Points In Phase Space Research Articles

Verifying single-mode nonclassicality beyond negativity in phase space

While negativity in phase space is a well-known signature of nonclassicality, a wide variety of nonclassical states require their characterization beyond negativity. We establish a framework of nonclassicality in phase space that addresses nonclassical states comprehensively with a direct experimental evidence. This includes the negativity of phase-space distribution as a special case and further analyzes quantum states with positive distributions effectively. We prove that it detects all nonclassical Gaussian states and all non-Gaussian states of arbitrary dimension remarkably by examining three phase-space points only. Our formalism also provides an experimentally accessible lower bound for a nonclassicality measure based on trace distance. Importantly, this foundational approach can be further adapted to constitute practical tests in two directions looking into particle and wave nature of bosonic systems, via an array of nonideal on-off detectors and coarse-grained homodyne measurement, respectively. All these tests are practically powerful in characterizing nonclasssical states reliably against noise, making a versatile tool for a broad range of quantum systems in quantum technologies.

Entanglement dynamics of spins using a few complex trajectories

In this paper, we consider two spins initially prepared in a product of coherent states and study their entanglement dynamics due to a general interacting Hamiltonian. We adopt an approach that allowed the derivation of a semiclassical formula for the linear entropy of the reduced density operator, assumed as an entanglement quantifier. The resulting expression depends on sets of four trajectories, originated from the underlying classical description, and having mutually connected final phase-space points. Such classical elements, which are capable to reproduce the quantum entanglement even for long values of propagation time, arise when we assume a proper analytical continuation of the classical phase space onto a complex domain. We apply this theory to a particular physical system, showing that taking into account only a few sets of complex trajectories is enough to get an excellent agreement between the semiclassical linear entropy of the reduced density operator and its quantum counterpart.

Cosmological evolution with quadratic gravity and nonideal fluids

Some cosmological models based on the gravitational theory f(R)=R+zeta R^2, and on fluids obeying to the equations of state of Redlich–Kwong, Berthelot, and Dieterici are proposed for describing smooth transitions between different cosmic epochs. A dynamical system analysis reveals that these models contain fixed points which correspond to an inflationary, a radiation dominated and a late-time accelerating epoch, and a nonsingular bouncing solution, the latter being an asymptotic fixed point of the compactified phase space. The infinity of the compactified phase space is interpreted as a region in which the non-ideal behaviors of the previously mentioned cosmic fluids are suppressed. Physical constraints on the adopted dimensionless variables are derived by demanding the theory to be free from ghost and tachyonic instabilities, and a novel cosmological interpretation of such variables is proposed through a cosmographic analysis. The different effects of the equation of state parameters on the number of equilibrium solutions and on their stability nature are clarified. Some generic properties of these models, which are not sensitive to the particular fluid considered, are identified, while differences are critically examined by showing that the Redlich–Kwong scenario admits a second radiation-dominated epoch and a Big Rip Singularity.

Updates to the one-loop provider NLOX

We describe the 1.2 update to NLOX, a computer program for calculations in high-energy particle physics. New features since the 1.0 release and other changes are described, along with usage documentation.

Phase space correspondence between Jordan and Einstein frames

Scalar–tensor theories of gravity can be formulated in the Einstein frame or in the Jordan frame (JF) which are related with each other by conformal transformations. Although the two frames describe the same physics and are equivalent, the stability of the field equations in the two frames is not the same. Here, we implement dynamical system and phase space approach as a robustness tool to investigate this issue. We concentrate on the Brans–Dicke theory in a Friedmann–Lemaitre–Robertson–Walker universe, but the results can easily be generalized. Our analysis shows that while there is a one-to-one correspondence between critical points in two frames and each critical point in one frame is mapped to its corresponds in another frame, however, stability of a critical point in one frame does not guarantee the stability in another frame. Hence, an unstable point in one frame may be mapped to a stable point in another frame. All trajectories between two critical points in phase space in one frame are different from their corresponding in other ones. This indicates that the dynamical behavior of variables and cosmological parameters is different in two frames. Hence, for those features of the study, which focus on observational measurements, we must use the JF where experimental data have their usual interpretation.

Basin of Attraction Analysis of New Memristor-Based Fractional-Order Chaotic System

Memristor is the fourth basic electronic element discovered in addition to resistor, capacitor, and inductor. It is a nonlinear gadget with memory features which can be used for realizing chaotic, memory, neural network, and other similar circuits and systems. In this paper, a novel memristor-based fractional-order chaotic system is presented, and this chaotic system is taken as an example to analyze its dynamic characteristics. First, we used Adomian algorithm to solve the proposed fractional-order chaotic system and yield a chaotic phase diagram. Then, we examined the Lyapunov exponent spectrum, bifurcation, SE complexity, and basin of attraction of this system. We used the resulting Lyapunov exponent to describe the state of the basin of attraction of this fractional-order chaotic system. As the local minimum point of Lyapunov exponential function is the stable point in phase space, when this stable point in phase space comes into the lowest region of the basin of attraction, the solution of the chaotic system is yielded. In the analysis, we yielded the solution of the system equation with the same method used to solve the local minimum of Lyapunov exponential function. Our system analysis also revealed the multistability of this system.

Non-ideal surface modelling approach for enabling uncertainty representation

Non-ideal models have gained considerable attention in the design stage for evaluating product functionality and performance considering the realistic conditions of the product surface. The performance evaluation generally involves deriving the statistical distributions of certain key characteristics, such as gaps and distances, based on numerous assembly process simulations, thus necessitating a large number of samples of non-ideal surfaces. The traditional practice for generating such samples is to model the systematic and random deviations of each point on the part surface. It is difficult, however, to represent the randomness of the real surface formation mechanism using this point-wise approach, thus leading to significant computational cost. As a result, this technique hinders the wider adoption of specified surface randomness representations in performance simulation and analysis.The deviation modes that feature non-ideal surfaces are dominated by manufacturing error factors, which could be systematically represented by corresponding mode functions. Furthermore, the variations in the manufacturing process inevitably cause randomness in surface deviations. In this paper, this randomness is defined as the uncertainty of non-ideal surfaces and reflected by the randomness of deviation mode function coefficients. A conceptual non-ideal surface error model that characterises the uncertainty of non-ideal surfaces through the random distributions of mode function coefficients as represented in phase space is therefore proposed. The phase space is a multi-dimensional space, in which each point denotes a group of mode function coefficients that determines a certain non-ideal surface sample. Following the distribution law of coefficients, a number of non-ideal surface samples can be generated by phase space sampling. This method reduces the problem from modelling an infinite number of surface points to modelling a finite coefficient dimension of points in the compact phase space. It thus enables the efficient modelling of non-ideal surfaces that reflect realistic errors resulting from the manufacturing process and facilitates the product performance analysis under the effect of surface randomness. To verify the method, a case study involving a tapered surface manufactured by the taper-turning process is presented. The phase space and error model are discussed with respect to typical manufacturing error factors in this process.

Ubiquitous quantum scarring does not prevent ergodicity

In a classically chaotic system that is ergodic, any trajectory will be arbitrarily close to any point of the available phase space after a long time, filling it uniformly. Using Born’s rules to connect quantum states with probabilities, one might then expect that all quantum states in the chaotic regime should be uniformly distributed in phase space. This simplified picture was shaken by the discovery of quantum scarring, where some eigenstates are concentrated along unstable periodic orbits. Despite that, it is widely accepted that most eigenstates of chaotic models are indeed ergodic. Our results show instead that all eigenstates of the chaotic Dicke model are actually scarred. They also show that even the most random states of this interacting atom-photon system never occupy more than half of the available phase space. Quantum ergodicity is achievable only as an ensemble property, after temporal averages are performed.

Sequential discontinuities of Feynman integrals and the monodromy group

We generalize the relation between discontinuities of scattering amplitudes and cut diagrams to cover sequential discontinuities (discontinuities of discontinuities) in arbitrary momentum channels. The new relations are derived using time-ordered perturbation theory, and hold at phase-space points where all cut momentum channels are simultaneously accessible. As part of this analysis, we explain how to compute sequential discontinuities as monodromies and explore the use of the monodromy group in characterizing the analytic properties of Feynman integrals. We carry out a number of cross-checks of our new formulas in polylogarithmic examples, in some cases to all loop orders.

Combustion performance of a low NOx gas turbine combustor using urea addition into liquid fuel

The present work provides an experimental investigation on the effects of the urea addition in water emulsified fuels in terms of combustion performance and emissions. Experiments have been carried out using a test rig equipped with a 300-kW Jet-A1 fueled swirling burner operating under lean conditions. Different equivalence ratios and various urea in water percent concentrations have been tested. Measurements of temperature and emissions have been performed, in combination with high-speed flame imaging in visible and ultraviolet spectral ranges.Results have shown that the use of 2.5 wt% (percent concentration by weight) water-2 wt% urea solution into Jet-A1 fuel represents a promising combustion control strategy, since it leads to the same nitrogen oxides (NOx) reduction of a leaner neat fuel flame (about 30% less than the neat fuel case), but without significant penalty on the overall combustion performance, viz. combustion temperature, thermal and combustion efficiencies and pollutant emissions. The snapshot Proper Orthogonal Decomposition (POD) of the broadband flame emission images, methylidyne radical and hydroxyl radical chemiluminescence allowed to characterize of the flame dynamics and the flame stability. Differences between the flame regimes were also investigated by the POD mode 3 eigenstructure. The reduced flame stability for the leaner operating condition was confirmed by the energy increase of the first POD modes. The more unstable flame dynamics was confirmed by the frequency distribution of the phase points in the POD phase-space of modes 1 and 2 that highlighted a significant rise of the phase angles occurrence in the range [−π/6, π/6].

Twist-and-store entanglement in bimodal and spin-1 Bose-Einstein condensates

A scheme for dynamical stabilization of entanglement quantified by the quantum Fisher information is analyzed numerically and analytically for bimodal and spin-1 Bose-Einstein condensates in the context of atomic interferometry. The scheme consists of twisting dynamics followed by a single rotation of a state which limits further evolution around stable center fixed points in the mean-field phase space. The resulting level of entanglement is of the order or larger than at the moment of rotation. It is demonstrated that the readout measurement of parity quantifies the level of entanglement during entire evolution.

Robust control of unstable nonlinear quantum systems

Adiabatic passage is a standard tool for achieving robust transfer in quantum systems. We show that, in the context of driven nonlinear Hamiltonian systems, adiabatic passage becomes highly non-robust when the target is unstable. We show this result for a generic (1:2) resonance, for which the complete transfer corresponds to a hyperbolic fixed point in the classical phase space featuring an adiabatic connectivity strongly sensitive to small perturbations of the model. By inverse engineering, we devise high-fidelity and robust partially non-adiabatic trajectories. They localize at the approach of the target near the stable manifold of the separatrix, which drives the dynamics towards the target in a robust way. These results can be applicable to atom-molecule Bose-Einstein condensate conversion and to nonlinear optics.

First order comprehensive adjoint sensitivity analysis methodology (1st-casam) for computing efficiently the exact response sensitivities for physical systems with imprecisely known boundaries and parameters: General theory and illustrative paradigm applications

This work presents the novel first-order comprehensive adjoint sensitivity analysis methodology (1st-CASAM) for computing efficiently, exactly, and exhaustively, the first-order response sensitivities for nonlinear physical systems characterized by imprecisely known (uncertain) parameters model and domain boundary parameters. This novel methodology is designed for generic operator-valued model responses, treating scalar-valued responses as particular cases. The 1st-CASAM highlights the conclusion that response sensitivities to the imprecisely known domain boundaries can arise both from the definition of the system’s response as well as from the equations and boundary conditions defining the model on its imprecisely known domain. The application of this new methodology is illustrated on two paradigm models, each admitting exact closed-form expressions for the sensitivities of typical model responses to the model’s imprecisely known (uncertain) model and boundary parameters, as follows: (a) a nonlinear heat conduction equation; and (b) a neutron diffusion equation modeling a subcritical reactor with an external source. Very importantly, the diffusion equation is subject to vacuum boundary conditions on the extrapolated system boundary, thus illustrating the computation of response sensitivities to boundary parameters involving not only imprecisely known geometrical dimensions but also imprecisely known nuclear data. For the nonlinear heat conduction equation, the value of the temperature that would be measured at a point in phase-space is chosen as a representative scalar-valued response, while the temperature distribution in the entire phase-space, having uncertain boundaries, of the model’s independent variable has been chosen as a representative function-valued response. For the neutron diffusion equation, a reaction-rate within the imprecisely known domain boundaries has been chosen as a representative scalar-valued response, and the neutron flux distribution within the imprecisely known domain boundaries has been chosen as a representative function-valued response. The application of the 1st-CASAM for the operator-valued responses is illustrated using: (a) a pseudo-spectral collocation representation for the sensitivities of the temperature distribution response; and (b) a spectral Fourier representation for the sensitivities of the neutron flux distribution response. These paradigm examples highlight the details of applying the general 1st-CASAM methodology, underscoring the effects of imprecisely known domain boundaries on both the system’s state functions and responses. By enabling, in premiere, the exact computations of function-valued response sensitivities to boundary parameters and conditions, the novel adjoint sensitivity analysis methodology presented in this work enables the quantification of the effects of manufacturing tolerances on the responses of physical and engineering systems.

Exploring 2D Synthetic Quantum Hall Physics with a Quasiperiodically Driven Qubit.

Quasiperiodically driven quantum systems are predicted to exhibit quantized topological properties, in analogy with the quantized transport properties of topological insulators. We use a single nitrogen-vacancy center in diamond to experimentally study a synthetic quantum Hall effect with a two-tone drive. We measure the evolution of trajectories of two quantum states, initially prepared at nearby points in synthetic phase space. We detect the synthetic Hall effect through the predicted overlap oscillations at a quantized fundamental frequency proportional to the Chern number, which characterizes the topological phases of the system. We further observe half-quantization of the Chern number at the transition between the synthetic Hall regime and the trivial regime, and the associated concentration of local Berry curvature in synthetic phase space. Our Letter opens up the possibility of using driven qubits to design and study higher-dimensional topological insulators and semimetals in synthetic dimensions.

Probing nonclassicality with matrices of phase-space distributions

We devise a method to certify nonclassical features via correlations of phase-space distributions by unifying the notions of quasiprobabilities and matrices of correlation functions. Our approach complements and extends recent results that were based on Chebyshev's integral inequality \cite{BA19}. The method developed here correlates arbitrary phase-space functions at arbitrary points in phase space, including multimode scenarios and higher-order correlations. Furthermore, our approach provides necessary and sufficient nonclassicality criteria, applies to phase-space functions beyonds-parametrized ones, and is accessible in experiments. To demonstrate the power of our technique, the quantum characteristics of discrete- and continuous-variable, single- and multimode, as well as pure and mixed states are certified only employing second-order correlations and Husimi functions, which always resemble a classical probability distribution. Moreover, nonlinear generalizations of our approach are studied. Therefore, a versatile and broadly applicable framework is devised to uncover quantum properties in terms of matrices of phase-space distributions.

Coupling a Gaussian input beam to a quadrupole mass filter with round rods

We consider an input ion beam that has Gaussian distributions of the initial ion coordinates and transverse velocities at the entrance of a quadrupole mass filter (QMF) with round rods. To calculate the statistical QMF acceptance, an ion trajectory method has been used. The probability density functions of accepted points on phase planes of transverse ion positions and velocities in the X and Y directions are used to visualize the distributions of phase space points. To determine the statistical acceptances we calculate the second moments of the probability density functions. We define the statistical acceptance of the quadrupole mass filter as the area of an ellipse on the phase plane determined by the second moments. The values of the second moments characterize the efficiency of the coupling of the Gaussian ion source and the acceptance of a quadrupole rod set with round rods. The above approach is a supplement to a previously published theoretical model of QMF fields with circular electrodes.

Hydrodynamic Attractors in Phase Space.

Hydrodynamic attractors have recently gained prominence in the context of early stages of ultrarelativistic heavy-ion collisions at the RHIC and LHC. We critically examine the existing ideas on this subject from a phase space point of view. In this picture the hydrodynamic attractor can be seen as a special case of the more general phenomenon of dynamical dimensionality reduction of phase space regions. We quantify this using principal component analysis. Furthermore, we adapt the well known slow-roll approximation to this setting. These techniques generalize easily to higher dimensional phase spaces, which we illustrate by a preliminary analysis of a dataset describing the evolution of a five-dimensional manifold of initial conditions immersed in a 16-dimensional representation of the phase space of the Boltzmann kinetic equation in the relaxation time approximation.

Traveling Wave Solution of the Olver–Rosenau Equation Solved by Dynamics System

Olver–Rosenau equations presented by Olver and Rosenau can be rewritten to the dynamic system by the wave transformation. The system is a Hamiltonian system with the first integral, and its phase-space and equilibrium point analysis are given in different parameter spaces in detail. On this basis, we can derive various solutions of the original equation relating these orbits in different phase-space planes, and the theoretical basis of the numerical solution is provided for engineering application and production practice.

Method of Topological Roughness in Tasks of Research and Control of Synergetic Systems

The method of a research of roughness of dynamic systems based on a concept of roughness according to Andronov—Pontryagin and named with "method of topological roughness" is considered. The concepts "roughness" and "bifurcation" of dynamic systems formulated at a dawn of formation of the scientific direction of mathematics — topology, are given by the great French scientist A. Poincare. Also the concept of roughness according to Andronov—Pontryagin is formulated and conditions of accessibility of required roughness of a dynamic system are defined. The definitions of concepts of the maximum roughness and minimum non roughness of dynamic systems entered by the author earlier are given. The corresponding theorems of necessary and sufficient conditions of accessibility of the maximum roughness and the minimum non roughness and also emergence of bifurcations of topological structures of dynamic systems, which were proved in the fundamental works of the author given in the list of references are formulated. At the same time it is claimed that sets of rough and not rough systems make roughnesses of a set, continuous on an indicator. As a roughness indicator in a method the number of conditionality of a matrix of reduction to a diagonal (quasidiagonal) type of a matrix of Jacobi in special points of phase space of a system is used. The method allows to controling roughness of control systems on the basis of the theorem formulated with use of the matrix equation of Sylvester and proved in works of the author which is also provided in this work. The main stages of researches of roughness and bifurcations of systems by means of the considered method are formulated in the form of the corresponding algorithm. In work questions of synergetic systems and chaos (strange attractors) in them, founders of science of synergetics — H. Haken, I. Prigozhin are briefly stated. The method can be used for researches of roughness and bifurcations of dynamic systems and also synergetic systems and chaos of the different physical nature. In works of the author the method is approved on examples of many synergetic systems, such as Lorenz attractors and R ö ssler, Belousov-Zhabotinsky’s systems, Chua, "predator-prey", Henon, Hopf’s bifurcations, etc. In this work of a possibility of a method are illustrated on examples of synergetic system Chua and also a technical system in the form of the nonlinear servomechanism.

Non-Hessian method for normal coordinate calculations: Application to evaluate vibrational spectra

Normal mode analysis has been employed to characterize fundamental phenomena in diverse scientific disciplines, including the identification of molecular vibrations in chemistry and their peak assignments in infrared and Raman spectra. The vibrational modes are traditionally determined from the potential energy surface under the harmonic approximation. Here, we introduce a methodology called Expanded Moment of Inertia Tensor, EMIT, which can extract a complete set of normal coordinates from the molecular geometry at any point in phase space. Its principle lies in the conservative expansion of the angular momentum within the rigidbody framework. This method provides normal modes that resemble those obtained from group theory and show their efficacy by accurately reproducing all characteristic frequencies in the vibrational spectra of NO2- and NO3- in aqueous solution. Not only this geometry-based method of normal mode determination looks promising in many fields, but it also inspires us to view known molecular quantities in physics differently.