Nondegenerate System Research Articles

The role of constraints within generalized nonextensive statistics

The Gibbs–Jaynes path for introducing statistical mechanics is based on the adoption of a specific entropic form S and of physically appropriate constraints. For instance, for the usual canonical ensemble, one adopts (i) S 1=−k∑ ip i ln p i , (ii) ∑ i p i =1, and (iii) ∑ ip i ε i=U 1 ({ ε i }≡ eigenvalues of the Hamiltonian; U 1≡ internal energy). Equilibrium consists in optimizing S 1 with regard to { p i } in the presence of constraints (ii) and (iii). Within the recently introduced nonextensive statistics, (i) is generalized into S q = k[1−∑ i p i q ]/[ q−1] ( q→1 reproduces S 1), (ii) is maintained, and (iii) is generalized in a manner which might involve p i q . In the present effort, we analyze the consequences of some special choices for (iii), and their formal and practical implications for the various physical systems that have been studied in the literature. To illustrate some mathematically relevant points, we calculate the specific heat respectively associated with a nondegenerate two-level system as well as with the classical and quantum harmonic oscillators.

Field theory for coherent optical pulse propagation

We introduce a notion of ``matrix potential'' to nonlinear optical systems. In terms of a matrix potential $g$, we present a gauge-field-theoretic formulation of the Maxwell-Bloch equation that provides a semiclassical description of the propagation of optical pulses through resonant multilevel media. We show that the Bloch part of the equation can be solved identically through $g$ and the remaining Maxwell equation becomes a second-order differential equation with a reduced set of variables due to the gauge invariance of the system. Our formulation clarifies the (non-Abelian) symmetry structure of the Maxwell-Bloch equations for various multilevel media in association with symmetric spaces $G/H$. In particular, we associate the nondegenerate two-level system for self-induced transparency with $G/H=\mathrm{SU}(2)/\mathrm{U}(1)$ and three-level $\ensuremath{\Lambda}$ or $V$ systems with $G/H=\mathrm{SU}(3)/\mathrm{U}(2)$. We give a detailed analysis for the two-level case in the matrix potential formalism, and address various properties of the system including soliton numbers, effective potential energy, gauge and discrete symmetries, modified pulse area, conserved topological, and nontopological charges. The nontopological charge measures the amount of self-detuning of each pulse. Its conservation law leads to a different type of pulse stability analysis that explains earlier numerical results.

Canonical forms for the invariant tensors and A-B-C-cohomologies of integrable Hamiltonian systems

The canonical forms for the tensors, , that are invariant with respect to a Liouville-integrable non-degenerate Hamiltonian system on a symplectic manifold are derived. It is proved that the characteristic polynomial of any invariant (1,1) tensor is a perfect square; therefore its eigenvalues have even multiplicities. Any invariant metric is indefinite and has signature . The derived canonical forms are applied to the calculation of the A-B-C-cohomologies of Liouville-integrable Hamiltonian systems.

Plasmon excitations in nondegenerate quasi-one-dimensional electron systems

The many-body dielectric formalism is applied for the laterally confined nondegenerate electron system over the liquid helium surface. The density-density response function dependent on the wave number and frequency for the multisubband quasi-one-dimensional (Q1D) system is evaluated within the random-phase approximation for arbitrary values of wave number and subband index. The spectra of both intrasubband and intersubband plasmons are obtained in the limit of small wave numbers and low temperatures. The intrasubband plasmon dispersion is almost soundlike and similar to that of intrasubband plasmons in degenerate Q1D systems and in the Q1D electron chain. The intersubband mode is optical-like with frequency close to the one-electron frequency related to the lateral parabolic confinement. Intra- and intersubband magnetoplasmon frequencies are also derived. Our results are compared with recent spectroscopic experiment.

Stability of nondegenerate canonical oscillating system near to integrable ones

Stability of nondegenerate canonical oscillating system near to integrable ones

Photoemission from a system of localized pairs on a cuprate lattice

Abstract We postulate that the anisotropic normal-state gap seen recently in cuprates may be the result of pre-formed pairs and use a one-step Green function method to analyse a model in which the normal state consists of a non-degenerate system of such pairs. We observe that, for a completely localized pair band, an angularly independent peak is obtained with a width equal to the single-electron bandwidth. The peak, rather than corresponding to a point on the single-electron energy surface, is in fact a result of the collective contribution from all pairs and is mathematically similar to a Van Hove singularity. The relation of this calculation to the available angle resolved, photoemission spectroscopy data and proposals for further research are discussed.

A criterion for orbital equivalence of integrable Hamiltonian systems in the vicinity of elliptic orbits. An orbital invariant in the Lagrange problem

In this paper we obtain a criterion for the continuous and smooth orbital equivalence of integrable Hamiltonian systems with degrees of freedom in the vicinity of compact elliptic orbits. Moreover, we construct a complete orbital invariant for a non-degenerate integrable Hamiltonian system with two degrees of freedom in a neighbourhood of an elliptic singular point, and propose a rule from which to compute this orbital invariant. The orbital invariant is computed for integrable Lagrange systems in rigid body dynamics. In this way we find an explicit decomposition of all Lagrange systems into classes of orbitally equivalent ones in the vicinity of equilibria.

Electron polarization in nonrelativistic nondegenerate magnetized electron gas

For a nondegenerate nonrelativistic electron system, taking account of the minimum of its total energy, the population of the Landau electron levels by electrons with intrinsic magnetic moments parallel and antiparallel to the external magnetic field is determined, as a function of the external magnetic field strength H. The results are then compared with analogous data obtained by other authors without taking the minimum of the system’s total energy into account.

On the master symmetries related to certain classes of integrable Hamiltonian systems

We present the complete classification of master symmetries related to a non-degenerate Hamiltonian system that is integrable in the Arnol'd - Liouville's sense. It is shown that a -integrable Hamiltonian system cannot have generators of degree greater than 2. Specific properties of the master symmetries classified in terms of the action - angle variables are investigated.

Transient degenerate four-wave mixing in molecular systems

The dependence of the degenerate four-wave mixing (DFWM) signal on laser intensity, molecular transition moment and molecular concentration were examined experimentally by taking the iodine molecule (l2) as a testing example. The experimental results will be interpreted by a non-steady-state DFWM theory, an extension of the non-degenerate two-level system previously proposed. The distinctive physical feature of this mechanism is that the present observation is the result of non-stationary evolution of the excited-state population, in contrast to the conventional theory based on the steady-state treatment. In this study, we have also taken into account the magnetic sub-levels of molecular rovibronic quantum states. The non-steady-state model we developed in this paper is applicable to fast pump and slow relaxation conditions, thus being more suitable for gas-phase systems than the condensed-phase where fast relaxations usually take place.

Theory of tensor invariants of integrable Hamiltonian systems. I. Incompatible Poisson structures

This paper develops a new theory of tensor invariants of a completely integrable non-degenerate Hamiltonian system on a smooth manifoldM n. The central objects in this theory are supplementary invariant Poisson structuresP c which are incompatable with the original Poisson structureP 1 for this Hamiltonian system. A complete classification of invariant Poisson structures is derived in a neighbourhood of an invariant toroidal domain. This classification resolves the well-known Inverse Problem that was brought into prominence by Magri's 1978 paper deveoted to the theory of compatible Poisson structures. Applications connected with the KAM theory, with the Kepler problem, with the basic integrable problem of celestial mechanics, and with the harmonic oscillator are pointed out. A cohomology is defined for dynamical systems on smooth manifolds. The physically motivated concepts of dynamical compatibility and strong dynamical compatibility of pairs of Poisson structures are introduced to study the diversity of pairs of Poisson structures incompatible in Magri's sense. It is proved that if a dynamical systemV preserves two strongly dynamically compatible Poisson structuresP 1 andP 2 in a general position then this system is completely integrable. Such a systemV generates a hierarchy of integrable dynamical systems which in general are not Hamiltonian neither with respect toP 1 nor with respect toP 2. Necessary conditions for dynamical compatibility and for strong dynamical compatibility are derived which connect these global properties with new local invariants of an arbitrary pair of incompatible Poisson structures.

Symmetry breaking of the admittance of a classical two-dimensional electron system in a magnetic field.

The symmetry properties under magnetic-field reversal are reported for the elements of the admittance matrix of a set of probes that couple capacitively to a nondegenerate two-dimensional electron system, which in itself is not connected to an electron reservoir. Strong asymmetries are observed in multiterminal measurement configurations (three probes or more), whereas two-terminal measurements are symmetric. The asymmetries are explained in terms of classical edge magnetoplasmons and satisfy the generalized Onsager-Casimir symmetry requirements.

Effects of electron correlation on doping in conjugated polymers — a study of the SSHH and BKH models using the DMRG method

We adopt the Su-Schrieffer-Heeger (SSH) Hamiltonian supplemented by a Hubbard term, which we call the SSHH model, for systems with a degenerate ground state, and the Brazovskii-Kirova (BK) Hamiltonian supplemented by a Hubbard term (BKH model) for systems with a nondegenerate ground state, to study the effect of electron correlation on doping in conjugated polymers. The density matrix renormalization group (DMRG) technique developed by White has proved to be a very powerful tool for treating low-dimensional systems with strong correlation. This method is applied to the above models. We first study the metal-insulator transition within the SSHH model. With the BKH model, we investigate the formation of bipolarons versus separated polarons in a nondegenerate system. Optical signatures of polarons are also calculated.

Asymptotic analysis of the degenerate Boltzmann-Poisson system for semiconductors

Abstract In semiconductor physics, the kinetic Boltzmann equation with a quadratic collision operator coupled with the Poisson equation are used to model degeneracy phenomena (which appear in highly doped devices). Assuming the electron distribution is small, we make an asymptotic analysis of the Boltzmann-Poisson system. We obtain the weak convergence, local in time, of the degenerate Boltzmann-Poisson system to the nondegenerate Boltzmann-Poisson system (with a linear collision operator).

Effects of strongly coupled nondegenerate primary shear systems on LEDS formation during martensitic transformations in b.c.c. alloys

Martensitic transformations in b.c.c. metals and alloys can proceed by three different type degenerate primary shear systems which are strongly and weakly coupled to the principal strains. These give rise to different habit planes, orientation relationships, and twin domain structures. In addition to this predicted behaviour, a strongly coupled nondegenerate primary shear system is available which cannot maintain the mean field and consequently cannot consummate the martensitic transformation. However, because of its strong coupling to the principal strains it will nucleate spontaneously at dissociated dislocations above M s , and its long-range stress fields will tend to clamp operation of the degenerate primary shear systems. The effect is a lower transformation temperature and a wide hysteresis in the transformation behavior. The clamping is avoided and the mean field maintained by alternating lattice-variant shears which give rise to the metastable ω-phase in Ti- and Zr-based alloys. Alloying lowers the energy of the collapsed (111) b planes and makes ω energetically favourable. Dislocation mechanisms for this transformation are described and the resulting morphology of ω shown to be consistent with the reported soft mode behavior, phonon dispersion, and streaking in diffraction patterns

ON THE EXISTENCE AND UNIQUENESS OF TRANSIENT SOLUTIONS OF A DEGENERATE NONLINEAR DRIFT-DIFFUSION MODEL FOR SEMICONDUCTORS

We analyze the degenerate transient multi-dimensional quasi-hydrodynamic model for semiconductors with general recombination rate. We present existence results for general nonlinear diffusivities for the nondegenerate and the degenerate Dirichlet-Neumann mixed boundary value problem. Uniqueness of solutions of the nondegenerate system can be proved in the Dirichlet boundary case. Concerning the degenerate problem uniqueness can only be shown under some conditions on the initial and boundary data or on the electric field.

Many-electron effects in the magnetoresistivity of a non-degenerate two-dimensional electron gas

Abstract The influence of many-electron effects on the magnetoresistivity pxx of a non-degenerate two-dimensional electron system is discussed, using the Einstein diffusion relation to demonstrate the physical principles involved. For independent electrons the two limiting theories are the Drude model (classical electrons) and the self-consistent Born approximation (quantized Landau levels), but electron-electron interactions modify the kinetics of the semiclassical cyclotron orbits both in classically strong and in quantizing magnetic fields. In both cases, pxx depends on the electron density, and calculations of pxx are presented for a typical experimental system.

Many-electron magnetotransport in 2D electrons on liquid helium

A many-electron theory of magnetotransport of a nondegenerate 2D electron system is presented along with experimental data in classically strong and quantising magnetic fields. Due to the electron—electron interaction each individual electron is driven by a fluctuating electric field E which converts the discrete Landau levels for non-interacting electrons, spacing ħω c, into a continuous spectrum. Hence the classical magnetoresistance is very small (compared to the single-electron theory) until eER c < ħω c where R c is the classical Larmor radius. At high magnetic fields, ħω c ⪢ KT > EEl( l=(ħ/ mω c 1 2 for eEl/ħ less than the electron momentum scattering rate, the single-electron theory becomes valid. The onset field B 0 for magnetoresistance lies in the range 0.3-1.0 T for electrons on liquid helium. These many-electron effects have been observed experimentally near 1 K where the electron mobility is high and limited by 4He vapor atoms and below 1 K in the ripplon scattering regime.

Magnetoconductance of electrons on liquid helium in the extreme quantum limit

Measurements of the longitudinal magnetoconductivity σxx of the nondegenerate two-dimensional system of electrons on the surface of liquid helium are presented for magnetic fields up to 22T in the temperature range 0.4K < T < 2.1 K. For temperatures above 1.6 K our experimental results are in good agreement with a quantum magnetotransport theory of short-range scattering within broadened Landau levels. If the temperature is reduced below 1.5 K, a deviation of the experimental data from the theoretical calculations is observed for magnetic fields between 2 T and 10 T. At temperatures T < 0.8K where the conductvity of the surface electrons is determined by the scattering between electrons and ripplons, the magnetic field dependence of our measurements of 1/σxx(B) for high fields is rather poorly reproduced by the existing theories.

Gain in a three-level Lambda system driven by a single pump.

We calculate the absorption spectrum of a weak probe interacting with a \ensuremath{\Lambda}-shaped three-level system driven by a single pump. We find that there are four types of terms that contribute to the probe absorption: population-difference and two-photon coherence terms which derive solely from interaction with the pump (dc terms) and population-difference and two-photon coherence terms that pulsate at the pump-probe frequency offset. We find that the main features of the probe spectrum derive from the pulsating terms whereas the dc terms either interfere destructively with each other or with the pulsating terms. This is a clear extension to the three-level system of the concept of light amplification by coherence. We calculate the spectrum for three cases: (i) a degenerate population-trapped system, (ii) a degenerate system in which both transitions are population inverted, and (iii) a nondegenerate population-trapped system. The results of case (i) are similar to those for the two-level system but the population-trapped system has the advantage of avoiding pump absorption. We show that the Rabi frequency of the nonabsorbed pump can be found by determining the probe frequencies at which probe nonabsorption occurs. In (ii), the results are compared with the case where two pumps interact resonantly with a nondegenerate system. In (iii), we find that the populations in the lower states behave dispersively as a function of the pump detuning. For Rabi frequencies that are small compared to the separation between the lower levels, the lower level nearer to resonance is less populated than the other level. However, at higher Rabi frequencies, the situation is reversed. The probe spectrum contains two absorption peaks when the pump Rabi frequency is small and two emission peaks when it is large.