effective potential method in order to show the stability of the QBIC wave function in comparison with that of the other eigenstates.Comment: Around 20 pages, 50 total figure