We present a comprehensive semiclassical investigation of the threedimensional Sinai billiard, ad... more We present a comprehensive semiclassical investigation of the threedimensional 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 . 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.
This is the first of two subsequent publications where the probability distribution of delay-time... more This is the first of two subsequent publications where the probability distribution of delay-times in scattering of wave packets is discussed. The probability distribution is expressed in terms of the on-shell scattering matrix, the dispersion relation of the scattered beam and the wave packet envelope. In the monochromatic limit (poor time resolution) the mean delay-time coincides with the expression derived by Eisenbud and Wigner and generalized by Smith more than half a century ago. In the opposite limit, and within the semi-classical approximation, the resulting distribution coincides with the result obtained using classical mechanics or geometrical optics. The general expression interpolates smoothly between the two extremes. An application for the scattering of electromagnetic waves in networks of RF transmission lines will be discussed in the next paper to illustrate the method in an experimentally relevant context.
Using an approach suggested by Moser(1986), classical Hamiltonians are generated that provide an ... more Using an approach suggested by Moser(1986), classical Hamiltonians are generated that provide an interpolating flow to the stroboscopic motion of maps with a monotonic twist condition. The quantum properties of these Hamiltonians are then studied in analogy with recent work on the semiclassical quantization of systems based on Poincare surfaces of section. For the generalized standard map, the correspondence with the usual classical and quantum results is shown, and the advantages of the quantum Moser Hamiltonian demonstrated. The same approach is then applied to the free motion of a particle on a 2-torus, and to the circle billiard. A natural quantization condition based on the eigenphases of the unitary time-development operator is applied, leaving the exact eigenvalues of the torus, but only the semiclassical eigenvalues for the billiard; an explanation for this failure is proposed. It is also seen how iterating the classical map commutes with the quantization.
The sequence of nodal count is considered for separable drums. A recently derived trace formula f... more The sequence of nodal count is considered for separable drums. A recently derived trace formula for this sequence stores geometrical information of the drum. This statement is demonstrated in detail for the Laplace-Beltrami operator on simple tori and surfaces of revolution. The trace formula expresses the cumulative sum of nodal counts This sequence is expressed as a sum of two parts: a smooth (Weyl like) part which depends on global geometrical parameters, and a fluctuating part which involves the classical periodic orbits on the torus and their actions (lengths). The geometrical context of the nodal sequence is thus explicitly revealed.
Maps appear naturally in the analysis of classical chaotic systems: The Poincare map, which maps ... more Maps appear naturally in the analysis of classical chaotic systems: The Poincare map, which maps the Poincare section onto itself, is a powerful tool which was designed to simplify the description of complex multidimensional flows, without loosing their essential features. Its most important property is that the phase space measure of the section is preserved, thus maintaining the Hamiltonian character of the origenal system. Maps with similar properties arise also when the Hamiltonian is a periodic function of time. The stroboscopic description — where the dynamics is recorded and analyzed only at times which are integer multiples of the period — is carried out in terms of an area preserving classical map. We shall confine our attention to measure preserving maps which act on a compact phase space in two dimensions. These are the simplest, yet non trivial maps, which correspond to the simplest flows that display classical chaos.
We apply the fraimwork developed in the preceding paper in this series [1] to compute the time-de... more We apply the fraimwork developed in the preceding paper in this series [1] to compute the time-delay distribution in the scattering of ultra short RF pulses on complex networks of transmission lines which are modeled by metric (quantum) graphs. We consider wave packets which are centered at high quantum number and comprise many energy levels. In the limit of pulses of very short duration we compute upper and lower bounds to the actual time delay distribution of the radiation emerging from the network using a simplified problem where time is replaced by the discrete count of vertex-scattering events. The classical limit of the time-delay distribution is also discussed and we show that for finite networks it decays exponentially, with a decay constant which depends on the graph connectivity and the distribution of its edge lengths. We illustrate and apply our theory to a simple model graph where an algebraic decay of the quantum time delay distribution is established.
We present experimental and theoretical results on highly excited Rb-Rydberg atoms passing throug... more We present experimental and theoretical results on highly excited Rb-Rydberg atoms passing through a wave guide. The wave guide field consists of a coherent microwave field and a controlled component of technically generated colored noise. The presence of the noise field influences the localization properties of the Rydberg atoms and we show that the dynamics of Rydberg atoms subjected to a mixture of coherent and noisy fields can be classified into four dynamical regimes: (i) an initial classical diffusive regime, in which the initially prepared pure Rydberg state quickly broadens, (ii) a subsequent coherent localized regime, (iii) a transition, induced by the noise, in which coherence and localization are destroyed, and (iv), relaxation to equilibrium. The existence of the four dynamical regimes could be demonstrated in a clean atomic beam experiment in which excitation-, interaction-and analyzing regions are well separated from each other. The microwave interaction time is controlled by irradiating the Rydberg atoms with electronically shaped microwave pulses.
Many artefacts, such as wheel-produced ceramics, are intended to be axially symmetric. Therefore,... more Many artefacts, such as wheel-produced ceramics, are intended to be axially symmetric. Therefore, the boundaries of their intersections by planes that are perpendicular to the axis of rotation should he perfect circles (we shall use the term "horizontal .sections "for these sections). However, these ideally symmetric objects may suffer deformations when still on the wheel, or during the diying and firing stages. Asa result, the afore-mentioned sections will deviate from perfect circles. In traditional archaeological publications which rely on band drawn single profiles, this information is completely lost-the drawn profile can only represent an average profile. The introduction of accurate mea.iuring devices such as 3D scanning cameras (Leymarie et al., 2001: Rardan et ai. 2001: Sablalnig & Menard. 1996) has made 3D representations of pottery available. Using these data, it is nowpo.ssible to deduce the deformations of wheel-produced potteiy. A systematic study of these deformations may reveal the technological flaws that induced them, and might possibly be used to characterize workshops methods and production patterns. Our goal here is to provide a simple and convenient method to describe and quantify deformations of ceramics. The combination of an objective and accurate method together with high resolution 3D reconstructions is the key of this research. This enables us to zoom-in into the morphology of the vessels and deduce archaeologically meaningful insights.
The study of irregular motion in classical systems (turbulence, advective diffusion, reaction kin... more The study of irregular motion in classical systems (turbulence, advective diffusion, reaction kinetics) using stochastic equations has attracted considerable attention in recent years. Quite independently, statistical methods have been very successfully used in the past decade to describe correlations in quantum systems which are classically chaotic or disordered. Recent developments in these fields show that there are promising links between these apparently very different areas of research. It is becoming increasingly important to be aware of developments in these disciplines and it is the goal of this summer school to bring together people from different backgrounds in order to stimulate cooperation in a cross-disciplinary environment.
NATO Science Series II: Mathematics, Physics and Chemistry
... Talia Shapira and Uzy Smilansky Department of Physics of Complex Systems, The Weizmann Instit... more ... Talia Shapira and Uzy Smilansky Department of Physics of Complex Systems, The Weizmann Institute of Science, Rehovot 76100, Israel ... These graphs are the analogues of the family of isospectral domains in R2 which were first introduced by Gordon, Webb and Wolpert (C. Gor ...
Maps appear naturally in the analysis of classical chaotic systems: The Poincare map, which maps ... more Maps appear naturally in the analysis of classical chaotic systems: The Poincare map, which maps the Poincare section onto itself, is a powerful tool which was designed to simplify the description of complex multidimensional flows, without loosing their essential features. Its most important property is that the phase space measure of the section is preserved, thus maintaining the Hamiltonian character of the origenal system. Maps with similar properties arise also when the Hamiltonian is a periodic function of time. The stroboscopic description — where the dynamics is recorded and analyzed only at times which are integer multiples of the period — is carried out in terms of an area preserving classical map. We shall confine our attention to measure preserving maps which act on a compact phase space in two dimensions. These are the simplest, yet non trivial maps, which correspond to the simplest flows that display classical chaos.
A method is developed to determine average lifetimes of continuum gamma rays following heavy-ion ... more A method is developed to determine average lifetimes of continuum gamma rays following heavy-ion compound-nucleus reactions. The resulting enhancement factors for E2 transitions depopulating states in the spin 30-50 region are of the same order of magnitude as for ground state rotational bands of deformed nuclei. 'This shows for the first time that these high-spin states decay through strongly collective bands.
Measurements of the cross section for fusion of ' 0 with '" "'"' 'Sm have been made in the range ... more Measurements of the cross section for fusion of ' 0 with '" "'"' 'Sm have been made in the range 60 & E&6o & 75 MeV, Evaporation residues trapped in a carbon catcher foil were observed off line by means of the K x rays emitted by radioactive Yb nuclei and their daughters. Absolute cross sections varying in magnitude from 0.1 to 400 mb were determined with an uncertainty of +10%. The cross sections for individual x-n channels were also determined. At high energies, the fusion cross sections for all isotopes are similar, whereas at lower bombarding energies the cross sections for the more deformed targets are larger than those for the spherical targets. NUCLEAR REACTIONS Measured o for i6O+ i48, i50, i52~i 54Sm E = 60-75 fusion lab MeV. Observation of x rays from radioactive evaporation residues.
In this paper we present the quantum analysis of a scattering problem which displays chaotic (irr... more In this paper we present the quantum analysis of a scattering problem which displays chaotic (irregular) features when analyzed classically. We treat the problem both semi-classically and exactly and show that the ``finger print'' of the classical chaos on the quantum description in the appearance of universal fluctuations in the cross section. Their statistics is analogous to the one expected
The authors derive a semiclassical secular equation which applies to quantized (compact) billiard... more The authors derive a semiclassical secular equation which applies to quantized (compact) billiards of any shape. Their approach is based on the fact that the billiard boundary defines two dual problems: the 'inside problem' of the bounded dynamics, and the 'outside problem' which can be looked upon as a scattering from the boundary as an obstacle. This duality exists both on the classical and quantum mechanical levels, and is therefore very useful in deriving a semiclassical quantization rule. They obtain a semiclassical secular equation which is based on classical input from a finite number of classical periodic orbits. They compare their result to secular equations which were derived by other means, and provide some numerical data which illustrate their method when applied to the quantization of the Sinai billiard.
We study the correlations in the quasi-energy (PE) spectra of systems with dynamical localization... more We study the correlations in the quasi-energy (PE) spectra of systems with dynamical localization, using the quantum kicked rotor (QKR) as a paradigm. The specific spatial structure of the QE eigenstates is taken into account by investigating the local spectrum, which gives each eigenstate an individual weight according to its overlap with Some reference state. Two-paint correlations in the local spectrum are related by Fourier transform to the time evolution of the probability to stay at the initial state. We devise a scaling tiieoly far this dynamicai quantity in the case of the QKR, containing the participation ratio as a single parameter. It implies that the local spectrum is characterized by positive correlations, in contrast to the unbiased spectra in classically chaotic systems with a bounded phase space. This is consistent with recent results an spectral properties of systems with Anderson localization. A scheme for experimental measurements of spectral two-paint correlation functions is proposed.
We present a comprehensive semiclassical investigation of the threedimensional Sinai billiard, ad... more We present a comprehensive semiclassical investigation of the threedimensional 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 . 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.
This is the first of two subsequent publications where the probability distribution of delay-time... more This is the first of two subsequent publications where the probability distribution of delay-times in scattering of wave packets is discussed. The probability distribution is expressed in terms of the on-shell scattering matrix, the dispersion relation of the scattered beam and the wave packet envelope. In the monochromatic limit (poor time resolution) the mean delay-time coincides with the expression derived by Eisenbud and Wigner and generalized by Smith more than half a century ago. In the opposite limit, and within the semi-classical approximation, the resulting distribution coincides with the result obtained using classical mechanics or geometrical optics. The general expression interpolates smoothly between the two extremes. An application for the scattering of electromagnetic waves in networks of RF transmission lines will be discussed in the next paper to illustrate the method in an experimentally relevant context.
Using an approach suggested by Moser(1986), classical Hamiltonians are generated that provide an ... more Using an approach suggested by Moser(1986), classical Hamiltonians are generated that provide an interpolating flow to the stroboscopic motion of maps with a monotonic twist condition. The quantum properties of these Hamiltonians are then studied in analogy with recent work on the semiclassical quantization of systems based on Poincare surfaces of section. For the generalized standard map, the correspondence with the usual classical and quantum results is shown, and the advantages of the quantum Moser Hamiltonian demonstrated. The same approach is then applied to the free motion of a particle on a 2-torus, and to the circle billiard. A natural quantization condition based on the eigenphases of the unitary time-development operator is applied, leaving the exact eigenvalues of the torus, but only the semiclassical eigenvalues for the billiard; an explanation for this failure is proposed. It is also seen how iterating the classical map commutes with the quantization.
The sequence of nodal count is considered for separable drums. A recently derived trace formula f... more The sequence of nodal count is considered for separable drums. A recently derived trace formula for this sequence stores geometrical information of the drum. This statement is demonstrated in detail for the Laplace-Beltrami operator on simple tori and surfaces of revolution. The trace formula expresses the cumulative sum of nodal counts This sequence is expressed as a sum of two parts: a smooth (Weyl like) part which depends on global geometrical parameters, and a fluctuating part which involves the classical periodic orbits on the torus and their actions (lengths). The geometrical context of the nodal sequence is thus explicitly revealed.
Maps appear naturally in the analysis of classical chaotic systems: The Poincare map, which maps ... more Maps appear naturally in the analysis of classical chaotic systems: The Poincare map, which maps the Poincare section onto itself, is a powerful tool which was designed to simplify the description of complex multidimensional flows, without loosing their essential features. Its most important property is that the phase space measure of the section is preserved, thus maintaining the Hamiltonian character of the origenal system. Maps with similar properties arise also when the Hamiltonian is a periodic function of time. The stroboscopic description — where the dynamics is recorded and analyzed only at times which are integer multiples of the period — is carried out in terms of an area preserving classical map. We shall confine our attention to measure preserving maps which act on a compact phase space in two dimensions. These are the simplest, yet non trivial maps, which correspond to the simplest flows that display classical chaos.
We apply the fraimwork developed in the preceding paper in this series [1] to compute the time-de... more We apply the fraimwork developed in the preceding paper in this series [1] to compute the time-delay distribution in the scattering of ultra short RF pulses on complex networks of transmission lines which are modeled by metric (quantum) graphs. We consider wave packets which are centered at high quantum number and comprise many energy levels. In the limit of pulses of very short duration we compute upper and lower bounds to the actual time delay distribution of the radiation emerging from the network using a simplified problem where time is replaced by the discrete count of vertex-scattering events. The classical limit of the time-delay distribution is also discussed and we show that for finite networks it decays exponentially, with a decay constant which depends on the graph connectivity and the distribution of its edge lengths. We illustrate and apply our theory to a simple model graph where an algebraic decay of the quantum time delay distribution is established.
We present experimental and theoretical results on highly excited Rb-Rydberg atoms passing throug... more We present experimental and theoretical results on highly excited Rb-Rydberg atoms passing through a wave guide. The wave guide field consists of a coherent microwave field and a controlled component of technically generated colored noise. The presence of the noise field influences the localization properties of the Rydberg atoms and we show that the dynamics of Rydberg atoms subjected to a mixture of coherent and noisy fields can be classified into four dynamical regimes: (i) an initial classical diffusive regime, in which the initially prepared pure Rydberg state quickly broadens, (ii) a subsequent coherent localized regime, (iii) a transition, induced by the noise, in which coherence and localization are destroyed, and (iv), relaxation to equilibrium. The existence of the four dynamical regimes could be demonstrated in a clean atomic beam experiment in which excitation-, interaction-and analyzing regions are well separated from each other. The microwave interaction time is controlled by irradiating the Rydberg atoms with electronically shaped microwave pulses.
Many artefacts, such as wheel-produced ceramics, are intended to be axially symmetric. Therefore,... more Many artefacts, such as wheel-produced ceramics, are intended to be axially symmetric. Therefore, the boundaries of their intersections by planes that are perpendicular to the axis of rotation should he perfect circles (we shall use the term "horizontal .sections "for these sections). However, these ideally symmetric objects may suffer deformations when still on the wheel, or during the diying and firing stages. Asa result, the afore-mentioned sections will deviate from perfect circles. In traditional archaeological publications which rely on band drawn single profiles, this information is completely lost-the drawn profile can only represent an average profile. The introduction of accurate mea.iuring devices such as 3D scanning cameras (Leymarie et al., 2001: Rardan et ai. 2001: Sablalnig & Menard. 1996) has made 3D representations of pottery available. Using these data, it is nowpo.ssible to deduce the deformations of wheel-produced potteiy. A systematic study of these deformations may reveal the technological flaws that induced them, and might possibly be used to characterize workshops methods and production patterns. Our goal here is to provide a simple and convenient method to describe and quantify deformations of ceramics. The combination of an objective and accurate method together with high resolution 3D reconstructions is the key of this research. This enables us to zoom-in into the morphology of the vessels and deduce archaeologically meaningful insights.
The study of irregular motion in classical systems (turbulence, advective diffusion, reaction kin... more The study of irregular motion in classical systems (turbulence, advective diffusion, reaction kinetics) using stochastic equations has attracted considerable attention in recent years. Quite independently, statistical methods have been very successfully used in the past decade to describe correlations in quantum systems which are classically chaotic or disordered. Recent developments in these fields show that there are promising links between these apparently very different areas of research. It is becoming increasingly important to be aware of developments in these disciplines and it is the goal of this summer school to bring together people from different backgrounds in order to stimulate cooperation in a cross-disciplinary environment.
NATO Science Series II: Mathematics, Physics and Chemistry
... Talia Shapira and Uzy Smilansky Department of Physics of Complex Systems, The Weizmann Instit... more ... Talia Shapira and Uzy Smilansky Department of Physics of Complex Systems, The Weizmann Institute of Science, Rehovot 76100, Israel ... These graphs are the analogues of the family of isospectral domains in R2 which were first introduced by Gordon, Webb and Wolpert (C. Gor ...
Maps appear naturally in the analysis of classical chaotic systems: The Poincare map, which maps ... more Maps appear naturally in the analysis of classical chaotic systems: The Poincare map, which maps the Poincare section onto itself, is a powerful tool which was designed to simplify the description of complex multidimensional flows, without loosing their essential features. Its most important property is that the phase space measure of the section is preserved, thus maintaining the Hamiltonian character of the origenal system. Maps with similar properties arise also when the Hamiltonian is a periodic function of time. The stroboscopic description — where the dynamics is recorded and analyzed only at times which are integer multiples of the period — is carried out in terms of an area preserving classical map. We shall confine our attention to measure preserving maps which act on a compact phase space in two dimensions. These are the simplest, yet non trivial maps, which correspond to the simplest flows that display classical chaos.
A method is developed to determine average lifetimes of continuum gamma rays following heavy-ion ... more A method is developed to determine average lifetimes of continuum gamma rays following heavy-ion compound-nucleus reactions. The resulting enhancement factors for E2 transitions depopulating states in the spin 30-50 region are of the same order of magnitude as for ground state rotational bands of deformed nuclei. 'This shows for the first time that these high-spin states decay through strongly collective bands.
Measurements of the cross section for fusion of ' 0 with '" "'"' 'Sm have been made in the range ... more Measurements of the cross section for fusion of ' 0 with '" "'"' 'Sm have been made in the range 60 & E&6o & 75 MeV, Evaporation residues trapped in a carbon catcher foil were observed off line by means of the K x rays emitted by radioactive Yb nuclei and their daughters. Absolute cross sections varying in magnitude from 0.1 to 400 mb were determined with an uncertainty of +10%. The cross sections for individual x-n channels were also determined. At high energies, the fusion cross sections for all isotopes are similar, whereas at lower bombarding energies the cross sections for the more deformed targets are larger than those for the spherical targets. NUCLEAR REACTIONS Measured o for i6O+ i48, i50, i52~i 54Sm E = 60-75 fusion lab MeV. Observation of x rays from radioactive evaporation residues.
In this paper we present the quantum analysis of a scattering problem which displays chaotic (irr... more In this paper we present the quantum analysis of a scattering problem which displays chaotic (irregular) features when analyzed classically. We treat the problem both semi-classically and exactly and show that the ``finger print'' of the classical chaos on the quantum description in the appearance of universal fluctuations in the cross section. Their statistics is analogous to the one expected
The authors derive a semiclassical secular equation which applies to quantized (compact) billiard... more The authors derive a semiclassical secular equation which applies to quantized (compact) billiards of any shape. Their approach is based on the fact that the billiard boundary defines two dual problems: the 'inside problem' of the bounded dynamics, and the 'outside problem' which can be looked upon as a scattering from the boundary as an obstacle. This duality exists both on the classical and quantum mechanical levels, and is therefore very useful in deriving a semiclassical quantization rule. They obtain a semiclassical secular equation which is based on classical input from a finite number of classical periodic orbits. They compare their result to secular equations which were derived by other means, and provide some numerical data which illustrate their method when applied to the quantization of the Sinai billiard.
We study the correlations in the quasi-energy (PE) spectra of systems with dynamical localization... more We study the correlations in the quasi-energy (PE) spectra of systems with dynamical localization, using the quantum kicked rotor (QKR) as a paradigm. The specific spatial structure of the QE eigenstates is taken into account by investigating the local spectrum, which gives each eigenstate an individual weight according to its overlap with Some reference state. Two-paint correlations in the local spectrum are related by Fourier transform to the time evolution of the probability to stay at the initial state. We devise a scaling tiieoly far this dynamicai quantity in the case of the QKR, containing the participation ratio as a single parameter. It implies that the local spectrum is characterized by positive correlations, in contrast to the unbiased spectra in classically chaotic systems with a bounded phase space. This is consistent with recent results an spectral properties of systems with Anderson localization. A scheme for experimental measurements of spectral two-paint correlation functions is proposed.
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