Momentum Space Information Research Articles

Direct determination of 2D momentum space from 2D spatial coherence of light using a modified Michelson interferometer.

Momentum space distribution of photons coming out of any light emitting material/device provides critical information about their underlying physical origin. Conventional methods of determining such properties impose specific instrumentational difficulties for probing samples kept within a low temperature cryostat. There were past studies to measure a one-dimensional coherence function, which could then be used for extracting momentum space information, as well as reports of measurements of just a two-dimensional (2D) coherence function. However, all of those are associated with additional experimental complexities. So, here we propose a simpler, modified Michelson interferometer based optical setup that is kept at room temperature and placed outside the low temperature cryostat at a distance away from it. We initially measure the 2D coherence function of emitted light, which can then be used to directly estimate the 2D in-plane momentum space distribution by calculating its fast Fourier transform. We also discuss how this experimental method can overcome instrumentational difficulties encountered in the past. Similar instrumentations can also be extended for momentum space resolved astronomical studies and telecommunications involving distant light sources.

Analytical Solutions of the Klein-Gordon equation and Fisher information with Inversely Quadratic Hellman potential

The inversely quadratic Hellman (IQH) potential has gained recognition in recent times. This paper is focused on the analytical solutions of the Klein-Gordon equation with the IQH potential and its application to Fisher information. Using the parametric form of the Nikiforov-Uvarov method, the relativistic and non-relativistic energies of the IQH potential are obtained, along with the wave functions. Furthermore, the Fisher information measure and uncertainty of the system are obtained. Our results show that the momentum space probability density function is more localized. This is confirmed to be true by the high values of the Fisher information in momentum space. Furthermore, the occurrence of squeezed states is observed. The uncertainty results and the Fisher information show that the system is stable.

Quantum information entropy and squeezing of 𝒫𝒯-symmetric potential

The information theoretic concepts are crucial to study the quantum mechanical systems. In this paper, the information densities of [Formula: see text]-symmetric potential have been demonstrated and their properties deeply analyzed. The position space and momentum space information entropy is obtained and Bialynicki-Birula–Mycielski inequality is saturated for different parameters of the potential. Some interesting features of information entropy have been discussed. The variation in these entropies is described which gets saturated for specific values of the parameter. These have also been analyzed for the [Formula: see text]-symmetry breaking case. Further, the entropy squeezing phenomenon has been investigated in position space as well as momentum space. Interestingly, [Formula: see text] phase transition conjectures the entropy squeezing in position space and momentum space.

Tripartite information, scrambling, and the role of Hilbert space partitioning in quantum lattice models

For the characterization of the dynamics in quantum many-body systems the question how information spreads and becomes distributed over the constituent degrees of freedom is of fundamental interest. The delocalization of information under many-body dynamics has been dubbed scrambling and out-of-time-order correlators were proposed to probe this behavior. In this work we investigate the time-evolution of tripartite information as a natural operator-independent measure of scrambling, which quantifies to which extent the initially localized information can only be recovered by global measurements. Studying the dynamics of quantum lattice models with tunable integrability breaking we demonstrate that in contrast to quadratic models generic interacting systems scramble information irrespective of the chosen partitioning of the Hilbert space, which justifies the characterization as scrambler. Without interactions the dynamics of tripartite information in momentum space reveals unambiguously the absence of scrambling.

Application of information theory in atoms, nuclei and atomic clusters

The position- and momentum-space information entropies of the electron distributions of atomic clusters are calculated using a Woods-Saxon single particle potential. The same entropies are also calculated for nuclear distributions according to the Skyrme parametrization of the nuclear mean field. It turns out that a similar functional form S = α + Μη Ν for the entropy as function of the number of particles Ν holds approximately for atoms, nuclei and atomic clusters. It is conjectured that this is a universal property of a many-fermion system in a mean field.

A machine learning study to identify spinodal clumping in high energy nuclear collisions

The coordinate and momentum space configurations of the net baryon number in heavy ion collisions that undergo spinodal decomposition, due to a first-order phase transition, are investigated using state-of-the-art machine-learning methods. Coordinate space clumping, which appears in the spinodal decomposition, leaves strong characteristic imprints on the spatial net density distribution in nearly every event which can be detected by modern machine learning techniques. On the other hand, the corresponding features in the momentum distributions cannot clearly be detected, by the same machine learning methods, in individual events. Only a small subset of events can be systematically differ- entiated if only the momentum space information is available. This is due to the strong similarity of the two event classes, with and without spinodal decomposition. In such sce- narios, conventional event-averaged observables like the baryon number cumulants signal a spinodal non-equilibrium phase transition. Indeed the third-order cumulant, the skewness, does exhibit a peak at the beam energy (Elab = 3–4 A GeV), where the transient hot and dense system created in the heavy ion collision reaches the first-order phase transition.

Reply to “Comments on ‘Statistical complexity and Fisher–Shannon information measure of [formula omitted]’ [Phys. Lett. A 372 (13) (2008) 2271–2273

Clarifications about the source of error in [Phys. Lett. A 371 (2008) 2271] of momentum-space information measures are given along with the revised estimates which satisfy all the points commented upon by Astashkevich. We note that the position-space quantities calculated previously remain unchanged. Our main conclusion, that the inter-nuclear distance, R, dependence of statistical complexity and Fisher–Shannon measure derived from the simple Coulson wave function contain significant information on molecular bonding in H2+, remains qualitatively valid in the light of calculations using more accurate wave functions as reported by Astashkevich.

On the realization of quantum Fisher information

With special attention to the role of information theory in physical sciences we present analytical results for the coordinate- and momentum-space Fisher information of some important one-dimensional quantum systems which differ in spacing of their energy levels. The studies envisaged allow us to relate the coordinate-space information () with the familiar energy levels of the quantum system. The corresponding momentum-space information () does not obey such a simple relationship with the energy spectrum. Our results for the product () depend quadratically on the principal quantum number n and satisfy an appropriate uncertainty relation derived by Dehesa et al (2007 J. Phys. A: Math. Theor. 40 1845)

Information entropies for the Morse potential using the J-matrix method

Information entropies, such as Shannon, Fisher and Rényi entropies, with their associated quantities, are calculated and discussed for the Morse potential in the position (r) and momentum (p) spaces using the J-matrix method. The entropies, for the s-wave, are numerically obtained for the diatomic molecules H2 and LiH. It is the first information entropy calculation for real diatomic molecules using the Morse potential. Interesting characteristic features of the entropy densities are shown, where the position and momentum space information entropies satisfy the uncertainty relations.

Information entropic measures of a quantum harmonic oscillator in symmetric and asymmetric confinement within an impenetrable box

Information‐based uncertainty measures like Shannon entropy, Onicescu energy and Fisher information (in position and momentum space) are employed to understand the effect of symmetric and asymmetric confinement in a quantum harmonic oscillator. Also, the transformation of the Hamiltonian into a dimensionless form gives an idea of the composite effect of force constant and confinement length (xc). In the symmetric case, a wide range of xc has been taken up, whereas asymmetric confinement is dealt with by shifting the minimum of the potential from the origin keeping box length and boundary fixed. Eigenvalues and eigenvectors for these systems are obtained quite accurately via an imaginary‐time propagation scheme. For asymmetric confinement, a variation‐induced exact diagonalization procedure is also introduced, which produces very high quality results. One finds that, in symmetric confinement, after a certain characteristic xc, all these properties converge to respective values of a free harmonic oscillator. In the asymmetric situation, excited‐state energies always pass through a maximum. For this potential, the classical turning point decreases, whereas well depth increases with the strength of asymmetry. A study of these uncertainty measures reveals that localization increases with an increase of the asymmetry parameter. image

Quantum information entropy of Eckart potential

AbstractIn this work, the position and momentum space information densities of the Eckart potential are graphically demonstrated and their properties are studied. The position space information densities have quite an asymmetric shape depending on the values of quantum numbers. The information entropy is obtained and Bialynicki‐Birula and Mycielski inequality is numerically saturated for some parameters of the potential. It is shown that the inequality is saturated with increasing potential depth.

Wigner Functions and Quark Orbital Angular Momentum

Wigner distributions contain combined position and momentum space information of the quark distributions and are related to both generalized parton distributions (GPDs) and transverse momentum dependent parton distributions (TMDs). We report on a recent model calculation of the Wigner distributions for the quark and their relation to the orbital angular momentum.

Wigner Distributions and Orbital Angular Momentum of Quarks

We present a recent model calculation of the Wigner distributions for the quarks and the orbital angular momentum carried by the quarks. These Wigner distributions contain combined position and momentum space information of the quark distributions and are related to both generalized parton distributions (GPDs) and transverse momentum dependent parton distributions (TMDs).

Systematic Study of Information Measures, Statistical Complexity and Atomic Structure Properties

We present a comparative study of several information and statistical complexity measures in order to examine a possible correlation with certain experimental properties of atomic structure. Comparisons are also carried out quantitatively using Pearson correlation coefficient. In particular, it is shown that Fisher information in momentum space is very sensitive to shell effects. It is also seen that three measures expressed in momentum space that is Fisher information, Fisher-Shannon plane and LMC complexity are associated with atomic radius, ionization energy, electronegativity, and atomic dipole polarizability. Our results indicate that a momentum space treatment of atomic periodicity is superior to a position space one. Finally we present a relation that emerges between Fisher information and the second moment of the probability distribution in momentum space i.e. an energy functional of interest in (e,2e) experiments.

Quark Wigner distributions and orbital angular momentum in light-front dressed quark model

We calculate the Wigner functions for a quark target dressed with a gluon. These give a combined position and momentum space information of the quark distributions and are related to both generalized parton distributions (GPDs) and transverse momentum dependent parton distributions (TMDs). We calculate and compare the different definitions of quark orbital angular momentum in this model. We compare our results with other model calculations.

Quantum information entropies of ultracold atomic gases in a harmonic trap

The position and momentum space information entropies of weakly interacting trapped atomic Bose–Einstein condensates and spin-polarized trapped atomic Fermi gases at absolute zero temperature are evaluated. We find that sum of the position and momentum space information entropies of these quantum systems containing N atoms confined in a D( ≤ 3)-dimensional harmonic trap has a universal form as $ S_\mathrm{t}^{(D)} = N(a D - b \ln N) $ , where a ≃ 2.332 and b = 2 for interacting bosonic systems and a ≃ 1.982 and b = 1 for ideal fermionic systems. These results obey the entropic uncertainty relation given by Beckner, Bialynicki-Birula and Myceilski.

On the quantum information entropies and squeezing associated with the eigenstates of the isotonic oscillator

In this paper we calculate the position and momentum space information entropies for the quantum states associated with a particular physical system, i.e. the isotonic oscillator Hamiltonian. We present our results for its ground states, as well as for its excited states. We observe that the lower bound of the sum of the position and momentum entropies expressed by the Beckner, Bialynicki-Birula and Mycielski (BBM) inequality is satisfied. Moreover, there exist eigenstates that exhibit squeezing in the position information entropy. In fact, entropy squeezing, which occurs in position, will be compensated for by an increase in momentum entropy, such that the BBM inequality is guaranteed. To complete our study we investigate the amplitude squeezing in x and p-quadratures corresponding to the eigenstates of the isotonic oscillator and show that amplitude squeezing, again in x, will be revealed as expected, while the Heisenberg uncertainty relationship is also satisfied. Finally, our numerical calculations of the entropy densities will be presented graphically.

Momentum spectra for dynamically assisted Schwinger pair production

Recently the dynamically assisted Schwinger mechanism, i.e., electron–positron pair production from vacuum by a combination of laser pulses with different time scales has been proposed. The corresponding results, which suggest that the rate of produced pairs is significantly enhanced by dynamical effects, are verified. Employing the framework of quantum kinetic theory intrinsically enables us to additionally provide momentum space information on the generated positron spectrum.

Heavy d-electron quasiparticle interference and real-space electronic structure of Sr3Ru2O7

As well as providing subatomic-scale real-space images of metals, the scanning tunnelling microscope also reveals momentum–space information. Now it is possible to use this technique to image a heavy-electron liquid and obtain information on orbital structures. The intriguing idea that strongly interacting electrons can generate spatially inhomogeneous electronic liquid-crystalline phases is over a decade old1,2,3,4,5, but these systems still represent an unexplored frontier of condensed-matter physics. One reason is that visualization of the many-body quantum states generated by the strong interactions, and of the resulting electronic phases, has not been achieved. Soft condensed-matter physics was transformed by microscopies that enabled imaging of real-space structures and patterns. A candidate technique for obtaining equivalent data in the purely electronic systems is spectroscopic imaging scanning tunnelling microscopy (SI-STM). The core challenge is to detect the tenuous but ‘heavy’ momentum (k)-space components of the many-body electronic state simultaneously with its real-space constituents. Sr3Ru2O7 provides a particularly exciting opportunity to address these issues. It possesses a very strongly renormalized ‘heavy’ d-electron Fermi liquid6,7 and exhibits a field-induced transition to an electronic liquid-crystalline phase8,9. Finally, as a layered compound, it can be cleaved to present an excellent surface for SI-STM.

Compton profile, (e,2e) studies, and “atoms in molecules”

By direct comparison of (e,2e) measurements with Compton lineshape determinations, we first elucidate whether the (e, 2e) measurement strongly perturbs the molecular system. Specifically, results are presented for N2 and H2O. Although the error bars are somewhat large, excellent agreement is found between the two types of experiment. In order to relate such momentum space information on chemical bonding to other properties of interest, we have then examined: (i) The first moment of the measured molecular Compton profile. We show from the Thomas-Fermi statistical theory for molecules that this moment yields an approximation to the Dirac-Slater exchange energy, and values are given for N2 and H2O. (ii) The asymptotic form far from nuclei of the r space electron density which follows from density functional theory, and the justification it gives for “atomic-like building blocks” at large r or equivalently small momentum p. As predicted by the theory, the small p behavior of the Compton profile can be related between the equilibrium molecule, when the diffraction factor is removed, and the united atom, the link being established via the appropriate ionization potentials in the two configurations.