Laguerre–Gaussian modes and the Wigner transform

Recent developments in laser physics have called renewed attention to Laguerre–Gaussian (LG) beams of paraxial light. In this paper we consider the corresponding LG modes for the two-dimensional harmonic oscillator, which appear in the transversal plane at the laser beam's waist. We see how they arise as Wigner transforms of Hermite–Gaussian modes, and we proceed to find a closed form for their own Wigner transforms, providing an alternative to the methods of Simon and Agarwal. Our main observation is that the Wigner transform intertwines the creation and annihilation operators for the two classes of modes.     

Some Operators for ANNA Calculations

This paper presents a series of operators for ANOVA (analysis of variance) calculations by regression methods. The operators, named CøDE, RøV, MøDEL, SCP, DECøMP, PATTER, SWP, and BAKSøL, are intended to be a subset of building blocks for a computing system to perform analyses of variance. The determination of parameter aliases during computation has been shown to be an important advantage of the regression approach. The regression approach has been made more feasible for balanced experimental designs by making use of structure in the balanced case in order to use mschine store efficiency. A compact model specification method has been developed which is based on three algebraic relations, + (ØR), * (CRØSS), and ← (NEST), between factors. Careful attention has been given to numerical questions. The operators are easily exported between computers with compatible FORTRAN IV facilities.     

Mössbauer study of oxo derivatives of iron in the Fe2O3-Na2O2 system

Various compositions of oxo derivatives of iron reacting with sodium peroxide have been studied by Mössbauer spectroscopy. We have examined several mathematical models of the measured spectra. The results obtained are inconsistent with hypotheses made previously that such conditions may lead to the formation of compounds of iron in oxidation states above (6+). We demonstrate that a large excess of an alkali peroxide leads, most likely, to the formation of at least two iron(V) derivatives in tetrahedral coordination. In their Mössbauer spectra, they have isomer shifts of ?0.45 and ?0.51 mm/s and unusually large quadrupole splittings: 1.32 and 1.94 mm/s (at room temperature).     

Conjunctions in Paris: Interactions between Rømer and Huygens

Christiaan Huygens (1692–1695) and Ole Rømer (1644–1710) closely interacted during the 1670s, when they were both in and around the Académie Royale des Sciences in Paris. They were part of a small group with a shared interest in precision instruments. In the course of their interactions Rømer played a decisive role in two of Huygens renowned achievements: the wave theory of light and the design of a planetarium. Rømer's discovery of the finite speed of light, confirmed Huygens main supposition that light propagates at finite speed. The news of the discovery renewed his interest in the nature of light, resulting in 1677 in the formulation of the principle of wave propagation. Rømer's subsequent criticism of Huygens' theory induced him to seek experimental verification of his claims which he successfully did in 1679. Huygens' planetarium of 1682, in the meantime, was a direct response to the instrument Rømer presented at the Académie in 1680. According to Huygens, his instrument provided a more economic and more truthful mechanical representation of celestial motions. Despite their affinity in matters of ingenuity, Huygens and Rømer never became really close. After their departure from Paris, contact between them virtually ceased. This may be explained by the socio‐cultural differences between both savants.     

Collective response to perturbations in a data-driven fish school model

Fish schools are able to display a rich variety of collective states and behavioural responses when they are confronted by threats. However, a school''s response to perturbations may be different depending on the nature of its collective state. Here we use a previously developed data-driven fish school model to investigate how the school responds to perturbations depending on its different collective states, we measure its susceptibility to such perturbations, and exploit its relation with the intrinsic fluctuations in the school. In particular, we study how a single or a small number of perturbing individuals whose attraction and alignment parameters are different from those of the main population affect the long-term behaviour of a school. We find that the responsiveness of the school to the perturbations is maximum near the transition region between milling and schooling states where the school exhibits multistability and regularly shifts between these two states. It is also in this region that the susceptibility, and hence the fluctuations, of the polarization order parameter is maximal. We also find that a significant school''s response to a perturbation only happens below a certain threshold of the noise to social interactions ratio.     

Quantum phase measurements and non-classical polarization states of light

Abstract

In our paper we consider the non-classical behaviour of both the Hermitian (observable) Stokes parameters of light and the phase difference of two modes that describe the quantum polarization states of optical field. To characterize the degree of polarization of light we introduce a new quantity taking into account the quantum properties of different quantum states of two orthogonally polarized modes. The problem of determination of the phase difference in two modes of optical field for the quantum polarization states of light is discussed. To describe in general such a quantum field we introduce two pairs of the phase operators: the phase angles for the Stokes parameters of light in a three-dimensional picture of the Poincaré sphere. We also consider a special type of the eight-port polarization interferometer (polarimeter) for simultaneous homodyne detection of both the Stokes parameters of light and the polarization phase operators and their fluctuations as well. Using an anisotropic (spatioperiodic) Kerr-like nonlinear medium associated with the polarization interferometer we could generate and also observe the polarization-squeezed phase states of light. The fluctuations in the phase difference between two orthogonally polarized modes for these non-classical states are less than the fluctuations for light in coherent state.     

The first iteration of Grover's algorithm using classical light with orbital angular momentum

We present an experimental realization of the first iteration in Grover's quantum algorithm using classical light and linear optical elements. The algorithm serves to find an entry marked by an oracle in an unstructured database. In our scheme, the quantum states encoding the database are represented by helical modes generated by means of a Spatial Light Modulator, while the marking corresponds to a π phase shift of the hidden mode. The optical implementation of Grover's algorithm then selectively amplifies the intensity of the marked mode such that it can be revealed by a modal decomposition. The core of the algorithm – a geometrical reflection of the point representing all database entries – is implemented in a single step independent of the size of the database. Moreover, we demonstrate experimentally that one iteration of the algorithm is enough to identify the marked entry, as a consequence of using classical states of light.     

Mössbauer Investigation of Superconductors LaFeO 0.85F 0.15As and High-Temperature Kondo Effect

We report the results of investigation of high-temperature superconductor (HTSC) LaFeO _(1?x)F _x As at x = 0.15. We applied the XRD method, the measurement of the hysteresis curve and resistivity, as well as the Mössbauer measurements at the temperature range 4.2–300 K. We show that in the given sample, owing to doping by fluorine, full suppression of antiferromagnetism is not accompanied by the rise of superconductivity. Instead, we observe a high-temperature (at T ≤ 40 K) Kondo effect, which indicates a strong coupling of electron’s spins with the magnetic moments of impurities. We also find a correlation between the revealed high-temperature Kondo effect and the appearance of a quadrupole splitting in the Mössbauer spectra below 40 K. The revealed effects can shed light on the mechanism of superconductivity in the iron-containing HTSC.     

Quantum-optical Phase and Canonical Conjugation

Abstract

We use a new limiting procedure, developed to study quantum-optical phase, to examine canonically conjugate operators in general. We find that Dirac's assumption that photon number and phase should be canonically conjugate variables, similar to momentum and position, is essentially correct. The difficulties with Dirac's approach are shown to arise through use of a form of the canonical commutator which, although the only possible form in the usual infinite Hilbert space approach, is not sufficiently general to be used as a model for a number-phase commutator. The approach in this paper unifies the theory of conjugate operators, which include photon number and phase, angular momentum and angle, and momentum and position as particular cases. The usual position-momentum commutator is regained from a more generally applicable expression by means of a domain restriction which cannot be used for the phase-number commutator.     

Solitary Waves of Maxwell's Equations in Nonlinear Anisotropic Media

Abstract

The (1 + 1)-D solitary wave solutions of Maxwell's equations in nonlinearity induced anisotropic media (in liquids such as carbon disulphide, and in crystals, etc.) are investigated. We find that there is no arbitrarily linearly polarized (in the x-y plane perpendicular to the propagation direction z) soliton solution from Maxwell's equations except that with linear polarization either in alignment with or orthogonal to the geometric axis of the light induced refractive index change. This contradicts the prediction of the vector nonlinear Schroedinger equation (an approximation of Maxwell's equations) which yields soliton solutions with an arbitrary linear polarization. However, Maxwell's equations are found to admit stable elliptically polarized solitary wave solutions which reduce to the stable circularly polarized solitary wave solutions of the vector nonlinear Schroedinger equation when the induced refractive index change approaches zero.     

Negative binomial states of the quantized radiation field

Abstract

Negative binomial states of a single field mode are introduced and their properties are compared with those of the (positive) binomial states. We find that the two types of state have similar properties if the roles of the creation and annihilation operators are interchanged.     

A mathematical structure relating situation knowledge to performance

The linguistic conjecture that a person's knowledge regarding a situation affects some aspects of that person's performance at a task is translated into a mathematical structure. The structure contains a symbolic representation of the conjecture, along with a symbolic definition of what type of knowledge constitutes ‘situation’ knowledge. The structure is applied to a short example to demonstrate its use. The translation of the conjecture into a mathematical structure enables researchers to use a consistent, objective framework for testing hypothesised relationships between one's situation knowledge and performance, as well as enables mathematical tools to be applied to the study of this conjecture.     

‘Senses and Hands to the Same Degree as Thought’—Ole Rømer's Mechanical Astronomy

The astronomer Ole Rømer emphasized the mechanical nature of the practice of astronomy and this paper attempts to unravel what Rømer meant by the close association between mechanics and astronomy. The point of departure is Rømer's work with Tycho Brahe's observations and his stay at the Royal Academy of the Sciences in Paris. Analyses of Rømer's letters and treatises show that he not only focused on direct presentations of observations and instruments, but demanded an independence of his results that went beyond the rhetoric of first‐hand knowledge. Rømer wanted his observations and instruments to demonstrate the physical theories themselves, rather than asking for theories to be accepted on his authority. His invention of the meridian circle with its facility for cross‐checking measurements is seen in connection with this drive to make what is observed ‘deserve credit in its own right rather than because of the observer.’     

Principles of calculating the dynamical response of misaligned complex resonant optical interferometers

In the long-baseline laser interferometers for measuring gravitational waves that are now under construction, understanding the dynamical response to small distortions such as angular alignment fluctuations presents a unique challenge. These interferometers comprise multiple coupled optical resonators with light storage times approaching 100 m. We present a basic formalism to calculate the frequency dependence of periodic variations in angular alignment and longitudinal displacement of the resonator mirrors. The electromagnetic field is decomposed into a superposition of higher-order spatial modes, Fourier frequency components, and polarization states. Alignment fluctuations and length variations of free-space propagation are represented by matrix operators that act on the multicomponent state vectors of the field.     

Effect of quantum interference on a three-level atom driven by two laser fields

Abstract

We study the effect of quantum interference on the population distribution and absorptive properties of a V-type three-level atom driven by two lasers of unequal intensities and different angular frequencies. Three coupling configurations of the lasers to the atom are analysed: (a) both lasers coupled to the same atomic transition, (b) each laser coupled to different atomic transition and (c) each laser coupled to both atomic transitions. Dressed states for the three coupling configurations are identified, and the population distribution and absorptive properties of the weaker field are interpreted in terms of transition dipole moments and transition frequencies among these dressed states. In particular, we find that in the first two cases there is no population inversion between the bare atomic states, but the population can be trapped in a superposition of the dressed states induced by quantum interference and the stronger field. We show, that the trapping of the population, which results from the cancellation of transition dipole moments, does not prevent the weaker field to be coupled to the cancelled (dark) transitions. As a result, the weaker field can be strongly amplified on transparent transitions. In the case of each laser coupled to both atomic transitions the population can be trapped in a linear superposition of the excited bare atomic states leaving the ground state unpopulated in the steady state. Moreover, we find that the absorption rate of the weaker field depends on the detuning of the strong field from the atomic resonances and the splitting between the atomic excited states. When the strong field is resonant to one of the atomic transitions a quasi-trapping effect appears in one of the dressed states. In the quasi-trapping situation all the transition dipole moments are different from zero, which allows the weaker field to be amplified on the inverted transitions. When the strong field is tuned halfway between the atomic excited states, the population is completely trapped in one of the dressed states and no amplification is found for the weaker field.     

Physical properties of a trapped two-level ion decaying by thermal and squeezed vacuum reservoirs

In this paper, we first study the interaction between a trapped two-level ion with a field reservoir in a general form. Then, we use thermal and squeezed vacuum reservoirs to continue our investigation. We try to find the equation of motion for the whole system density operator by applying the Weisskopf theory. Next, we evaluate the explicit form of the density matrix elements analytically. In the obtained density matrix elements, we arrive at some physical parameters of the trapped ion system as well as the applied field reservoirs by which one can control the transition probabilities between the trapped ionic states. We also introduce some hermitian operators for the trapped ion system and calculate the expectation values of their time evolution by considering the mentioned two reservoirs. In the continuation, we design a system for measuring the hermitian trapped ion’s operators theoretically. Finally, we find a way to transport the information of the trapped ion system by our proposed quantum processor.     

Searching for motifs in the behaviour of larval Drosophila melanogaster and Caenorhabditis elegans reveals continuity between behavioural states

We present a novel method for the unsupervised discovery of behavioural motifs in larval Drosophila melanogaster and Caenorhabditis elegans. A motif is defined as a particular sequence of postures that recurs frequently. The animal''s changing posture is represented by an eigenshape time series, and we look for motifs in this time series. To find motifs, the eigenshape time series is segmented, and the segments clustered using spline regression. Unlike previous approaches, our method can classify sequences of unequal duration as the same motif. The behavioural motifs are used as the basis of a probabilistic behavioural annotator, the eigenshape annotator (ESA). Probabilistic annotation avoids rigid threshold values and allows classification uncertainty to be quantified. We apply eigenshape annotation to both larval Drosophila and C. elegans and produce a good match to hand annotation of behavioural states. However, we find many behavioural events cannot be unambiguously classified. By comparing the results with ESA of an artificial agent''s behaviour, we argue that the ambiguity is due to greater continuity between behavioural states than is generally assumed for these organisms.     

Squeezing in Correlated Quantum Systems

Abstract

We discuss the connection between quantum correlations and squeezing in simple quantum optical systems. We illustrate this connection by a study of two-mode states of light produced by parametric down-conversion and similar two-photon processes. The intermode correlations in these systems are shown to be responsible for modifications in photon-number sum and difference operators, and for squeezing in the superpositions of the two modes. The disappearance of the diagonal coherent-state quasiprobability function P(α) when non-classical light properties are important is noted, and alternative and better-behaved Wigner functions and coherent-state expectation Q-functions for the two-mode system are developed.     

Complex dynamics of semantic memory access in reading

Understanding a word in context relies on a cascade of perceptual and conceptual processes, starting with modality-specific input decoding, and leading to the unification of the word''s meaning into a discourse model. One critical cognitive event, turning a sensory stimulus into a meaningful linguistic sign, is the access of a semantic representation from memory. Little is known about the changes that activating a word''s meaning brings about in cortical dynamics. We recorded the electroencephalogram (EEG) while participants read sentences that could contain a contextually unexpected word, such as ‘cold’ in ‘In July it is very cold outside’. We reconstructed trajectories in phase space from single-trial EEG time series, and we applied three nonlinear measures of predictability and complexity to each side of the semantic access boundary, estimated as the onset time of the N400 effect evoked by critical words. Relative to controls, unexpected words were associated with larger prediction errors preceding the onset of the N400. Accessing the meaning of such words produced a phase transition to lower entropy states, in which cortical processing becomes more predictable and more regular. Our study sheds new light on the dynamics of information flow through interfaces between sensory and memory systems during language processing.