Quantum Phase-space Distributions Research Articles

Classical treatments of quantum mechanical effects in collisions of weakly bound complexes

Abstract Classical and quantum simulations of Ne + Ar2 collision dynamics are performed in order to investigate where quantum mechanical effects are most important and where classical simulations provide good descriptions of the dynamics. It is found that when Ar2 is in a low-lying vibrational state, the differences between the results of quantum and quasiclassical simulations are profound. However, excellent agreement between the results of the quantum and classical simulations can be achieved when the initial conditions for the classical trajectories are sampled from the quantum phase space distribution given by the Wigner function. These effects are largest when collisions occur under constrained geometries or when Ar2 is in its ground vibrational state. The results of this work suggest that sampling the initial conditions using the Wigner function provides a straightforward way to incorporate the most important quantum mechanical effects in simulations of collisions involving very cold weakly bound complexes.

Quark imaging in the proton via quantum phase-space distributions

We develop the concept of quantum phase-space (Wigner) distributions for quarks and gluons in the proton. To appreciate their physical content, we analyze the contraints from special relativity on the interpretation of elastic form factors, and examine the physics of the Feynman parton distributions in the proton's rest frame. We relate the quark Wigner functions to the transverse-momentum dependent parton distributions and generalized parton distributions, emphasizing the physical role of the skewness parameter. We show that the Wigner functions allow us to visualize quantum quarks and gluons using the language of classical phase space. We present two examples of the quark Wigner distributions and point out some model-independent features.

Viewing the proton through “color” filters

While the form factors and parton distributions provide separately the shape of the proton in coordinate and momentum spaces, a more powerful imaging of the proton structure can be obtained through quantum phase-space distributions. Here we introduce the Wigner-type quark and gluon distributions which depict a full-3D proton at every fixed Feynman momentum, like what is seen through momentum(“color”)-filters. After appropriate reductions, the phase-space distributions are related to the generalized parton distributions (GPDs) and transverse-momentum dependent parton distributions measurable in high-energy experiments.

Quantum localization in the three-dimensional kicked Rydberg atom

We study the three-dimensional (3D) unidirectionally kicked Rydberg atom. For parabolic initial states elongated in the direction of the kicks we show that the ionization of the quantum system is suppressed as compared to the classical counterpart and that the quantum wave function is localized along all degrees of freedom, whereas the classical system is globally diffusive. We discuss the connection to the previously studied one-dimensional (1D) model of the kicked Rydberg atom and verify that the 1D model is a good approximation to the 3D quantum case in the limiting case of the most elongated initial states. We further study the quantum phase-space distribution (Husimi distribution) of the eigenstates of the period-one time-evolution (Floquet) operator and show that the eigenstates are localized in phase space. For the most elongated parabolic initial state, we are able to identify the unstable periodic orbits around which Floquet states localize. We discuss the possibility of observing quantum localization in high Rydberg states in $ng100.$

Evolution of classical and quantum phase-space distributions: A new trajectory approach for phase space hydrodynamics

Recently, Donoso and Martens described a method for evolving both classical and quantum phase-space distribution functions, W(q,p,t), that involves the propagation of an ensemble of correlated trajectories. The trajectories are linked into a unified whole by spatial and momentum derivatives of density dependent terms in the equations of motion. On each time step, these nonlocal terms were evaluated by fitting the density around each trajectory to an assumed functional form. In the present study, we develop a different trajectory method for propagating phase-space distribution functions. A hierarchy of coupled analytic equations of motion are derived for the q and p derivatives of the density and a truncated set of these are integrated along each trajectory concurrently with the equation of motion for the density. The advantage of this approach is that individual trajectories can be propagated, one at a time, and function fitting is not required to evaluate the nonlocal terms. Regional nonlocality can be incorporated at various levels of approximation to “dress” what would otherwise be “thin” locally propagating trajectories. This derivative propagation method is used to obtain trajectory solutions for the Klein–Kramers equation, the Husimi equation, and for a smoothed version of the Caldeira–Leggett equation derived by the Diosi. Trajectory solutions are obtained for the relaxation of an oscillator in contact with a thermal bath and for the decay of a metastable state.

Quantum phase-space function formulation of reactive flux theory

On the basis of a coherent-state representation of the quantum noise operator and an ensemble averaging procedure a scheme for quantum Brownian motion has been proposed recently [Banerjee et al., Phys. Rev. E 65, 021109 (2002); 66, 051105 (2002)]. We extend this approach to formulate reactive flux theory in terms of quantum phase space distribution functions and to derive a time-dependent quantum transmission coefficient—a quantum analog of the classical Kramers–Grote–Hynes coefficient in the spirit of Kohen and Tannor’s classical formulation. The theory is valid for arbitrary noise correlation and temperature. The specific forms of this coefficient in the Markovian as well as in the non-Markovian limits have been worked out in detail for the intermediate to strong damping regimes with an analysis of quantum effects. While the classical transmission coefficient is independent of temperature, its quantum counterpart has significant temperature dependence particularly in the low-temperature regime.

NEW APPLICATIONS OF <η| REPRESENTATION IN THERMAL FIELD STATISTICS

We derive <η| representation of density matrices ρ for radiation fields, where <η| is the newly established coherent thermal state [H.-Y. Fan and Y. Fan, Phys. Lett.A246, 242 (1998)] in thermal field dynamics. We then apply <η|ρ> to calculate thermal averages and develop the theory for the quantum phase space distributions. We also study the quantum master equations in the <η| representation and find it has similar evolving property to the coherent representation of a pure state.

On marginalization of phase-space distribution functions

We discuss marginalization procedures based on integration of quantum phase-space distribution functions over a family of phase-space manifolds. We show that under some conditions the resulting marginals are always nonnegative.

Quantum phase space distributions in thermofield dynamics

It is shown that the quantum phase space distributions corresponding to a density operator can be expressed, in thermofield dynamics, as overlaps between the state and `thermal' coherent states. The usefulness of this approach is brought out in the context of a master equation describing a nonlinear oscillator for which exact expressions for the quantum phase distributions for an arbitrary initial condition are derived.

Correlation coefficient for incompatible observables of the quantum harmonic oscillator

We explore, using elementary examples of harmonic-oscillator superposition states, the quantum-mechanical measures of covariance and correlation coefficient for two noncommuting observables. A Schrödinger cat state which provides perfect correlation or anticorrelation of the position and momentum operators is presented. We employ quantum and classical phase-space distributions as graphical illustrations of correlation coefficients for the two theories.

Generalized antinormally ordered quantum phase-space distribution functions.

A class of positive quantum phase-space distributions based on antinormal ordering of the squeezed-photon annihilation and creation operators is introduced. This class of distributions, referred to as the generalized antinormally ordered distributions, is shown to be generated when the Wigner distribution is smoothed with a squeezed Gaussian wave packet. The distribution function associated with generalized antinormal ordering can thus be identified as the Husimi distribution function. The usefulness of the distributions in studies of the quantum particle dynamics is illustrated with simple examples.

Quantum surfaces of section for the diamagnetic hydrogen atom: Husimi functions versus Wigner functions

We compare quantum phase space distributions of individual quantum states of the diamagnetic hydrogen atom obtained by means of Wigner functions with those given by Husimi functions. The comparison is carried out at effective h(cross) approximately 0.035 and at a fixed scaled energy ( epsilon =-0.316) which corresponds to an especially interesting mixed classical phase-space structure. The object of the comparison is to establish which of the two distributions best correlates with the classical Poincare surface of section for a representative set of single states. As expected, the states investigated display the strongest positive intensity on a single local invariant phase-space structure (such as a scar or torus) and at this level there is little difference between the Husimi and the Wigner function. However the weak structures (fringes of the Wigner function or zeros of the Husimi function) are shown here to be radically different for the Wigner relative to the Husimi representation. In both cases the weak structures permeate all of phase space. In addition they show different character for integrable and chaotic dynamics and so reflect the global structure of phase-space to a much greater extent than the localized strong structure. The Wigner functions of selected states are shown to have additional dynamically significant structures which are not apparent in the Husimi functions. These include negative intensity features seen in the Wigner-but not the Husimi-functions which are very well correlated with structures of the classical Poincare surfaces of section. Despite its complex oscillatory nature the Wigner function delineates better classical features, for example showing the outline of islands of stability avoided by states supported by chaotic regions with greater sharpness than the Husimi function.

Positivization and regularization of quantum phase space distributions

It is shown that, to any quasiprobability distribution corresponding to a given density operator, one can associate a class of positive distributions in an extended space. In situations where the quasiprobability distributions are singular, these distributions are shown to provide regularized versions thereof. The positive distribution introduced by Drummond and Gardiner (1981) in the context of the Glauber-Sudarshan P-representation is shown to arise as a special case. One can thus associate a positive distribution to the Wigner function such that its moments give the averages of Weyl-ordered operators with respect to the given density operator. It is also found that one can associate to the Glauber-Sudarshan P-function, a distribution in the extended space which, though not positive, interestingly, involves the Wigner function (1932). A measurement scheme which directly yields these positive distributions is presented.

Phase-space distributions and spectral properties for non-hydrogenic atoms in magnetic fields

The authors present the first study of the quantum phase-space behaviour (Wigner functions) for non-hydrogenic atoms in magnetic fields as well as a comprehensive study of spectral properties. They consider primarily an energy regime (scaled energy -0.5) where hydrogen is near-integrable and hydrogenic wavefunctions would be localized on tori. They find that the quantum energy level statistics for non-hydrogenic atoms are at the 'chaotic' (Wigner) limit. However, the quantum phase space distributions, contrary to what one would expect if the underlying classical motion were chaotic, remain dominated by torus-like structures. But the wavefunctions do explore a larger fraction of phase-space than in the hydrogenic case where, in the integrable regime, Wigner wavefunctions are generally localized on a single torus. Due to the non-semiclassical nature of the core they are localized on more than one torus; additional structures other than the tori are also present. Possible interpretations of the results in terms of models of the underlying classical dynamics are discussed.

Generic properties of a class of translation invariant quantum maps

The existence of delocalized regimes at all positive values of the Planck constant is proved to be a generic property of quantum maps of the 'kicked Harper' type. The qualitative properties of delocalized wavepacket propagation are illustrated by numerical computations of the quantum-phase space distributions.

NON-RELATIVISTIC FERMIONS, COADJOINT ORBITS OF W∞ AND STRING FIELD THEORY AT c=1

We apply the method of coadjoint orbits of W∞-algebra to the problem of non-relativistic fermions in one dimension. This leads to a geometric formulation of the quantum theory in terms of the quantum phase space distribution of the Fermi fluid. The action has an infinite series of expansion in the string coupling, which to leading order reduces to the previously discussed geometric action for the classical Fermi fluid based on the group w∞ of area-preserving diffeomorphisms. We briefly discuss the strong coupling limit of the string theory which, unlike the weak coupling regime, does not seem to admit a two-dimensional space-time picture. Our methods are equally applicable to interacting fermions in one dimension.