Phase Space Representation Research Articles

Optimal lossy quantum interferometry in phase space

We analyse the phase space representation of the optimal measurement of a phase shift in an interferometer with equal photon loss in both its arms. In the local phase estimation scenario with a fixed number of photons, we identify features of the spin Wigner function that warrant sub-shot-noise precision, and discuss their sensitivity to losses. We derive the asymptotic form of an integral kernel describing the process of photon loss in the phase space in the limit of large photon numbers. The analytic form of this kernel allows one to assess the ultimate precision limit for a lossy interferometer. We also provide a general lower bound on the quantum Fisher information in terms of spin Wigner functions.

Predicting Financial Market Crashes Using Ghost Singularities

We analyse the behaviour of a non-linear model of coupled stock and bond prices exhibiting periodically collapsing bubbles. By using the formalism of dynamical system theory, we explain what drives the bubbles and how foreshocks or aftershocks are generated. A dynamical phase space representation of that system coupled with standard multiplicative noise rationalises the log-periodic power law singularity pattern documented in many historical financial bubbles. The notion of ‘ghosts of finite-time singularities’ is introduced and used to estimate the end of an evolving bubble, using finite-time singularities of an approximate normal form near the bifurcation point. We test the forecasting skill of this method on different stochastic price realisations and compare with Monte Carlo simulations of the full system. Remarkably, the former is significantly more precise and less biased. Moreover, the method of ghosts of singularities is less sensitive to the noise realisation, thus providing more robust forecasts.

Automated identification of epileptic seizures in EEG signals based on phase space representation and statistical features in the CEEMD domain

Abstract Epileptic seizure detection based on visual inspection by expert physicians is burdensome, and subject to error and bias. In this work, we present a novel method for the automated identification of epileptic seizure using a single-channel EEG signal. We utilize the complete ensemble empirical mode decomposition with adaptive noise (CEEMDAN) technique to devise an effective feature extraction scheme for physiological signal analysis, and construct the corresponding growth curve. Then, various statistical features are extracted from the growth curve as the feature set, and this is fed to the random forest classifier for completing the detection. The suitability of the extracted features is established through statistical measures and graphical analysis. The proposed method is evaluated for the well-known problem of classifying epileptic seizure and seizure-free signals using a publically available EEG database from the University of Bonn. To assess the performance of the classification method, 10-fold cross-validation is performed. Compared to state-of-the-art algorithms, the numerical results confirm the superior algorithm performance of the proposed scheme in terms of accuracy, sensitivity, specificity, and Cohen’s Kappa statistics.

Local time of Lévy random walks: A path integral approach.

The local time of a stochastic process quantifies the amount of time that sample trajectories x(τ) spend in the vicinity of an arbitrary point x. For a generic Hamiltonian, we employ the phase-space path-integral representation of random walk transition probabilities in order to quantify the properties of the local time. For time-independent systems, the resolvent of the Hamiltonian operator proves to be a central tool for this purpose. In particular, we focus on the local times of Lévy random walks (Lévy flights), which correspond to fractional diffusion equations.

Wigner functions for gauge equivalence classes of unitary irreducible representations of noncommutative quantum mechanics

While Wigner functions forming phase space representation of quantum states is a well-known fact, their construction for noncommutative quantum mechanics (NCQM) remains relatively lesser known, in particular with respect to gauge dependencies. This paper deals with the construction of Wigner functions of NCQM for a system of 2-degrees of freedom using 2-parameter families of gauge equivalence classes of unitary irreducible representations (UIRs) of the Lie group $\g$ which has been identified as the kinematical symmetry group of NCQM in an earlier paper. This general construction of Wigner functions for NCQM, in turn, yields the special cases of Landau and symmetric gauges of NCQM.

Phase-space representations of symmetric informationally complete positive-operator-valued-measure fiducial states

The problem of finding symmetric informationally complete POVMs (SIC-POVMs) has been solved numerically for all dimensions $d$ up to 67 (A.J. Scott and M. Grassl, {\it J. Math. Phys.} 51:042203, 2010), but a general proof of existence is still lacking. For each dimension, it was shown that it is possible to find a SIC-POVM which is generated from a fiducial state upon application of the operators of the Heisenberg-Weyl group. We draw on the numerically determined fiducial states to study their phase-space features, as displayed by the characteristic function and the Wigner, Bargmann and Husimi representations, adapted to a Hilbert space of finite dimension. We analyze the phase-space localization of fiducial states, and observe that the SIC-POVM condition is equivalent to a maximal delocalization property. Finally, we explore the consequences in phase space of the conjectured Zauner symmetry. In particular, we construct an Hermitian operator commuting with this symmetry that leads to a representation of fiducial states in terms of eigenfunctions with definite semiclassical features.

Dispersion Operators Algebra and Linear Canonical Transformations

This work intends to present a study on relations between a Lie algebra called dispersion operators algebra, linear canonical transformation and a phase space representation of quantum mechanics that we have introduced and studied in previous works. The paper begins with a brief recall of our previous works followed by the description of the dispersion operators algebra which is performed in the framework of the phase space representation. Then, linear canonical transformations are introduced and linked with this algebra. A multidimensional generalization of the obtained results is given.

Coherent states in projected Hilbert spaces

Coherent states in a projected Hilbert space have many useful properties. When there are conserved quantities, a representation of the entire Hilbert space is not necessary. The same issue arises when conditional observations are made with postselected measurement results. In these cases, only a part of the Hilbert space needs to be represented, and one can define this restriction by way of a projection operator. Here coherent state bases and normally ordered phase-space representations are introduced for treating such projected Hilbert spaces, including existence theorems and dynamical equations. These techniques are very useful in studying novel optical or microwave integrated photonic quantum technologies, such as boson sampling or Josephson quantum computers. In these cases, states become strongly restricted due to inputs, nonlinearities, or conditional measurements. This paper focuses on coherent phase states, which have especially simple properties. Practical applications are reported on calculating recurrences in anharmonic oscillators, the effects of arbitrary phase noise on Schr\"odinger cat fringe visibility, and on boson sampling interferometry for large numbers of modes.

WavePacket: A Matlab package for numerical quantum dynamics. I: Closed quantum systems and discrete variable representations

WavePacket is an open-source program package for the numerical simulation of quantum-mechanical dynamics. It can be used to solve time-independent or time-dependent linear Schrödinger and Liouville–von Neumann-equations in one or more dimensions. Also coupled equations can be treated, which allows to simulate molecular quantum dynamics beyond the Born–Oppenheimer approximation. Optionally accounting for the interaction with external electric fields within the semiclassical dipole approximation, WavePacket can be used to simulate experiments involving tailored light pulses in photo-induced physics or chemistry. The graphical capabilities allow visualization of quantum dynamics ‘on the fly’, including Wigner phase space representations. Being easy to use and highly versatile, WavePacket is well suited for the teaching of quantum mechanics as well as for research projects in atomic, molecular and optical physics or in physical or theoretical chemistry. The present Part I deals with the description of closed quantum systems in terms of Schrödinger equations. The emphasis is on discrete variable representations for spatial discretization as well as various techniques for temporal discretization. The upcoming Part II will focus on open quantum systems and dimension reduction; it also describes the codes for optimal control of quantum dynamics. The present work introduces the MATLAB version of WavePacket 5.2.1 which is hosted at the Sourceforge platform, where extensive Wiki-documentation as well as worked-out demonstration examples can be found. Program summaryProgram Title: WavePacketProgram Files doi:http://dx.doi.org/10.17632/tc5mrhx9sb.1Licensing provisions: GPLv3Programming language:MatlabNature of problem:Schrödingerś equations are of fundamental importance in non-relativistic quantum mechanics of distinguishable particles. The solutions of the time-independent equation (TISE) are wavefunctions in coordinate space, the absolute squares of which are usually interpreted as probability density. The time-dependent equation (TDSE) describes the dynamics of a quantum system evolving in time. It plays a crucial role for the simulation, understanding, and prediction of modern experiments in atomic, molecular and optical physics where systems are driven by temporally shaped external fields.Solution method:All numerical methods compiled in WavePacket are based on discrete variable representations. Currently implemented are Gauss–Hermite, Gauss–Legendre and FFT-based schemes. The TISE is solved either by direct diagonalization or by propagation in imaginary time. For the TDSE there is a choice between second order differencing, operator splitting and Chebychev polynomial methods.Additional comments including Restrictions and Unusual features:The WavePacket program package is rather easy and intuitive to use, providing visualization of quantum dynamics ‘on the fly’. It is mainly intended for low-dimensional systems, typically not exceeding three to five degrees of freedom. Detailed user guides and reference manuals are available through numerous Wiki pages hosted at the SourceForge platform where also a large number of well documented demonstration examples can be found.

Grassmann phase-space methods for fermions: Uncovering the classical probability structure

The phase-space description of bosonic quantum systems has numerous applications in such fields as quantum optics, trapped ultracold atoms, and transport phenomena. Extension of this description to the case of fermionic systems leads to formal Grassmann phase-space quasiprobability distributions and master equations. The latter are usually considered as not possessing probabilistic interpretation and as not directly computationally accessible. Here, we describe how to construct $c$-number interpretations of Grassmann phase-space representations and their master equations. As a specific example, the Grassmann $B$ representation is considered. We discuss how to introduce $c$-number probability distributions on Grassmann algebra and how to integrate them. A measure of size and proximity is defined for Grassmann numbers, and the Grassmann derivatives are introduced which are based on infinitesimal variations of function arguments. An example of $c$-number interpretation of formal Grassmann equations is presented.

Zonal-flow dynamics from a phase-space perspective

The wave kinetic equation (WKE) describing drift-wave (DW) turbulence is widely used in the studies of zonal flows (ZFs) emerging from DW turbulence. However, this formulation neglects the exchange of enstrophy between DWs and ZFs and also ignores effects beyond the geometrical-optics limit. We derive a modified theory that takes both of these effects into account, while still treating DW quanta (“driftons”) as particles in phase space. The drifton dynamics is described by an equation of the Wigner–Moyal type, which is commonly known in the phase-space formulation of quantum mechanics. In the geometrical-optics limit, this formulation features additional terms missing in the traditional WKE that ensure exact conservation of the total enstrophy of the system, in addition to the total energy, which is the only conserved invariant in previous theories based on the WKE. Numerical simulations are presented to illustrate the importance of these additional terms. The proposed formulation can be considered as a phase-space representation of the second-order cumulant expansion, or CE2.

Phase space representation of neutron monitor count rate and atmospheric electric field in relation to solar activity in cycles 21 and 22.

Heliospheric modulation of galactic cosmic rays links solar cycle activity with neutron monitor count rate on earth. A less direct relation holds between neutron monitor count rate and atmospheric electric field because different atmospheric processes, including fluctuations in the ionosphere, are involved. Although a full quantitative model is still lacking, this link is supported by solid statistical evidence. Thus, a connection between the solar cycle activity and atmospheric electric field is expected. To gain a deeper insight into these relations, sunspot area (NOAA, USA), neutron monitor count rate (Climax, Colorado, USA), and atmospheric electric field (Lisbon, Portugal) are presented here in a phase space representation. The period considered covers two solar cycles (21, 22) and extends from 1978 to 1990. Two solar maxima were observed in this dataset, one in 1979 and another in 1989, as well as one solar minimum in 1986. Two main observations of the present study were: (1) similar short-term topological features of the phase space representations of the three variables, (2) a long-term phase space radius synchronization between the solar cycle activity, neutron monitor count rate, and potential gradient (confirmed by absolute correlation values above ~0.8). Finally, the methodology proposed here can be used for obtaining the relations between other atmospheric parameters (e.g., solar radiation) and solar cycle activity.

Wigner function of a relativistic particle in a time-dependent linear potential

We construct phase-space representations for a relativistic particle in both a constant and a time-dependent linear potential. We obtain explicit expressions for the Wigner distribution functions for these systems and find the correct nonrelativistic limit and free-particle limit for these functions. We derive the relativistic dynamical equation governing the time development of the Wigner distribution function and relativistic equation for the Wigner distribution function of stationary states and also calculate the amplitudes of transitions between energy states.

Direct measurement of time-frequency analogs of sub-Planck structures

Exploiting the correspondence between Wigner distribution function and a frequency-resolved optical gating (FROG) measurement, we experimentally demonstrate existence of the chessboard-like interference patterns with a time-bandwidth product smaller than that of a transform-limited pulse in the phase space representation of compass states. Using superpositions of four electric pulses as realization of compass states, we have shown via direct measurements that displacements leading to orthogonal states can be smaller than limits set by uncertainty relations. In the experiment we observe an exactly chronocyclic correspondence to the sub-Planck structure in the interference pattern appearing for superposition of two Sch\"{o}dinger-cat-like states in a position-momentum phase space.

Phase Space Framework for Coherence: Towards Quantum Computing System Design

This work aims to propose a framework for the phase space representation of quantum coherence in matter. The central thesis of this work is that quantum coherence could result from weak time-periodic forcing in the absence of other types of interactions (or measurements) from the environment. Therefore the Floquet formalism is employed to depict the coherent quantum system. A phase space representation for coherent quantum systems is formulated and the resulting three uncertainty principles are presented. The mathematical structure for quantum coherent transport is constructed via a non-equilibrium thermodynamic approach. The framework presented in this paper is targeted towards the development of nanosystems and other elements; employed to design systems for quantum computing.

Ray tracing method in phase space for two-dimensional optical systems.

Ray tracing is a forward method to calculate the photometric variables at the target of a non-imaging optical system. In this paper, a new ray tracing technique is presented to improve the accuracy and to reduce the computational time of the classical ray tracing approach. The method is based on the phase space representation of the source and the target of the optical system, and it is applied to a two-dimensional TIR-collimator. The strength of the method lies in tracing fewer rays through the system. Only rays that lie in the meridional plane are considered. A procedure that disregards rays in smooth regions in phase space, where the luminance is continuous, is implemented and only the rays close to discontinuities are traced. The efficiency of the method is demonstrated by numerical simulations that compare the new method with Monte Carlo ray tracing. The results show that the phase space approach is faster and more accurate than the already existing ray tracing method; moreover the phase space method converges as one over the number of rays traced unlike Monte Carlo ray tracing in which the speed of convergence is proportional to one over the square root of the number of rays.

Phase-space representation of a non-Hermitian system withPTsymmetry

We present a phase space study of non-Hermitian Hamiltonian with $\mathcal{PT}$-symmetry based on the Wigner distribution function. For an arbitrary complex potential, we derive a generalized continuity equation for the Wigner function flow and calculate the related circulation values. Studying vicinity of an exceptional point, we show that a $\mathcal{PT}$-symmetric phase transition from an unbroken $\mathcal{PT}$-symmetry phase to a broken one is a second-order phase transition.

Mixed semiclassical-classical propagators for the Wigner phase space representation.

We formulate mixed semiclassical-classical (SC-Cl) propagators by adding a further approximation to the phase-space SC propagators, which have been formulated in our previous paper [S. Koda, J. Chem. Phys. 143, 244110 (2015)]. We first show that the stationary phase approximation over the operation of the phase-space van Vleck propagator on initial distribution functions results in the classical mechanical time propagation. Then, after dividing the degrees of freedom (DOFs) of the total system into the semiclassical DOFs and the classical DOFs, the SC-Cl van Vleck propagator and the SC-Cl Herman-Kluk (HK) propagator are derived by performing the stationary phase approximation only with respect to the classical DOFs. These SC-Cl propagators are naturally decomposed to products of the phase-space SC propagators and the classical mechanical propagators when the system does not have any interaction between the semiclassical and the classical DOFs. In addition, we also numerically compare the original phase-space HK (full HK) propagator and the SC-Cl HK propagator in terms of accuracy and efficiency to find that the accuracy of the SC-Cl HK propagator can be comparable to that of the full HK propagator although the latter is more accurate than the former in general. On the other hand, we confirm that the convergence speed of the SC-Cl HK propagator is faster than that of the full HK propagator. The present numerical tests indicate that the SC-Cl HK propagator can be more accurate than the full HK propagator when they use a same and finite number of classical trajectories due to the balance of the accuracy and the efficiency.

Quantum de Finetti theorems and mean-field theory from quantum phase space representations

We introduce the number-conserving quantum phase space description as a versatile tool to address fundamental aspects of quantum many-body systems. Using phase space methods we prove two alternative versions of the quantum de Finetti theorem for finite-dimensional bosonic quantum systems, which states that a reduced density matrix of a many-body quantum state can be approximated by a convex combination of product states where the error is proportional to the inverse particle number. This theorem provides a formal justification for the mean-field description of many-body quantum systems, as it shows that quantum correlations can be neglected for the calculation of few-body observables when the particle number is large. Furthermore we discuss methods to derive the exact evolution equations for quantum phase space distribution functions as well as upper and lower bounds for the ground state energy. As an important example, we consider the Bose–Hubbard model and show that the mean-field dynamics is given by a classical phase space flow equivalent to the discrete Gross–Pitaevskii equation.

Finite linear spaces, plane geometries, Hilbert spaces and finite phase space

Finite plane geometry is associated with finite dimensional Hilbert space. The association allows mapping of q-number Hilbert space observables to the c-number formalism of quantum mechanics in phase space. The mapped entities reflect geometrically based line–point interrelation. Particularly simple formulas are involved when use is made of mutually unbiased bases representations for the Hilbert space entries. The geometry specifies a point–line interrelation. Thus underpinning d-dimensional Hilbert space operators (resp. states) with geometrical points leads to operators termed “line operators” underpinned by the geometrical lines. These “line operators”, \(\hat{L}_j;\) (j designates the line) form a complete orthogonal basis for Hilbert space operators. The representation of Hilbert space operators in terms of these operators form the phase space representation of the d-dimensional Hilbert space. Examples for the use of the “line operators” in mapping (finite dimensional) Hilbert space operators onto finite dimensional phase space functions are considered. These include finite dimensional Wigner function and Radon transform and a geometrical interpretation for the involvement of parity in the mappings of Hilbert space onto phase space. Two d-dimensional particles product states are underpinned with geometrical points. The states, \(|L_j\rangle \) underpinned with the corresponding geometrical lines are maximally entangled states (MES). These “line states” provide a complete \(d^2\) dimensional orthogonal MES basis for for the two d-dimensional particles. The complete \(d^2\) dimensional MES i.e. the “line states” are shown to provide a transparent geometrical interpretation to the so-called Mean King Problem and its variant. The “line operators” (resp. “line states”) are studied in detail. The paper aims at self sufficiency and to this end all relevant notions are explained herewith.