In this paper [published in Phys. Rev. E 102, 042210 (2020)], a new method for the calculation of excited chaotic eigenfunctions in arbitrary energy windows is presented. We demonstrate the feasibility of using wavefunctions localized on unstable periodic orbits as efficient basis sets for this task in classically chaotic systems. The number of required localized wavefunctions is only of the order of the ratio t H /t E , with t H the Heisenberg time and t E the Ehrenfest time. As an illustration, we present convincing results for a coupled two-dimensional quartic oscillator with chaotic dynamics.
A hypothesis about the average phase-space distribution of resonance eigenfunctions in chaotic systems with escape through an opening is proposed. Eigenfunctions with decay rate $γ$ are described by a classical measure that $(i)$ is conditionally invariant with classical decay rate $γ$ and $(ii)$ is uniformly distributed on sets with the same temporal distance to the quantum resolved chaotic saddle. This explains the localization of fast-decaying resonance eigenfunctions classically. It is found to occur in the phase-space region having the largest distance to the chaotic saddle. We discuss th…
Physical systems are often neither completely closed nor completely open, but instead they are best described by dynamical systems with partial escape or absorption. In this paper we introduce classical measures that explain the main properties of resonance eigenfunctions of chaotic quantum systems with partial escape. We construct a family of conditionally-invariant measures with varying decay rates by interpolating between the natural measures of the forward and backward dynamics. Numerical simulations in a representative system show that our classical measures correctly describe the main fe…
In most realistic models for quantum chaotic systems, the Hamiltonian matrices in unperturbed bases have a sparse structure. We study correlations in eigenfunctions of such systems and derive explicit expressions for some of the correlation functions with respect to energy. The analytical results are tested in several models by numerical simulations. An application is given for a relation between transition probabilities.
Classical-quantum correspondence for conservative chaotic Hamiltonians is investigated in terms of the structure of the eigenfunctions and the local density of states, using as a model a 2D rippled billiard in the regime of global chaos. The influence of the observed localized and sparsed states in the quantum-classical correspondence is discussed.
We present a semiclassical approach to eigenfunction statistics in chaotic and weakly disordered quantum systems which goes beyond Random Matrix Theory, supersymmetry techniques, and existing semiclassical methods. The approach is based on a generalization of Berry's Random Wave Model, combined with a consistent semiclassical representation of spatial two-point correlations. We derive closed expressions for arbitrary wavefunction averages in terms of universal coefficients and sums over classical paths, which contain, besides the supersymmetry results, novel oscillatory contributions. Their ph…
We study the resonance (or Gamow) eigenstates of open chaotic systems in the semiclassical limit, distinguishing between left and right eigenstates of the non-unitary quantum propagator, and also between short-lived and long-lived states. The long-lived left (right) eigenstates are shown to concentrate as $\hbar\to 0$ on the forward (backward) trapped set of the classical dynamics. The limit of a sequence of eigenstates $\{ψ(\hbar)\}_{\hbar\to 0}$ is found to exhibit a remarkably rich structure in phase space that depends on the corresponding limiting decay rate. These results are illustrated …
We study the statistical distribution of components in the non-perturbative parts of energy eigenfunctions (EFs), in which main bodies of the EFs lie. Our numerical simulations in five models show that deviation of the distribution from the prediction of random matrix theory (RMT) is useful in characterizing the process from nearly-integrable to chaotic, in a way somewhat similar to the nearest-level-spacing distribution. But, the statistics of EFs reveals some more properties, as described below. (i) In the process of approaching quantum chaos, the distribution of components shows a delay fea…
A general program to show quantum-classical correspondence for bound conservative integrable and chaotic systems is described. The method is applied to integrable systems and the nature of the approach to the classical limit, the cancellation of essential singularities, is demonstrated. The application to chaotic systems requires an understanding of classical Liouville eigenfunctions and a Liouville spectral decomposition, developed herein. General approaches to the construction of these Liouville eigenfunctions and classical spectral projectors in quantum and classical mechanics are discussed…
We study eigenstates of chaotic billiards in the momentum representation and propose the radially integrated momentum distribution as useful measure to detect localization effects. For the momentum distribution, the radially integrated momentum distribution, and the angular integrated momentum distribution explicit formulae in terms of the normal derivative along the billiard boundary are derived. We present a detailed numerical study for the stadium and the cardioid billiard, which shows in several cases that the radially integrated momentum distribution is a good indicator of localized eigen…
Statistical structural properties of many-body chaotic eigenfunctions and applications
University of Science and Technology of China（中国科学技术大学）；CAS Key Laboratory of Microscale Magnetic Resonance, University of Science and Technology of China（中国科学院微观磁学共振重点实验室，中国科学技术大学）；Anhui Center for fundamental sciences in theoretical physics（安徽基础科学前沿中心）；University of Osnabrück（奥斯纳布吕克大学）
We study the statistics of the experimental eigenfunctions of chaotic and disordered microwave billiards in terms of the moments of their spatial distributions, such as the Inverse Participation Ratio (IPR) and density-density auto-correlation. A path from chaos to disorder is described in terms of increasing IPR. In the chaotic, ballistic limit, the data correspond well with universal results from random matrix theory. Deviations from universal distributions are observed due to disorder induced localization, and for the weakly disordered case the data are well-described by including finite co…
We introduce a single-channel opening in a random Hamiltonian and a quantized chaotic map: localization on the opening occurs as a sensible deviation of the wavefunction statistics from the predictions of random matrix theory, even in the semiclassical limit. Increasing the coupling to the open channel in the quantum model, we observe a similar picture to resonance trapping, made of few fast-decaying states, whose left (right) eigenfunctions are entirely localized on the (preimage of the) opening, and plentiful long-lived states, whose probability density is instead suppressed at the opening. …
We give an introduction to some of the numerical aspects in quantum chaos. The classical dynamics of two--dimensional area--preserving maps on the torus is illustrated using the standard map and a perturbed cat map. The quantization of area--preserving maps given by their generating function is discussed and for the computation of the eigenvalues a computer program in Python is presented. We illustrate the eigenvalue distribution for two types of perturbed cat maps, one leading to COE and the other to CUE statistics. For the eigenfunctions of quantum maps we study the distribution of the eigen…
Given a Dirichlet eigenfunction of a 2D quantum billiard, the boundary domain count is the number of intersections of the nodal lines with the boundary. We study the integer sequence defined by these numbers, sorted according to the energies of the eigenfunctions. Based on a variant of Berry's random wave model, we derive a semi-classical trace formula for the sequence of boundary domain counts. The formula consists of a Weyl-like smooth part, and an oscillating part which depends on classical periodic orbits and their geometry. The predictions of this trace formula are supported by numerical …
We study quantum kicked rotator in the classically fully chaotic regime, in the domain of the semiclassical behaviour. We use Izrailev's N-dimensional model for various N<=4000, which in the limit N-> infinity tends to the quantized kicked rotator, not only for K=5 as studied previously, but for many different values of the classical kick parameter 5<=K<=35, and also of the quantum parameter k. We describe the dynamical localization of chaotic eigenstates as a paradigm for other both time-periodic and time-independent fully chaotic or/and mixed type Hamilton systems. We generalize the localiza…
We study chaotic eigenfunctions in wedge-shaped and rectangular regions using a generalization of Berry's conjecture. An expression for the two-point correlation function is derived and verified numerically.
We present a method to efficiently compute the eigenfunctions of classically chaotic systems. The key point is the definition of a modified Gram-Schmidt procedure which selects the most suitable elements from a basis set of scar functions localized along the shortest periodic orbits of the system. In this way, one benefits from the semiclassical dynamical properties of such functions. The performance of the method is assessed by presenting an application to a quartic two dimensional oscillator whose classical dynamics are highly chaotic. We have been able to compute the eigenfunctions of the s…
We study the time evolution of occupation numbers for interacting Fermi-particles in the situation when exact compound states are "chaotic". This situation is generic for highly excited many-particles states in heavy nuclei, complex atoms, quantum dots, spin systems and quantum computer models. Numerical data show perfect agreement with a simple theory for the onset of thermalization in close systems of interacting particles.
How Are Quantum Eigenfunctions of Hydrogen Atom Related To Its Classical Elliptic Orbits?
International Center for Quantum Materials, School of Physics, Peking University（北京大学物理学院量子材料国际中心）；Wilczek Quantum Center, Shanghai Institute for Advanced Studies（上海高等研究院吴文俊理论数学中心（斐泽克量子中心））；Hefei National Laboratory（合肥国家实验室）
We study statistical properties of excited levels of the E x (b_1+b_2) Jahn-Teller model. The multitude of avoided crossings of energy levels is generally claimed to be a testimony of quantum chaos. We found that apart from two limiting cases (E x e and Holstein model) the distribution of nearest-neighbor spacings is rather stable as to the change of parameters and different from the Wigner one. This limiting distribution assumably shows scaling ~$\sqrt{S}$ at small S and resembles the semi-Poisson law P(S)= 4S \exp (-2 S) at S> 1. The latter is believed to be universal and characteristic, e.g…
We study the time evolution of an initially excited many-body state in a finite system of interacting Fermi-particles in the situation when the interaction gives rise to the ``chaotic'' structure of compound states. This situation is generic for highly excited many-particle states in quantum systems, such as heavy nuclei, complex atoms, quantum dots, spin systems, and quantum computers. For a strong interaction the leading term for the return probability $W(t)$ has the form $W(t)\simeq \exp (-Δ_E^2t^2)$ with $Δ_E^2$ as the variance of the strength function. The conventional exponential linear …
Excited-state quantum phase transitions and chaos in a three-level Lipkin model
Escuela Politécnica Superior, Universidad CEU Fernando III, CEU Universities（CEU费尔南多三世大学高等理工学院）；Departamento de Física Aplicada III, Escuela Técnica Superior de Ingeniería, Universidad de Sevilla（塞维利亚大学高等工程学院应用物理系）；Multidisciplinary Unit for Energy Science, Universidad de Sevilla（塞维利亚大学能源科学多学科单元）；Departamento de Física Atómica, Molecular y Nuclear, Facultad de Física, Universidad de Sevilla（塞维利亚大学物理学院原子、分子和核物理系）
