Periodic Phase Space Research Articles

Translations and reflections on the torus: identities for discrete Wigner functions and transforms

A finite Hilbert space can be associated to a periodic phase space, that is, a torus. Then a subgroup of operators corresponding to reflections and translations on the torus form respectively the basis for the discrete Weyl representation, including the Wigner function, and for its Fourier conjugate, the chord representation. They are invariant under Clifford transformations and obey analogous product rules to the continuous representations, so allowing for the calculation of expectations and correlations for observables. We here import new identities from the continuum for products of pure state Wigner and chord functions, involving, for instance the inverse phase space participation ratio and correlations of a state with its translate. New identities are derived involving transition Wigner or chord functions of transition operators . Extension of the reflection and translation operators to a doubled torus phase space leads to the representation of superoperators and so to the construction of the propagator of Wigner functions from the Weyl representation of the evolution operator.

Quantum Rabi model in the Brillouin zone with ultracold atoms

The quantum Rabi model describes the interaction between a two-level quantum system and a single bosonic mode. We propose a method to perform a quantum simulation of the quantum Rabi model introducing a novel implementation of the two-level system, provided by the occupation of Bloch bands in the first Brillouin zone by ultracold atoms in tailored optical lattices. The effective qubit interacts with a quantum harmonic oscillator implemented in an optical dipole trap. Our realistic proposal allows to experimentally investigate the quantum Rabi model for extreme parameter regimes, which are not achievable with natural light-matter interactions. Furthermore, we also identify a generalized version of the quantum Rabi model in a periodic phase space.

Ergodic theory and visualization. II. Fourier mesochronic plots visualize (quasi)periodic sets.

We present an application and analysis of a visualization method for measure-preserving dynamical systems introduced by I. Mezić and A. Banaszuk [Physica D 197, 101 (2004)], based on frequency analysis and Koopman operator theory. This extends our earlier work on visualization of ergodic partition [Z. Levnajić and I. Mezić, Chaos 20, 033114 (2010)]. Our method employs the concept of Fourier time average [I. Mezić and A. Banaszuk, Physica D 197, 101 (2004)], and is realized as a computational algorithms for visualization of periodic and quasi-periodic sets in the phase space. The complement of periodic phase space partition contains chaotic zone, and we show how to identify it. The range of method's applicability is illustrated using well-known Chirikov standard map, while its potential in illuminating higher-dimensional dynamics is presented by studying the Froeschlé map and the Extended Standard Map.

Constructing Periodic Phase Space Orbits from ab Initio Molecular Dynamics Trajectories to Analyze Vibrational Spectra: Case Study of the Zundel (H5O2+) Cation

A method of analysis is introduced to probe the spectral features obtained from ab initio molecular dynamics simulations. Here, the instantaneous mass-weighted velocities are projected onto irreducible representations constructed from discrete time translation groups comprising operations that invoke the time-domain symmetries (or periodic phase space orbits) reflected in the spectra. The projected velocities are decomposed using singular value decomposition (SVD) to construct a set of "modes" pertaining to a given frequency domain. These modes now include all anharmonicities, as sampled during the dynamics simulations. In this approach, the underlying motions are probed in a manner invariant with respect to coordinate transformations, operations being performed along the time axis rather than coordinate axes, making the analysis independent of choice of reference frame. The method is used to probe the underlying motions responsible for the doublet at ∼1000 cm(-1) in the vibrational spectrum of the H5O2(+), Zundel cation. The associated analysis results are confirmed by projecting the Fourier transformed velocities onto the harmonic normal mode coordinates and a set of mass-weighted, symmetrized Jacobi coordinates. It is found that the two peaks of the doublet are described and differentiated by their respective contributions from the proton transfer, water-water stretch, and water wag coordinates, as these are defined. Temperature dependent effects are also briefly noted.

Worldline instantons, vacuum pair production and Gutzwiller's trace formula**Plenary talk at QFEXT07 (Quantum Fields Under External Conditions), Leipzig, September 2007.

In this talk, I describe a new application of the Gutzwiller trace formula formalism, to give a compact expression for the semiclassical vacuum pair production rate in quantum electrodynamics, for general inhomogeneous electromagnetic background fields. The semiclassical rate is expressed in terms of geometric properties of classical periodic phase space orbits for the associated Euclidean problem of propagation of a charged particle in the inhomogeneous background field, parametrized by the proper time.

Polarization fields and phase space densities in storage rings: Stroboscopic averaging and the ergodic theorem

A class of orbital motions with volume preserving flows and with vector fields periodic in the “time” parameter θ is defined. Spin motion coupled to the orbital dynamics is then defined, resulting in a class of spin–orbit motions which are important for storage rings. Phase space densities and polarization fields are introduced. It is important, in the context of storage rings, to understand the behavior of periodic polarization fields and phase space densities. Due to the 2 π time periodicity of the spin–orbit equations of motion the polarization field, taken at a sequence of increasing time values θ , θ + 2 π , θ + 4 π , … , gives a sequence of polarization fields, called the stroboscopic sequence. We show, by using the Birkhoff ergodic theorem, that under very general conditions the Cesàro averages of that sequence converge almost everywhere on phase space to a polarization field which is 2 π -periodic in time. This fulfills the main aim of this paper in that it demonstrates that the tracking algorithm for stroboscopic averaging, encoded in the program SPRINT and used in the study of spin motion in storage rings, is mathematically well-founded. The machinery developed is also shown to work for the stroboscopic average of phase space densities associated with the orbital dynamics. This yields a large family of periodic phase space densities and, as an example, a quite detailed analysis of the so-called betatron motion in a storage ring is presented.

Quantum fidelity decay in quasi-integrable systems

We show, via numerical simulations, that the fidelity decay behavior of quasi-integrable systems is strongly dependent on the location of the initial coherent state with respect to the underlying classical phase space. In parallel to classical fidelity, the quantum fidelity generally exhibits Gaussian decay when the perturbation affects the frequency of periodic phase space orbits and power-law decay when the perturbation changes the shape of the orbits. For both behaviors the decay rate also depends on initial state location. The spectrum of the initial states in the eigenbasis of the system reflects the different fidelity decay behaviors. In addition, states with initial Gaussian decay exhibit a stage of exponential decay for strong perturbations. This elicits a surprising phenomenon: a strong perturbation can induce a higher fidelity than a weak perturbation of the same type.

Spectrum of charged black holes—the big fix mechanism revisited

Following an earlier suggestion of the authors (Barvinsky A and Kunstatter G 1997 Mass spectrum for black holes in generic 2-D dilaton gravity Proc. 2nd International A D Sakharov Conference on Physics ed I M Dremin and A M Seminkhatov (Singapore: World Scientific) pp 210–15), we use some basic properties of Euclidean black hole thermodynamics and the quantum mechanics of systems with periodic phase space coordinate to derive the discrete two-parameter area spectrum of generic charged spherically symmetric black holes in any dimension. For the Reissner–Nordstrom black hole we get A/4Gℏ = π(2n + p + 1), where the integer p = 0, 1, 2,… gives the charge spectrum, with Q = ± √ ℏp. The quantity π(2n + 1), n = 0, 1,…, gives a measure of the excess of the mass/energy over the critical minimum (i.e. extremal) value allowed for a given fixed charge Q. The classical critical bound cannot be saturated due to vacuum fluctuations of the horizon, so that generically extremal black holes do not appear in the physical spectrum. Consistency also requires the black hole charge to be an integer multiple of any fundamental elementary particle charge: Q = ±me, m = 0, 1, 2,…. As a by-product this yields a relation between the fine structure constant and integer parameters of the black hole—a kind of the Coleman big fix mechanism induced by black holes. In four dimensions, this relationship is e2/ℏ = p/m2 and requires the fine structure constant to be a rational number. Finally, we prove that the horizon area is an adiabatic invariant, as has been conjectured previously.

On the Aubry-Mather Theory in Statistical Mechanics

We generalize Aubry-Mather theory for configurations on the line to general sets with a group action. Cocycles on the group play the role of rotation numbers. The notion of Birkhoff configuration can be generalized to this setting. Under mild conditions on the group, we show how to find Birkhoff ground states for many-body interactions which are ferromagnetic, invariant under the group action and having periodic phase space.

Nonlinear dynamics in periodic phase space

Regular and chaotic dynamics of a system with periodic phase space perturbed by an alternating external field is considered. It is relevant for the electronic motion in two dimensions in the presence of a uniform magnetic field and a perpendicular alternating electric field. The phase space is divided into cells embedded in a chaotic mesh. Bifurcations of resonances within the cells are studied. Transport takes place in the chaotic mesh. It is analyzed in the framework of the separatrix map. Accelerator modes are found for some values of parameters and their bifurcations are investigated. Their effects on transport in phase space are discussed.