A Bohmian approach to quantum fractals

A quantum fractal is a wavefunction with a real and an imaginary part continuous everywhere, but differentiable nowhere. This lack of differentiability has been used as an argument to deny the general validity of Bohmian mechanics (and other trajectory--based approaches) in providing a complete interpretation of quantum mechanics. Here, this assertion is overcome by means of a formal extension of Bohmian mechanics based on a limiting approach. Within this novel formulation, the particle dynamics is always satisfactorily described by a well defined equation of motion. In particular, in the case of guidance under quantum fractals, the corresponding trajectories will also be fractal.Comment: 19 pages, 3 figures (revised version

Reply to Comment by Galapon on 'Almost-periodic time observables for bound quantum systems'

In a recent paper [1] (also at http://lanl.arxiv.org/abs/0803.3721), I made several critical remarks on a 'Hermitian time operator' proposed by Galapon [2] (also at http://lanl.arxiv.org/abs/quant-ph/0111061). Galapon has correctly pointed out that remarks pertaining to 'denseness' of the commutator domain are wrong [3]. However, the other remarks still apply, and it is further noted that a given quantum system can be a member of this domain only at a set of times of total measure zero.Comment: 3 page

Effects of various assumptions on the calculated liquid fraction in isentropic saturated equilibrium expansions

The saturated equilibrium expansion approximation for two phase flow often involves ideal-gas and latent-heat assumptions to simplify the solution procedure. This approach is well documented by Wegener and Mack and works best at low pressures where deviations from ideal-gas behavior are small. A thermodynamic expression for liquid mass fraction that is decoupled from the equations of fluid mechanics is used to compare the effects of the various assumptions on nitrogen-gas saturated equilibrium expansion flow starting at 8.81 atm, 2.99 atm, and 0.45 atm, which are conditions representative of transonic cryogenic wind tunnels. For the highest pressure case, the entire set of ideal-gas and latent-heat assumptions are shown to be in error by 62 percent for the values of heat capacity and latent heat. An approximation of the exact, real-gas expression is also developed using a constant, two phase isentropic expansion coefficient which results in an error of only 2 percent for the high pressure case

Heisenberg-style bounds for arbitrary estimates of shift parameters including prior information

A rigorous lower bound is obtained for the average resolution of any estimate of a shift parameter, such as an optical phase shift or a spatial translation. The bound has the asymptotic form k_I/ where G is the generator of the shift (with an arbitrary discrete or continuous spectrum), and hence establishes a universally applicable bound of the same form as the usual Heisenberg limit. The scaling constant k_I depends on prior information about the shift parameter. For example, in phase sensing regimes, where the phase shift is confined to some small interval of length L, the relative resolution \delta\hat{\Phi}/L has the strict lower bound (2\pi e^3)^{-1/2}/, where m is the number of probes, each with generator G_1, and entangling joint measurements are permitted. Generalisations using other resource measures and including noise are briefly discussed. The results rely on the derivation of general entropic uncertainty relations for continuous observables, which are of interest in their own right.Comment: v2:new bound added for 'ignorance respecting estimates', some clarification

Stochastic Heisenberg limit: Optimal estimation of a fluctuating phase

The ultimate limits to estimating a fluctuating phase imposed on an optical beam can be found using the recently derived continuous quantum Cramer-Rao bound. For Gaussian stationary statistics, and a phase spectrum scaling asymptotically as 1/omega^p with p>1, the minimum mean-square error in any (single-time) phase estimate scales as N^{-2(p-1)/(p+1)}, where N is the photon flux. This gives the usual Heisenberg limit for a constant phase (as the limit p--> infinity) and provides a stochastic Heisenberg limit for fluctuating phases. For p=2 (Brownian motion), this limit can be attained by phase tracking.Comment: 5+4 pages, to appear in Physical Review Letter

Surprises in the suddenly-expanded infinite well

I study the time-evolution of a particle prepared in the ground state of an infinite well after the latter is suddenly expanded. It turns out that the probability density $|\Psi(x, t)|^{2}$ shows up quite a surprising behaviour: for definite times, {\it plateaux} appear for which $|\Psi(x, t)|^{2}$ is constant on finite intervals for $x$. Elements of theoretical explanation are given by analyzing the singular component of the second derivative $\partial_{xx}\Psi(x, t)$. Analytical closed expressions are obtained for some specific times, which easily allow to show that, at these times, the density organizes itself into regular patterns provided the size of the box in large enough; more, above some critical time-dependent size, the density patterns are independent of the expansion parameter. It is seen how the density at these times simply results from a construction game with definite rules acting on the pieces of the initial density.Comment: 24 pages, 14 figure

Cost-effective aperture arrays for SKA Phase 1: single or dual-band?

An important design decision for the first phase of the Square Kilometre Array is whether the low frequency component (SKA1-low) should be implemented as a single or dual-band aperture array; that is, using one or two antenna element designs to observe the 70-450 MHz frequency band. This memo uses an elementary parametric analysis to make a quantitative, first-order cost comparison of representative implementations of a single and dual-band system, chosen for comparable performance characteristics. A direct comparison of the SKA1-low station costs reveals that those costs are similar, although the uncertainties are high. The cost impact on the broader telescope system varies: the deployment and site preparation costs are higher for the dual-band array, but the digital signal processing costs are higher for the single-band array. This parametric analysis also shows that a first stage of analogue tile beamforming, as opposed to only station-level, all-digital beamforming, has the potential to significantly reduce the cost of the SKA1-low stations. However, tile beamforming can limit flexibility and performance, principally in terms of reducing accessible field of view. We examine the cost impacts in the context of scientific performance, for which the spacing and intra-station layout of the antenna elements are important derived parameters. We discuss the implications of the many possible intra-station signal transport and processing architectures and consider areas where future work could improve the accuracy of SKA1-low costing.Comment: 64 pages, 23 figures, submitted to the SKA Memo serie

Quantum properties of classical Fisher information

The Fisher information of a quantum observable is shown to be proportional to both (i) the difference of a quantum and a classical variance, thus providing a measure of nonclassicality; and (ii) the rate of entropy increase under Gaussian diffusion, thus providing a measure of robustness. The joint nonclassicality of position and momentum observables is shown to be complementary to their joint robustness in an exact sense.Comment: 16 page