Phase Space Representation Research Articles

The Wigner function of the relativistic finite-difference oscillator in an external field

The phase-space representation for a relativistic linear oscillator in a homogeneous external field expressed through the finite-difference equation is constructed. Explicit expressions of the relativistic oscillator Wigner quasi-distribution function for the stationary states as well as for the states of thermodynamical equilibrium are obtained and their correct limits are shown.

Definition and invariance properties of the complex degree of spatial coherence

Within the framework of the phase-space representation of random electromagnetic fields provided by electromagnetic spatial coherence wavelets, and by using the Fresnel-Arago laws for interference and polarization as an analysis tool, the meaning of the spatial coherence-polarization tensor and its invariance under transformations is studied. The results give new insight into the definition and properties of the complex degree of spatial coherence by showing that its invariance is not required for properly describing the behavior of random electromagnetic fields within the scope of physically measurable quantities.

Bohmian Total Potential View to Quantum Effects III. Tunnelling in Phase Space

The tunnel effect in one-dimensional collision problems is analyzed in terms of numerical wavepacket time propagations, in position, momentum, and phase space representations of the problem. Bohm's total potential is used to provide a classical-like description of tunnelling quantum dynamics, under a time- and density-dependent potential. Wigner, Husimi, as well as Bohm phase space distributions are used in the present study, with their relative performances for the present problem having been analyzed.

The Poincaré Group in a Demisemidirect Product with a Non-associative Algebra with Representations that Include Particles and Quarks—II

The quarks and particles’ mass and mass/spin relations are provided with coordinates in configuration space and/or momentum space by means of the marriage of ordinary Poincare group representations with a non-associative algebra made through a demisemidirect product, in the notation of Leibniz algebras. Thus, we circumvent the restriction that the Poincare group cannot be extended to a larger group by any means (including the (semi)direct product) to get even the mass relations. Finally, we will discuss a connection between the phase space representations of the Poincare group and the phase space representations of the associated Leibniz algebra.

Moyal phase-space analysis of nonlinear optical Kerr media

Nonlinear optical media of Kerr type are described by a particular version of an anharmonic quantum harmonic oscillator. The dynamics of this system can be described using the Moyal equations of motion, which correspond to a quantum phase-space representation of the Heisenberg equations of motion. For the Kerr system we derive exact solutions of the Moyal equations for an irreducible set of observables. These Moyal solutions incorporate the asymptotics of the classical limit in a simple, explicit form. An unusual feature of these solutions is that they exhibit periodic singularities in the time variable. These singularities are removed by the phase-space averaging required to construct the expectation value for an arbitrary initial state. Nevertheless, for strongly number-squeezed initial states the effects of the singularity remain observable.

Dynamic programming revisited: a generalized formalism for arbitrary ray trajectories in inhomogeneous optical media with radial dependence

We present a formalism based upon dynamic programming (DP), to characterize light propagation in particular GRIN (gradient index) media by analyzing ray trajectories associated with skew-type rays. We study the conditions for the formation of periodic trajectories and stability of the system. We perform a comparative study with the classical formalism based on the Hamilton-Jacobi equation. The DP formalism allows representation in phase (momentum) space.

Entangled state for constructing a generalized phase-space representation and its statistical behavior

We construct a generalized phase-space representation (GPSR) based on the idea of Einstein-Podolsky-Rosen quantum entanglement, i.e., we generalize the Torres-Vega-Frederick phase-space representation to the entangled case, which is characteristic of the features when two particles' relative coordinate, total momentum operators, and their conjugative variables, respectively, operate on the GPSR. This representation is complete and nonorthogonal. The Weyl-ordered form of the density operator of GPSR is derived, and its identification with the generalized Husimi operator is recognized, which clearly exhibit its statistical behavior. The minimum uncertainty relation obeyed by the GPSR is also demonstrated.

Localization of Resonance Eigenfunctions on Quantum Repellers

We introduce a new phase space representation for open quantum systems. This is a very powerful tool to help advance in the study of the morphology of their eigenstates. We apply it to two different versions of a paradigmatic model, the baker map. This allows us to show that the long-lived resonances are strongly scarred along the shortest periodic orbits that belong to the classical repeller. Moreover, the shape of the short-lived eigenstates is also analyzed. Finally, we apply an antiunitary symmetry measure to the resonances that allows us to quantify their localization on the repeller.

On a special picture of dynamical evolution of nonlinear quantum systems in the phase-space representation

A special picture of quantum evolution in the phase-space representation is derived. In this picture, both the symbols of operators as well as the distribution density, carry part of the time dependence. The symbols of operators evolve classically, whereas the distribution density carries the remaining time dependence, which is needed to recover the full time dependence of quantum averages. Such representation is free of semiclassical expansions or a mixed classical-quantum description. The spin-boson Hamiltonian system is explicitly studied; its phase-space structure is established by obtaining analytical solutions for the three stable regions. Quantum averages are calculated.

Mapping approach for quantum-classical time correlation functions

The calculation of quantum canonical time correlation functions is considered in this paper. Transport properties, such as diffusion and reaction rate coefficients, can be determined from time integrals of these correlation functions. Approximate quantum-classical expressions for correlation functions, which are amenable to simulation, are derived. These expressions incorporate the full quantum equilibrium structure of the system but approximate the dynamics by quantum-classical evolution where a quantum subsystem is coupled to a classical environment. The main feature of the formulation is the use of a mapping basis where the subsystem quantum states are represented by fictitious harmonic oscillator states. This leads to a full phase space representation of the dynamics that can be simulated without appeal to surface-hopping methods. The results in this paper form the basis for new simulation algorithms for the computation of quantum transport properties of large many-body systems.

Moshinsky atom and density functional theory — A phase space view

The problem of two particles in a common harmonic oscillator potential interacting through harmonic oscillator forces is discussed in the Weyl–Wigner phase-space representation. The Wigner function of the system is an ordinary function of the phase-space constants of the motion. The density functional description of the system is touched upon. The functional that determines the confining harmonic potential from the particle density is also an ordinary function.

Quantum phase-space distributions with compact support

Phase-space representations of quantum distributions such as the Wigner function and Kadanoff–Baym Green's functions do not have compact support which leads to difficulties for numerical and perturbative schemes in quantum transport simulations of nano-devices. We demonstrate that a useful complex-valued quantum phase-space distribution with compact support can be constructed. In many practical applications only the real part is required to obtain useful dynamical averages.

Framed Hilbert space: hanging the quasi-probability pictures of quantum theory

Building on earlier work, we further develop a formalism based on the mathematical theory of frames that defines a set of possible phase-space or quasi-probability representations of finite-dimensional quantum systems. We prove that an alternate approach to defining a set of quasi-probability representations, based on a more natural generalization of a classical representation, is equivalent to our earlier approach based on frames, and therefore is also subject to our no-go theorem for a non-negative representation. Furthermore, we clarify the relationship between the contextuality of quantum theory and the necessity of negativity in quasi-probability representations and discuss their relevance as criteria for non-classicality. We also provide a comprehensive overview of known quasi-probability representations and their expression within the frame formalism.

Dynamical properties of dipolar Fermi gases

In this paper, we investigate the dynamical properties of a one-component Fermi gas with dipole–dipole interaction between particles. Using a variational function based on the Thomas–Fermi density distribution in phase space representation, the total energy is described as a function of deformation parameters in both real and momentum spaces. Various thermodynamic quantities of a uniform dipolar Fermi gas are derived, and then instability of this system is discussed. For a trapped dipolar Fermi gas, the collective oscillation frequencies are derived with the energy-weighted sum rule method. The frequencies for the monopole, quadrupole, radial and axial modes are calculated, and softening against collapse is shown as the dipolar strength approaches the critical value. Finally, we investigate the expansion dynamics of the Fermi gas and show how the dipolar interaction manifests itself in the shape of an expanded cloud.

Pulse propagation and classification in time/frequency and position/wave number phase space.

In dispersive pulse propagation, different frequencies travel at different velocities, and, hence, the pulse is nonstationary in that it changes over time and distance. Joint phase space representations, such as the Wigner distribution and the spectrogram, are often employed to study nonstationary signals, and, hence, it is natural to consider their application to dispersive propagation. We consider position/wave number and time/frequency representations of a pulse propagating with dispersion and damping. We give the Wigner distribution of the pulse at a particular time/position in terms of the Wigner distribution of the initial pulse. Approximations of this result are derived, which are accurate and remarkably simple to apply, as compared to the stationary phase approximation. The approximations provide insights as well, in that they show how each point in phase space propagates at the group velocity, and lead to new features for classification, based on local phase space moments, that are invariant to t...

Phase-space representation of electromagnetic radiometry

The phase-space representation of electromagnetic radiometry is founded on the electromagnetic generalized radiance tensors, which allow overcoming the limitations due to the scalar electromagnetic generalized radiances. The fundamental quantities of both scalar generalized radiometry and classical radiometry or photometry become particular cases. The transport of measurable radiometric quantities by the electromagnetic field is described in terms of the propagation of the contributions from individual radiators and their redistribution over each wavefront on propagation. A physical meaning is given to the negative values of the generalized radiance, which gives new insights into Poynting's theory of energy transport.

Qubit phase space:SU(n)coherent-statePrepresentations

We introduce a phase-space representation for qubits and spin models. The technique uses an $\mathrm{SU}(n)$ coherent-state basis and can equally be used for either static or dynamical simulations. We review previously known definitions and operator identities, and show how these can be used to define an off-diagonal, positive phase-space representation analogous to the positive-$P$ function. As an illustration of the phase-space method, we use the example of the Ising model, which has exact solutions for the finite-temperature canonical ensemble in two dimensions. We show how a canonical ensemble for an Ising model of arbitrary structure can be efficiently simulated using SU(2) or atomic coherent states. The technique utilizes a transformation from a canonical (imaginary-time) weighted simulation to an equivalent unweighted real-time simulation. The results are compared to the exactly soluble two-dimensional case. We note that Ising models in one, two, or three dimensions are potentially achievable experimentally as a lattice gas of ultracold atoms in optical lattices. The technique is not restricted to canonical ensembles or to Ising-like couplings. It is also able to be used for real-time evolution and for systems whose time evolution follows a master equation describing decoherence and coupling to external reservoirs. The case of $\mathrm{SU}(n)$ phase space is used to describe $n$-level systems. In general, the requirement that time evolution be stochastic corresponds to a restriction to Hamiltonians and master equations that are quadratic in the group generators or generalized spin operators.

Landau–Zener tunnelling in dissipative circuit QED

We investigate the influence of temperature and dissipation on the Landau–Zener transition probability in circuit QED. Dissipation is modelled by coupling the transmission line to a bath of harmonic oscillators, and the reduced density operator is treated within Bloch–Redfield theory. A phase-space representation allows an efficient numerical implementation of the resulting master equation. It provides reliable results which are valid even for rather low temperatures. We find that the spin-flip probability as a function of temperature and dissipation strength exhibits a non-monotonic behaviour. Our numerical results are complemented by analytical solutions for zero temperature and for vanishing dissipation strength.

Quantumness witnesses

A recently proposed test of quantumness Alicki and Van Ryn (2008 J. Phys. A: Math. Theor. 41 062001) is put into a broader mathematical and physical perspective. The notion of quantumness witnesses is introduced, in analogy to entanglement witnesses, and is illustrated by examples of single qubit and many-body systems with additive observables. We also compare our proposal with the quantumness test based on quantum correlations (entanglement) and Bell inequalities, and go on to discuss a class of quantumness witnesses associated with the phase-space representation of quantum mechanics.

Generalized squeezing operators, bipartite Wigner functions and entanglement via Wehrl's entropy functionals

We introduce a new class of unitary transformations based on the Lie algebra that generalizes, for certain particular representations of its generators, well-known squeezing transformations in quantum optics. To illustrate our results, we focus on the two-mode bosonic representation and show how the parametric amplifier model can be modified in order to generate such a generalized squeezing operator. Furthermore, we obtain a general expression for the bipartite Wigner function which allows us to identify two distinct sources of entanglement, here labelled dynamical and kinematical entanglement. We also establish a quantitative estimate of entanglement for bipartite systems through some basic definitions of entropy functionals in continuous phase-space representations.