Husimi Distribution Research Articles

The Classical-Quantum Limit

The standard notion of a classical limit, represented schematically by ℏ→0, provides a method for approximating a quantum system by a classical one. In this work, we explain why the standard classical limit fails when applied to subsystems, and show how one may resolve this by explicitly modeling the decoherence of a subsystem by its environment. Denoting the decoherence time by τ, we demonstrate that a double scaling limit in which ℏ→0 and τ→0 such that the ratio Ef=ℏ/τ remains fixed leads to an irreversible open-system evolution with well-defined classical and quantum subsystems. The main technical result is showing that, for arbitrary Hamiltonians, the generators of partial versions of the Wigner, Husimi, and Glauber-Sudarshan quasiprobability distributions may all be mapped in the above-mentioned double scaling limit to the same completely positive classical-quantum generator. This provides a regime in which one can study effective and consistent classical-quantum dynamics. Published by the American Physical Society 2024

Dynamics of phase space quasi-probability coherence of one of two moving atoms trapped in a cavity coherent field

This study explores the dynamics of atomic mixedness and phase space quasi-probability information of one of two moving atoms trapped in a cavity filled by a coherent field. The atom-photon interactions are considered with the resonance case through nonlinear intensity-dependent coupling. We investigate the impacts of the atom-photon and atomic motion couplings on the dynamics of the atomic Wehrl entropy's mixedness and phase space Husimi distribution information. By helping Wehrl entropy's mixedness dynamics, the dynamics of acceleration and enhancement of the generation of the atomic phase space quasi-probability coherence are examined, and the Husimi distributions of the generated atomic states of one of the two moving atoms are explored. The results are shown that the atomic Husimi distribution and Wehrl entropy's mixedness are susceptible to an increase in the atom-photon and the atomic motion couplings. Therefore, the atomic phase space Husimi distribution coherence can be improved by increasing the atomic motion and nonlinear atom-photon couplings.

Semiclassical Husimi Distributions of Schur Vectors in Non-Hermitian Quantum Systems.

We construct a semiclassical phase-space density of Schur vectors in non-Hermitian quantum systems. Each Schur vector is associated to a single Planck cell. The Schur states are organized according to a classical norm landscape on phase space-a classical manifestation of the lifetimes which are characteristic of non-Hermitian systems. To demonstrate the generality of this construction we apply it to a highly nontrivial example: a PT-symmetric kicked rotor in the regimes of mixed and chaotic classical dynamics.

Phase-space quantum distributions and information theory

The local structure present in Wigner and Husimi phase-space distributions and their marginals are studied and quantified via information-theoretic quantities. Shannon, R\'enyi, and cumulative residual entropies of the Wigner and Husimi distributions are examined in the ground and excited states of a harmonic oscillator. The entropies of the Wigner function marginals are lower than the corresponding entropies of the Husimi function marginals, which illustrates how the nodal structure present in the Wigner function is lost upon consideration of the Husimi function. Shannon and cumulative residual entropies of the Wigner function yield entropies which are complex valued. Absolute values and real components of these quantities increase with quantum number, displaying a behavior which is consistent with their real-valued Husimi function counterparts. This comportment is also similar to the real-valued R\'enyi entropies of the Wigner function. The entropies of the Wigner function are observed to be lower than the corresponding Husimi function ones, in agreement with the results for the marginals. The real components of the Wigner function entropies are seen to be closer to the uncertainty relation bound compared to the corresponding Husimi function entropies. These real components are also closer to the bound when contrasted to the entropic sum of the marginal densities. The R\'enyi or collision entropy of the Wigner function sits exactly on the bound. Related statistical correlation measures show that the position-momentum correlation is larger in the Wigner function compared to the Husimi function and increases with quantum number.

Husimi Dynamics Generated by non-Hermitian Hamiltonians.

The dynamics generated by non-Hermitian Hamiltonians are often less intuitive than those of conventional Hermitian systems. Even for models as simple as a complexified harmonic oscillator, the dynamics for generic initial states shows surprising features. Here we analyze the dynamics of the Husimi distribution in a semiclassical limit, illuminating the foundations of the full quantum evolution. The classical Husimi evolution is composed of two factors: (i)the initial Husimi distribution evaluated along phase-space trajectories and (ii)the final value of the norm corresponding to each phase-space point. Both factors conspire to lead to intriguing dynamical behaviors. We demonstrate how the full quantum dynamics unfolds on top of the classical Husimi dynamics for two instructive examples.

Classical and Quantum chaos in a spin-1 atomic Bose–Einstein condensate via Floquet driving

We investigate both the classical and quantum dynamics of a spin-1 Bose–Einstein condensate (BEC) system subject to periodic driving of quadratic Zeeman splitting. Without driving, the system is integrable. Here, we found that increasing the driving strength will drive the system to chaos. Classically, we demonstrate chaotic dynamics by examining the Poincaré section and Lyapunov exponent. While in the quantum regime, the level statistic and Husimi distribution are studied, and they also clearly indicate the onset of quantum chaos. The dynamics of atom number and out-of-time-ordered correlator (OTOC) are also studied. In the integrable region, the variance of atom number and OTOC both remain very small during the time evolution; while in the chaos region, a quick increase is observed in the early time. This work presents a quantum many-body system showing chaotic behavior and can be readily realized in experiment.

The Husimi function of a semiconfined harmonic oscillator model with a position-dependent effective mass

In this paper, the phase space representation for a semiconfined harmonic oscillator model with a position-dependent effective mass is constructed. We have found the Husimi distribution function for the stationary states of the oscillator model under consideration for both cases without and with the applied external homogeneous field. The obtained function is expressed through the double sum of the parabolic cylinder function. Different special cases and the limit relations are discussed, too.

Coherent state representation of thermal correlation functions with applications to rate theory.

A coherent state phase space representation of operators, based on the Husimi distribution, is used to derive an exact expression for the symmetrized version of thermal correlation functions. In addition to the time and temperature independent phase space representation of the two operators whose correlation function is of interest, the integrand includes a non-negative distribution function where only one imaginary time and one real time propagation are needed to compute it. The methodology is exemplified for the flux side correlation function used in rate theory. The coherent state representation necessitates the use of a smeared Gaussian flux operator whose coherent state phase space representation is identical to the classical flux expression. The resulting coherent state expression for the flux side correlation function has a number of advantages as compared to previous formulations. Since only one time propagation is needed, it is much easier to converge it with a semiclassical initial value representation. There is no need for forward-backward approximations, and in principle, the computation may be implemented on the fly. It also provides a route for analytic semiclassical approximations for the thermal rate, as exemplified by a computation of the transmission factor through symmetric and asymmetric Eckart barriers using a thawed Gaussian approximation for both imaginary and real time propagations. As a by-product, this example shows that one may obtain "good" tunneling rates using only above barrier classical trajectories even in the deep tunneling regime.

Observation of quantum phase synchronization in a nuclear-spin system

We report an experimental study of phase synchronization in a pair of interacting nuclear spins subjected to an external drive in nuclear magnetic resonance architecture. A weak transition-selective radio-frequency field applied on one of the spins is observed to cause phase localization, which is experimentally established by measuring the Husimi distribution function under various drive conditions. To this end, we have developed a general interferometric technique to directly extract values of the Husimi function via the transverse magnetization of the undriven nuclear spin. We further verify the robustness of synchronization to detuning in the system by studying the Arnold tongue behavior.

Quantum signatures of chaos in relativistic quantum billiards with shapes of circle- and ellipse-sectors* *Dedicated to the memory of Fritz Haake. Fritz introduced me (Barbara) into the field of quantum chaos. He supervised my diploma and doctoral thesis and since then has been a mentor, friend and helpful advisor during all the stages of my scientific career. I take his

According to the Berry–Tabor conjecture, the spectral properties of typical nonrelativistic quantum systems with an integrable classical counterpart agree with those of Poissonian random numbers. We investigate to what extend it applies to relativistic neutrino billiards (NBs) consisting of a spin-1/2 particle confined to a bounded planar domain by imposing suitable boundary conditions (BCs). In distinction to nonrelativistic quantum billiards (QBs), NBs do not have a well-defined classical counterpart. However, the peaks in the length spectra, that is, the modulus of the Fourier transform of the spectral density from wave number to length, of NBs are just like for QBs at the lengths of periodic orbits of the classical billiard (CB). This implies that there must be a connection between NBs and the dynamic of the CB. We demonstrate that NBs with shapes of circle- and ellipse-sectors with an integrable classical dynamic, obtained by cutting the circle and ellipse NB along symmetry lines, have no common eigenstates with the latter and that, indeed, their spectral properties can be similar to those of classically chaotic QBs. These features orginate from the intermingling of symmetries of the spinor components and the discontinuity in the BCs leading to contradictory conditional equations at corners connecting curved and straight boundary parts. To corroborate the necessity of the curved boundary part in order to generate GOE-like behavior, we furthermore consider the right-angled triangle NB constructed by halving the equilateral-triangle NB along a symmetry axis. For an understanding of these findings in terms of purely classical quantities we use the semiclassical approach recently developed for massive NBs, and Poincaré–Husimi distributions of the eigenstates in classical phase space. The results indicate, that in the ultrarelativistic limit these NBs do not show the behavior expected for classically chaotic QBs.

Phase-space representation of diffraction in time: analytic results

Diffraction in time manifests itself as the appearance of probability-density fringes when a matter wave passes through an opaque screen with abrupt temporal variations of transmission properties. Here we analytically describe the phase-space structure of diffraction-in-time fringes for a class of smooth time gratings. More precisely, we obtain an analytic expression for the Husimi distribution representing the state of the system in the case of time gratings comprising a succession of Lorentzian-like slits. In particular, for a double-slit scenario, we derive a simple and intuitive expression that accurately captures the position of interference fringes in phase space.

DENSITY OPERATOR REPRESENTATION IN MULTI-MATRIX VECTOR COHERENT STATES: LANDAU PROBLEM IN A HARMONIC POTENTIAL BACKGROUND

The density operator representation in the context of multi-matrix vector coherent states basis is performed and applied to Landau levels of an electron in an electromagnetic field coupled to an isotropic harmonic potential. Main relevant statistical properties such as the Mandel Q -parameter and the signal-to-quantum-noise ratio are derived and discussed. The Q -Husimi distribution is also investigated from the viewpoint of the Glauber-Sudarshan- P representation of the density operator.

Spin coherent states, Bell states, spin Hamilton operators, entanglement, Husimi distribution, uncertainty relation and Bell inequality

Abstract We investigate spin Hamilton operators and compare spin coherent states and Bell states concerning entanglement, Husimi distributions, uncertainty relation and Bell inequality. The distances between spin coherent states and Bell states are derived. The Rayleigh quotients of spin Hamilton operators for spin coherent states and Bell states are evaluated and compared.

$\mathcal {P}\mathcal {T}$-symmetry in Compact Phase Space for a Linear Hamiltonian

We study the time evolution of a PT-symmetric, non-Hermitian quantum system for which the associated phase space is compact. We focus on the simplest non-trivial example of such a Hamiltonian, which is linear in the angular momentum operators. In order to describe the evolution of the system, we use a particular disentangling decomposition of the evolution operator, which remains numerically accurate even in the vicinity of the Exceptional Point. We then analyze how the non-Hermitian part of the Hamiltonian affects the time evolution of two archetypical quantum states, coherent and Dicke states. For that purpose we calculate the Husimi distribution or Q function and study its evolution in phase space. For coherent states, the characteristics of the evolution equation of the Husimi function agree with the trajectories of the corresponding angular momentum expectation values. This allows to consider these curves as the trajectories of a classical system. For other types of quantum states, e.g. Dicke states, the equivalence of characteristics and trajectories of expectation values is lost.

Hadronization of correlated gluon fields

Following an explicit example, we present the chain of steps required for an event-by-event description of hadron production in high energy hadronic and nuclear collisions. We start from incoming nuclei, described in the color glass condensate effective theory, whose collision creates the gluon fields of the glasma. Individual gluons are then sampled from the gluon fields' Husimi (smeared Wigner) distributions and clustered using a new spacetime based algorithm. Clusters are fed into the Herwig event generator, which performs the hadronization, conserving energy and momentum. We discuss the physical implications of smearing and problems with the quasiparticle picture for the studied processes. We compute spectra of charged hadrons and identified particles and their azimuthal momentum anisotropies, and address systematic uncertainties on observables, resulting from the general lack of detailed knowledge of the hadronization mechanism.

Chaos and order in librating quantum planar elastic pendulum

We investigate, numerically and analytically, the quantum mechanics of planar elastic pendulum in libration. The classical phase space of the system consists of both regular Kolmogorov–Arnold–Moser trajectories and chaotic trajectories. We systematically study the former by using the canonical perturbation theory along with the Einstein–Brillouin–Keller quantization rule and the latter using out-of-time-ordered-correlator among other tools and techniques, like the Husimi distribution and energy level spacing distribution. We find that the deviations in the energy level statistics, due to the presence of nearly degenerate levels, disappear by considering the states with same symmetry. We also propose an eight-point-out-of-time-ordered-correlator whose growth rate between the exponential decay time (around which nonlinearity sets in) and the Ehrenfest time (after which quantum effect washes out the classical growth) can be used to relate the quantum correlator to the second largest Lyapunov exponent of the corresponding classical system.

A non-Hermitian PT-symmetric kicked top

A non-Hermitian PT-symmetric version of the kicked top is introduced to study the interplay of quantum chaos with balanced loss and gain. The classical dynamics arising from the quantum dynamics of the angular momentum expectation values are derived. It is demonstrated that the presence of PT-symmetry can lead to ‘stable’ mixed regular chaotic behaviour without sinks or sources for subcritical values of the gain–loss parameter. This is an example of what is known in classical dynamical systems as reversible dynamical systems. For large values of the kicking strength a strange attractor is observed that also persists if PT-symmetry is broken. The intensity dynamics of the classical map is investigated, and found to provide the main structure for the Husimi distributions of the subspaces of the quantum system belonging to certain ranges of the imaginary parts of the quasienergies. Classical structures are also identified in the quantum dynamics. Finally, the statistics of the eigenvalues of the quantum system are analysed and it is shown that if most of the eigenvalues are complex (which is the case already for fairly small non-Hermiticity parameters) the nearest-neighbour distances of the (unfolded) quasienergies follow a two-dimensional Posisson distribution when the classical dynamics is regular. In the chaotic regime, on the other hand they are in line with recently identified universal complex level spacing distributions for non-Hermitian systems, with transpose symmetry ÂT = Â. It is demonstrated how breaking this symmetry (by introducing an extra term in the Hamiltonian) recovers the more familiar universality class for non-Hermitian systems given by the complex Ginibre ensemble. Both universality classes display cubic level repulsion. The PT-symmetry of the system does not seem to influence the complex level spacings. Similar behaviour is also observed for the spectrum of a PT-symmetric extension of the triadic Baker map.

Statistical properties of the localization measure of chaotic eigenstates in the Dicke model.

The quantum localization is one of the remarkable phenomena in the studies of quantum chaos and plays an important role in various contexts. Thus, an understanding of the properties of quantum localization is essential. In spite of much effort dedicated to investigating the manifestations of localization in the time-dependent systems, the features of localization in time-independent systems are still less explored, particularly in quantum systems which correspond to the classical systems with smooth Hamiltonian. In this work, we present such a study for a quantum many-body system, namely, the Dicke model. The classical counterpart of the Dicke model is given by a smooth Hamiltonian with two degrees of freedom. We examine the signatures of localization in its chaotic eigenstates. We show that the entropy localization measure, which is defined in terms of the information entropy of Husimi distribution, behaves linearly with the participation number, a measure of the degree of localization of a quantum state. We further demonstrate that the localization measure probability distribution is well described by the β distribution. We also find that the averaged localization measure is linearly related to the level repulsion exponent, a widely used quantity to characterize the localization in chaotic eigenstates. Our findings extend the previous results in billiards to the quantum many-body system with classical counterpart described by a smooth Hamiltonian, and they indicate that the properties of localized chaotic eigenstates are universal.

Coherent representation of fields and deformation quantization

Motivated by some well-known results in the phase space description of quantum optics and quantum information theory, we aim to describe the formalism of quantum field theory by explicitly considering the holomorphic representation for a scalar field within the deformation quantization program. Notably, the symbol of a symmetric ordered operator in terms of holomorphic variables may be straightforwardly obtained by the quantum field analogue of the Husimi distribution associated with a normal ordered operator. This relation also allows to establish a [Formula: see text]-equivalence between the Moyal and the normal star-products. In addition, by writing the density operator in terms of coherent states we are able to directly introduce a series representation of the Wigner functional distribution, which may be convenient in order to calculate probability distributions of quantum field observables without performing formal phase space integrals at all.

Nonclassical Effects Based on Husimi Distributions in Two Open Cavities Linked by an Optical Waveguide.

Nonclassical effects are investigated in a system formed by two quantum wells, each of which is inside an open cavity. The cavities are spatially separated, linked by a fiber, and filled with a linear optical medium. Based on Husimi distributions (HDs) and Wehrl entropy, we explore the effects of the physical parameters on the generation and the robustness of the mixedness and HD information in the phase space. The generated quantum coherence and the HD information depend crucially on the cavity-exciton and fiber cavity couplings as well as on the optical medium density. The HD information and purity are lost due to the dissipation. This loss may be inhibited by increasing the optical susceptibility as well as the couplings of the exciton-cavity and the fiber-cavity. These parameters control the regularity, amplitudes, and frequencies of the generated mixedness.