Position Momentum Research Articles

On quantum states for angular position and angular momentum of light

In the present paper we construct a properly defined quantum state ψ(θ) expressed in terms of elliptic Jacobi theta functions for the self-adjoint observables angular position θ and the corresponding angular momentum operator L=−id/dθ in units of ℏ=1 . The quantum uncertainties Δθ and ΔL for the state are well-defined and are shown to give a lower value of the uncertainty product ΔθΔL in contrast to the so called minimal uncertainty states as discussed in Franke-Arnold et al (2004 New J. Phys. 6 103-1-8). The mean value ⟨L⟩ of the state ψ(θ) is not required to be an integer. In the case of any half-integer mean value ⟨L⟩ the state constructed exhibits a remarkable critical behavior with upper and lower bounds Δθ=π2/3−2 and ΔL=1/2 .

Mathematical simulation of À LA Einstein loss of the non-separable character of a function for two particles at the quantum-classical boundary

In this article, we present mathematical simulations of non-separable functions (those that would “correspond” to two entangled quantum particles) that lose this character only as a result of approaching the quantum-classical frontier. No mathematical representation of the action of deteriorating agents of quantum entanglement was included in the simulation. Such loss manifests itself both from the point of view of position space and momentum space. For certain limits, compatible with the space considered, the non-separable functions defined here transform into separable functions or cancel each other out at this boundary, thus erasing the (mathematical representation of) the quantum characteristic with no equivalent in the classical world. These simulations do not concern the loss of a physical property or characteristic, but rather the loss of a mathematical characteristic of a function for two quantum particles. The “ghostly action at a distance”, colloquially expressed by Prof. A. Einstein, has a “spatially limited and non-instantaneous action” as it’s opposite, which mathematically takes place in simulations of non-separable quantum functions, as shown here.

Fisher Information for a System Composed of a Combination of Similar Potential Models

The solutions to the radial Schrödinger equation for a pseudoharmonic potential and Kratzer potential have been studied separately in the past. Despite different reports on the Kratzer potential, the fundamental theoretical quantities such as Fisher information have not been reported. In this study, we obtain the solution to the radial Schrödinger equation for the combination of the pseudoharmonic and Kratzer potentials in the presence of a constant-dependent potential, utilizing the concepts and formalism of the supersymmetric and shape invariance approach. The position expectation value and momentum expectation value are calculated employing the Hellmann–Feynman Theory. These expectation values are then used to calculate the Fisher information for both position and momentum spaces in both the absence and presence of the constant-dependent potential. The results obtained revealed that the presence of the constant-dependent potential leads to an increase in the energy eigenvalue, as well as in the position and momentum expectation values. Additionally, the constant-dependent potential increases the Fisher information for both position and momentum spaces. Furthermore, the product of the position expectation value and the momentum expectation value, along with the product of the Fisher information, satisfies both Fisher’s inequality and Cramer–Rao’s inequality.

Agreement between a markerless and a marker-based motion capture systems for balance related quantities

Balance studies usually focus on quantities describing the global body motion. Assessing such quantities using classical marker-based approach can be tedious and modify the participant’s behaviour. The recent development of markerless motion capture methods could bypass the issues related to the use of markers. This work compared dynamic balance related quantities obtained with markers and videos. Sixteen young healthy participants performed four different motor tasks: walking at self-selected speed, balance loss, walking on a narrow beam and countermovement jumps. Their movements were recorded simultaneously by marker-based and markerless motion capture systems. Videos were processed using a commercial markerless pose estimation software, Theia3D. The centre of mass position (CoM) was computed, and the associated extrapolated centre of mass position (XCoM) and whole-body angular momentum (WBAM) were derived. Bland-Altman analysis was performed and root mean square difference (RMSD) and coefficient of correlation were computed to compare the results obtained with marker-based and markerless methods. Bias remained of the magnitude of a few mm for CoM and XCoM positions, and RMSD of CoM and XCoM was around 1 cm. RMSD of the WBAM was less than 10 % of the total amplitude in any direction, and bias was less than 1 %. Results suggest that outcomes of balance studies will be similar whether marker-based or markerless motion capture system are used. Nevertheless, one should be careful when assessing dynamic movements such as jumping, as they displayed the biggest differences (both bias and RMSD), although it is unclear whether these differences are due to errors in markerless or marker-based motion capture system.

Popper’s conjecture with angular slits and twisted light

Uncertainty relations are core to both classical and quantum physics, and lend themselves to tests across many degrees of freedom, with structured light emerging as a vibrant tool to harness these degrees of freedom. Here, we test Popper’s conjecture by replacing the traditional spatial and momentum states with angular position and orbital angular momentum (OAM) states of photons, showing that the OAM spectrum for an entangled photon passing through a virtual slit differs from that of a photon passing through a physical slit. To achieve this, we produce two OAM entangled photons, one of which is sent to a slit encoded as a digital hologram, thereby localising its angular position, all the while measuring the OAM of the other. We show that the measured OAM spectrum is limited to that of the initial SPDC photons, independent of the OAM encoded into the slit, consistent with Popper’s viewpoint. Our approach allows us to overcome prior limitations imposed by physical slits and linear momentum, and offers a versatile toolbox for further probes of quantum systems.

Six-dimensional light-front Wigner distribution of the pion

We study the recently defined six-dimensional light-front Wigner distributions for the pion case. By using the light-cone spectator model, we calculate the Wigner distributions of the unpolarized, longitudinal-polarized, and transversely-polarized quark in the pion. We present the numerical results of six-dimensional light-front Wigner distributions by fixing certain coordinates. The Wigner distributions exhibit a diffraction pattern in the longitudinal position space, in similar to the single slit diffraction in optics. We also present the numerical results of the three-dimensional distributions in the subspace of longitudinal position and transverse position or transverse momentum coordinates. The six-dimensional light-front Wigner distributions not only contain the most complete information of parton distributions in the hadron, but also provide new physical quantities for future measurements.

Monotone complexity measures of multidimensional quantum systems with central potentials

In this work, we explore the (inequality-type) properties of the monotone complexity-like measures of the internal complexity (disorder) of multidimensional non-relativistic electron systems subject to a central potential. Each measure quantifies the combined balance of two spreading facets of the electron density of the system. We show that the hyperspherical symmetry (i.e., the multidimensional spherical symmetry) of the potential allows Cramér–Rao, Fisher–Shannon, and Lopez-Ruiz, Mancini, Calbet–Rényi complexity measures to be expressed in terms of the space dimensionality and the hyperangular quantum numbers of the electron state. Upper bounds, mutual complexity relationships, and complexity-based uncertainty relations of position–momentum type are also found by means of the electronic hyperangular quantum numbers and, at times, the Heisenberg–Kennard relation. We use a methodology that includes a variational approach with a covariance matrix constraint and some algebraic linearization techniques of hyperspherical harmonics and Gegenbauer orthogonal polynomials.

Quasiperiodic Oscillations and Dynamics of Test Particles around Quasi- and Non-Schwarzschild Black Holes

One of the open problems in black hole physics is testing spacetime around black holes through astrophysical observations in the strong field regime. In fact, black holes cannot produce radiation themselves in the electromagnetic spectrum. However, a black hole’s gravity plays an important role in the production of the radiation of the accretion disc around it. One may obtain valuable information from the electromagnetic radiation of accretion discs about the gravitational properties of the spacetime around black holes. In this work, we study particle dynamics in the spacetime of quasi- and non-Schwarzschild black holes. We compare the gravitational effects of the spacetime deformation parameters of both black hole solutions on the innermost stable circular orbit (ISCO) radius, position, energy, and angular momentum of test particles at the ISCO, together with the energy efficiency of the accretion disc in the thin Novikov–Thorn model. Furthermore, we study the frequencies of particle oscillations in the radial and angular directions along circular stable orbits around both deformed black holes. Furthermore, we investigate quasiperiodic oscillations around the black holes in the relativistic precession model. We show the dependence of the deviation parameters on the orbits of twin peak QPOs with the frequency ratio 3:2. In the obtained results, we compare the gravitational effects of deviation parameters with the spin of a rotating Kerr black hole. Finally, we obtain constraints on the values of the deviation parameter of the spacetime around the black hole at the center of the microquasars GRO J1655-40 and GRS 1915-105 and their mass, using the χ2 method.

Topological localized region of Goos-Hänchen shifts in reflection and transmission

In this manuscript, the Goos-Hänchen shift of the transmitted and reflected probe beam is controlled and modified. Topologically localized regions are reported in the G-H shift by introducing the position and orbital angular momentum dependency in the Rabi frequencies of the control fields. The system parameters along with the azimuthal quantum number are used to control and modify G-H shift as well as the topological regions. Crater and peak type topological regions are reported in the G-H shift of the reflected as well as transmitted beam. The number of localized regions are reported to strictly follow the general formula 2l2. The maximum shift in the reflected spectrum is reported to be Sr/λ=−140, where as in the transmitted spectrum it is only St/λ=−15. The modified results might have significant applications in optical waveguide switches and sensors.

Improving Mechanical Oscillator Cooling in a Double-Coupled Cavity Optomechanical System with an Optical Parametric Amplifier

We investigate the cooling phenomenon of a mechanical oscillator in a double-coupled cavity optomechanical system. Our model includes two single-mode optical cavities. The left cavity is an optomechanical system with an optical parametric amplifier, and the right cavity is a standard optical cavity. The two optical cavities couple with each other by exchanging photons. The optomechanical system is effectively driven by an input laser field. By solving the linear quantum Langevin equation of the system under a steady-state condition, we can obtain the position fluctuation spectrum and momentum fluctuation spectrum of the mechanical oscillator, and then, the expression of its effective temperature is obtained. Through numerical analysis, we find the change in the effective temperature of the mechanical oscillator under different physical parameters. The results show that the cooling of the mechanical oscillator can be significantly improved in the presence of optical parameter amplification and adjustment of optical cavity parameters. Our cooling solutions have potential applications for the preparation of nonclassical states of mechanical oscillators, high-precision measurements, and quantum information processing.

Fisher information of a modified trigonometric inversely quadratic potential

• A new form of potential was proposed • The solution of the Schrödinger equation with the potential was obtained. • Using expectation values, the Fisher information was calculated. • The Cramer-Rao inequality was verified • The results of some special cases of the potential were obtained. The solution of the non-relativistic wave equation for modified trigonometric inversely quadratic potential (MTIAP) is obtained. The energy equation obtained, and the Hamiltonian which comprises the inverse squared term and the potential function were used to obtain the various radial expectation values. These expectation values were used to estimate the Fisher information for position configuration and momentum space. Numerical results were computed for Fisher information-based uncertainty relation with m = 0 and m = 1 . Some useful potentials like the isotropic harmonic oscillator and hydrogen atom were deduced from the modified trigonometric inversely quadratic potential and the Fisher information for the two potentials was also studied.

Operational classical mechanics: holonomic systems

We construct an operational formulation of classical mechanics without presupposing previous results from analytical mechanics. In doing so, we rediscover several results from analytical mechanics from an entirely new perspective. We start by expressing the position and velocity of point particles as the eigenvalues of self-adjoint operators acting on a suitable Hilbert space. The concept of holonomic constraint is shown to be equivalent to a restriction to a linear subspace of the free Hilbert space. The principal results we obtain are: (1) the Lagrange equations of motion are derived without the use of D’Alembert or Hamilton principles, (2) the constraining forces are obtained without the use of Lagrange multipliers, (3) the passage from a position–velocity to a position–momentum description of the movement is done without the use of a Legendre transformation, (4) the Koopman–von Neumann theory is obtained as a result of our ab initio operational approach, (5) previous work on the Schwinger action principle for classical systems is generalized to include holonomic constraints.

Physics aware analytics for accurate state prediction of dynamical systems

It has been successfully demonstrated that synchronisation of physical prior, like conservation laws with a conventional neural network significantly decreases the amount of training necessary to learn the dynamics of non-linear physical systems. Recent research shows how parameterisation of Lagrangian and Hamiltonian using conventional Neural Network’s weights and biases are achieved, and then executing the Euler-Lagrangian and Hamilton’s equation of motion for prediction of future state vectors proved to be more efficient than conventional Neural Networks in predicting non-linear dynamical systems. In classical mechanics, the Lagrangian formalism predicts the trajectory by solving the second-order Euler-Lagrangian equation following all the symmetries of the system while, the Hamiltonian mechanics explicitly uses the Poisson bracket of canonical position and canonical momentum to arrive at two independent first-order equations of motion. For a physical system that follows energy conservation, we obtain the momentum taking the first derivative of the Lagrangian which supplements the energy conservation via Hamiltonian Mechanics. The dynamics of the physical system converges when the generalised coordinates transform to suitable action angle variables using Poisson bracket formalism conserving the energy. In this work, we have demonstrated the prediction by integrating Hamiltonian and Lagrangian neural network leading to faster convergence with the actual dynamics from less number of training epochs. Hence, integration of HNN(Hamiltonian Neural Network) and LNN(Lagrangian Neural Network) outperforms their autonomous implementation. We have tested our proposed model in three conservative non-linear dynamical systems and found promising results.

Experimental investigation of the uncertainty principle for radial degrees of freedom

While the uncertainty principle for linear position and linear momentum, and more recently for angular position and angular momentum, is well established, its radial equivalent has so far eluded researchers. Here we exploit the logarithmic radial position, ln r , and hyperbolic momentum, P H , to formulate a rigorous uncertainty principle for the radial degree of freedom of transverse light modes. We show that the product of their uncertainties is bounded by Planck’s constant, Δ ln r · Δ P H ≥ ℏ / 2 , and identify a set of radial intelligent states that satisfy the equality. We illustrate the radial uncertainty principle for a variety of intelligent states, by preparing transverse light modes with suitable radial profiles. We use eigenmode projection to measure the corresponding hyperbolic momenta, confirming the minimum uncertainty bound. Optical systems are most naturally described in terms of cylindrical coordinates, and our radial uncertainty relation provides the missing piece in characterizing optical quantum measurements, providing a new platform for the fundamental tests and applications of quantum optics.

Path Integral Action for a Resonant Detector of Gravitational Waves in the Generalized Uncertainty Principle Framework

The Heisenberg uncertainty principle is modified by the introduction of an observer-independent minimal length. In this work, we have considered the resonant gravitational wave detector in the modified uncertainty principle framework, where we have used the position momentum uncertainty relation with a quadratic order correction only. We have then used the path integral approach to calculate an action for the bar detector in the presence of a gravitational wave and then derived the Lagrangian of the system, leading to the equation of motion for the configuration-space position coordinate in one dimension. We then find a perturbative solution for the coordinate of the detector for a circularly polarized gravitational wave, leading to a classical solution of the same for the given initial conditions. Using this classical form of the coordinate of the detector, we finally obtain the classical form of the on-shell action describing the harmonic oscillator–gravitational wave system. Finally, we have obtained the free particle propagator containing the quantum fluctuation term considering gravitational wave interaction.

Measurement of two-photon position–momentum Einstein–Podolsky–Rosen correlations through single-photon intensity measurements

The measurement of the position–momentum Einstein–Podolsky–Rosen (EPR) correlations of a two-photon state is important for many quantum information applications ranging from quantum key distribution to coincidence imaging. However, all the existing techniques for measuring the position–momentum EPR correlations involve coincidence detection and thus suffer from issues that result in less accurate measurements. In this article, we propose and demonstrate an experimental scheme that does not require coincidence detection for measuring the EPR correlations. Our technique works for two-photon states that are pure, irrespective of whether the state is separable or entangled. We theoretically show that if the pure two-photon state satisfies a certain set of conditions then the position–momentum EPR correlations can be obtained by doing the intensity measurements on only one of the photons. We experimentally demonstrate this technique for pure two-photon states produced by type-I spontaneous parametric down-conversion, and to the best of our knowledge, we report the most accurate experimental measurement of position–momentum EPR correlations.

Relativistic ro‐vibrational feature of electron in Bohr's orbits of hydrogen‐like atoms in Heisenberg picture

AbstractThe relativistic effects have been widely reported to affect the chemical and physical properties of the heavy elements such as lanthanides (), actinides (), and transactinides (). This effect is definitely weakened by reducing the atomic number in lighter elements such as hydrogen‐like atoms (HLAs). The aim of present paper is to investigate the relativistic effects of electron motion in Bohr orbits on the chemical and physical properties of HLAs. The theoretical model is based on the Heisenberg (rather than Schrodinger) picture where the relativistic vibrational Hamiltonian (RVH) is expanded as a power series of harmonic oscillator Hamiltonian for the first time. By applying the first‐order RVH (correct to ) to the Heisenberg equation, a pair of coupled equations is obtained for the relativistic position and linear momentum of electron. A simple comparison of the first‐order relativistic and nonrelativistic equations reveals that the relativistic natural frequency of an HLA (like entropy) is slowly raised by increasing beyond . In general, RVH plays a fundamental role because it specifies the temporal relativistic variations of position, velocity, and linear momentum of the oscillating electron. The results are finally verified by demonstrating energy conservation.

Uncertainty Relations in the Madelung Picture.

Madelung showed how the complex Schrödinger equation can be rewritten in terms of two real equations, one for the phase and one for the amplitude of the complex wave function, where both equations are not independent of each other, but coupled. Although these equations formally look like classical hydrodynamic equations, they contain all the information about the quantum system. Concerning the quantum mechanical uncertainties of position and momentum, however, this is not so obvious at first sight. We show how these uncertainties are related to the phase and amplitude of the wave function in position and momentum space and, particularly, that the contribution from the phase essentially depends on the position–momentum correlations. This will be illustrated explicitly using generalized coherent states as examples.

Controlling quantum correlations in optical-angle–orbital-angular-momentum variables

The entanglement of optical angular position and orbital angular momentum has become one of the important degrees of freedom for quantum information science. Quantum control of the two conjugate variables may provide a gate to control the production of the entanglement. Here, we theoretically and experimentally show that by controlling the transverse spatial coherence of the pump beam, one can modulate the strength of the quantum correlations between the angular position and orbital angular momentum in down-converted photon pairs. By determining the relation of the Einstein, Podolsky, and Rosen correlations with the transverse spatial coherence of the pump, the boundary of the entanglement can be determined. Our work shows that the transverse spatial coherence of the pump can act as a switch to control the production of quantum entanglement both for discrete and continuous conjugate variables and may provide another platform for the quantum control research field.

High Dimensional Atomic States of Hydrogenic Type: Heisenberg-like and Entropic Uncertainty Measures.

High dimensional atomic states play a relevant role in a broad range of quantum fields, ranging from atomic and molecular physics to quantum technologies. The D-dimensional hydrogenic system (i.e., a negatively-charged particle moving around a positively charged core under a Coulomb-like potential) is the main prototype of the physics of multidimensional quantum systems. In this work, we review the leading terms of the Heisenberg-like (radial expectation values) and entropy-like (Rényi, Shannon) uncertainty measures of this system at the limit of high D. They are given in a simple compact way in terms of the space dimensionality, the Coulomb strength and the state’s hyperquantum numbers. The associated multidimensional position–momentum uncertainty relations are also revised and compared with those of other relevant systems.