Quantum nonlinear dynamics of continuously measured systems

Classical dynamics is formulated as a Hamiltonian flow in phase space, while quantum mechanics is formulated as unitary dynamics in Hilbert space. These different formulations have made it difficult to directly compare quantum and classical nonlinear dynamics. Previous solutions have focused on computing quantities associated with a statistical ensemble such as variance or entropy. However a more diner comparison would compare classical predictions to the quantum predictions for continuous simultaneous measurement of position and momentum of a single system, in this paper we give a theory of such measurement and show that chaotic behavior in classical systems fan be reproduced by continuously measured quantum systems.

Model-based procedure for scale-up of wet, overflow ball mills - Part III: Validation and discussion

A new ball mill scale-up procedure is developed. This procedure has been validated using seven sets of Ml-scale ball mil data. The largest ball mills in these data have diameters (inside liners) of 6.58m. The procedure can predict the 80% passing size of the circuit product to within +/-6% of the measured value, with a precision of +/-11% (one standard deviation); the re-circulating load to within +/-33% of the mass-balanced value (this error margin is within the uncertainty associated with the determination of the re-circulating load); and the mill power to within +/-5% of the measured value. This procedure is applicable for the design of ball mills which are preceded by autogenous (AG) mills, semi-autogenous (SAG) mills, crushers and flotation circuits. The new procedure is more precise and more accurate than Bond's method for ball mill scale-up. This procedure contains no efficiency correction which relates to the mill diameter. This suggests that, within the range of mill diameter studied, milling efficiency does not vary with mill diameter. This is in contrast with Bond's equation-Bond claimed that milling efficiency increases with mill diameter. (C) 2001 Elsevier Science Ltd. All rights reserved.

Derivation of the probability distribution function for the local density of states of a disordered quantum wire via the replica trick and supersymmetry

We consider the statistical properties of the local density of states of a one-dimensional Dirac equation in the presence of various types of disorder with Gaussian white-noise distribution. It is shown how either the replica trick or supersymmetry can be used to calculate exactly all the moments of the local density of states.' Careful attention is paid to how the results change if the local density of states is averaged over atomic length scales. For both the replica trick and supersymmetry the problem is reduced to finding the ground state of a zero-dimensional Hamiltonian which is written solely in terms of a pair of coupled spins which are elements of u(1, 1). This ground state is explicitly found for the particular case of the Dirac equation corresponding to an infinite metallic quantum wire with a single conduction channel. The calculated moments of the local density of states agree with those found previously by Al'tshuler and Prigodin [Sov. Phys. JETP 68 (1989) 198] using a technique based on recursion relations for Feynman diagrams. (C) 2001 Elsevier Science B.V. All rights reserved.

Stochastic gauges in quantum dynamics for many-body simulations

Quantum dynamics simulations can be improved using novel quasiprobability distributions based on non-orthogonal Hermitian kernel operators. This introduces arbitrary functions (gauges) into the stochastic equations. which can be used to tailor them for improved calculations. A possible application to full quantum dynamic simulations of BEC's is presented. (C) 2001 Elsevier Science B.V. All rights reserved.

Quantum criticality

We investigate the theory of quantum fluctuations in non-equilibrium systems having large critical fluctuations. This allows us to treat the limits imposed by nonlinearities to quantum squeezing and noise reduction, and also to envisage future tests of quantum theory in regions of macroscopic quantum fluctuations. A long-term objective of this research is to identify suitable physical systems in which macroscopic 'Schrodinger cat'-like behaviour may be observed. We investigate two systems in particular of much current experimental interest, namely the degenerate parametric oscillator near threshold, and the evaporatively cooled (BEC). We compare the results obtained in the positive-P representation, as a fully quantum mechanical calculation, with the truncated Wigner phase space equation, also known as semi-classical theory. We show when these results agree and differ in calculations taken beyond the linearized approximation. In the region where the largest quantum fluctuations and Schrodinger cat-like behaviour might be expected, we find that the quantum predictions correspond very closely to the semi-classical theory. Nature abhors observing a Schrodinger car.

Generalized quasi mode theory of macroscopic canonical quantization in cavity quantum electrodynamics and quantum optics I. Theory

The quasi mode theory of macroscopic quantization in quantum optics and cavity QED developed by Dalton, Barnett and Knight is generalized. This generalization allows for cases in which two or more quasi permittivities, along with their associated mode functions, are needed to describe the classical optics device. It brings problems such as reflection and refraction at a dielectric boundary, the linear coupler, and the coupling of two optical cavities within the scope of the theory. For the most part, the results that are obtained here are simple generalizations of those obtained in previous work. However the coupling constants, which are of great importance in applications of the theory, are shown to contain significant additional terms which cannot be 'guessed' from the simpler forms. The expressions for the coupling constants suggest that the critical factor in determining the strength of coupling between a pair of quasi modes is their degree of spatial overlap. In an accompanying paper a fully quantum theoretic derivation of the laws of reflection and refraction at a boundary is given as an illustration of the generalized theory. The quasi mode picture of this process involves the annihilation of a photon travelling in the incident region quasi mode, and the subsequent creation of a photon in either the incident region or transmitted region quasi modes.

Generalized quasi mode theory of macroscopic canonical quantization in cavity quantum electrodynamics and quantum optics II. Application to reflection and refraction at a dielectric interface

The generalization of the quasi mode theory of macroscopic quantization in quantum optics and cavity QED presented in the previous paper, is applied to provide a fully quantum theoretic derivation of the laws of reflection and refraction at a boundary. The quasi mode picture of this process involves the annihilation of a photon travelling in the incident region quasi mode, and the subsequent creation of a photon in either the incident region or transmitted region quasi modes. The derivation of the laws of reflection and refraction is achieved through the dual application of the quasi mode theory and a quantum scattering theory based on the Heisenberg picture. Formal expressions from scattering theory are given for the reflection and transmission coefficients. The behaviour of the intensity for a localized one photon wave packet coming in at time minus infinity from the incident direction is examined and it is shown that at time plus infinity, the light intensity is only significant where the classical laws of reflection and refraction predict. The occurrence of both refraction and reflection is dependent upon the quasi mode theory coupling constants between incident and transmitted region quasi modes being nonzero, and it is seen that the contributions to such coupling constants come from the overlap of the mode functions in the boundary layer region, as might be expected from a microscopic theory.

Theory of pseudomodes in quantum optical processes

This paper deals with non-Markovian behavior in atomic systems coupled to a structured reservoir of quantum electromagnetic field modes, with particular relevance to atoms interacting with the field in high-Q cavities or photonic band-gap materials. In cases such as the former, we show that the pseudomode theory for single-quantum reservoir excitations can be obtained by applying the Fano diagonalization method to a system in which the atomic transitions are coupled to a discrete set of (cavity) quasimodes, which in turn are coupled to a continuum set of (external) quasimodes with slowly varying coupling constants and continuum mode density. Each pseudomode can be identified with a discrete quasimode, which gives structure to the actual reservoir of true modes via the expressions for the equivalent atom-true mode coupling constants. The quasimode theory enables cases of multiple excitation of the reservoir to now be treated via Markovian master equations for the atom-discrete quasimode system. Applications of the theory to one, two, and many discrete quasimodes are made. For a simple photonic band-gap model, where the reservoir structure is associated with the true mode density rather than the coupling constants, the single quantum excitation case appears to be equivalent to a case with two discrete quasimodes.

A systematic approach to error isolation in computerized wastewater simulation models

Activated sludge models are used extensively in the study of wastewater treatment processes. While various commercial implementations of these models are available, there are many people who need to code models themselves using the simulation packages available to them, Quality assurance of such models is difficult. While benchmarking problems have been developed and are available, the comparison of simulation data with that of commercial models leads only to the detection, not the isolation of errors. To identify the errors in the code is time-consuming. In this paper, we address the problem by developing a systematic and largely automated approach to the isolation of coding errors. There are three steps: firstly, possible errors are classified according to their place in the model structure and a feature matrix is established for each class of errors. Secondly, an observer is designed to generate residuals, such that each class of errors imposes a subspace, spanned by its feature matrix, on the residuals. Finally. localising the residuals in a subspace isolates coding errors. The algorithm proved capable of rapidly and reliably isolating a variety of single and simultaneous errors in a case study using the ASM 1 activated sludge model. In this paper a newly coded model was verified against a known implementation. The method is also applicable to simultaneous verification of any two independent implementations, hence is useful in commercial model development.

Characterizing mixing and measurement in quantum mechanics

What fundamental constraints characterize the relationship between a mixture rho = Sigma (i)p(i)rho (i) of quantum states, the states rho (i) being mixed, and the probabilities p(i)? What fundamental constraints characterize the relationship between prior and posterior states in a quantum measurement? In this paper we show that then are many surprisingly strong constraints on these mixing and measurement processes that can be expressed simply in terms of the eigenvalues of the quantum states involved. These constraints capture in a succinct fashion what it means to say that a quantum measurement acquires information about the system being measured, and considerably simplify the proofs of many results about entanglement transformation.