Distribution Of Resonance Widths Research Articles

Resonance distribution in the quantum random Lorentz gas

The multiple scattering model of a quantum particle in a random Lorentz gas consisting of fixed point scatterers is considered in arbitrary dimension. An efficient method is developed to numerically compute the map of the density of scattering resonances in the complex plane of the wavenumber without finding them one by one. The method is applied to two collision models for the individual scatterers, namely a resonant model, and a non-resonant hard-sphere model. The results obtained with the former are compared to the literature. In particular, the spiral arms surrounding the single-scatterer resonance are identified as proximity resonances. Moreover, the hard-sphere model is used to reveal previously unknown structures in the resonance density. Finally, it is shown how Anderson localization affects the distribution of resonance widths, especially in the one-dimensional case.

Modification of the Porter-Thomas Distribution by Rank-One Interaction.

The Porter-Thomas (PT) distribution of resonance widths is one of the oldest and simplest applications of statistical ideas in nuclear physics. Previous experimental data confirmed it quite well, but recent and more careful investigations show clear deviations from this distribution. To explain these discrepancies, Volya, Weidenmüller, and Zelevinsky [Phys. Rev. Lett. 115, 052501 (2015)PRLTAO0031-900710.1103/PhysRevLett.115.052501] argued that to get a realistic model of nuclear resonances is not enough to consider one of the standard random matrix ensembles which leads immediately to the PT distribution, but it is necessary to add a rank-one interaction which couples resonances to decay channels. The purpose of this Letter is to solve this model analytically and to find explicitly the modifications of the PT distribution due to such an interaction. Resulting formulas are simple, in good agreement with numerics, and could explain experimental results.

Spectral statistics of molecular resonances in erbium isotopes: How chaotic are they?

We perform a comprehensive analysis of the spectral statistics of the molecular resonances in (166)Er and (168)Er observed in recent ultracold collision experiments [Frisch et al., Nature (London) 507, 475 (2014)] with the aim of determining the chaoticity of this system. We calculate different independent statistical properties to check their degree of agreement with random matrix theory (RMT), and analyze if they are consistent with the possibility of having missing resonances. The analysis of the short-range fluctuations as a function of the magnetic field points to a steady increase of chaoticity until B∼30 G. The repulsion parameter decreases for higher magnetic fields, an effect that can be interpreted as due to missing resonances. The analysis of long-range fluctuations allows us to be more quantitative and estimate a 20%-25% fraction of missing levels. Finally, a study of the distribution of resonance widths provides additional evidence supporting missing resonances of small width compared with the experimental magnetic field resolution. We conclude that further measurements with increased resolution will be necessary to give a final answer to the problem of missing resonances and the agreement with RMT.

Resonance width distribution in RMT: Weak-coupling regime beyond Porter-Thomas

We employ the random matrix theory (RMT) framework to revisit the distribution of resonance widths in quantum chaotic systems weakly coupled to the continuum via a finite number M of open channels. In contrast to the standard first-order perturbation theory treatment we do not a priori assume the resonance widths being small compared to the mean level spacing. We show that to the leading order in weak coupling the perturbative distribution of the resonance widths (in particular, the Porter-Thomas distribution at M = 1) should be corrected by a factor related to a certain average of the ratio of square roots of the characteristic polynomial (“spectral determinant”) of the underlying RMT Hamiltonian. A simple single-channel expression is obtained that properly approximates the width distribution also at large resonance overlap, where the Porter-Thomas result is no longer applicable.

Cooperative effects and disorder: A scaling analysis of the spectrum of the effective atomic Hamiltonian

We study numerically the spectrum of the non-Hermitian effective Hamiltonian that describes the dipolar interaction of a gas of $N\ensuremath{\gg}1$ atoms with the radiation field. We analyze the interplay between cooperative effects and disorder for both scalar and vectorial radiation fields. We show that for dense gases, the resonance width distribution follows, both in the scalar and vectorial cases, a power law $P(\ensuremath{\Gamma})\ensuremath{\sim}{\ensuremath{\Gamma}}^{\ensuremath{-}4/3}$ that originates from cooperative effects between more than two atoms. This power law is different from the $P(\ensuremath{\Gamma})\ensuremath{\sim}{\ensuremath{\Gamma}}^{\ensuremath{-}1}$ behavior, which has been considered as a signature of Anderson localization of light in random systems. We show that in dilute clouds, the center of the energy distribution is described by Wigner's semicircle law in the scalar and vectorial cases. For dense gases, this law is replaced in the vectorial case by the Laplace distribution. Finally, we show that in the scalar case the degree of resonance overlap increases as a power law of the system size for dilute gases, but decays exponentially with the system size for dense clouds.

Resonance width distribution for open quantum systems

Recent measurements of resonance widths for low-energy neutron scattering off heavy nuclei show large deviations from the Porter-Thomas distribution. We propose a ``standard'' width distribution based on the random matrix theory for a chaotic quantum system with a single open decay channel. Two methods of derivation lead to a single analytical expression that recovers, in the limit of very weak continuum coupling, the Porter-Thomas distribution. The parameter defining the result is the ratio of typical widths $\ensuremath{\Gamma}$ to the energy level spacing $D$. Compared to the Porter-Thomas distribution, the new distribution suppresses small widths and increases the probabilities of larger widths. We show also that it is necessary to take into account the $\ensuremath{\gamma}$ channels.

Porter-Thomas distribution in unstable many-body systems

We use the continuum shell model approach to explore the resonance width distribution in unstable many-body systems. The single-particle nature of a decay, the few-body character of the interaction Hamiltonian, and the collectivity that emerges in nonstationary systems due to the coupling to the continuum of reaction states are discussed. Correlations between the structures of the parent and daughter nuclear systems in the common Fock space are found to result in deviations of decay width statistics from the Porter-Thomas distribution.

Distribution of Resonance Widths and Dynamics of Continuum Coupling

We analyze the statistics of resonance widths in a many-body Fermi system with open decay channels. Depending on the strength of continuum coupling, such a system reveals growing deviations from the standard chi-square (Porter-Thomas) width distribution. The deviations emerge from the process of increasing interaction of intrinsic states through common decay channels; in the limit of perfect coupling this process leads to the superradiance phase transition. The width distribution depends also on the intrinsic dynamics (chaotic versus regular). The results presented here are important for understanding the recent experimental data concerning the width distribution for neutron resonances in nuclei.

Truncations of random orthogonal matrices

Statistical properties of nonsymmetric real random matrices of size M, obtained as truncations of random orthogonal N×N matrices, are investigated. We derive an exact formula for the density of eigenvalues which consists of two components: finite fraction of eigenvalues are real, while the remaining part of the spectrum is located inside the unit disk symmetrically with respect to the real axis. In the case of strong nonorthogonality, M/N=const, the behavior typical to real Ginibre ensemble is found. In the case M=N-L with fixed L, a universal distribution of resonance widths is recovered.

Resonance width distribution for high-dimensional random media

We study the distribution of resonance widths $\mathcal{P}(\ensuremath{\Gamma})$ for three-dimensional (3D) random scattering media and analyze how it changes as a function of the randomness strength. We are able to identify in $\mathcal{P}(\ensuremath{\Gamma})$ the system-inherent fingerprints of the metallic, localized, and critical regimes. Based on the properties of resonance widths, we also suggest a criterion for determining and analyzing the metal-insulator transition. Our theoretical predictions are verified numerically for the prototypical 3D tight-binding Anderson model.

State-selected dynamics of the complex-forming bimolecular reaction Cl- +CH3 Cl'-->ClCH3+Cl'-: a four-dimensional quantum scattering study.

Time-independent quantum scattering calculations have been carried out on the Walden inversion S(N)2 reaction Cl(-)+CH(3)Cl(')(v(1),v(2),v(3))-->ClCH(3)(v(1) ('),v(2) ('),v(3) ('))+Cl('-). The two C-Cl stretching modes (quantum numbers v(3) and v(3) (')) and the totally symmetric internal modes of the methyl group (C-H stretching vibration, v(1) and v(1) ('), and inversion bending vibration, v(2) and v(2) (')) are treated explicitly. A four-dimensional coupled cluster potential energy surface is employed. The scattering problem is formulated in hyperspherical coordinates using the exact Hamiltonian and exploiting the full symmetry of the problem. Converged state-selected reaction probabilities and product distributions have been calculated up to 6100 cm(-1) above the vibrational ground state of CH(3)Cl, i.e., up to initial vibrational excitation (2,0,0). In order to extract all scattering resonances, the energetic grid was chosen to be very fine, partly down to a resolution of 10(-12) cm(-1). Up to 2500 cm(-1) translational energy, initial excitation of the umbrella bending vibration, (0,1,0), is more efficient for reaction than exciting the C-Cl stretching mode, (0,0,1). The combined excitation of both vibrations results in a synergic effect, i.e., a considerably higher reaction probability than expected from the sum of both independent excitations, even higher than (0,0,2) up to 1500 cm(-1) translational energy. Product distributions show that the umbrella mode is strongly coupled to the C-Cl stretching mode and cannot be treated as a spectator mode. The reaction probability rises almost linearly with increasing initial excitation of the umbrella bending mode. The effect with respect to the C-Cl stretch is five times larger for more than two quanta in this mode, and in agreement with previous work saturation is found. Exciting the high-frequency C-H stretching mode, (1,0,0), yields a large increase for small energies [more than two orders of magnitude larger than (0,0,0)], while for translational energies higher than 2000 cm(-1), it becomes a pure spectator mode. For combined initial excitations including the symmetric C-H stretch, the spectator character of the latter is even more pronounced. However, up to more than 1500 cm(-1) translational energy, the C-H vibration does not behave adiabatically during the course of reaction, because only 20% of the initial energy is found in the same mode of the product molecule. The distribution of resonance widths and peak heights is discussed, and it is found that individual resonances pertinent to intermediate complexes Cl(-)...CH(3)Cl show product distributions independent of the initial vibrational state of the reactant molecule. The relatively high reactivity, of resonance states with respect to excitation of any mode, found in previous work is confirmed in the present calculations. However, reactivity of intermediate states and reactivity with respect to initial vibrational excitation have to be distinguished. There is a strong mixing between the vibrational states reflected in numerous avoided crossings of the hyperspherical adiabatic curves.

Probing Anderson localization of light via decay rate statistics.

We have studied the distribution of resonance widths P(Gamma) in one-, two-, and three-dimensional multiple light scattering systems. P(Gamma) should follow a universal power law P(Gamma) approximately Gamma(-1) in the localized regime as confirmed by extensive numerical calculations. This behavior can be interpreted as an unambiguous signature of exponential Anderson localization of light in open systems.

Isolated Resonances in Conductance Fluctuations in Ballistic Billiards

We study numerically quantum transport through a billiard with a classically mixed phase space. In particular, we calculate the conductance and Wigner delay time by employing a recursive Green's function method. We find sharp, isolated resonances with a broad distribution of resonance widths in both the conductance and the Wigner time, in contrast to the well-known smooth conductance fluctuations of completely chaotic billiards. In order to elucidate the origin of the isolated resonances, we calculate the associated scattering states as well as the eigenstates of the corresponding closed system. As a result, we find a one-to-one correspondence between the resonant scattering states and eigenstates of the closed system. The broad distribution of resonance widths is traced to the structure of the classical phase space. Husimi representations of the resonant scattering states show a strong overlap either with the regular regions in phase space or with the hierarchical parts surrounding the regular regions. We are thus lead to a classification of the resonant states into regular and hierarchical, depending on their phase space portrait.

Distribution of resonance widths in localized tight-binding models

We numerically analyze the distribution of scattering resonance widths in one- and quasi-one dimensional tight binding models, in the localized regime. We detect and discuss an algebraic decay of the distribution, similar, though not identical, to recent theoretical predictions.

Signature of dynamical localization in the resonance width distribution of wave-chaotic dielectric cavities

We consider the effect of dynamical localization on the widths of the resonances in open wave-chaotic dielectric cavities. We show that dynamical localization leads to a log-normal distribution of the resonance widths which scales with the localization length in excellent agreement with the results of numerical calculations for open rough microcavities.

Quantum Transport through Ballistic Cavities: Soft vs Hard Quantum Chaos

We study transport through a two-dimensional billiard attached to two infinite leads by numerically calculating the Landauer conductance and the Wigner time delay. In the generic case of a mixed phase space we find a power-law distribution of resonance widths and a power-law dependence of conductance increments apparently reflecting the classical dwell time exponent, in striking difference to the case of a fully chaotic phase space. Surprisingly, these power laws appear on energy scales below the mean level spacing, in contrast to semiclassical expectations.

Time-delay correlations and resonances in one-dimensional disordered systems

The frequency dependent time delay correlation function $K(\Omega)$ is studied analytically for a particle reflected from a finite one-dimensional disordered system. In the long sample limit $K(\Omega)$ can be used to extract the resonance width distribution $\rho(\Gamma)$. Both quantities are found to decay algebraically as $\Gamma^{-\nu}$, and $\Omega^{-\nu}$, $\nu\simeq 1.25$ in a large range of arguments. Numerical calculations for the resonance width distribution in 1D non-Hermitian tight-binding model agree reasonably with the analytical formulas.

Scar and antiscar quantum effects in open chaotic systems.

We predict and numerically observe strong periodic orbit effects in the properties of weakly open quantum systems with a chaotic classical limit. Antiscars lead to a large number of exponentially narrow isolated resonances when the single-channel (or tunneling) opening is located on a short unstable orbit of the closed system; the probability to remain in the system at long times is thus exponentially enhanced over the random matrix theory prediction. The distribution of resonance widths and the probability to remain are quantitatively given in terms of only the stability matrix of the orbit on which the opening is placed. The long-time remaining probability density is nontrivially distributed over the available phase space; it can be enhanced or suppressed near orbits other than the one on which the lead is located, depending on the periods and classical actions of these other orbits. These effects of the short periodic orbits on quantum decay rates have no classical counterpart, and first appear on times scales much larger than the Heisenberg time of the system. All the predictions are quantitatively compared with numerical data.

Quantum chaos in [formula omitted] microstructures

We report distinct signatures of ‘quantum chaos’ in both open and closed systems. Quantum chaos refers to the quantum signature of systems which classically exhibit chaotic dynamics. In the open case, we measure the line shape of the negative magneto-resistance peak in ballistic cavities fabricated on very high quality GaAs Al x Ga 1 − x As heterostructures. We find a striking difference in the line shape for chaotic, stadium cavities versus non-chaotic, circle cavities: Lorentzian versus triangular. In the case of closed systems, we study the distribution of Coulomb blockade conductance peak heights in individual quantum dots. We find a strongly non-Gaussian distribution and a significant difference between the zero and non-zero magnetic fields. While the zero magnetic field distribution is related to the well-known Porter-Thomas distribution of resonance widths in the scattering of neutrons from nuclei, the non-zero magnetic field distribution is completely new and previously unobserved.

Statistics of resonance poles, phase shifts and time delays in quantum chaotic scattering: Random matrix approach for systems with broken time-reversal invariance

Assuming the validity of random matrices for describing the statistics of a closed chaotic quantum system, we study analytically some statistical properties of the S-matrix characterizing scattering in its open counterpart. In the first part of the paper we attempt to expose systematically ideas underlying the so-called stochastic (Heidelberg) approach to chaotic quantum scattering. Then we concentrate on systems with broken time-reversal invariance coupled to continua via Mopen channels; a=1,2,…,M. A physical realization of this case corresponds to the chaotic scattering in ballistic microstructures pierced by a strong enough magnetic flux. By using the supersymmetry method we derive an explicit expression for the density of S-matrix poles (resonances) in the complex energy plane. When all scattering channels are considered to be equivalent our expression describes a crossover from the χ2 distribution of resonance widths (regime of isolated resonances) to a broad power-like distribution typical for the regime of overlapping resonances. The first moment is found to reproduce exactly the Moldauer–Simonius relation between the mean resonance width and the transmission coefficient. Under the same assumptions we derive an explicit expression for the parametric correlation function of densities of eigenphases θa of the S-matrix (taken modulo 2π). We use it to find the distribution of derivatives τa=∂θa/∂E of these eigenphases with respect to the energy (“partial delay times”) as well as with respect to an arbitrary external parameter. We also find the parametric correlations of the Wigner–Smith time delay τw(E)=(1/M)∑a ∂θa/∂E at two different energies E−Ω/2 and E+Ω/2 as well as at two different values of the external parameter. The relation between our results and those following from the semiclassical approach as well as the relevance to experiments are briefly discussed.