Harel Primack

SEMICLASSICAL QUANTIZATION OF BILLIARDS WITH MIXED BOUNDARY CONDITIONS

The semiclassical theory for billiards with mixed boundary conditions is developed and explicit expressions for the smooth and the oscillatory parts of the spectral density are derived. The parametric dependence of the spectrum on the boundary condition is shown to be a very useful diagnostic tool in the semiclassical analysis of the spectrum of billiards. It is also used to check in detail some recently proposed parametric spectral statistics. The methods are illustrated in the analysis of the spectrum of the Sinai billiard and its parametric dependence on the boundary condition on the dispersing arc.

The quantum three-dimensional Sinai billiard – a semiclassical analysis

Abstract We present a comprehensive semiclassical investigation of the three-dimensional Sinai billiard, addressing a few outstanding problems in “quantum chaos”. We were mainly concerned with the accuracy of the semiclassical trace formula in two and higher dimensions and its ability to explain the universal spectral statistics observed in quantized chaotic systems. For this purpose we developed an efficient KKR algorithm to compute an extensive and accurate set of quantal eigenvalues. We also constructed a systematic method to compute millions of periodic orbits in a reasonable time. Introducing a proper measure for the semiclassical error and using the quantum and the classical databases for the Sinai billiards in two and three dimensions, we concluded that the semiclassical error (measured in units of the mean level spacing) is independent of the dimensionality, and diverges at most as log ℏ . This is in contrast with previous estimates. The classical spectrum of lengths of periodic orbits was studied and shown to be correlated in a way which induces the expected (random matrix) correlations in the quantal spectrum, corroborating previous results obtained in systems in two dimensions. These and other subjects discussed in the report open the way to extending the semiclassical study to chaotic systems with more than two freedoms.

Quantal–Classical Duality and the Semiclassical Trace Formula☆

Abstract We consider Hamiltonian systems which can be described both classically and quantum mechanically. Trace formulas establish links between the energy spectra in the quantum description and the spectrum of actions of periodic orbits in the Newtonian description. This is the duality which we investigate in the present paper. The duality holds for chaotic as well as for integrable systems. Billiard systems are a very convenient paradigm and we use them for most of our discussions. However, we also show how to transcribe the results to general Hamiltonian systems. In billiards, it is natural to think of the quantal spectrum (eigenvalues of the Helmholtz equation) and the classical spectrum (lengths of periodic orbits) as two manifestations of the properties of the billiard boundary. The trace formula express this link since it can be thought of as a Fourier transform relation between the classical and the quantum spectral densities. It follows that the two-point statistics of the quantal spectrum is related to the two-point statistics of the classical spectrum via a double Fourier transform. The universal correlations of the quantal spectrum are well known; consequently one can deduce the classical universal correlations. In particular, an explicit expression for the scale of the classical correlations is derived and interpreted. This allows a further extension of the formalism to the case of complex billiard systems, and in particular to the most interesting case of diffusive system. The effects of symmetry and symmetry-breaking are also discussed. The concept of classical correlations allows a better understanding of the so-called diagonal approximation and its breakdown. It also paves the way towards a semiclassical theory that is capable of global description of spectral statistics beyond the breaktime. An illustrative application is the derivation of the disorder-limited breaktime in the case of a disordered chain, thus obtaining a semiclassical theory for localization. We also discuss other applications such as the two-cell systems, periodic chains, and localization theory in more than one dimension. A numerical study of classical correlations in the case of the 3D Sinai billiards is presented. Here it is possible to test some assumptions and conjectures that underlie our formulation. In particular we gain a direct understanding of specific statistical properties of the classical spectrum, as well as their semiclassical manifestation in the quantal spectrum. We also analyze the spectral duality for integrable systems, and show that the Poissonian statistics of both the classical and the quantum spectra can be traced to the same origin.

On the accuracy of the semiclassical trace formula

The semiclassical trace formula provides the basic construction from which one derives the semiclassical approximation for the spectrum of quantum systems which are chaotic in the classical limit. When the dimensionality of the system increases, the mean level spacing decreases as , while the semiclassical approximation is commonly believed to provide an accuracy of order , independently of d. If this was true, the semiclassical trace formula would be limited to systems in only. In this work we set out to define proper measures of the semiclassical spectral accuracy, and to propose theoretical and numerical evidence to the effect that the semiclassical accuracy, measured in units of the mean level spacing, depends only weakly (if at all) on the dimensionality. Detailed and thorough numerical tests were performed for the Sinai billiard in two and three dimensions, substantiating the theoretical arguments.

Penumbra diffraction in the semiclassical quantization of concave billiards

The semiclassical description of billiard spectra is extended to include the diffractive contributions from orbits which are nearly tangent to a concave part of the boundary. The leading correction for an unstable isolated orbit is of the same order as the standard Gutzwiller expression itself. The importance of the diffraction corrections is further emphasized by an estimate which shows that for any large fixed k almost all contributing periodic orbits are affected. The theory is tested numerically using the annulus and Sinai billiard. For the Sinai billiard, the investigation of the spectral density is complemented by an analysis which is based on the scattering approach to quantization. The merits of this approach as a tool to investigate refined semiclassical theories are discussed and demonstrated.

Quantal consequences of perturbations which destroy structurally unstable orbits in chaotic billiards

Non-generic contributions to the quantal level density from parallel segments in billiards are investigated. These contributions are due to the existence of marginally stable families of periodic orbits, which are structurally unstable, in the sense that small perturbations, such as a slight tilt of one of the segments, destroy them completely. We investigate the effects of such perturbations on the corresponding quantum spectra, and demonstrate them for the stadium billiard.

Synchrotron radiation of crystallized beams.

We study the modifications of synchrotron radiation of charges in a storage ring as they are cooled. The pair correlation lengths between the charges are manifest in the synchrotron radiation and coherence effects exist for wavelengths longer than the coherence lengths between the charges. Therefore, the synchrotron radiation can be used as a diagnostic tool to determine the state (gas, liquid, crystal) of the charged plasma in the storage ring. We show also that the total power of the synchrotron radiation is significantly reduced for crystallized beams, both coasting and bunched. This opens the possibility of accelerating particles to ultrarelativistic energies using small-sized cyclic accelerators.

DIAGNOSTIC CRITERION FOR CRYSTALLIZED BEAMS

Based on a cooling hysteresis first observed in connection with small ion crystals in a Paul trap, we propose the following diagnostic criterion for establishing the presence of a crystallized beam in a storage ring: Absence of heating following reduction of the cooling power. The validity and applicability of the criterion is discussed in detail and confirmed with the help of detailed numerical simulations.

SUPPRESSION OF SYNCHROTRON RADIATION DUE TO BEAM CRYSTALLIZATION

Abstract. With respect to a “hot”, non-crystallized beam the synchrotron radiation of a cold crystallized beam is considerably modified. We predict suppression of synchrotron radiation emitted by a crystallized beam in a storage ring. We also propose experiments to detect this effect.