Software fault-freeness and reliability predictions

Many software development practices aim at ensuring that software is correct, or fault-free. In safety critical applications, requirements are in terms of probabilities of certain behaviours, e.g. as associated to the Safety Integrity Levels of IEC 61508. The two forms of reasoning - about evidence of correctness and about probabilities of certain failures -are rarely brought together explicitly. The desirability of using claims of correctness has been argued by many authors, but not been taken up in practice. We address how to combine evidence concerning probability of failure together with evidence pertaining to likelihood of fault-freeness, in a Bayesian framework. We present novel results to make this approach practical, by guaranteeing reliability predictions that are conservative (err on the side of pessimism), despite the difficulty of stating prior probability distributions for reliability parameters. This approach seems suitable for practical application to assessment of certain classes of safety critical systems

Amplitude-mode dynamics of polariton condensates

We study the stability of collective amplitude excitations in non-equilibrium polariton condensates. These excitations correspond to renormalized upper polaritons and to the collective amplitude modes of atomic gases and superconductors. They would be present following a quantum quench or could be created directly by resonant excitation. We show that uniform amplitude excitations are unstable to the production of excitations at finite wavevectors, leading to the formation of density-modulated phases. The physical processes causing the instabilities can be understood by analogy to optical parametric oscillators and the atomic Bose supernova.Comment: 4 pages, 2 figure

Mixed Symmetry Solutions of Generalized Three-Particle Bargmann-Wigner Equations in the Strong-Coupling Limit

Starting from a nonlinear isospinor-spinor field equation, generalized three-particle Bargmann-Wigner equations are derived. In the strong-coupling limit, a special class of spin 1/2 bound-states are calculated. These solutions which are antisymmetric with respect to all indices, have mixed symmetries in isospin-superspin space and in spin orbit space. As a consequence of this mixed symmetry, we get three solution manifolds. In appendix \ref{b}, table 2, these solution manifolds are interpreted as the three generations of leptons and quarks. This interpretation will be justified in a forthcoming paper.Comment: 17 page

Numerical time propagation of quantum systems in radiation fields

Atoms, molecules or excitonic quasiparticles, for which excitations are induced by external radiation fields and energy is dissipated through radiative decay, are examples of driven open quantum systems. We explain the use of commutator-free exponential time-propagators for the numerical solution of the associated Schr\"odinger or master equations with a time-dependent Hamilton operator. These time-propagators are based on the Magnus series but avoid the computation of commutators, which makes them suitable for the efficient propagation of systems with a large number of degrees of freedom. We present an optimized fourth order propagator and demonstrate its efficiency in comparison to the direct Runge-Kutta computation. As an illustrative example we consider the parametrically driven dissipative Dicke model, for which we calculate the periodic steady state and the optical emission spectrum.Comment: 23 pages, 11 figure

Mixtures of fermionic atoms in an optical lattice

A mixture of light and heavy spin-polarized fermionic atoms in an optical lattice is considered. Tunneling of the heavy atoms is neglected such that they are only subject to thermal fluctuations. This results in a complex interplay between light and heavy atoms caused by quantum tunneling of the light atoms. The distribution of the heavy atoms is studied. It can be described by an Ising-like distribution with a first-order transition from homogeneous to staggered order. The latter is caused by an effective nonlocal interaction due to quantum tunneling of the light atoms. A second-order transition is also possible between an ordered and a disordered phase of the heavy atoms

Energy evolution in time-dependent harmonic oscillator

The theory of adiabatic invariants has a long history, and very important implications and applications in many different branches of physics, classically and quantally, but is rarely founded on rigorous results. Here we treat the general time-dependent one-dimensional harmonic oscillator, whose Newton equation $\ddot{q} + \omega^2(t) q=0$ cannot be solved in general. We follow the time-evolution of an initial ensemble of phase points with sharply defined energy $E_0$ at time $t=0$ and calculate rigorously the distribution of energy $E_1$ after time $t=T$, which is fully (all moments, including the variance $\mu^2$) determined by the first moment $\bar{E_1}$. For example, $\mu^2 = E_0^2 [(\bar{E_1}/E_0)^2 - (\omega (T)/\omega (0))^2]/2$, and all higher even moments are powers of $\mu^2$, whilst the odd ones vanish identically. This distribution function does not depend on any further details of the function $\omega (t)$ and is in this sense universal. In ideal adiabaticity $\bar{E_1} = \omega(T) E_0/\omega(0)$, and the variance $\mu^2$ is zero, whilst for finite $T$ we calculate $\bar{E_1}$, and $\mu^2$ for the general case using exact WKB-theory to all orders. We prove that if $\omega (t)$ is of class ${\cal C}^{m}$ (all derivatives up to and including the order $m$ are continuous) $\mu \propto T^{-(m+1)}$, whilst for class ${\cal C}^{\infty}$ it is known to be exponential $\mu \propto \exp (-\alpha T)$.Comment: 26 pages, 5 figure

Distribution of roots of random real generalized polynomials

The average density of zeros for monic generalized polynomials, $P_n(z)=\phi(z)+\sum_{k=1}^nc_kf_k(z)$, with real holomorphic $\phi ,f_k$ and real Gaussian coefficients is expressed in terms of correlation functions of the values of the polynomial and its derivative. We obtain compact expressions for both the regular component (generated by the complex roots) and the singular one (real roots) of the average density of roots. The density of the regular component goes to zero in the vicinity of the real axis like $|\hbox{\rm Im}\,z|$. We present the low and high disorder asymptotic behaviors. Then we particularize to the large $n$ limit of the average density of complex roots of monic algebraic polynomials of the form $P_n(z) = z^n +\sum_{k=1}^{n}c_kz^{n-k}$ with real independent, identically distributed Gaussian coefficients having zero mean and dispersion $\delta = \frac 1{\sqrt{n\lambda}}$. The average density tends to a simple, {\em universal} function of $\xi={2n}{\log |z|}$ and $\lambda$ in the domain $\xi\coth \frac{\xi}{2}\ll n|\sin \arg (z)|$ where nearly all the roots are located for large $n$.Comment: 17 pages, Revtex. To appear in J. Stat. Phys. Uuencoded gz-compresed tarfile (.66MB) containing 8 Postscript figures is available by e-mail from [email protected]

Dipolar superfluidity in electron-hole bilayer systems

Bilayer electron-hole systems, where the electrons and holes are created via doping and confined to separate layers, undergo excitonic condensation when the distance between the layers is smaller than typical distance between particles within a layer. We argue that the excitonic condensate is a novel dipolar superfluid in which the phase of the condensate couples to the {\it gradient} of the vector potential. We predict the existence of dipolar supercurrent which can be tuned by an in-plane magnetic field and detected by independent contacts to the layers. Thus the dipolar superfluid offers an example of excitonic condensate in which the {\it composite} nature of its constituent excitons is manifest in the macroscopic superfluid state. We also discuss various properties of this superfluid including the role of vortices.Comment: 5 pages, 1 figure, minor changes and added few references; final published versio