Just as a coherent state may be considered as a quantum point, its restriction to a factor space of the full Hilbert space can be interpreted as a quantum plane. The overlap of such a factor coherent state with a full pure state is akin to a quantum section. It defines a reduced pure state in the cofactor Hilbert space. The collection of all the Wigner functions corresponding to a full set of parallel quantum sections defines the Husimi-Wigner reresentation. It occupies an intermediate ground between drastic suppression of nonclassical features, characteristic of Husimi functions, and the daunting complexity of higher dimensional Wigner functions. After analysing these features for simpler states, we exploit this new representation as a probe of numerically computed eigenstates of chaotic Hamiltonians. The individual two-dimensional Wigner functions resemble those of semiclassically quantized states, but the regular ring pattern is broken by dislocations.
The projector onto single quantum map eigenstates is written only in terms of powers of the evolution operator, up to half the Heisenberg time, and its traces. These powers are semiclassically approximated, by a complex generating function, giving the Husimi distribution of the eigenstates. The results are tested on the Cat and Baker maps.
For the representation of eigenstates on a Poincaré section at the boundary of a billiard different variants have been proposed. We compare these Poincaré Husimi functions, discuss their properties and based on this select one particularly suited definition. For the mean behaviour of these Poincaré Husimi functions an asymptotic expression is derived, including a uniform approximation. We establish the relation between the Poincaré Husimi functions and the Husimi function in phase space from which a direct physical interpretation follows. Using this, a quantum ergodicity theorem for the Poinca…
We study the efficiency of quantum algorithms which aim at obtaining phase space distribution functions of quantum systems. Wigner and Husimi functions are considered. Different quantum algorithms are envisioned to build these functions, and compared with the classical computation. Different procedures to extract more efficiently information from the final wave function of these algorithms are studied, including coarse-grained measurements, amplitude amplification and measure of wavelet-transformed wave function. The algorithms are analyzed and numerically tested on a complex quantum system sh…
Local parametric statistics of zeros of Husimi representations of quantum eigenstates are introduced. It is conjectured that for a classically fully chaotic systems one should use the model of parametric statistics of complex roots of Gaussian random polynomials which is exactly solvable as demonstrated below. For example, the velocities (derivatives of zeros of Husimi function with respect to an external parameter) are predicted to obey a universal (non-Maxwellian) distribution ${d P(v)}/{dv^2} = 2/(πσ^2)(1 + |v|^2/σ^2)^{-3},$ where $σ^2$ is the mean square velocity. The conjecture is demonst…
We construct a semiclassical expression for the Husimi function of autonomous systems in one degree of freedom, by smoothing with a Gaussian function an expression that captures the essential features of the Wigner function in the semiclassical limit. Our approximation reveals the "center and chord" estructure that the Husimi function inherits from the Wigner function, down to very shallow "valleys", where lie the Husimi zeroes. This explanation for the distribution of zeroes along curves relies on the geometry of the classical torus, rather than the complex analytical properties of the WKB me…
Using the Bargmann-Husimi representation of quantum mechanics on a torus phase space, we study analytically eigenstates of quantized cat maps. The linearity of these maps implies a close relationship between classically invariant sublattices on the one hand, and the patterns (or `constellations') of Husimi zeros of certain quantum eigenstates on the other hand. For these states, the zero patterns are crystals on the torus. As a consequence, we can compute explicit families of eigenstates for which the zero patterns become uniformly distributed on the torus phase space in the limit $\hbar\to 0$…
Band Husimi distributions (BHDs) are introduced in the quantum-chaos problem on a toral phase space. In the framework of this phase space, a quantum state must satisfy Bloch boundary conditions (BCs) on a torus and the spectrum consists of a finite number of levels for given BCs. As the BCs are varied, a level broadens into a band. The BHD for a band is defined as the uniform average of the Husimi distributions for all the eigenstates in the band. The generalized BHD for a set of adjacent bands is the average of the BHDs associated with these bands. BHDs are shown to be closer, in several aspe…
We demonstrate that the energy or quasienergy level spacing distribution in dynamically localized chaotic eigenstates is excellently described by the Brody distribution, displaying the fractional power law level repulsion. This we show in two paradigmatic systems, namely for the fully chaotic eigenstates of the kicked rotator at K=7, and for the chaotic eigenstates in the mixed-type billiard system (Robnik 1983), after separating the regular and chaotic eigenstates by means of the Poincaré Husimi function, at very high energies with great statistical significance (587654 eigenstates, starting …
Using coherent-state representations of quantum mechanics (Bargmann, Husimi, and stellar representations), we describe analytically the phase-space structure of the general eigenstates corresponding to a 1-dimensional bilinear hyperbolic Hamiltonian, H=pq or equivalently H=1/2(P^2-Q^2). Their semi-classical behaviour is discussed for eigenvalues either near or away from the separatrix energy {H=0}, especially in the phase-space vicinity of the saddle-point (q,p)=(0,0).
The zeroes of the Husimi function provide a minimal description of individual quantum eigenstates and their distribution is of considerable interest. We provide here a numerical study for pseudo- integrable billiards which suggests that the zeroes tend to diffuse over phase space in a manner reminiscent of chaotic systems but nevertheless contain a subtle signature of pseudo-integrability. We also find that the zeroes depend sensitively on the position and momentum uncertainties with the classical correspondence best when the position and momentum uncertainties are equal. Finally, short range …
It is shown that the Husimi representations of chaotic eigenstates are strongly correlated along classical trajectories. These correlations extend across the whole system size and, unlike the corresponding eigenfunction correlations in configuration space, they persist in the semiclassical limit. A quantitative theory is developed on the basis of Gaussian wavepacket dynamics and random-matrix arguments. The role of symmetries is discussed for the example of time-reversal invariance.
We develop a semiclassical approximation for the spectral Wigner and Husimi functions in the neighbourhood of a classically unstable periodic orbit of chaotic two dimensional maps. The prediction of hyperbolic fringes for the Wigner function, asymptotic to the stable and unstable manifolds, is verified computationally for a (linear) cat map, after the theory is adapted to a discrete phase space appropriate to a quantized torus. The characteristic fringe patterns can be distinguished even for quasi-energies where the fixed point is not Bohr-quantized. The corresponding Husimi function dampens t…
We study the quantum mechanics of a billiard (Robnik 1983) in the regime of mixed-type classical phase space (the shape parameter λ=0.15) at very high-lying eigenstates, starting at about 1.000.000th eigenstate and including the consecutive 587654 eigenstates. By calculating the normalized Poincaré Husimi functions of the eigenstates and comparing them with the classical phase space structure, we introduce the overlap criterion which enables us to separate with great accuracy and reliability the regular and chaotic eigenstates, and the corresponding energies. The chaotic eigenstates appear all…
We study the correspondence between phase-space localization of quantum (quasi-)energy eigenstates and classical correlation decay, given by Ruelle-Pollicott resonances of the Frobenius-Perron operator. It will be shown that scarred (quasi-)energy eigenstates are correlated: Pairs of eigenstates strongly overlap in phase space (scar in same phase-space regions) if the difference of their eigenenergies is close to the phase of a leading classical resonance. Phase-space localization of quantum states will be measured by $L^2$ norms of their Husimi functions.
We study individual eigenstates of quantized area-preserving maps on the 2-torus which are classically chaotic. In order to analyze their semiclassical behavior, we use the Bargmann-Husimi representations for quantum states, as well as their stellar parametrization, which encodes states through a minimal set of points in phase space (the constellation of zeros of the Husimi density). We rigorously prove that a semiclassical uniform distribution of Husimi densities on the torus entails a similar equidistribution for the corresponding constellations. We deduce from this property a universal beha…
The phenomenon of quantum localization in classically chaotic eigenstates is one of the main issues in quantum chaos (or wave chaos), and thus plays an important role in general quantum mechanics or even in general wave mechanics. In this work we propose two different localization measures characterizing the degree of quantum localization, and study their relation to another fundamental aspect of quantum chaos, namely the (energy) spectral statistics. Our approach and method is quite general, and we apply it to billiard systems. One of the signatures of the localization of chaotic eigenstates …
A generalization of Brillouin-Wigner perturbation theory is applied numerically to the Wigner Band Random Matrix model. The perturbation theory tells that a perturbed energy eigenstate can be divided into a perturbative part and a non-perturbative part with the perturbative part expressed as a perturbation expansion. Numerically it is found that such a division is important in understanding many properties of both eigenstates and the so-called local spectral density of states (LDOS). For the average shape of eigenstates, its central part is found to be composed of its non-perturbative part and…
The properties of the resonant Gamow states are studied numerically in the semiclassical limit for the quantum Chirikov standard map with absorption. It is shown that the number of such states is described by the fractal Weyl law and their Husimi distributions closely follow the strange repeller set formed by classical orbits nonescaping in future times. For large matrices the distribution of escape rates converges to a fixed shape profile characterized by a spectral gap related to the classical escape rate.
Band distributions (BDs) are introduced describing quantization in a toral phase space. A BD is the uniform average of an eigenstate phase-space probability distribution over a band of toral boundary conditions. A general explicit expression for the Wigner BD is obtained. It is shown that the Wigner functions for {\em all} of the band eigenstates can be reproduced from the Wigner BD. Also, BDs are shown to be closer to classical distributions than eigenstate distributions. Generalized BDs, associated with sets of adjacent bands, are used to extend in a natural way the Chern-index characterizat…
The localization measures $A$ (based on the information entropy) of localized chaotic eigenstates in the Poincaré-Husimi representation have a distribution on a compact interval $[0,A_0]$, which is well approximated by the {\em beta distribution}, based on our extensive numerical calculations. The system under study is the Bunimovich' stadium billiard, which is a classically ergodic system, also fully chaotic (positive Lyapunov exponent), but in the regime of a slightly distorted circle billiard (small shape parameter $\varepsilon$) the diffusion in the momentum space is very slow. The paramet…
Eigenstates in finite systems such as nuclei, atoms, atomic clusters and quantum dots with few excited particles are chaotic superpositions of shell model basis states. We study criterion for the equilibrium distribution of basis components (ergodicity, or Quantum Chaos), effects of level density variation and transition from the Breit-Wigner to the Gaussian shape of eigenstates and strength functions. In the model of $n$ interacting particles distributed over $m$ orbitals, the shape is given by the Breit-Wigner function with the width in the form of gaussian dependence on energy.
There is no unique way to quantify the degree of delocalization of quantum states in unbounded continuous spaces. In this work, we explore a recently introduced localization measure that quantifies the portion of the classical phase space occupied by a quantum state. The measure is based on the $α$-moments of the Husimi function and is known as the Rényi occupation of order $α$. With this quantity and random pure states, we find a general expression to identify states that are maximally delocalized in phase space. Using this expression and the Dicke model, which is an interacting spin-boson mo…
When a regular classical system is perturbed, non-linear resonances appear as prescribed by the KAM and Poincarè-Birkhoff theorems. Manifestations of this classical phenomena to the morphologies of quantum wave functions are studied in this letter. We reveal a systematic formation of an universal structure of localized wave functions in systems with mixed classical dynamics. Unperturbed states that live around invariant tori are mixed when they collide in an avoided crossing if their quantum numbers differ in a multiple to the order of the classical resonance. At the avoided crossing eigenstat…
The boundary of the lemon billiards is defined by the intersection of two circles of equal unit radius with the distance 2B between their centers, as introduced by Heller and Tomsovic in Phys. Today 46 38 (1993). We study two classical and quantum lemon billiards, for the cases B = 0.1953, 0.083, which are mixed-type billiards with complex structure of phase space, without significant stickiness regions. A preliminary study of their spectra was published recently (Physics 1 1-14 (2021)). We calculate a great number $10^6$ of consecutive eigenstates and their Poincaré-Husimi (PH) functions, and…
We have studied statistical properties of the values of the Wigner function W(x) of 1D quantum maps on compact 2D phase space of finite area V. For this purpose we have defined a Wigner function probability distribution P(w) = (1/V) int delta(w-W(x)) dx, which has, by definition, fixed first and second moment. In particular, we concentrate on relaxation of time evolving quantum state in terms of W(x), starting from a coherent state. We have shown that for a classically chaotic quantum counterpart the distribution P(w) in the semi-classical limit becomes a Gaussian distribution that is fully de…
The unitary representation of exact quantum Poincare mapping is constructed. It is equivalent to the compact representation in a sense that it yields equivalent quantization condition with important advantage over the compact version: since it preserves the probability it can be literally interpreted as the quantum Poincare mapping which generates quantum time evolution at fixed energy between two successive crossings with surface of section (SOS). SOS coherent state representation (SOS Husimi distribution) of arbitrary (either stationary or evolving) quantum SOS state (vector from the Hilbert…
For generic 4D symplectic maps we propose the use of 3D phase-space slices which allow for the global visualization of the geometrical organization and coexistence of regular and chaotic motion. As an example we consider two coupled standard maps. The advantages of the 3D phase-space slices are presented in comparison to standard methods like 3D projections of orbits, the frequency analysis, and a chaos indicator. Quantum mechanically, the 3D phase-space slices allow for the first comparison of Husimi functions of eigenstates of 4D maps with classical phase space structures. This confirms the …
We show that the energy averaged entropy localization length for the eigenstates of the Ce atom is well approximated by the prediction of the Wigner ensemble.
