Table of contents

If an atomic system is being observed, and its observer is also observed, would the first observer be in a superposition of states and evolve deterministically before being observed? In this paper, a simple model of non-demolition measurement is analyzed in order to elucidate the so-called 'Wigner's friend' paradox. The model illustrates the decoherence of an atomic system and its observer (the 'friend') as the latter is being observed (by Wigner).

This review is presented on modern research to achieve in a laboratory experiment the new level of shock-wave pressure of a few hundred or even thousands of Mbar when a substance is exposed to a stream of laser-accelerated fast electrons. The applications associated with the use of ultra-power shock waves as the ignition driver of inertial fusion targets as well as the tool in studying the equation of a state of a matter are discussed.

Physical and biological observation methods provide a variety of bilayer membranes' shapes and their transformations. Besides, the topological and geometrical methods allow us to deduce a classification of all possible membrane surfaces. This double-sided approach leads to a deeper insight into membranes properties. Our goal is to apply an appropriate mathematical technique for classifying vesicles (closed surfaces in mathematical terminology) and for their transformation ways. The problem turned out to be an intricate one, and to our knowledge no mathematical techniques have been applied to its solution. We find that all vesicles can be decomposed in a small number of universality classes generated by a few 'bricks': a torus, a screwed torus, and the real projective plane. We consider several ways of transforming membrane surfaces, bearing in mind that they possess an additional extremal property. Our method exploits different constructions of minimal surfaces in S³. We estimate energetic barrier for transformation of minimal membrane surfaces using the so-called doubling procedure. This problem is far from being a pure theoretical exercise. For instance, almost all cells' biological functions, or tumor progression, are accompanied by apparently singular cell membrane transformations.

We use an optical analog of a quantum optomechanical system to show that non-trivial effective couplings, such as column isolation and diagonal coupling can be engineered in a two-dimensional array of nearest-neighbour-coupled waveguides by an adequate selection of refractive indices and nearest-neighbour couplings.

We analyze in detail the creation of fermions and bosons from a vacuum by an electric field that exponentialy decreases in time. In our calculations, we use quantum electrodynamics (QED) and mainly consider the particle creation effect in a homogeneous electric field. To this end, we find complete sets of exact solutions of the d-dimensional Dirac equation in the exponentially decreasing electric field, and we use them to calculate all the characteristics of the effect, and specifically the total number of created particles and the probability that a vacuum will remain a vacuum. Note that the latter quantities were derived in the case under consideration for the first time. All possible asymptotic regimes are discussed in detail. In addition, switching on and switching off effects are studied.

I present a derivation of form factors in the algebraic cluster model for an arbitrary number of identical clusters. The form factors correspond to representation matrix elements which are derived in closed form for the harmonic oscillator and deformed oscillator limits. These results are relevant for applications in nuclear, molecular and hadronic physics.

In this article we present a review of the Radon transform and the instability of the tomographic reconstruction process. We show some new mathematical results in tomography obtained by a variational formulation of the reconstruction problem based on the minimization of a Mumford–Shah type functional. Finally, we exhibit a physical interpretation of this new technique and discuss some possible generalizations.

A new integral representation is obtained for the star product corresponding to the s-ordering of the creation and annihilation operators. This parametric ordering convention introduced by Cahill and Glauber enables one to vary the type of ordering in a continuous way from normal order to antinormal order. Our derivation of the corresponding integral representation is based on using reproducing formulas for analytic and antianalytic functions. We also discuss a different representation whose kernel is a generalized function and compare the properties of this kernel with those of the kernels of another family of star products which are intermediate between the ${qp}$- and pq-quantization.

The low and higher energy limits of the electroweak model are obtained from the first principles of gauge theory. Both limits are given by the same contraction of the gauge group, but for the different consistent rescalings of the field space. Mathematical contraction parameter in both cases is interpreted as energy. Very weak neutrino–matter interactions are explained by zero tending contraction parameter, which depends on neutrino energy. The second consistent rescaling corresponds to the higher energy limit of the electroweak model. At the infinite energy all particles lose mass, electroweak interactions become long-range and are mediated by neutral currents. The limit model represents the development of the early Universe from the big bang up to the end of the first second.

We present several ideas in the direction of physical interpretation of q- and f-oscillators as nonlinear oscillators. First we show that an arbitrary one-dimensional integrable system in action-angle variables can be naturally represented as a classical and quantum f-oscillator. As an example, the semi-relativistic oscillator as a descriptive of the Landau levels for relativistic electron in magnetic field is solved as an f-oscillator. By using dispersion relation for q-oscillator we solve the linear q-Schrödinger equation and corresponding nonlinear complex q-Burgers equation. The same dispersion allows us to construct integrable q-NLS model as a deformation of cubic NLS in terms of recursion operator of NLS hierarchy. A peculiar property of the model is to be completely integrable at any order of expansion in deformation parameter around q = 1. As another variation on the theme, we consider hydrodynamic flow in bounded domain. For the flow bounded by two concentric circles we formulate the two circle theorem and construct the solution as the q-periodic flow by non-symmetric q-calculus. Then we generalize this theorem to the flow in the wedge domain bounded by two arcs. This two circular-wedge theorem determines images of the flow by extension of q-calculus to two bases: the real one, corresponding to circular arcs and the complex one, with q as a primitive root of unity. As an application, the vortex motion in annular domain as a nonlinear oscillator in the form of classical and quantum f-oscillator is studied. Extending idea of q-oscillator to two bases with the golden ratio, we describe Fibonacci numbers as a special type of q-numbers with matrix Binet formula. We derive the corresponding golden quantum oscillator, nonlinear coherent states and Fock–Bargman representation. Its spectrum satisfies the triple relations, while the energy levels' relative difference approaches asymptotically to the golden ratio and has no classical limit.

The effects of general noncommutativity of operators on producing deformed coherent squeezed states is examined in phase space. A two-dimensional noncommutative (NC) quantum system supported by a deformed mathematical structure, similar to that of Hadamard billiard, is obtained and the components behaviour is monitored in time. It is assumed that the independent degrees of freedom are two free 1D harmonic oscillators (HOs), so the system Hamiltonian does not contain interaction terms. Through the NC deformation parameterized by a Seiberg–Witten transform on the original canonical variables, one gets the standard commutation relations for the new ones, such that the obtained, new, Hamiltonian represents two interacting 1D HOs. By admitting that one HO is inverted relatively to the other, we show that their effective interaction induces a squeezing dynamics for initial coherent states imaged in the phase space. A suitable pattern of logarithmic spirals is obtained and some relevant properties are discussed in terms of Wigner functions, which are essential to put in evidence the effects of the noncommutativity.

We consider a simple nonlinear (quartic in the fields) gauge-invariant modification of classical electrodynamics, to show that it possesses a regularizing ability sufficient to make the field energy of a point charge finite. The model is exactly solved in the class of static central-symmetric electric fields. Collation with quantum electrodynamics (QED) results in the total field energy of a point elementary charge about twice the electron mass. The proof of the finiteness of the field energy is extended to include any polynomial selfinteraction, thereby the one that stems from the truncated expansion of the Euler–Heisenberg local Lagrangian in QED in powers of the field strength.

Recent work in the literature has studied rigid Casimir cavities in a weak gravitational field, or in de Sitter spacetime, or yet other spacetime models. The present review paper studies the difficult problem of direct evaluation of scalar Green functions for a Casimir-type apparatus in de Sitter spacetime. Working to first order in the small parameter of the problem, i.e. twice the gravity acceleration times the plates' separation divided by the speed of light in vacuum, suitable coordinates are considered for which the differential equations obeyed by the zeroth- and first-order Green functions can be solved in terms of special functions. This result can be used, in turn, to obtain, via the point-split method, the regularized and renormalized energy–momentum tensor both in the scalar case and in the physically more relevant electromagnetic case.

We present an extension of the spin-adapted configuration-interaction method (SACI) for the computation of four electrons in a quasi two-dimensional quantum dot. By a group-theoretical decomposition of the basis set and working with relative and center-of-mass (cm) coordinates we obtain an analytical identification of all spurious cm states of the Coulomb-interacting electrons. We find a substantial reduction in the basis set used for numerical computations. At the same time we increase the accuracy compared to the standard SACI due to the absence of distortions caused by an unbalanced cut-off of cm excitations.

We propose a simple scheme to generate deterministic entanglement between two movable end mirrors in a Fabry–Perot cavity using a single photon superposition state. We derive analytically the expressions of the generated entangled states and the degree of entanglement for each state. We show that strong entanglement can be obtained either in the single-photon strong coupling regime deterministically or in the single-photon weak coupling regime conditionally.

We consider a spin ring with an arbitrary number of spins on the ring and one spin in its center in a strong external magnetic field. The spins on the ring are connected by the secular dipole–dipole interactions and interact with the central spin through the Heisenberg zz-interaction. We show that the quantum discord, describing quantum correlations between the ring and the central spin, can be obtained analytically for an arbitrary number of the spins in the high-temperature approximation. We demonstrate the evolution of quantum correlations at different numbers of the spins. The contributions of longitudinal and transversal spin interactions to the quantum discord are discussed.

Inter-relation between quantum and classical probability models is one of the most fundamental problems of quantum foundations. Nowadays this problem also plays an important role in quantum technologies, in quantum cryptography and the theory of quantum random generators. In this letter, we compare the viewpoint of Richard Feynman that the behavior of quantum particles cannot be described by classical probability theory with the viewpoint that quantum–classical inter-relation is more complicated (cf, in particular, with the tomographic model of quantum mechanics developed in detail by Vladimir Man'ko). As a basic example, we consider the two-slit experiment, which played a crucial role in quantum foundational debates at the beginning of quantum mechanics (QM). In particular, its analysis led Niels Bohr to the formulation of the principle of complementarity. First, we demonstrate that in complete accordance with Feynman's viewpoint, the probabilities for the two-slit experiment have the non-Kolmogorovian structure, since they violate one of basic laws of classical probability theory, the law of total probability (the heart of the Bayesian analysis). However, then we show that these probabilities can be embedded in a natural way into the classical (Kolmogorov, 1933) probability model. To do this, one has to take into account the randomness of selection of different experimental contexts, the joint consideration of which led Feynman to a conclusion about the non-classicality of quantum probability. We compare this embedding of non-Kolmogorovian quantum probabilities into the Kolmogorov model with well-known embeddings of non-Euclidean geometries into Euclidean space (e.g., the Poincaré disk model for the Lobachvesky plane).

Within the f-deformed oscillator formalism, we derive a Markovian master equation for the description of the damped dynamics of nonlinear systems that interact with their environment. The applicability of this treatment to the particular case of a Morse-like oscillator interacting with a thermal field is illustrated, and the decay of quantum coherence in such a system is analyzed in terms of the evolution on phase space of its nonlinear coherent states via the Wigner function.

We present a tomographic approach to the study of quantum nonlocality in multipartite systems. Bell inequalities for tomograms belonging to a generic tomographic scheme are derived by exploiting tools from convex geometry. Then, possible violations of these inequalities are discussed in specific tomographic realizations providing some explicit examples.

Dynamics of entangled states of two independent single-mode cavities in a correlated (squeezed) reservoir are investigated in the context of matching the correlations contained in the entangled states to those contained in the reservoir. We illustrate our considerations by examining the time evolution of entanglement of initial single and double excitation NOON and EPR states, and a comparison is made of when each cavity is coupled to its own reservoir or both cavities are coupled to a common reservoir. It is shown that the evolution of the initial entanglement and transfer of entanglement from the squeezed reservoir to the cavity modes depend crucially on the matching of the initial correlations to those contained in the squeezed reservoir. In particular, it is found that initially entangled modes with correlations different from the reservoir correlations prevent the transfer of the correlations from the squeezed field to the modes. In addition, we find that the transient entanglement exhibits several features unique to the quantum nature of squeezing. In particular, we show that in the case of separate squeezed reservoirs the initial entanglement disappears at a finite time, which for the so-called classically squeezed field remains almost the same as in the case of a thermal field. In the case of a common reservoir a recurrence of entanglement occurs and we find that this feature also results from the reservoir correlations unique to quantum correlations. There is no revival of the entanglement when the modes interact with a classically correlated field.

We consider the optimal approximation of certain quantum states of a harmonic oscillator with the superposition of a finite number of coherent states in phase space placed either on an ellipse or on a certain lattice. These scenarios are currently experimentally feasible. The parameters of the ellipse and the lattice and the coefficients of the constituent coherent states are optimized numerically, via a genetic algorithm, in order to obtain the best approximation. It is found that for certain quantum states the obtained approximation is better than the ones known from the literature thus far.

Non-commutative tomography is a technique originally developed and extensively used by Professors M A Man'ko and V I Man'ko in quantum mechanics. Because signal processing deals with operators that, in general, do not commute with time, the same technique has a natural extension to this domain. Here, a review is presented of the theory and some applications of non-commutative tomography for time series as well as some new results on signal processing on graphs.

We propose a new method for the derivation of Husimi symbols, for operators that are given in the form of products of an arbitrary number of coordinates, and momentum operators, in an arbitrary order. For such an operator, in the standard approach, one expresses coordinate and momentum operators as a linear combination of the creation and annihilation operators, and then uses the antinormal ordering to obtain the final form of the symbol. In our method, one obtains the Husimi symbol in a much more straightforward fashion, departing directly from operator explicit form without transforming it through creation and annihilation operators. With this method the mean values of some operators are found. It is shown how the Heisenberg and the Schrödinger–Robertson uncertainty relations, for position and momentum, are transformed under scale transformation $(q;p)\to (\lambda q;\lambda p)$. The physical sense of some states which can be constructed with this transformation is also discussed.

The characterization of physical systems requires a comprehensive understanding of quantum effects. One aspect is a proper quantification of the strength of such quantum phenomena. Here, a general convex ordering of quantum states will be introduced which is based on the algebraic definition of classical states. This definition resolves the ambiguity of the quantumness quantification using topological distance measures. Classical operations on quantum states will be considered to further generalize the ordering prescription. Our technique can be used for a natural and unambiguous quantification of general quantum properties whose classical reference has a convex structure. We apply this method to typical scenarios in quantum optics and quantum information theory to study measures which are based on the fundamental quantum superposition principle.

On the base of symplectic quantum tomogram we define a probability distribution on the plane ${\mathbb{R}}^{2}$. The dual map transfers all observables which are polynomials of the position and momentum operators to the set of polynomials of two variables. In this representation the average values of observables can be calculated by means of integration over all the plane.

We study the structure of the phase diagram for systems consisting of two- and three-level particles dipolarly interacting with a one-mode electromagnetic field, inside a cavity, paying particular attention to the case of a finite number of particles, and showing that the divergences that appear in other treatments are a consequence of the mathematical approximations employed and can be avoided by studying the system in an exact manner quantum-mechanically or via a catastrophe formalism with variational trial states that satisfy the symmetries of the appropriate Hamiltonians. These variational states give an excellent approximation not only to the exact quantum phase space, but also to the energy spectrum and the expectation values of the atomic and field operators. Furthermore, they allow for analytic expressions in many of the cases studied. We find the loci of the transitions in phase space from one phase to the other, and the order of the quantum phase transitions are determined explicitly for each of the configurations, with and without detuning. We also derive the critical exponents for the various systems, and the phase structure at the triple point present in the Ξ-configuration of three-level systems is studied.

We address quantum phase channels, i.e communication schemes where information is encoded in the phase-shift imposed to a given signal, and analyze their performances in the presence of phase diffusion. We evaluate mutual information for coherent and phase-coherent signals, and for both ideal and realistic phase receivers. We show that coherent signals offer better performances than phase-coherent ones, and that realistic phase channels are effective ones in the relevant regime of low energy and large alphabets.

The persistence of coherence of a system of superconducting flux-qubits, in the presence of an ensemble of electrons, is analysed. The time evolution of the system is solved exactly. We study the time evolution of the Wigner function of the reduced density matrix of the qubits, in order to determine the rate of decoherence induced by the coupling with the electrons. We found that the rate of induced decoherence may be reduced, depending on the interactions with the electrons. Also, we discuss the squeezing properties of the system.

In continuous-variable quantum information, non-Gaussian entangled states that are obtained from Gaussian entangled states via photon subtraction are known to contain more entanglement. This makes them better resources for quantum information processing protocols, such as, quantum teleportation. We discuss the teleportation of non-Gaussian, non-classical Schrödinger-cat states of light using two-mode squeezed vacuum light that is made non-Gaussian via subtraction of a photon from each of the two modes. We consider the experimentally realizable cat states produced by subtracting a photon from the single-mode squeezed vacuum state. We discuss two figures of merit for the teleportation process, (a) the fidelity, and (b) the maximum negativity of the Wigner function at the output. We elucidate how the non-Gaussian entangled resource lowers the requirements on the amount of squeezing necessary to achieve any given fidelity of teleportation, or to achieve negative values of the Wigner function at the output.

We discuss the polarization of paraxial and nonparaxial classical light fields by resorting to a multipole expansion of the corresponding polarization matrix. It turns out that only a dipolar term contributes when one considers SU(2) (paraxial) or SU(3) (nonparaxial) as fundamental symmetries. In this latter case, one can alternatively expand in SU(2) multipoles, and then both a dipolar and a quadrupolar component contribute, which explains the richer structure of this nonparaxial instance. These multipoles uniquely determine Wigner functions, in terms of which we examine some intriguing hallmarks arising in this classical scenario.

We show a sample of some relevant developments in classical and quantum tomography that have taken place over the last twenty years. We will present a general conceptual framework that provides a simple unifying mathematical picture for them and, as an effective use of it, three subjects have been chosen that offer a wide panorama of the scope of classical and quantum tomography: tomography along lines and submanifolds, coherent state tomography and tomography in the abstract algebraic setting of quantum systems.

I briefly review the tomographic approach to quantum cosmology, and the results and perspectives following from it.

A method is suggested for quickly and easily estimating multiple ionization (MI) cross sections of heavy atoms colliding with highly charged ions, using the independent-particle model (IPM). One-electron ionization probabilities p(b) are calculated using the geometrical model for $p(0)$ values at zero impact parameter b and the relativistic Born approximation for one-electron ionization cross sections. Numerical results of MI cross sections are presented for Ne and Ar atoms colliding with Ar⁸⁺, Fe²⁰⁺, Au²⁴⁺, Bi⁶⁷⁺ and U⁹⁰⁺ ions at energies 1 MeV u${}^{-1}$–10 GeV u${}^{-1}$and compared with available experimental data and CTMC (classical trajectory Monte Carlo) calculations. The present method of calculation describes experimental dependencies of MI cross sections on the number of ejected electrons m within a factor of two to three. Numerical calculations show that at intermediate ion energies E = 1 − 10 MeV u${}^{-1}$, the contribution of MI cross sections to the total, i.e. summed over all m values, is quite large ∼35% and decreases with increasing energy.

Randomness is fundamental in quantum theory, with many philosophical and practical implications. In this paper we discuss the concept of algorithmic randomness, which provides a quantitative method to assess the Borel normality of a given sequence of numbers, a necessary condition for it to be considered random. We use Borel normality as a tool to investigate the randomness of ten sequences of bits generated from the differences between detection times of photon pairs generated by spontaneous parametric downconversion. These sequences are shown to fulfil the randomness criteria without difficulties. As deviations from Borel normality for photon-generated random number sequences have been reported in previous work, a strategy to understand these diverging findings is outlined.

The conventional theory of superconducting alloys does not take into account the discrete character of impurities. Experimental data for superfluid ³He in aerogel and for some high-T_c superconductors reveal a significant discrepancy between the observed temperatures, T_c, of their transitions in the superfluid or superconducting states and those predicted theoretically. Here a theoretical scheme is presented for finding corrections to the T_c originating from spatial correlations between impurities. Analysis is limited to the Ginzburg and Landau temperature region. The shift of T_c with respect to the pure material is represented as a series in concentration of the impurities, x. In the first order on x, the conventional mean-field result for lowering T_c is recovered. The contribution of correlations enters the second-order term. It is expressed via the structure factor of the ensemble of impurities. For superfluid ³He in a silica aerogel, the sign of the correction corresponds to an enhancement of the T_c, so that the resulting pair-breaking effect of impurities is weakened. When the correlation radius of the impurities, R, exceeds the coherence length of the superfluid, ${\xi }_{0},$ the contribution of correlations to the shift of T_c acquires a factor, $\sim {(R/{\xi }_{0})}^{2},$ and the weakening of the pair-breaking effect becomes appreciable. The presented scheme is applied to the superfluid ³He in aerogel.

The introduction of operator states and of observables in various fields of quantum physics has raised questions about the mathematical structures of the corresponding spaces. In the framework of third quantization it had been conjectured that we deal with Hilbert spaces although the mathematical background was not entirely clear, particularly, when dealing with bosonic operators. This in turn caused some doubts about the correct way to combine bosonic and fermionic operators or, in other words, regular and Grassmann variables. In this paper we present a formal answer to the problems on a simple and very general basis. We illustrate the resulting construction by revisiting the Bargmann transform and finding the known connection between ${{\mathcal{L}}}^{2}({\mathbb{R}})$ and the Bargmann–Hilbert space. We pursue this line of thinking one step further and discuss the representations of complex extensions of linear canonical transformations as isometries between dual Hilbert spaces. We then use the formalism to give an explicit formulation for Fock spaces involving both fermions and bosons thus solving the problem at the origin of our considerations.

The dipole cutoff behavior for the proton form factor has been and still is one of the major issues in high-energy physics. It is shown that this dipole behavior comes from the coherence between the Lorentz contraction of the proton size and the decreasing wavelength of the incoming photon signal. The contraction rates are the same for both cases. This form of coherence is studied also in the momentum–energy space. The coherence effect in this space can be explained in terms of two overlapping wave functions.

We investigate the effect of slow spring-constant drifts of the trap used to shuttle two ions of different mass. We design transport protocols to suppress or mitigate the final excitation energy by applying invariant-based inverse engineering, perturbation theory, and a harmonic dynamical normal-mode approximation. A simple, explicit trigonometric protocol for the trap trajectory is found to be robust with respect to the spring-constant drifts.

We explore two ways of quantizing constraints through integral quantizations based on the Weyl–Heisenberg group. The first one is based on the quantization of distributions on phase space expressing geometric constraints, like Dirac or Heaviside distributions. The second one is implemented in the spirit of Dirac. In both cases we complete the study with a semiclassical analysis through the use of lower symbols. For the sake of simplicity, we restrict our study to the complex plane viewed as the phase space of the motion on the line, or as the phase space for classical electromagnetic radiation (quadratures). Despite the fact that the two procedures do not in general yield identical results, we show interesting features in exploring aspects of their quantum and semiclassical versions.

An interaction free evolving state of a closed bipartite system composed of two interacting subsystems is a generally mixed state evolving as if the interaction were a c-number. In this paper we find the characteristic equation of states possessing similar properties for a bipartite systems governed by a linear dynamical equation whose generator is sum of a free term and an interaction term. In particular in the case of a small system coupled to its environment, we deduce the characteristic equation of decoherence free states namely mixed states evolving as if the interaction term were effectively inactive. Several examples illustrate the applicability of our theory in different physical contexts.

In this paper we examine the decay of quantum correlations for the radiation field in a two-mode squeezed thermal state in contact with local thermal reservoirs. Two measures of the evolving quantum correlations between the modes are compared: the entanglement of formation and the quantum discord. We derive analytic expressions of the entanglement-death time in two special cases: when the reservoirs for each mode are identical, as well as when a single reservoir acts on the first mode only. In the latter configuration, we show that all the pure Gaussian states lose their entanglement at the same time, determined solely by the field-reservoir coupling. Also investigated is the evolution of the Gaussian quantum discord for the same choices of thermal baths. We notice that the discord can increase in time above its initial value in a special situation, namely, when the input state is mixed, and local measurements are performed on the attenuated mode. This enhancement of discord is stronger for zero-temperature reservoirs and increases with the input degree of mixing.

A function of positive type can be defined as a positive functional on a convolution algebra of a locally compact group. In the case where the group is abelian, by Bochner's theorem a function of positive type is, up to normalization, the Fourier transform of a probability measure. Therefore, considering the group of translations on phase space, a suitably normalized phase-space function of positive type can be regarded as a realization of a classical state. Thus, it may be called a function of classical positive type. Replacing the ordinary convolution on phase space with the twisted convolution, one obtains a noncommutative algebra of functions whose positive functionals we may call functions of quantum positive type. In fact, by a quantum version of Bochner's theorem, a continuous function of quantum positive type is, up to normalization, the (symplectic) Fourier transform of a Wigner quasi-probability distribution; hence, it can be regarded as a phase-space realization of a quantum state. Playing with functions of positive type—classical and quantum—one is led in a natural way to consider a class of semigroups of operators, the classical-quantum semigroups. The physical meaning of these mathematical objects is unveiled via quantization, so obtaining a class of quantum dynamical semigroups that, borrowing terminology from quantum information science, may be called classical-noise semigroups.

A Hubbard–Luttinger model is developed for qualitative description of one-dimensional motion of interacting Pi-conductivity-electrons in carbon single-wall nanotubes at low temperatures. The low-lying excitations in one-dimensional electron gas are described in terms of interacting bosons. The Bogolyubov transformation allows one to describe the system as an ensemble of non-interacting quasi-bosons. Operators of Fermi excitations and Green functions of fermions are introduced. The electric current is derived as a function of potential difference on the contact between a nanotube and a normal metal. Deviations from Ohm law produced by electron–electron short-range repulsion as well as by the transverse quantization in single-wall nanotubes are discussed. The results are compared with experimental data.

We study semiclassical dynamics of the resonant Dicke model under the rotation wave approximation (RWA) for initial vacuum field state and all excited atoms in the asymptotic limit of a large number of atoms. We develop a new approach for description of the evolution of such unstable states by combining semiclassical and quantum approaches.

We discuss some of the main features of a recently-generated form of hybrid entanglement between discrete- and continuous-variable states of light. Ideally, such a kind of entanglement should involve single-photon and coherent states as key representatives of the respective categories of states. Here we investigate the characteristics and limits of a scheme that, relying on a superposition of photon-creation operators onto two distinct modes, realizes the above ideal form of hybrid entanglement in an approximate way.

We revise the Lewis–Riesenfeld invariant method for solving the quantum time-dependent harmonic oscillator in light of the quantum Arnold transformation previously introduced and its recent generalization to the quantum Arnold–Ermakov–Pinney transformation. We prove that both methods are equivalent and show the advantages of the quantum Arnold–Ermakov–Pinney transformation over the Lewis–Riesenfeld invariant method. We show that, in the quantum time-dependent and damped harmonic oscillator, the invariant proposed by Dodonov and Man'ko is more suitable, and provide some examples to illustrate it, focusing on the damped case.

Spin torque majority gate (STMG) is one of the promising options for beyond-complementary metal–oxide–semiconductor non-volatile logic circuits for normally-off computing. Modeling of prior schemes demonstrated logic completeness using majority operation and nonlinear transfer characteristics. However significant problems arose with cascade-ability and input output isolation manifesting as domain walls (DWs) stopping, reflecting off ends of wires or propagating back to the inputs. We introduce a new scheme to enable cascade-ability and isolation based on (a) in-plane DW automotion in interconnects, (b) exchange coupling of magnetization between two FM layers, and (c) 'round-about' topology for the majority gate. We performed micro-magnetic simulations that demonstrate switching operation of this STMG scheme. These circuits were verified to enable isolation of inputs from output signals and to be cascade-able without limitations.

Schmidt modes of non-collinear biphoton angular wave functions are found analytically. The experimentally realizable procedure for their separation is described. Parameters of the Schmidt decomposition are used to evaluate the degree of the biphoton's angular entanglement.

We give a short review of known exact inequalities that can be interpreted as 'energy–time' and 'frequency–time' uncertainty relations. In particular we discuss a precise form of signals minimizing the physical frequency–time uncertainty product. Also, we calculate the 'stationarity time' for mixed Gaussian states of a quantum harmonic oscillator, showing explicitly that pure quantum states are 'more fragile' than mixed ones with the same value of the energy dispersion. The problems of quantum evolution speed limits, time operators and measurements of energy and time are briefly discussed, too.

It is shown that a nonlinear reformulation of time-dependent and time-independent quantum mechanics in terms of Riccati equations not only provides additional information about the physical system, but also allows for formal comparison with other nonlinear theories. This is demonstrated for the nonlinear Burgers and Korteweg–de Vries equations with soliton solutions. As Riccati equations can be linearized to corresponding Schrödinger equations, this also applies to the Riccati equations that can be obtained by integrating the nonlinear soliton equations, resulting in a time-independent Schrödinger equation with Rosen–Morse potential and its supersymmetric partner. Because both soliton equations lead to the same Riccati equation, relations between the Burgers and Korteweg–de Vries equations can be established. Finally, a connection with the inverse scattering method is mentioned.

The results of measurements of the DD-reaction yields from the Pd/PdO:D_x and the Ti/TiO₂:D_x heterostructures in the energy range of 10–25 keV are presented. The neutron and proton fluxes are measured using a neutron detector based on ³He-counters and a CR-39 plastic track detector. Comparisons with calculations show the significant effect of DD-reaction yield amplification. It was first shown that the impact of H⁺(protons) and Ne⁺ ion beams in the energy range of 10–25 keV at currents of 0.01–0.1 mA on the deuterated heterostructure results in appreciable DD-reaction yield enhancement.

The single-qudit state tomograms are shown to have the no-signaling property. The known and new entropic and information inequalities for Shannon, von Neumann, and q-entropies of the composite and noncomposite systems characterizing correlations in these systems are discussed. The spin tomographic probability distributions determining the single qudit states are demonstrated to satisfy the strong subadditivity condition for Tsallis q-entropy. Examples of the new entropic inequalities for q-entropy are considered for qudits with $j=5/2$, $j=7/2$.

A new solution is proposed to the longstanding problem of describing the quantum phase of a harmonic oscillator. In terms of an 'exponential phase operator', defined by a new 'polar decomposition' of the quantized amplitude of the oscillator, a regular phase operator is constructed in the Hilbert–Fock space as a strongly convergent power series. It is shown that the eigenstates of the new 'exponential phase operator' are SU(1,1) coherent states associated to the Holstein–Primakoff realization. In terms of these eigenstates the diagonal representation of phase densities and a generalized spectral resolution of the regular phase operator are derived, which are very well suited to our intuitive pictures of classical phase-related quantities.

In oceanography and meteorology, it is important to know not only where water or air masses are headed for, but also where they came from as well. For example, it is important to find unknown sources of oil spills in the ocean and of dangerous substance plumes in the atmosphere. It is impossible with the help of conventional ocean and atmospheric numerical circulation models to extrapolate backward from the observed plumes to find the source because those models cannot be reversed in time. We review here recently elaborated backward-in-time numerical methods to identify and study mesoscale eddies in the ocean and to compute where those waters came from to a given area. The area under study is populated with a large number of artificial tracers that are advected backward in time in a given velocity field that is supposed to be known analytically or numerically, or from satellite and radar measurements. After integrating advection equations, one gets positions of each tracer on a fixed day in the past and can identify from known destinations a particle positions at earlier times. The results provided show that the method is efficient, for example, in estimating probabilities to find increased concentrations of radionuclides and other pollutants in oceanic mesoscale eddies. The backward-in-time methods are illustrated in this paper with a few examples. Backward-in-time Lagrangian maps are applied to identify eddies in satellite-derived and numerically generated velocity fields and to document the pathways by which they exchange water with their surroundings. Backward-in-time trapping maps are used to identify mesoscale eddies in the altimetric velocity field with a risk to be contaminated by Fukushima-derived radionuclides. The results of simulations are compared with in situ mesurement of caesium concentration in sea water samples collected in a recent research vessel cruise in the area to the east of Japan. Backward-in-time latitudinal maps and the corresponding material-line techniques are applied to document transport of water masses across strong currents. Backward-in-time drift maps are shown to be useful in identifying the Lagrangian fronts favorable for fishery.

The radiation–pressure interaction between electromagnetic fields and mechanical resonators can be used to efficiently entangle two light fields coupled to the same mechanical mode. We analyze the performance of this process under realistic conditions, and we determine the effectiveness of the resulting entanglement as a resource for quantum teleportation of continuous-variable light signals over large distances, mediated by concatenated swap operations. We study the sensitiveness of the protocol to the quality factor of the mechanical systems, and its performance in non-ideal situations in which losses and reduced detection efficiencies are taken into account.

The SU(2) group is used in two different fields of quantum optics, the quantum polarization and quantum interferometry. Quantum degrees of polarization may be based on distances of a polarization state from the set of unpolarized states. The maximum polarization is achieved in the case where the state is pure and then the distribution of the photon-number sums is optimized. In quantum interferometry, the SU(2) intelligent states have also the property that the Fisher measure of information is equal to the inverse minimum detectable phase shift on the usual simplifying condition. Previously, the optimization of the Fisher information under a constraint was studied. Now, in the framework of constraint optimization, states similar to the SU(2) intelligent states are treated.

If the state of a quantum system is sampled out of a suitable ensemble, the measurement of some observables will yield (almost) always the same result. This leads us to the notion of quantum typicality: for some quantities the initial conditions are immaterial. We discuss this problem in the framework of Bose–Einstein condensates.

Properties of weak spatio-spectral twin beams in paraxial approximation are analyzed using the decomposition into appropriate paired modes. Numbers of paired modes as well as numbers of modes in the signal (or idler) field in the transverse wave-vector and spectral domains are analyzed as functions of pump-beam parameters. Spatial and spectral coherence of weak twin beams is described by auto- and cross-correlation functions. Relation between the numbers of modes and coherence is discussed.

A brief review of the principal ideas in respect of high frequency gravitational radiation generated and detected in laboratory conditions is presented. Interaction of electromagnetic and gravitational waves in a strong magnetic field is considered as a more promising variant of the laboratory GW-Hertz experiment. The formulae of the direct and inverse Gertsenshtein–Zeldovich effect are derived. Numerical estimates are given and a discussion of the possibility of observation of these effects in a laboratory is carried out.

In this work we present the simplest generic form of the propagator for the time-dependent quadratic Hamiltonian. We manifest the simplicity of our method by giving explicitly the propagators for a free particle in time-dependent electric field and the Paul trap. Exact transition amplitudes and uncertainties are calculated analytically for the Paul trap and harmonic oscillator (HO). The results show that near the instability regions very large quantum mechanical uncertainties are obtained as demonstrated in a special figure. The method is also applied to calculating the trajectory of a classical forced time-dependent HO.

A class of circulant stochastic matrices and positive maps in matrix algebras displaying circulant structure is analyzed. It is shown how the spectral properties of matrices and linear maps are related to positivity conditions. An interesting interplay between the classical and quantum case is revealed.

Some recent results concerning a particle confined in a one-dimensional box with moving walls are briefly reviewed. By exploiting the same techniques used for the 1D problem, we investigate the behavior of a quantum particle confined in a two-dimensional box (a 2D billiard) whose walls are moving by recasting the relevant mathematical problem with moving boundaries in the form of a problem with fixed boundaries and time-dependent Hamiltonian. Changes of the shape of the box are shown to be important, as it clearly emerges from the comparison between the 'pantographic' case (same shape of the box through the entire process) and the case with deformation.

Starting from the characteristic function of an operator, we investigate cumulant expansions in quantum optics and apply them to two-dimensional distributions for the canonical variables of the phase space in the case of one degree of freedom (Wigner quasiprobability and its Fourier transform, uncertainty matrix) and to one-dimensional distributions (phase operator, time evolution operator to Hamiltonian). In the relations between cumulants and moments, we make emphasis on the central moments of an operator. It is shown that the determinant of the uncertainty matrix (modified uncertainty product) is invariant with respect to rotation and squeezing of the state in the phase space, whereas the uncertainty sum is only invariant with respect to rotations. We examine some problems for exponentials of the phase operator and show how mean values and variances are connected with the cumulants. The Hilbert–Schmidt distance of a state during time evolution to an initial state is discussed by cumulants.

In this study, we provide a way to describe the dynamics of quantum tomography in an open system with a generalized master equation, considering a case where the relevant system under tomographic measurement is influenced by the environment. We apply this to spin tomography because such situations typically occur in μSR (muon spin rotation/relaxation/resonance) experiments where microscopic features of the material are investigated by injecting muons as probes. As a typical example to describe the interaction between muons and a sample material, we use a spin-boson model where the relevant spin interacts with a bosonic environment. We describe the dynamics of a spin tomogram using a time-convolutionless type of generalized master equation that enables us to describe short time scales and/or low-temperature regions. Through numerical evaluation for the case of Ohmic spectral density with an exponential cutoff, a clear interdependency is found between the time evolution of elements of the density operator and a spin tomogram. The formulation in this paper may provide important fundamental information for the analysis of results from, for example, μSR experiments on short time scales and/or in low-temperature regions using spin tomography.

We critically analyze the concept of photon helicity and its connection with the Pauli–Lubański vector from the viewpoint of the complex electromagnetic field, ${\bf{E}}+{\rm{i}}{\bf{H}},$ sometimes attributed to Riemann but studied by Weber, Silberstein, and Minkowski. To this end, a complex covariant form of Maxwell's equations is used. Weyl's two-component wave equation for massless neutrinos is also briefly discussed.

We show that the different values 1, 2 and 3 of the normalized second-order correlation function ${g}^{(2)}(0)$ corresponding to a coherent state, a thermal state and a highly squeezed vacuum originate from the different dimensionality of these states in phase space. In particular, we derive an exact expression for ${g}^{(2)}(0)$ in terms of the ratio of the moments of the classical energy evaluated with the Wigner function of the quantum state of interest and corrections proportional to the reciprocal of powers of the average number of photons. In this way we establish a direct link between ${g}^{(2)}(0)$ and the shape of the state in phase space. Moreover, we illuminate this connection by demonstrating that in the semi-classical limit the familiar photon statistics of a thermal state arise from an area in phase space weighted by a two-dimensional Gaussian, whereas those of a highly squeezed state are governed by a line-integral of a one-dimensional Gaussian.