Fields In Phase Space Research Articles

Entropic distinguishability of quantum fields in phase space

We present a general way of quantifying the entropic uncertainty of quantum field configurations in phase space in terms of entropic distinguishability with respect to the vacuum. Our approach is based on the functional Husimi Q-distribution and a suitably chosen relative entropy, which we show to be non-trivially bounded from above by the uncertainty principle. The resulting relative entropic uncertainty relation is as general as the concept of coherent states and thus holds for quantum fields of bosonic and fermionic type. Its simple form enables diverse applications, among which we present a complete characterization of the uncertainty surplus of arbitrary states in terms of the total particle number for a scalar field and the fermionic description of the Ising model. Moreover, we provide a quantitative interpretation of the role of the uncertainty principle for quantum phase transitions.

Noncommutativity in Configuration Space Induced by a Conjugate Magnetic Field in Phase Space

An external magnetic field in configuration space coupled to quantum dynamics induces noncommutativity in its velocity momentum space. By phase space duality, an external vector potential in the conjugate momentum sector of the system induces noncommutativity in its configuration space. Such a rationale for noncommutativity is explored herein for an arbitrary configuration space of Euclidean geometry. Ordinary quantum mechanics with a commutative configuration space is revisited first. Through the introduction of an arbitrary positive definite ∗-product, a one-to-one correspondence between the Hilbert space of abstract quantum states and that of the enveloping algebra of the position quantum operators is identified. A parallel discussion is then presented when configuration space is noncommutative, and thoroughly analysed when the conjugate magnetic field is momentum independent and nondegenerate. Once again the space of quantum states may be identified with the enveloping algebra of the noncommutative position quantum operators. Furthermore, when the positive definite ∗-product is adapted to the conjugate magnetic field, the coordinate operators span a Fock algebra of which the coherent states are the analogues of the structureless points in a commutative configuration space. These results generalise and justify a posteriori within ordinary canonical quantisation the heuristic approach to quantum mechanics in the noncommutative Euclidean plane as constructed and developed by F. G. Scholtz and his collaborators.

Controllable phase-dependent Wigner-function negativity at steady state via parametric driving and feedback

Generating the negative Wigner functions where the corresponding Wigner states are nonclassical has been recognized as a powerful tool for successfully performing quantum information and computing protocols beyond the scope of classical computers. Here, we present the possibility to generate and engineer the negative Wigner function at a steady state using parametric (two-photon) driving and homodyne-based feedback in a quantum van der Pol (vdP) oscillator. Specifically, we employ a quantum master equation approach for calculating the Wigner function of the vdP oscillator field in phase space and, furthermore, quantifying its negativity content. We clearly show that the negative-value magnitudes, regions, and shapes of the Wigner function can be effectively tuned by the parametric driving phase and the parametric driving amplitude, as well as the feedback coefficient within a large range. We identify different contributions of these involved parameters to the Wigner-function negativity. In the present scheme, more complex quantum coherence and interference phenomena are introduced via the parametric driving and feedback, which stabilizes the phase of the vdP oscillator field and renders the capability to generate the negative Wigner function. Therefore, the enhanced Wigner-function negativity can be achieved under these optimized system parameters. Our in-depth study provides insight into the formation and in situ control of the desirable Wigner nonclassical states. The obtained results are not limited to the vdP oscillator systems and should be generally applicable to other coherent coupled systems within the reach of modern experimental facilities.

Positivity preserving schemes for the fractional Klein-Kramers equation with boundaries

The fractional Klein-Kramers equation describes the process of subdiffusion in the presence of an external force field in phase space and incorporates a fractional operator in time of order α, 0 < α < 1. We present a family of finite volume schemes for the fractional Klein-Kramers equation, that includes first or second-order schemes in phase space, and implicit or explicit schemes in time with an order of accuracy that can change between α and 2−α. It is proved, for the open domain, that the schemes satisfy the positivity preserving property. The positivity preserving property for the explicit schemes imposes a strong condition in the relation between time step, space step and phase step, for small values of α, highlighting the advantage of using implicit schemes in these cases. For a bounded domain in space, two types of boundary conditions are considered, absorbing boundary conditions and reflecting boundary conditions. The inclusion of boundary conditions leads to some technical complications that require changes in the schemes near the boundary. The positivity preserving property holds for the new formulation and the overall accuracy is ensured with the use of non-uniform meshes. Numerical tests are presented in the end to show the convergence of the finite volume schemes.

Spin-1/2 Particles in Phase Space: Casimir Effect and Stefan-Boltzmann Law at Finite Temperature

The Dirac field, spin 1/2 particles, is investigated in phase space. The Dirac propagator is defined. The Thermo Field Dynamics (TFD) formalism is used to introduce finite temperature. The energy-momentum tensor is calculated at finite temperature. The Stefan-Boltzmann law is established, and the Casimir effect is calculated for the Dirac field in phase space at zero and finite temperature. A comparative analysis with these results in standard quantum mechanics space is realized.

Proposal of a Computational Approach for Simulating Thermal Bosonic Fields in Phase Space

When a quantum field is in contact with a thermal bath, the vacuum state of the field may be generalized to a thermal vacuum state, which takes into account the thermal noise. In thermo field dynamics, this is realized by doubling the dimensionality of the Fock space of the system. Interestingly, the representation of thermal noise by means of an augmented space is also found in a distinctly different approach based on the Wigner transform of both the field operators and density matrix, which we pursue here. Specifically, the thermal noise is introduced by augmenting the classical-like Wigner phase space by means of Nosé–Hoover chain thermostats, which can be readily simulated on a computer. In this paper, we illustrate how this may be achieved and discuss how non-equilibrium quantum thermal distributions of the field modes can be numerically simulated.

Non-abelian Gauge Symmetry for Fields in Phase Space: a Realization of the Seiberg-Witten Non-abelian Gauge Theory

The Seiberg-Witten formalism has been realized as an electrodynamics in phase space (associated to the Dirac equation written in phase space) and this fact is explored here with non-abelian gauge group. First, a physically heuristic presentation of the Seiberg-Witten approach is carried out for non-abelian gauge in order to guide the calculation procedures. These results are realized by starting with the Lagrangian density for the free Dirac field in phase space. Then a field strength is derived, where the non-abelian gauge group is the SU(2), corresponding to an isospin (non-abelian) field theory in phase space. An application to nucleon is then discussed.

On scalar electromagnetism in phase space

In this paper, the interaction of a scalar field and the electromagnetic field in phase space is analyzed. The scattering process is calculated up to first order in the Planck constant which is obtained by an expansion of the Moyal product in phase space. The transition amplitude is calculated in the same context.

Stefan-Boltzmann Law and Casimir Effect for the Scalar Field in Phase Space at Finite Temperature

We use the scalar field constructed in phase space to analyze the analogous Stefan-Boltzmann law and Casimir effect, both of them at finite temperature. The temperature is introduced by Thermo Field Dynamics (TFD) formalism and the quantities are analyzed once projected in the space of coordinates. We show that using the framework of phase space it is possible to introduce a thermal energy which is related to temperature as it vanishes when the temperature tends to zero. In fact given such a correlation the formalism of TFD is equivalent when project is in momenta space when compared to coordinates space.

Realization of the noncommutative Seiberg–Witten gauge theory by fields in phase space

Representations of the Poincaré symmetry are studied by using a Hilbert space with a phase space content. The states are described by wave functions (quasi-amplitudes of probability) associated with Wigner functions (quasi-probability density). The gauge symmetry analysis provides a realization of the Seiberg–Witten gauge theory for noncommutative fields.

Photon-number parity oscillations in the resonant Jaynes–Cummings model

We study the time evolution of photon number parity operator for a single-mode quantized cavity field interacting with a two-level atom as described by the resonant Jaynes–Cummings model. With the field prepared in a coherent state and the atom prepared in its excited state, we find that the photon number parity undergoes oscillations at the Rabi frequency but only during the period between the collapse and revival of the Rabi oscillations of the atomic inversion. We explain this in terms of the global parity of the Jaynes–Cummings model. The phenomenon is related to the interference of the counter-rotating components of the field in phase space, which in turn gives rise to occurrence of highly oscillatory photon number probability distributions near half the revival time of the atomic inversion noted by Bužek et al. (Phys. Rev. A 45, 1992, p. 8190). Detection of the photon number oscillations would amount to a detection of this interference effect.

Testing nonclassicality and non-gaussianity in phase space.

We theoretically propose and experimentally demonstrate a nonclassicality test of a single-mode field in phase space, which has an analogy with the nonlocality test proposed by Banaszek and Wódkiewicz [Phys. Rev. Lett. 82, 2009 (1999)]. Our approach to deriving the classical bound draws on the fact that the Wigner function of a coherent state is a product of two independent distributions as if the orthogonal quadratures (position and momentum) in phase space behave as local realistic variables. Our method detects every pure nonclassical Gaussian state, which can also be extended to mixed states. Furthermore, it sets a bound for all Gaussian states and their mixtures, thereby providing a criterion to detect a genuine quantum non-Gaussian state. Remarkably, our phase-space approach with invariance under Gaussian unitary operations leads to an optimized test for a given non-Gaussian state. We experimentally show how this enhanced method can manifest quantum non-Gaussianity of a state by simply choosing phase-space points appropriately, which is essentially equivalent to implementing a squeezing operation on a given state.

Discrete flow mapping: transport of phase space densities on triangulated surfaces

Energy distributions of high-frequency linear wave fields are often modelled in terms of flow or transport equations with ray dynamics given by a Hamiltonian vector field in phase space. Applications arise in underwater and room acoustics, vibroacoustics, seismology, electromagnetics and quantum mechanics. Related flow problems based on general conservation laws are used, for example, in weather forecasting or in molecular dynamics simulations. Solutions to these flow equations are often large-scale, complex and high-dimensional, leading to formidable challenges for numerical approximation methods. This paper presents an efficient and widely applicable method, called discrete flow mapping , for solving such problems on triangulated surfaces. An application in structural dynamics, determining the vibroacoustic response of a cast aluminium car body component, is presented.

Propagation of femtosecond terawatt laser pulses in N2 gas including higher-order Kerr effects

Propagation characteristic of femtosecond terawatt laser pulses in N2 gas with higher-order Kerr effect (HOKE) is investigated. Theoretical analysis shows that HOKE acting as Hamiltonian perturbation can destroy the coherent structure of a laser field and result in the appearance of incoherent patterns. Numerical simulations show that in this case two different types of complex structures can appear. It is found that the high-order focusing terms in HOKE can cause continuous phase shift and off-axis evolution of the laser fields when irregular homoclinic orbit crossings of the field in phase space take place. As the laser propagates, small-scale spatial structures rapidly appear and the evolution of the laser field becomes chaotic. The two complex patterns can switch between each other quasi-periodically. Numerical results show that the two complex patterns are associated with the stochastic evolution of the energy contained in the higher-order shorter-wavelength Fourier modes. Such complex patterns, associated with small-scale filaments, may be typical for laser propagation in a HOKE medium.

Detecting variation in chaotic attractors

If the output of an experiment is a chaotic signal, it may be useful to detect small changes in the signal, but there are a limited number of ways to compare signals from chaotic systems, and most known methods are not robust in the presence of noise. One may calculate dimension or Lyapunov exponents from the signal, or construct a synchronizing model, but all of these are only useful in low noise situations. I introduce a method for detecting small variations in a chaotic attractor based on directly calculating the difference between vector fields in phase space. The differences are found by comparing close strands in phase space, rather than close neighbors. The use of strands makes the method more robust to noise and more sensitive to small attractor differences.

Schrödinger Equations for Delta Potential Barrier in Quantum Phase Space

The collision with the delta potential barrier and the bound state in the delta potential well for a particle are both typical problems in quantum mechanics. We solve strictly the eigenequations of a particle in the delta potential fields in phase space in order to offer another soluble example for physical systems. We define the Schrödinger equations of the delta potential fields in the phase space.

Phase space methods for multi-arrival wavefronts

Most body wave seismic imaging schemes only exploit information contained in the first arrival of a seismic record. Later arrivals in the wavetrain, however, contain additional structural information, as the corresponding rays tend to sample slower regions of a medium that are often avoided by first-arrival raypaths. Here we investigate a Lagrangian (ray-based) and an Eulerian (grid-based) approach for the calculation of later arrivals. The Eulerian approach is based on the level set method, which implicitly evolves a wavefront by solving a pair of PDEs over a gridded velocity field in phase space. Our Lagrangian solver also uses phase space and represents the wavefront by a set of points, which are progressively moved through the velocity field using local ray tracing, with linear interpolation used to maintain a constant density of points. We compare the two methods using a velocity model of the subduction zone in the Tonga region. In theory both approaches can provide traveltimes for later arrivals. Our results clearly show that the Lagrangian approach is currently superior to the Eulerian scheme for the prediction of multi-arrival traveltimes when computation speed, ease of implementation, and accuracy are considered. In our experiments the Lagrangian solver is up to 6000 times faster, and successfully predicts later arrivals for our source receiver configuration in the subduction zone example. We then demonstrate the robustness and efficiency of the Lagrangian solver by tracking later arrivals in a smoothed version of the Marmousi model. By placing source points in certain parts of this model, we are able to find more than 60 secondary arrivals at surface receivers.

Toward a phase-space cartography of the short- and medium-range predictability of weather regimes

We compare the short- and medium-range predictability of weather regimes of a quasigeostrophicmodel as defined by a hierarchical cluster algorithm and a Lyapunov-based clusteringmethod recently introduced in the literature. Both procedures lead to weather regimesdisplaying very different predictability properties on the short and medium range bases. Whilethe former does not distinguish between stable and unstable weather regimes, the latter leadsto clusters which do not display a good medium range predictability. We introduce a newclustering method taking advantages of the two previous techniques. Its application in thecontext of the quasi-geostrophic model gives rise to regimes possessing at the same time a goodmedium range skill and well separated instability properties, indicating the possibility to builda systematic cartography of the short-term predictability of weather fields in phase space.

Rigorous solutions of a particle in δ potential fields in phase space

Within the framework of the quantum phase-space representation established by Torres-Vega and Frederick, the rigorous solutions of the Schrodinger equations of a particle in the δ potential fields are solved, and the Heisenberg uncertainty principle is interpreted in the phase space. Then it is found that, for the bound state of a particle in the δ potential well, the Wigner distribution function is the special result of the quantum phase-space probability density function with parameter k approaching infinity.

Phase-Space Quantization of Field Theory

In this lecture, a limited introduction of gauge invariance in phase-space is provided, predicated on canonical transformations in quantum phase-space. Exact characteristic trajectories are also specified for the time-propagating Wigner phase-space distribution function: they are especially simple--indeed, classical--for the quantized simple harmonic oscillator. This serves as the underpinning of the field theoretic Wigner functional formulation introduced. Scalar field theory is thus reformulated in terms of distributions in field phase-space. This is a pedagogical selection from work published and reported at the Yukawa Institute Workshop ''Gauge Theory and Integrable Models'', 26-29 January, 1999.