Phase Space Representation Research Articles

Electron–hole coherent states for the Bogoliubov–de Gennes equation

We construct a new set of generalized coherent states, the electron–hole coherent states, for a (quasi-)spin particle on the infinite line. The definition is inspired by applications to the Bogoliubov–de Gennes equations where the quasi-spin refers to electron- and hole-like components of electronic excitations in a superconductor. Electron–hole coherent states generally entangle the space and the quasi-spin degrees of freedom. We show that the electron–hole coherent states allow a resolution of unity to be obtained and form minimum uncertainty states for position and velocity where the velocity operator is defined using the Bogoliubov–de Gennes Hamiltonian. The usefulness and the limitations of electron–hole coherent states and the phase space representations built from them are discussed in terms of basic applications to the Bogoliubov–de Gennes equation such as Andreev reflection.

Initiation and termination of intraseasonal oscillations in nonlinear Laplacian spectral analysis-based indices

AbstractWe present a statistical analysis of the initiation and termination of boreal winter and boreal summer intraseasonal oscillations (ISOs). This study uses purely convection (infrared brightness temperature) data over a 23-year time interval from 1984–2006. The indices are constructed via the nonlinear Laplacian spectral analysis (NLSA) method and display high intermittency and non-Gaussian statistics. We first define primary, terminal, and full events in the NLSA-based indices, and then examine their statistics through the associated two-dimensional phase space representations. Roughly one full event per year was detected for the Madden-Julian oscillation (MJO), and 1.3 full events per year for the boreal summer ISO.We also find that 91%of the recovered full MJO events are circumnavigating and exhibit very little to no retrograde (westward) propagation. The Indian Ocean emerges as the most active region in terms of both the onset and decay of events, however relevant activity occurs over all phases, consistent with previous work.

Schrödinger like equation for wavelets

An explicit phase space representation of the wave function is build based on a wavelet transformation. The wavelet transformation allows us to understand the relationship between s − ordered Wigner function, (or Wigner function when s = 0), and the Torres-Vega-Frederick’s wave functions. This relationship is necessary to find a general solution of the Schrödinger equation in phase-space.

Complex phase space and Weyl’s commutation relations

We revisit von Neumann’s determination of the representations of the canonical commutation relations in Weyl form. We present an exposition of the original insights set within the convenient notational framework of symplectic structures. We study von Neumann’s projection operator and show how the complex phase space representation arises.

Initial-value semiclassical propagators for the Wigner phase space representation: Formulation based on the interpretation of the Moyal equation as a Schrödinger equation.

We formulate various semiclassical propagators for the Wigner phase space representation from a unified point of view. As is shown in several studies, the Moyal equation, which is an equation of motion for the Wigner distribution function, can be regarded as the Schrödinger equation of an extended Hamiltonian system where its "position" and "momentum" correspond to the middle point of two points of the original phase space and the difference between them, respectively. Then we show that various phase-space semiclassical propagators can be formulated just by applying existing semiclassical propagators to the extended system. As a result, a phase space version of the Van Vleck propagator, the initial-value Van Vleck propagator, the Herman-Kluk propagator, and the thawed Gaussian approximation are obtained. In addition, we numerically compare the initial-value phase-space Van Vleck propagator, the phase-space Herman-Kluk propagator, and the classical mechanical propagation as approximation methods for the time propagation of the Wigner distribution function in terms of both accuracy and convergence speed. As a result, we find that the convergence speed of the Van Vleck propagator is far slower than others as is the case of the Hilbert space, and the Herman-Kluk propagator keeps its accuracy for a long period compared with the classical mechanical propagation while the convergence speed of the latter is faster than the former.

Thermodynamics of trajectories and local fluctuation theorems for harmonic quantum networks

We present a general method to undertake a thorough analysis of the thermodynamics of the quantum jump trajectories followed by an arbitrary quantum harmonic network undergoing linear and bilinear dynamics. The approach is based on the phase-space representation of the state of a harmonic network. The large deviation function associated with this system encodes the full counting statistics of exchange and also allows one to deduce fluctuation theorems (FTs) obeyed by the dynamics. We illustrate the method showing the validity of a local FT about the exchange of excitations between a restricted part of the environment (i.e., a local bath) and a harmonic network coupled with different schemes.

Analysis of High-Dimensional Phase Space via Poincaré Section for Patient-Specific Seizure Detection.

In this paper, the performance of the phase space representation in interpreting the underlying dynamics of epileptic seizures is investigated and a novel patient-specific seizure detection approach is proposed based on the dynamics of EEG signals. To accomplish this, the trajectories of seizure and nonseizure segments are reconstructed in a high dimensional space using time-delay embedding method. Afterwards, Principal Component Analysis (PCA) was used in order to reduce the dimension of the reconstructed phase spaces. The geometry of the trajectories in the lower dimensions is then characterized using Poincaré section and seven features were extracted from the obtained intersection sequence. Once the features are formed, they are fed into a two-layer classification scheme, comprising the Linear Discriminant Analysis (LDA) and Naive Bayesian classifiers. The performance of the proposed method is then evaluated over the CHB-MIT benchmark database and the proposed approach achieved 88.27% sensitivity and 93.21% specificity on average with 25% training data. Finally, we perform comparative performance evaluations against the state-of-the-art methods in this domain which demonstrate the superiority of the proposed method.

Fock expansion of multimode pure Gaussian states

The Fock expansion of multimode pure Gaussian states is derived starting from their representation as displaced and squeezed multimode vacuum states. The approach is new and appears to be simpler and more general than previous ones starting from the phase-space representation given by the characteristic or Wigner function. Fock expansion is performed in terms of easily evaluable two-variable Hermite–Kampé de Fériet polynomials. A relatively simple and compact expression for the joint statistical distribution of the photon numbers in the different modes is obtained. In particular, this result enables one to give a simple characterization of separable and entangled states, as shown for two-mode and three-mode Gaussian states.

Gaussian fidelity distorted by external fields

Gaussian state decoherence aspects due to interacting magnetic-like and gravitational fields are quantified through the quantum fidelity and Shannon entropy in the scope of the phase-space representation of elementary quantum systems. For Gaussian Wigner functions describing harmonic oscillator states, an interacting external field destroys the quantum fidelity and introduces a quantum beating behavior. Likewise, it introduces harmonic profiles for free particle systems. Some aspects of quantum decoherence for the quantum harmonic oscillator and for the free particle limit are also quantified through the Shannon entropy. For the gravitational quantum well, the effect of a magnetic-like field on the quantum fidelity is suppressed by the linear term of the gravitational potential. To conclude, one identifies a fine formal connection of the quantum decoherence aspects discussed here with the noncommutative quantum mechanics.

Using the small alignment index chaos indicator to characterize the vibrational dynamics of a molecular system: LiNC-LiCN.

A study of the dynamical characteristics of the phase space corresponding to the vibrations of the LiNC-LiCN molecule using an analysis based on the small alignment index (SALI) is presented. SALI is a good indicator of chaos that can easily determine whether a given trajectory is regular or chaotic regardless of the dimensionality of the system, and can also provide a wealth of dynamical information when conveniently implemented. In two-dimensional (2D) systems SALI maps are computed as 2D phase space representations, where the SALI asymptotic values are represented in color scale. We show here how these maps provide full information on the dynamical phase space structure of the LiNC-LiCN system, even quantifying numerically the volume of the different zones of chaos and regularity as a function of the molecule excitation energy.

Ray-wave correspondence in chaotic dielectric billiards.

Based on the reformulation of the boundary integral equations recently derived by Creagh, Hamdin, and Tanner [J. Phys. A: Math. Theor. 46, 435203 (2013)] together with semiclassical (short wavelength) approximation, we theoretically show that low-loss resonances of a fully chaotic dielectric billiard can be related with ray dynamical orbits whose intensities are weighted by the Fresnel reflection and transmission coefficients. In addition, it is revealed that intensity localization spots observed in the phase-space representation of an individual resonance wave function are ray-dynamically correlated.

Continuous sampling of the squeezed-state nonclassicality

We report the direct -- continuous in phase -- sampling of a regularized $P$ function, the so-called nonclassicality quasiprobability, for squeezed light. Through their negativities, the resulting phase-space representation uncovers the quantum character of the state. In contrast to discrete phase-locked measurements, our approach allows an unconditional verification of nonclassicality by getting rid of interpolation errors due to fixed phases. To realize the equal phase distribution of measured quadratures, a data selection is implemented with quantum random numbers created by measuring the vacuum noise. The continuously measured squeezed field was generated in an optical parametric amplifier. Suitable pattern functions for obtaining the regularized $P$ function are investigated. The significance of detecting negativities in our application is determined. The sampling of nonclassicality quasiprobabilities is shown to be a powerful and universal method to visualize quantum effects within arbitrary quantum states.

A Gaussian wave packet phase-space representation of quantum canonical statistics.

We present a mapping of quantum canonical statistical averages onto a phase-space average over thawed Gaussian wave-packet (GWP) parameters, which is exact for harmonic systems at all temperatures. The mapping invokes an effective potential surface, experienced by the wave packets, and a temperature-dependent phase-space integrand, to correctly transition from the GWP average at low temperature to classical statistics at high temperature. Numerical tests on weakly and strongly anharmonic model systems demonstrate that thermal averages of the system energy and geometric properties are accurate to within 1% of the exact quantum values at all temperatures.

Conditions for Multi-functionality in a Rhythm Generating Network Inspired by Turtle Scratching.

Rhythmic behaviors such as breathing, walking, and scratching are vital to many species. Such behaviors can emerge from groups of neurons, called central pattern generators, in the absence of rhythmic inputs. In vertebrates, the identification of the cells that constitute the central pattern generator for particular rhythmic behaviors is difficult, and often, its existence has only been inferred. For example, under experimental conditions, intact turtles generate several rhythmic scratch motor patterns corresponding to non-rhythmic stimulation of different body regions. These patterns feature alternating phases of motoneuron activation that occur repeatedly, with different patterns distinguished by the relative timing and duration of activity of hip extensor, hip flexor, and knee extensor motoneurons. While the central pattern generator network responsible for these outputs has not been located, there is hope to use motoneuron recordings to deduce its properties. To this end, this work presents a model of a previously proposed central pattern generator network and analyzes its capability to produce two distinct scratch rhythms from a single neuron pool, selected by different combinations of tonic drive parameters but with fixed strengths of connections within the network. We show through simulation that the proposed network can achieve the desired multi-functionality, even though it relies on hip unit generators to recruit appropriately timed knee extensor motoneuron activity, including a delay relative to hip activation in rostral scratch. Furthermore, we develop a phase space representation, focusing on the inputs to and the intrinsic slow variable of the knee extensor motoneuron, which we use to derive sufficient conditions for the network to realize each rhythm and which illustrates the role of a saddle-node bifurcation in achieving the knee extensor delay. This framework is harnessed to consider bistability and to make predictions about the responses of the scratch rhythms to input changes for future experimental testing.

Dynamics of two-cluster systems in phase space

We present a phase-space representation of quantum state vectors for two-cluster systems. Density distributions in the Fock–Bargmann space are constructed for bound and resonance states of 6,7Li and 7,8Be, provided that all these nuclei are treated within a microscopic two-cluster model. The density distribution in the phase space is compared with those in the coordinate and momentum representations. Bound states realize themselves in a compact area of the phase space, as also do narrow resonance states. We establish the quantitative boundaries of this region in the phase space for the nuclei under consideration. Quantum trajectories are demonstrated to approach their classical limit with increasing energy.

Quasiprobability distributions in stochastic wave-function methods

Quasiprobability distributions emerging in the stochastic wave-function method of Carusotto et al. [Phys. Rev. A 63, 023606 (2001)] are investigated. We show that there are actually two types of quasiprobabilities. The first one, the ``diagonal Hartree-Fock state projection'' representation, is useful in representing the initial conditions for stochastic simulation in the most compact form. It defines antinormally ordered expansion of the density operator and normally ordered mapping of the observables to be averaged. We completely characterize the equivalence classes of this phase-space representation. The second quasiprobability distribution, the ``nondiagonal Hartree-Fock state projection'' representation, extends the first one in order to achieve stochastic representation of the quantum dynamics. We demonstrate how the differential identities of the stochastic ansatz generate the automorphisms of this phase-space representation. These automorphisms turn the stochastic representation into a gauge theory. The gauge transformations of the quasiprobability master equation are described. In particular, it is the analyticity of the stochastic ansatz that allows one to transform the master equation into the genuine Fokker-Planck equation. We demonstrate how the different variants of the stochastic wave-function method can be constructed, first by choosing a certain optimality criteria or constraints, and then by satisfying these criteria with a suitable choice of gauge. The problem of boundary terms is considered. It is demonstrated that the simple scheme with Fock states of Carusotto et al. is not subjected to this problem.

Quantum Bochner’s theorem for phase spaces built on projective representations

Bochner’s theorem gives the necessary and sufficient conditions on a function such that its Fourier transform corresponds to a true probability density function. In the Wigner phase space picture, quantum Bochner’s theorem gives the necessary and sufficient conditions on a function such that it is a quantum characteristic function of a valid (and possibly mixed) quantum state and such that its Fourier transform is a true probability density. We extend this theorem to discrete phase space representations which possess enough symmetry. More precisely, we show that discrete phase space representations that are built on projective unitary representations of abelian groups, with a slight restriction on admissible two-cocycles, enable a quantum Bochner’s theorem.

Quantum dynamics in a tiered non-Markovian environment

We introduce a new analytical method for studying the open quantum systems problem of a discrete system weakly coupled to an environment of harmonic oscillators. Our approach is based on a phase space representation of the density matrix for a system coupled to a two-tiered environment. The dynamics of the system and its immediate environment are resolved in a non-Markovian way, and the environmental modes of the inner environment can themselves be damped by a wider ‘universe’. Applying our approach to the canonical cases of the Rabi and spin-boson models we obtain new analytical expressions for an effective thermalization temperature and corrections to the environmental response functions as direct consequences of considering such a tiered environment. A comparison with exact numerical simulations confirms that our approximate expressions are remarkably accurate, while their analytic nature offers the prospect of deeper understanding of the physics which they describe. A unique advantage of our method is that it permits the simultaneous inclusion of a continuous bath as well as discrete environmental modes, leading to wide and versatile applicability.

Phase-space representation of completely positive quantum operation: invertible subdynamics of two-mode squeezed states.

Completely positive quantum operations are frequently discussed in the contexts of statistical mechanics and quantum information. They are customarily given by maps forming positive operator-values measures. To intuitively understand the physical meanings of such abstract operations, the method of phase-space representations is examined. This method enables one to grasp the operations in terms of the classical statistical notions. As an example of physical importance, here, the phase-space representation of the completely positive quantum operation arising from the single-mode subdynamics of the two-mode squeezed vacuum state, which maps from the vacuum state at vanishing temperature to mixed states with perfect decoherence including the thermal state, is studied. It is found in the P representation that remarkably this operation is invertible, implying that coherence lost by the quantum operation can be recovered.

Study on Linear Canonical Transformation in a Framework of a Phase Space Representation of Quantum Mechanics

We present a study on linear canonical transformation in the framework of a phase space representation of quantum mechanics that we have introduced in our previous work [1]. We begin with a brief recall about the so called phase space representation. We give the definition of linear canonical transformation with the transformation law of coordinate and momentum operators. We establish successively the transformation laws of mean values, dispersions, basis state and wave functions.Then we introduce the concept of isodispersion linear canonical transformation.