Phase Space Representation Research Articles

Quantum-Classical Correspondence in Response Theory

The correspondence principle between the quantum commutator [A, B] and the classical Poisson brackets iota h{A, B} is examined in the context of response theory. The classical response function is obtained as the leading term of the expansion of the phase space representation of the response function in terms of Weyl-Wigner transformations and is shown to increase without bound at long times as a result of ignoring divergent higher-order contributions. Systematical inclusion of higher-order contributions improves the accuracy of the h expansion at finite times. Resummation of all the higher-order terms establishes the classical-quantum correspondence <v + n|alpha(t)v> <--> alpha n e iota n omega t|Jv + nh/2. The time interval of the validity of the simple classical limit [A(t), B(0)] --> iota h{A(t), B(0)} is estimated for quasiperiodic dynamics and is shown to be inversely proportional to anharmonicity.

Qubits in phase space: Wigner-function approach to quantum-error correction and the mean-king problem

We analyze and further develop a new method to represent the quantum state of a system of $n$ qubits in a phase space grid of $N\times N$ points (where $N=2^n$). The method, which was recently proposed by Wootters and co--workers (Gibbons {\it et al.}, quant-ph/0401155), is based on the use of the elements of the finite field $GF(2^n)$ to label the phase space axes. We present a self--contained overview of the method, we give new insights on some of its features and we apply it to investigate problems which are of interest for quantum information theory: We analyze the phase space representation of stabilizer states and quantum error correction codes and present a phase space solution to the so--called ``mean king problem''.

Entanglement in quantum catastrophes

We classify entanglement singularities for various two-mode bosonic systems in terms of catastrophe theory. Employing an abstract phase-space representation, we obtain exact results in limiting cases for the entropy in cusp, butterfly, and two-dimensional catastrophes. We furthermore use numerical results to extract the scaling of the entropy with the non-linearity parameter, and discuss the role of mixing entropies in more complex systems.

WE-B-T-6E-01: Source Modelling and Beam Commissioning for Monte Carlo Treatment Planning

Source modeling and beam commissioning are important components of a Monte Carlo based treatment planning system. In this presentation, we will review the current status of Monte Carlo simulations of clinical photon and electron beams and the theories and methodologies used in particle phase space representation and reconstruction for Monte Carlo dose calculation. We will discuss the sensitivity of beam characterization to simulation details, such as beam energy, angle, intensity, and details of the treatment head design. We will review different source models for photon and electron beam characterization and discuss the accuracy and efficiency tradeoffs between full phase space and simplified source models. We will describe the methods and software that have been developed for source modeling and beam commissioning for the clinical implementation of the Monte Carlo method for treatment planning and beam delivery verification. We will present different methods for source parameterization based on simulated phase space data and a standard set of measured beam data including in-air and in-phantom output factors and in-phantom dose distributions. We will discuss problems associated with CT scanning and phantoms that can affect Monte Carlo accuracy. We will describe requirements and procedures for commissioning and acceptance testing of Monte Carlo dose calculation software for radiotherapy treatment planning. Educational Objectives: 1. Introduce the Monte Carlo method for clinical photon and electron beam simulations 2. Review theories and methodologies for phase space representation and reconstruction 3. Present different source models for Monte Carlo dose calculation 4. Describe different methods for source parameterization and beam commissioning 5. Provide requirements and guidelines for the clinical testing of Monte Carlo dose calculation software for radiotherapy treatment planning

Wigner function and Schrödinger equation in phase-space representation

We discuss a family of quasidistributions ($s$-ordered Wigner functions of Agarwal and Wolf [Phys. Rev. D 2, 2161 (1970); Phys. Rev. D 2, 2187 (1970); Phys. Rev. D 2, 2206 (1970)]) and its connection to the so-called phase space representation of the Schr\"odinger equation. It turns out that although Wigner functions satisfy the Schr\"odinger equation in phase space, they have a completely different interpretation.

A phase-space Gaussian beam summation representation of rough surface scattering

A Gaussian beam (GB) summation representation for rough surface scattering is introduced. In this scheme, the coherent and incoherent scattered fields are described by a phase-space summation of GBs that emanate from the rough surface at discrete set of points and directions. It thus involves stochastic GB2GB scattering matrices for the coherent and incoherent fields, and deterministic GB propagators. It benefits from the simplicity and accuracy of the latter, and can be used in applications involving propagation in complex scenarios comprising inhomogeneous media with rough surface boundaries. The GB2GB matrices are calculated from the statistical moments of the scattering amplitude, which are given either analytically or empirically. An analytical and numerical example for weakly rough surface is presented and discussed. Applications to the more complicated propagation scenario of doubly rough surface waveguide with multiple reflection phenomena will be presented in a follow-up publication.

Microscopic Theory for the Quantum to Classical Crossover in Chaotic Transport

We present a semiclassical theory for the scattering matrix S of a chaotic ballistic cavity at finite Ehrenfest time. Using a phase-space representation coupled with a multibounce expansion, we show how the Liouville conservation of phase-space volume decomposes S as S=S(cl) plus sign in circle S(qm). The short-time, classical contribution S(cl) generates deterministic transmission eigenvalues T=0 or 1, while quantum ergodicity is recovered within the subspace corresponding to the long-time, stochastic contribution S(qm). This provides a microscopic foundation for the two-phase fluid model, in which the cavity acts like a classical and a quantum cavity in parallel, and explains recent numerical data showing the breakdown of universality in quantum chaotic transport in the deep semiclassical limit. We show that the Fano factor of the shot-noise power vanishes in this limit, while weak localization remains universal.

Optics and squeeze transformations after Einstein

Journal of Optics B: Quantum and Semiclassical Optics will publish a special issue in connection with the 9th International Conference on Squeezed States and Uncertainty Relations, to be held in Besancon, France, on 2–6 May 2005. In 2005, the physics community celebrates the 100th anniversary of the publication of Einstein's theories of relativity and quantum physics. To celebrate these great contributions to physics, the conference will include sessions on Einstein's influence on modern optics and the foundations of quantum mechanics. Conference participants, as well as other researchers working in the field, are invited to submit research papers to this special issue of the journal. The topics to be covered include: • Superposition principle • Squeezed states • Uncertainty relations • Quantum state generation and characterization • Phase space and group representations in quantum physics • Quantum transforms in signal analysis • Information theory and quantum computing • Quantum interference, decoherence and entanglement measure • Quantum chaos and quantum control • Bell inequalities • Nonstationary Casimir effect • Quantum-like and mesoscopic systems Manuscripts should be submitted by 15 June 2005 as the special issue is scheduled for publication in December 2005. All papers will be peer reviewed and the normal refereeing standards of Journal of Optics B: Quantum and Semiclassical Optics will be maintained. The Editorial Division of IOP Publishing at the P N Lebedev Physical Institute in Moscow will oversee editorial procedures in association with the IOP Publishing office in Bristol. There are no page charges for publication. Submissions should preferably be in either standard LaTeX form or Microsoft Word. Advice on publishing your work in the journal, including specific information on figures, tables and references, may be found at www.iop.org/journals/authors. Manuscripts should be submitted by e-mail to the Guest Editors at IOPP@sci.lebedev.ru quoting the reference 'Special Issue/ST'. Authors are strongly encouraged to submit their work as soon as possible. Any subsequent change of address should be notified to the publishing office. If you have any questions, please do not hesitate to contact Claire Bedrock, Publisher (claire.bedrock@iop.org) or Margarita A Man'ko, Guest Editor (mmanko@sci.lebedev.ru). For further information on the journal, please visit our Website at www.iop.org/journals/jopb.

Particle Dynamics of a Phase Space Distribution Function

A hydrodynamic analogy for quantum mechanics is used to develop a phase-space representation in terms of a quasi-probability distribution function. Averages over phase space using this approach agree with the usual expectation values of quantum mechanics for a certain class of observables. We also derive the equations of motion that particles in an ensemble would have in phase space in order to mimic the time development of this probability distribution, thus giving the position and momentum of particles in the ensemble as a function of time. The equations of motion separate into position and momentum components. The position component reproduces the de Broglie-Bohm equation of motion. As a simple example, we calculate the phase space trajectories and entropy of a free particle wave packet.

Comment on “Redundancy of phase-space distribution functions in complex field recovery problems”

In a recent publication [Appl. Opt. 42, 1932 (2003)], redundancies of phase-space representations were studied. In particular, signal recovery from a single section of the ambiguity function was explored. It is shown that this signal-recovery method can be associated with interferometry.

Galilean-Covariant Clifford Algebras in the Phase-Space Representation

We apply the Galilean covariant formulation of quantum dynamics to derive the phase-space representation of the Pauli–Schrodinger equation for the density matrix of spin-1/2 particles in the presence of an electromagnetic field. The Liouville operator for the particle with spin follows from using the Wigner–Moyal transformation and a suitable Clifford algebra constructed on the phase space of a (4 + 1)-dimensional space–time with Galilean geometry. Connections with the algebraic formalism of thermofield dynamics are also investigated.

Smoothed Wigner functions: a tool to resolve semiclassical structures

The Wigner and Husimi distributions are the usual phase space representations of a quantum state. The Wigner distribution has structures of order ħ2. On the other hand, the Husimi distribution is a Gaussian smearing of the Wigner function on an area of size ħ and then, it only displays structures of size ħ. We have developed a phase space representation which results a Gaussian smearing of the Wigner function on an area of size ħσ, with σ≥1. Within this representation, the Husimi and Wigner functions are recovered when σ=1 and $ \sigma \gtrsim 2 $ respectively. We treat the application of this intermediate representation to explore the semiclassical limit of quantum mechanics. In particular we show how this representation uncover semiclassical hyperbolic structures of chaotic eigenstates.

Quantum-invariant processes in phase space

We show the formal equivalence between the phase-space representations of transformations and quantum states. We study invariant quantum input-output transformations in phase space and in Hilbert space. We show that all invariant processes are linear transformations while the converse is not true. Some relevant examples of application of these ideas are examined.

A multivariate, attractor-based approach to structural health monitoring

In this work, recent advances in the use of nonlinear time-series analysis for structural health monitoring are extended to incorporate multivariate data. Structural response data recorded at multiple locations are combined using a multivariate time delay embedding in order to reconstruct the structure's dynamical attractor. Using this approach, a global phase-space representation of the dynamics may be realized for spatially extended systems. A new attractor-based metric, chaotic amplification of attractor distortion (CAAD), is then introduced as a damage sensitive feature. The approach is implemented using data acquired from a composite beam, bolted at either end to steel plates. Degradation to the system is introduced as a loosening of the bolts at one end of the structure. Results based on multivariate attractor reconstruction show a clear ability to detect both the presence and magnitude of damage to the connection. Comparisons are then drawn between this approach and one where the same feature is extracted from attractors reconstructed using data acquired from the individual sensor locations. These features are combined “post-extraction” using a linear discriminant coordinant analysis. Performing the analysis separately at the individual sensor locations results in a significant reduction in discriminating power.

The Caldeira–Leggett quantum master equation in Wigner phase space: continued-fraction solution and application to Brownian motion in periodic potentials

The continued-fraction method to solve classical Fokker--Planck equations has been adapted to tackle quantum master equations of the Caldeira--Leggett type. This can be done taking advantage of the phase-space (Wigner) representation of the quantum density matrix. The approach differs from those in which some continued-fraction expression is found for a certain quantity, in that the full solution of the master equation is obtained by continued-fraction methods. This allows to study in detail the effects of the environment (fluctuations and dissipation) on several classes of nonlinear quantum systems. We apply the method to the canonical problem of quantum Brownian motion in periodic potentials both for cosine and ratchet potentials (lacking inversion symmetry).

Relationship between the Wigner function and the probability density function in quantum phase space representation

This paper discusses the relationship between the Wigner function, along with other related quasiprobability distribution functions, and the probability density distribution function constructed from the wave function of the Schr\odinger equation in quantum phase space, as formulated by Torres-Vega and Frederick (TF). At the same time, a general approach in solving the wave function of the Schr\odinger equation of TF quantum phase space theory is proposed. The relationship of the wave functions between the TF quantum phase space representation and the coordinate or momentum representation is thus revealed.

Quantum coherent tunneling between two atomic-molecular Bose-Einstein condensates

We study the quantum coherent tunneling dynamics of two weakly coupled atomic-molecular Bose-Einstein condensates (AMBEC). A weak link is supposed to be provided by a double-well trap. The regions of parameters where the macroscopic quantum localization of the relative atomic population occurs are revealed. The different dynamical regimes are found depending on the value of nonlinearity, namely, coupled oscillations of population imbalance of atomic and molecular condensate, including irregular oscillations regions, and macroscopic quantum self trapping regimes. Quantum means and quadrature variances are calculated for population of atomic and molecular condensates and the possibility of quadrature squeezing is shown via stochastic simulations within P-positive phase space representation method. Linear tunnel coupling between two AMBEC leads to correlations in quantum statistics.

Phase-space visualization of a metal–insulator transition

The Aubry–André model with its transition from a delocalized to a localized phase in one dimension is particularly well suited for a phase-space study of such a metal–insulator transition. The dependence of the Husimi function on the potential strength is discussed and described quantitatively by means of marginal distributions and the inverse participation ratio in phase space. The phase-space representation not only helps to visualize the metal–insulator transition but also sheds light on the question why such a transition is possible in a one-dimensional system. Differences and similarities between the Aubry–André model and the Anderson model in one and higher dimensions, respectively, will be pointed out.

Shot noise of a quantum shuttle.

We formulate a theory for shot noise in quantum nanoelectromechanical systems. As a specific example, the theory is applied to a quantum shuttle, and the zero-frequency noise, measured by the Fano factor F, is computed. F reaches very low values (F approximately 10(-2)) in the shuttling regime even in the quantum limit, confirming that shuttling is universally a low noise phenomenon. In approaching the semiclassical limit, the Fano factor shows a giant enhancement (F approximately 10(2)) at the shuttling threshold, consistent with predictions based on phase-space representations of the density matrix.

Exact solutions for nonlinear Schr dinger equation in phase space: applications to Bose–Einstein condensate

The stationary-state nonlinear Schrodinger equation, which models the dilute-gas Bose-Einstein condensate, is introduced within the framework of the quantum phase-space representation established by Torres-Vega and Frederick. The exact solutions of equation are obtained in the phase space, by means of the wave-mechanics method. The eigenfunctions in position and momentum spaces are obtained through the 'Fourier-like' projection transformation from the phase space eigenfunctions. The eigenfunction with a hypersecant part is discussed as an example.