Phase Space Representation Research Articles

Interference in phase space

The phase-space representation of interference based on the marginal power spectrum gives new insight on interference, enlarging its potential applications by means of the principle of spatial coherence modulation. Carrier and (0,pi)-rays produced by three different types of supports are introduced for describing interference as the result of adding the radiant energy propagated by the carriers and the modulating energy (which can be positive or negative) propagated by the (0,pi)-rays. Numerical examples are presented.

Scaling law and stability for a noisy quantum system

We show that a scaling law exists for the near-resonant dynamics of cold kicked atoms in the presence of a randomly fluctuating pulse amplitude. Analysis of a quasiclassical phase-space representation of the quantum system with noise allows a new scaling law to be deduced. The scaling law and associated stability are confirmed by comparison with quantum simulations and experimental data.

Weyl–Wigner formulation of noncommutative quantum mechanics

We address the phase-space formulation of a noncommutative extension of quantum mechanics in arbitrary dimension, displaying both spatial and momentum noncommutativities. By resorting to a covariant generalization of the Weyl–Wigner transform and to the Darboux map, we construct an isomorphism between the operator and the phase-space representations of the extended Heisenberg algebra. This map provides a systematic approach to derive the entire structure of noncommutative quantum mechanics in phase space. We construct the extended star product and Moyal bracket and propose a general definition of noncommutative states. We study the dynamical and eigenvalue equations of the theory and prove that the entire formalism is independent of the particular choice of the Darboux map. Our approach unifies and generalizes all the previous proposals for the phase-space formulation of noncommutative quantum mechanics. For concreteness we rederive these proposals by restricting our formalism to some two-dimensional spaces.

Phase-Space Representations in Optics: introduction to the feature issue

This special issue is devoted to a comprehensive cross section of current trends on the applications of phase space optics (PSO). The contributed papers in this issue cover a wide range of applications. In a way, these papers show that PSO is leaving its infant state to become a versatile tool for pushing the boundaries of applied optics.

Spatial coherence wavelets and phase-space representation of diffraction

The phase-space representation of the Fresnel-Fraunhofer diffraction of optical fields in any state of spatial coherence is based on the marginal power spectrum carried by the spatial coherence wavelets. Its structure is analyzed in terms of the classes of source pairs and the spot of the field, which is treated as the hologram of the map of classes. Negative values of the marginal power spectrum are interpreted as negative energies. The influence of the aperture edge on diffraction is stated in terms of the distortion of the supports of the complex degree of spatial coherence near it. Experimental results are presented.

Phase space representation of spatially partially coherent imaging

The phase space representation of imaging with optical fields in any state of spatial coherence is developed by using spatial coherence wavelets. It leads to new functions for describing the optical transfer and response of imaging systems when the field is represented by Wigner distribution functions. Specific imaging cases are analyzed in this context, and special attention is devoted to the imaging of two point sources.

Toward chemical applications of Heaviside Operational Ansatz: exact solution of radial Schrodinger equation for nonrelativistic N-particle system with pairwise $${\frac{1}{r_{ij}}}$$ radial potential in quantum phase space

This brief note presents the exact analytical solution of an example class of dynamical systems, realized by differential equations. These problems occur in quantum mechanics and their generalizations in theoretical physics, but they particularly find application to chemical systems. Addressed is the exact analytical solution of the radial Schrodinger equation for the nonrelativistic N-particle system with pairwise \({\frac{1}{r_{ij}}}\) radial potential dynamics in the quantum phase space representation (QPSR) (Torres-Vega, Frederick, J. Chem. Phys. 98 3103, 1993). Applying the formal symbolic solution scheme Heaviside Operational Ansatz (heretofore, HOA) put forward in (Electron. J. Theor. Phys. 1 , 10–16, 2004), reduces solution of the radial Schrodinger equation to exact quadratures.

Phase-space representation and polarization domains of random electromagnetic fields

The phase-space representation of stationary random electromagnetic fields is developed by using electromagnetic spatial coherence wavelets. The propagation of the field's power and states of spatial coherence and polarization results from correlations between the components of the field vectors at pairs of points in space. Polarization domains are theoretically predicted as the structure of the field polarization at the observation plane. In addition, the phase-space representation provides a generalization of the Poynting theorem. Theoretical predictions are examined by numerically simulating the Young experiment with electromagnetic waves. The experimental implementation of these results is a current subject of research.

Quantum Nonlocality and Entanglement of Pair Cat States with Phase Decoherence

We investigate the entanglement of pair cat states in the phase damping channel by adopting the log-negativity and then study the possible violations of Bell's inequalities for the pair cat states in terms of the Wigner representation in phase space based upon parity measurement and displacement operation.

Master Equation in Phase Space for a Uniaxial Spin System

A master equation, for the time evolution of the quasi-probability density function of spin orientations in the phase space representation of the polar and azimuthal angles is derived for a uniaxial spin system subject to a magnetic field parallel to the axis of symmetry. This equation is obtained from the reduced density matrix evolution equation (assuming that the spin-bath coupling is weak and that the correlation time of the bath is so short that the stochastic process resulting from it is Markovian) by expressing it in terms of the inverse Wigner-Stratonovich transformation and evaluating the various commutators via the properties of polarization operators and spherical harmonics. The properties of this phase space master equation, resembling the Fokker-Planck equation, are investigated, leading to a finite series (in terms of the spherical harmonics) for its stationary solution, which is the equilibrium quasi-probability density function of spin “orientations” corresponding to the canonical density matrix and which may be expressed in closed form for a given spin number. Moreover, in the large spin limit, the master equation transforms to the classical Fokker-Planck equation describing the magnetization dynamics of a uniaxial paramagnet.

Phase-space equilibrium distributions and their applications to spin systems with nonaxially symmetric Hamiltonians

The Fourier series representation of the equilibrium quasiprobability density function ${W}_{S}(\ensuremath{\vartheta},\ensuremath{\varphi})$ or Wigner function of spin ``orientations'' for arbitrary spin Hamiltonians in a representation (phase) space of the polar angles $(\ensuremath{\vartheta},\ensuremath{\varphi})$ (analogous to the Wigner function for translational motion) arising from the generalized coherent state representation of the density operator is evaluated explicitly for some nonaxially symmetric problems including a uniaxial paramagnet in a transverse external field, a biaxial, and a cubic system. It is shown by generalizing transition state theory to spins [i.e., calculating the escape rate using the equilibrium density function ${W}_{S}(\ensuremath{\vartheta},\ensuremath{\varphi})$ only] that one may evaluate the reversal time of the magnetization. The quantum corrections to the transition state theory escape rate equation for classical magnetic dipoles appear both in the prefactor and in the exponential part of the escape rate and exhibit a marked dependence on the spin number. Furthermore, the phase-space representation allows us to estimate the switching field curves and/or surfaces for spin systems because quantum effects in these fields can be estimated via Thiaville's geometrical method [Phys. Rev. B 61, 12221 (2000)] for the study of the magnetization reversal of single-domain ferromagnetic particles. The calculation is accomplished (just as the determination of the equilibrium quasiprobability distributions in the phase space of the polar angles) by calculating switching field curves and/or surfaces using the Weyl symbol ($c$-number representation) of the Hamiltonian operator for given magnetocrystalline-Zeeman energy terms. Examples of such calculations for various spin systems are presented. Moreover, the reversal time of the magnetization allows us to estimate thermal effects on the switching fields for spin systems.

Von Neumann Entropies Analysis in Hilbert Space for the Dissociation Processes of Homonuclear and Heteronuclear Diatomic Molecules

Quantum Information Theory is a new field with potential implications for the conceptual foundations of Quantum Mechanics through density matrices. In particular, information entropies in Hilbert space representation are highly advantageous in contrast with the ones in phase space representation since they can be eas- ily calculated for large systems. In this work, novel von Neumann conditional, mutual, and joint entropies are employed to analyze the dissociation process of small molecules, Cl2 and HCl, by using the spectral decomposition of the first reduced density matrix in natural atomic orbital-based representation which allows us to assure rota- tional invariance, N- and v-representability in the Atoms-in-Molecules (AIM) scheme. Quantum information entropies permit to analyze the dissociation process through quantum mechanics concepts such as electron correlation and entanglement, showing interesting critical points which are not present in the energy profile, such as charge depletion and accumulation, along with bond breaking regions.

Explicit state vector for Torres-Vega–Frederick phase space representation and its statistical behavior

We find the explicit state vector for Torres-Vega-Frederick phase space representation [Go. Torres-Vega and J. H. Frederick, J. Chem. Phys. 98, 3103 (1993)], denoted by Gamma. This set of states make up a complete and nonorthogonal representation. The Weyl ordered form of Gamma Gamma [see text for the sign] is derived, which can clearly exhibit the statistical behavior of marginal distribution of Gamma Gamma [see text for the sign]. The minimum uncertainty relation for mid R:Gamma is demonstrated, which shows it being a coherent squeezed state.

Quantum chaos in the mesoscopic device for the Josephson flux qubit

We show that the three-junction SQUID device designed for the Josephson flux qubit can be used to study quantum chaos when operated at high energies. In the parameter region where the system is classically chaotic we analyze the spectral statistics. The nearest neighbor distributions $P(s)$ are well fitted by the Berry Robnik theory employing as free parameters the pure classical measures of the chaotic and regular regions of phase space in the different energy regions. The phase space representation of the wave functions is obtained via the Husimi distributions and the localization of the states on classical structures is analyzed.

Master equation of the reduced statistical operator of an atom in a plasma

We use the nonequilibrium Liouville equation to derive the master equation for the reduced statistical operator in a heat bath represented by a many-particle environment. Focusing on the case of a weak system-bath coupling, we consider the Born-Markov approximation of the master equation and compare the result to different approaches. The master equation is elaborated for the special case of an atom as a reduced system in a plasma background. We find that the dynamical structure factor determines the effect of the plasma on the reduced system. We consider the operator equation in the atomic eigenstate and in the phase-space representation, which yields two limiting cases: quantum mechanical behavior similar to the isolated atom for the lower strongly bound levels and a semiclassical one for highly excited Rydberg levels.

Holography in phase space

Holography is reformulated by using the framework of phase-space optics. The Leith-Upatnieks off-axis reference hologram is compared with precursors, namely, single-sideband holography. The phase-space representation of complex amplitudes focuses on similarities between different holographic recording schemes and is particularly useful for investigating the degree of freedom and the space-bandwidth product of optical signals and systems. This allows one to include computer-generated holography and recent developments in digital holography in the discussion.

Phase space representations and perturbation theory for continuous-time histories

We consider two technical developments of the formalism of continuous-time histories. First, we provide an explicit description of histories of the simple harmonic oscillator on the classical history phase space, comparing and contrasting the Q, P, and Wigner representations; we conclude that a representation based on coherent states is the most appropriate. Second, we demonstrate a generic method for implementing a perturbative approach for interacting theories in the history formalism, using the quartic anharmonic oscillator. We make use of the identification of the closed-time-path generating functional with the decoherence functional to develop a perturbative expansion for the latter up to second order in the coupling constant. We consider both configuration space and phase space histories.

Detecting quantum light

The quantization of light is the basis for quantum optics and has led to the observation of a multitude of genuine quantum effects which cannot be explained by classical physics. Yet, there exist different views when we refer to the distinct quantum character of light as opposed to classical fields. A major impact on how we describe photonic states is caused by the detection method we use for the quantum state characterization. In the theoretical modelling measuring light always means the recording of photon statistics of some kind, though the way we interpret the results is quite different and depends mainly on the intensity of actual detected light. If we use conventional photodiodes and monitor bright light, we attribute the detected statistics to quadrature measurements in a phase space representation, or—in other words—we identify the quantum state by studying its uncertainties of its field amplitude and phase properties which correspond to conjugate quantum observables. For multi-photon states with very low intensity we have to employ avalanche photodiodes (APDs) to be able to see any signal, but photon number resolution seems difficult in this regime. Recent developments allow one to ascertain information about the photon statistics from APD measurements opening up new routes for characterizing photonic states. In this article we review the different methods for modern quantum state metrology where we include theoretical aspects as well as the description of the state-of-the-art technology for measuring photon statistics.

The Wigner function of a q-deformed harmonic oscillator model

The phase space representation for a q-deformed model of the quantum harmonic oscillator is constructed. We have found explicit expressions for both the Wigner and Husimi distribution functions for the stationary states of the q-oscillator model under consideration. The Wigner function is expressed as a basic hypergeometric series, related to the Al-Salam-Chihara polynomials. It is shown that, in the limit case h → 0 (q → 1), both the Wigner and Husimi distribution functions reduce correctly to their well-known non-relativistic analogues. Surprisingly, examination of both distribution functions in the q-deformed model shows that, when q ≪ 1, their behaviour in the phase space is similar to the ground state of the ordinary quantum oscillator, but with a displacement towards negative values of the momentum. We have also computed the mean values of the position and momentum using the Wigner function. Unlike the ordinary case, the mean value of the momentum is not zero and it depends on q and n. The ground-state-like behaviour of the distribution functions for excited states in the q-deformed model opens quite new perspectives for further experimental measurements of quantum systems in the phase space.

“Reduced” multidimensional Vlasov simulations, with applications to electrostatic structures in space plasmas

A class of two-dimensional open-boundary kinetic plasma simulations characterized as reduced Vlasov simulations have been developed and applied to the study of double layers and electron holes in Earth’s auroral downward current region. In these simulations, full Vlasov phase-space dynamics is followed only in the direction of the magnetic field or imposed current, with a limited representation of phase space employed in the perpendicular spatial dimension. These reduced methods include a dissipative moment closure for unmagnetized species and limitation of perpendicular velocity space to a ring of modes at a fixed v⊥ for weakly magnetized species. Simulations with weakly magnetized ions exhibit ion-Bernstein waves in association with electron holes and simulations with weakly magnetized or unmagnetized electrons result in electron holes elongated perpendicular to B. These results are in qualitative agreement with satellite observations and previous particle simulations.