We perform a detailed study of the chaotic component in mixed-type Hamiltonian systems on the example of a family of billiards [introduced by Robnik in J. Phys. A: Math. Gen. 16, 3971 (1983)]. The phase space is divided into a grid of cells and a chaotic orbit is iterated a large number of times. The structure of the chaotic component is discerned from the cells visited by the chaotic orbit. The fractal dimension of the border of the chaotic component for various values of the billiard shape parameter is determined with the box-counting method. The cell-filling dynamics is compared to a model of uncorrelated motion, the so-called random model [Robnik et al. J. Phys. A: Math. Gen. 30, L803 (1997)], and deviations attributed to sticky objects in the phase space are found. The statistics of the number of orbit visits to the cells is analyzed and found to be in agreement with the random model in the long run. The stickiness of the various structures in the phase space is quantified in terms of the cell recurrence times. The recurrence time distributions in a few selected cells as well as the mean and standard deviation of recurrence times for all cells are analyzed. The standard deviation of cell recurrence time is found to be a good quantifier of stickiness on a global scale. Three methods for determining the measure of the chaotic component are compared and the measure is calculated for various values of the billiard shape parameter. Lastly, the decay of correlations and the diffusion of momenta is analyzed.
For generic Hamiltonian systems we derive predictions for dynamical tunneling from regular to chaotic phase-space regions. In contrast to previous approaches, we account for the resonance-assisted enhancement of regular-to-chaotic tunneling in a non-perturbative way. This provides the foundation for future semiclassical complex-path evaluations of resonance-assisted regular-to-chaotic tunneling. Our approach is based on a new class of integrable approximations which mimic the regular phase-space region and its dominant nonlinear resonance chain in a mixed regular--chaotic system. We illustrate…
We present evidence that nonlinear resonances govern the tunneling process between symmetry-related islands of regular motion in mixed regular-chaotic systems.In a similar way as for near-integrable tunneling, such resonances induce couplings between regular states within the islands and states that are supported by the chaotic sea. On the basis of this mechanism, we derive a semiclassical expression for the average tunneling rate, which yields good agreement in comparison with the exact quantum tunneling rates calculated for the kicked rotor and the kicked Harper.
This paper presents a new generator of chaotic bit sequences with mixed-mode (continuous and discrete) inputs. The generator has an improved level of chaotic properties in comparison with the existing single source (input) digital chaotic bit generators. The 0-1 test is used to show the improved chaotic behavior of our generator having a chaotic continuous input (Chua, Rössler or Lorenz system) intermingled with a discrete input (logistic, Tinkerbell or Henon map) with various parameters. The obtained sequences of chaotic bits show some features of random processes with increased entropy level…
The dynamics in weakly chaotic Hamiltonian systems strongly depends on initial conditions and little can be affirmed about generic behaviors. Using two distinct Hamiltonian systems, namely one particle in an open rectangular billiard and four particles globally coupled on a discrete lattice, we show that in these models the transition from integrable motion to weak chaos emerges via chaotic stripes as the nonlinear parameter is increased. The stripes represent intervals of initial conditions which generate chaotic trajectories and increase with the nonlinear parameter of the system. In the bil…
We study the autocorrelation function of different types of eigenfunctions in quantum mechanical systems with either chaotic or mixed classical limits. We obtain an expansion of the autocorrelation function in terms of the correlation length. For localized states, like bouncing ball modes or states living on tori, a simple model using only classical input gives good agreement with the exact result. In particular, a prediction for irregular eigenfunctions in mixed systems is derived and tested. For chaotic systems, the expansion of the autocorrelation function can be used to test quantum ergodi…
We present a technique to control chaos in Hamiltonian systems which are close to integrable. By adding a small and simple control term to the perturbation, the system becomes more regular than the original one. We apply this technique to a model that reproduces turbulent ExB drift and show numerically that the control is able to drastically reduce chaotic transport.
Some non-Gaussian aspects of chaotic transport are investigated for a general class of two-dimensional area-preserving maps. Kurtosis, in particular, is calculated from the diffusion and the Burnett coefficients, which are obtained analytically. A characteristic time scale delimiting the onset of the Markovian regime for the master equation is established. Some explicit examples are discussed.
An effective characterization of chaotic conservative Hamiltonian systems in terms of the curvature associated with a Riemannian metric tensor derived from the structure of the Hamiltonian has been extended to a wide class of potential models of standard form through definition of a conformal metric. The geodesic equations reproduce the Hamilton equations of the original potential model through an inverse map in the tangent space. The second covariant derivative of the geodesic deviation in this space generates a dynamical curvature, resulting in (energy dependent) criteria for unstable behavi…
A chaotic autonomous Hamiltonian systems, perturbed by small damping and small external force, harmonically dependent on time, can acquire a strange attractor with properties similar to that of the canonical distribution - the Gibbs attractor. The evolution of the energy in such systems can be described as the energy diffusion. For the nonlinear Pullen - Edmonds oscillator with two degrees of freedom the properties of the Gibbs attractor and their dependence on parameters of the perturbation are studied both analytically and numerically.
In this paper, we show that "Labyrinth walks", the conservative version of "Labyrinth chaos" and member of the Thomas-Rössler class of systems, does not admit an autonomous Hamiltonian as a constant function in time, and as a consequence, does not admit a symplectic structure. However, it is conservative, and thus admits a vector potential, being at the same time chaotic. This exceptional set of properties makes "Labyrinth walks" an elegant example of a chaotic, conservative, non-Hamiltonian system, with only unstable stationary points in its phase space, arranged in a 3-dimensional grid. As a…
We present a comprehensive theory of resonance-assisted tunneling in quantum systems that exhibit a mixed regular-chaotic classical phase space structure. After general considerations, we specifically focus on quantum systems with one degree of freedom that are subject to a periodic sequence of kicks or to a periodic driving. Tunneling takes place between energetically degenerate quasimodes that are localized on symmetric regular islands within the stroboscopic Poincare surface of section. In contrast to previous theoretical descriptions of resonance-assisted tunneling, we derive a more precis…
Time dependent dynamics of the chaotic quantum-mechanical system has been studied. Irreversibility of the dynamics is shown. It is shown, that being in the initial moment in pure quantum-mechanical state, system makes irreversible transition into mixed state. Original mechanism of mixed state formation is offered. Quantum kinetic equation is obtained. Growth of the entropy during the evolution process is estimated.
The relaxation dynamics in mixed chaotic systems are believed to decay algebraically with a universal decay exponent that emerges from the hierarchical structure of the phase space. Numerical studies, however, yield a variety of values for this exponent. In order to reconcile these result we consider an ensemble of mixed chaotic systems approximated by rate equations, and analyze the fluctuations in the distribution of Poincare recurrence times. Our analysis shows that the behavior of these fluctuations, as function of time, implies a very slow convergence of the decay exponent of the relaxati…
In open chaotic systems the number of long-lived resonance states obeys a fractal Weyl law, which depends on the fractal dimension of the chaotic saddle. We study the generic case of a mixed phase space with regular and chaotic dynamics. We find a hierarchy of fractal Weyl laws, one for each region of the hierarchical decomposition of the chaotic phase-space component. This is based on our observation of hierarchical resonance states localizing on these regions. Numerically this is verified for the standard map and a hierarchical model system.
We study numerically statistical distributions of sums of chaotic orbit coordinates, viewed as independent random variables, in weakly chaotic regimes of three multi-dimensional Hamiltonian systems: Two Fermi-Pasta-Ulam (FPU-$β$) oscillator chains with different boundary conditions and numbers of particles and a microplasma of identical ions confined in a Penning trap and repelled by mutual Coulomb interactions. For the FPU systems we show that, when chaos is limited within "small size" phase space regions, statistical distributions of sums of chaotic variables are well approximated for surpri…
A general technique for the periodic orbit quantization of systems with near-integrable to mixed regular-chaotic dynamics is introduced. A small set of periodic orbits is sufficient for the construction of the semiclassical recurrence function up to, in principle, infinite length. As in our recent work the recurrence signal is inverted by means of a high resolution spectral analyzer (harmonic inversion) to obtain the semiclassical eigenenergies. The method is demonstrated for the hydrogen atom in a magnetic field. To our knowledge this is the first successful application of periodic orbit quan…
The chaotic diffusion for a family of Hamiltonian mappings whose angles diverge in the limit of vanishingly action is investigated by using the solution of the diffusion equation. The system is described by a two-dimensional mapping for the variables action, $I$, and angle, $θ$ and controlled by two control parameters: (i) $ε$, controlling the nonlinearity of the system, particularly a transition from integrable for $ε=0$ to non-integrable for $ε\ne0$ and; (ii) $γ$ denoting the power of the action in the equation defining the angle. For $ε\ne0$ the phase space is mixed and chaos is present in …
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 {\bf 46} 38 (1993). This paper is a continuation of our recent paper on classical and quantum ergodic lemon billiard ($B=0.5$) with strong stickiness effects published in Phys. Rev. E {\bf 103} 012204 (2021). Here we study the classical and quantum lemon billiards, for the cases $B=0.42,\;0.55,\; 0.6$, which are mixed-type billiards without stickiness regions and thus serve as ideal examples of s…
We explain the mechanism leading to directed chaotic transport in Hamiltonian systems with spatial and temporal periodicity. We show that a mixed phase space comprising both regular and chaotic motion is required and derive a classical sum rule which allows to predict the chaotic transport velocity from properties of regular phase-space components. Transport in quantum Hamiltonian ratchets arises by the same mechanism as long as uncertainty allows to resolve the classical phase-space structure. We derive a quantum sum rule analogous to the classical one, based on the relation between quantum t…
A method of chaos reduction for Hamiltonian systems is applied to control chaotic advection. By adding a small and simple term to the stream function of the system, the construction of invariant tori has a stabilization effect in the sense that these tori act as barriers to diffusion in phase space and the controlled Hamiltonian system exhibits a more regular behaviour.
We analyze energy spreading for a system that features mixed chaotic phase-space, whose control parameters (or slow degrees of freedom) vary quasi-statically. For demonstration purpose we consider the restricted 3~body problem, where the distance between the two central stars is modulated due to their Kepler motion. If the system featured hard-chaos, one would expect diffusive spreading with coefficient that can be estimated using linear-response (Kubo) theory. But for mixed phase space the chaotic sea is multi-layered. Consequently, it becomes a challenge to find a robust procedure that trans…
Transport in Hamiltonian systems with weak chaotic perturbations has been much studied in the past. In this paper, we introduce a new class of problems: transport in Hamiltonian systems with slowly changing phase space structure that are not order one perturbations of a given Hamiltonian. This class of problems is very important for many applications, for instance in celestial mechanics. As an example, we study a class of one-dimensional Hamiltonians that depend explicitly on time and on stochastic external parameters. The variations of the external parameters are responsible for a distortion …
The chaotic transport of charged particles in a turbulent electrostatic potential sets the conditions of a severe limitation to the plasma confinement in devices such as tokamaks. In this chapter, we consider the motion of impurities driven by the ExB velocity where a strong magnetic field B (which allows for the guiding center approximation) is uniform and constant, and a turbulent electric field is obtained from models or from numerical fluid codes. Hamiltonian dynamics rule the transport properties of these impurities. Therefore a technique to reduce chaotic diffusion in Hamiltonian systems…
We prove quantum-classical correspondence for bound conservative classically chaotic Hamiltonian systems. In particular, quantum Liouville spectral projection operators and spectral densities, and hence classical dynamics, are shown to approach their classical analogs in the $h\rightarrow 0$ limit. Correspondence is shown to occur via the elimination of essential singularities. In addition, applications to matrix elements of observables in chaotic systems are discussed.
We numerically show fractal Weyl law behavior in an open Hamiltonian system that is described by a smooth potential and which supports numerous above-barrier resonances. This behavior holds even relatively far away from the classical limit. The complex resonance wave functions are found to be localized on the fractal classical repeller.
We perform echo spectroscopy on ultra cold atoms in atom optics billiards, to study their quantum dynamics. The detuning of the trapping laser is used to change the ``perturbation'', which causes a decay in the echo coherence. Two different regimes are observed: First, a perturbative regime in which the decay of echo coherence is non-monotonic and partial revivals of coherence are observed. These revivals are more pronounced in traps with mixed dynamics as compared to traps where the dynamics is fully chaotic. Next, for stronger perturbations, the decay becomes monotonic and independent of the…
A phase-space semiclassical approximation valid to $O(\hbar)$ at short times is used to compare semiclassical accuracy for long-time and stationary observables in chaotic, stable, and mixed systems. Given the same level of semiclassical accuracy for the short time behavior, the squared semiclassical error in the chaotic system grows linearly in time, in contrast with quadratic growth in the classically stable system. In the chaotic system, the relative squared error at the Heisenberg time scales linearly with $\hbar_{\rm eff}$, allowing for unambiguous semiclassical determination of the eigenv…
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…
Designing a pseudorandom number generator (PRNG) is a difficult and complex task. Many recent works have considered chaotic functions as the basis of built PRNGs: the quality of the output would indeed be an obvious consequence of some chaos properties. However, there is no direct reasoning that goes from chaotic functions to uniform distribution of the output. Moreover, embedding such kind of functions into a PRNG does not necessarily allow to get a chaotic output, which could be required for simulating some chaotic behaviors. In a previous work, some of the authors have proposed the idea of …
We prove nonintegrability of a model Hamiltonian system defined on the Lie algebra $\mathfrak{su}_3$ suitable for investigation of connections between classical and quantum characteristics of chaos.
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.
We report on the application of chaos control to the irregular motion of an electron under the combined influence of a Coulomb and a magnetic field, the so-called ``diamagnetic Kepler problem'' (DKP). We show how to stabilize the classical chaotic orbits into regular motion and present a new method to follow the unstable orbits as the energy is increased from the regular to the chaotic regime.
We present a comprehensive account of directed transport in one-dimensional Hamiltonian systems with spatial and temporal periodicity. They can be considered as Hamiltonian ratchets in the sense that ensembles of particles can show directed ballistic transport in the absence of an average force. We discuss general conditions for such directed transport, like a mixed classical phase space, and elucidate a sum rule that relates the contributions of different phase-space components to transport with each other. We show that regular ratchet transport can be directed against an external potential g…
A signal processing method designed for the detection of linear (coherent) behaviors among random fluctuations is presented. It is dedicated to the study of data recorded from nonlinear physical systems. More precisely the method is suited for signals having chaotic variations and sporadically appearing regular linear patterns, possibly impaired by noise. We use time-frequency techniques and the Fractional Fourier transform in order to make it robust and easily implementable. The method is illustrated with an example of application: the analysis of chaotic trajectories of advected passive part…
Temporal evolutions toward thermal equilibria are numerically investigated in a Hamiltonian system with many degrees of freedom which has second order phase transition. Relaxation processes are studied through local order parameter, and slow relaxations of power type are observed at the critical energy of phase transition for some initial conditions. Numerical results are compared with results of a phenomenological theory of statistical mechanics. At the critical energy, the maximum Lyapunov exponent takes the largest value. Temporal evolutions and probability distributions of local Lyapunov e…
We study quantum-mechanical tunneling in mixed dynamical systems between symmetry-related phase space tori separated by a chaotic layer. Considering e.g. the annular billiard we decompose tunneling-related energy splittings and shifts into sums over paths in phase space. We show that tunneling transport is dominated by chaos-assisted paths that tunnel into and out of the chaotic layer via the ``beach'' regions sandwiched between the regular islands and the chaotic sea. Level splittings are shown to fluctuate on two scales as functions of energy or an external parameter: they display a dense se…
The phase space of an area-preserving map typically contains infinitely many elliptic islands embedded in a chaotic sea. Orbits near the boundary of a chaotic region have been observed to stick for long times, strongly influencing their transport properties. The boundary is composed of invariant circles, called "Boundary circles." We investigate the distribution of rotation numbers of boundary circles for the Henon quadratic map and show that the probability of occurrence of small elements of their continued fraction expansions is larger than would be expected for a number chosen at random. Ho…
We study quantum localization phenomena in chaotic systems with a parameter. The parametric motion of energy levels proceeds without crossing any other and the defined avoided crossings quantify the interaction between states. We propose the elimination of avoided crossings as the natural mechanism to uncover localized structures. We describe an efficient method for the elimination of avoided crossings in chaotic billiards and apply it to the stadium billiard. We find many scars of short periodic orbits revealing the skeleton on which quantum mechanics is built. Moreover, we have observed stro…
It is shown that a relevant control of Hamiltonian chaos is possible through suitable small perturbations whose form can be explicitly computed. In particular, it is possible to control (reduce) the chaotic diffusion in the phase space of a Hamiltonian system with 1.5 degrees of freedom which models the diffusion of charged test particles in a ``turbulent'' electric field across the confining magnetic field in controlled thermonuclear fusion devices. Though still far from practical applications, this result suggests that some strategy to control turbulent transport in magnetized plasmas, in pa…
Weak chaos in high-dimensional conservative systems can be characterized through sticky effect induced by invariant structures on chaotic trajectories. Suitable quantities for this characterization are the higher cummulants of the finite time Lyapunov exponents (FTLEs) distribution. They gather the {\it whole} phase space relevant dynamics in {\it one} quantity and give informations about ordered and random states. This is analyzed here for discrete Hamiltonian systems with local and global couplings. It is also shown that FTLEs plotted {\it versus} initial condition (IC) and the nonlinear par…
Friedel oscillations of electron densities near step edges have an analog in microwave billiards. A random plane wave model, normally only appropriate for the eigenfunctions of a purely chaotic system, can be applied and is tested for non-purely-chaotic dynamical systems with measurements on pseudo-integrable and mixed dynamics geometries. It is found that the oscillations in the pseudo-integrable microwave cavity matches the random plane-wave modeling. Separating the chaotic from the regular states for the mixed system requires incorporating an appropriate phase space projection into the mode…
Random matrix theory (RMT) provides a framework to study the spectral fluctuations in physical systems. RMT is capable of making predictions for the fluctuations only after the removal of the secular properties of the spectrum. Spectral unfolding procedure is used to separate the local level fluctuations from overall energy dependence of the level separation. The unfolding procedure is not unique. Several studies showed that statistics of long-term correlation in the spectrum are very sensitive to the choice of the unfolding function in polynomial unfolding. This can give misleading results re…
We investigate the relation between the chaotic dynamics and the hierarchical phase-space structure of generic Hamiltonian systems. We demonstrate that even in ideal situations when the phase space is dominated by an exactly self-similar structure, the long-time dynamics is {\it not} dominated by this structure. This has consequences for the power-law decay of correlations and Poincaré recurrences.
In open Hamiltonian systems, the escape from a bounded region of phase space according to an exponential decay law is frequently associated with the existence of hyperbolic dynamics in such a region. Furthermore, exponential decay laws based on the ergodic hypothesis are used to describe escapes in these systems. However, we uncover that the presence of the set that governs the hyperbolic dynamics, commonly known as the chaotic saddle, invalidates the assumption of ergodicity. For the paradigmatic Hénon-Heiles system, we use both theoretical and numerical arguments to show that the escaping dy…
Recently, a novel mixed-synchronization phenomenon is observed in counter-rotating nonlinear coupled oscillators. In mixed-synchronization state: some variables are synchronized in-phase, while others are out-of-phase. We have experimentally verified the occurrence of mixed-synchronization states in coupled counter-rotating chaotic piecewise Rossler oscillator. Analytical discussion on approximate stability analysis and numerical confirmation on the experimentally observed behavior is also given.
In this article we present an application of a method of control of Hamiltonian systems to the chaotic velocity diffusion of a cold electron beam interacting with electrostatic waves. We numerically show the efficiency and robustness of the additional small control term in restoring kinetic coherence of the injected electron beam.
The amplitude equation of Gierer-Mainhardt model has been actually derived near the boundary abuot which Turing and Hopf modes exist. In a parameter region where Hopf-Turing mixed mode solution is stable, a chaotic state that generally results from interaction between mixed modes, is observed. This chaotic region follows a strong selection of a spatially periodic order followed by a local, resonant, very large frequency temporal oscillation. A spatio-temporal forcing, responsible for what obseved, has been identified.
We study the closed Hamiltonian dynamics of a free particle moving on a ring, over one section of which it interacts linearly with a single harmonic oscillator. On the basis of numerical and analytical evidence, we conjecture that at small positive energies the phase space of our model is completely chaotic except for a single region of complete integrability with a smooth sharp boundary showing no KAM-type structures of any kind. This results in the cleanest mixed phase space structure possible, in which motions in the integrable region and in the chaotic region are clearly separated and inde…
We compute universal distributions for the transition probabilities of a Markov model for transport in the mixed phase space of area-preserving maps and verify that the survival probability distribution for trajectories near an infinite island-around-island hierarchy exhibits, on average, a power law decay with exponent $γ= 1.57$. This exponent agrees with that found from simulations of the Hénon and Chirikov-Taylor maps. This provides evidence that the Meiss-Ott Markov tree model describes the transport for mixed systems.
We present a comprehensive phase-space treatment of the motion of charged particles in electrodynamic traps. Focusing on five-wire surface-electrode Paul traps, we study the details of integrable and chaotic motion of a single ion. We introduce appropriate phase-space measures and give a universal characterization of the trap effectiveness as a function of the parameters. We rigorously derive the commonly used (time-independent) pseudopotential approximation, quantify its regime of validity and analyze the mechanism of its breakdown within the time-dependent potential. The phase space approach…
We investigate the statistical properties of the complexness parameter which characterizes uniquely complexness (biorthogonality) of resonance eigenstates of open chaotic systems. Specifying to the regime of isolated resonances, we apply the random matrix theory to the effective Hamiltonian formalism and derive analytically the probability distribution of the complexness parameter for two statistical ensembles describing the systems invariant under time reversal. For those with rigid spectra, we consider a Hamiltonian characterized by a picket-fence spectrum without spectral fluctuations. Then…
We show that the Ulam method applied to dynamical symplectic maps generates Ulam networks which belong to the class of small world networks appearing for social networks of people, actors, power grids, biological networks and Facebook. We analyze the small world properties of Ulam networks on examples of the Chirikov standard map and the Arnold cat map showing that the number of degrees of separation, or the Erdös number, grows logarithmically with the network size for the regime of strong chaos. This growth is related to the Lyapunov instability of chaotic dynamics. The presence of stability …
We propose a new method for determining the stochastic or ordered nature of trajectories in non-integrable Hamiltonian dynamical systems. The method consists of constructing a time-series from the divergence of nearby trajectories and then performing a power spectrum analysis of the series. Ordered trajectories present a spectrum that consists of a few spikes while the spectrum of stochastic trajectories is continuous. A test of the method with three different systems, a 2-D mapping as well as a 2-D and a 3-D Hamiltonian, shows that the method is fast and efficient, even in the case of sticky …
We discuss recent results obtained for the Hamiltonian Mean Field model. The model describes a system of N fully-coupled particles in one dimension and shows a second-order phase transition from a clustered phase to a homogeneous one when the energy is increased. Strong chaos is found in correspondence to the critical point on top of a weak chaotic regime which characterizes the motion at low energies. For a small region around the critical point, we find anomalous (enhanced) diffusion and Lévy walks in a transient temporal regime before the system relaxes to equilibrium.
In the present work we investigate phase correlations by recourse to the Shannon entropy. Using theoretical arguments we show that the entropy provides an accurate measure of phase correlations in any dynamical system, in particular when dealing with a chaotic diffusion process. We apply this approach to different low dimensional maps in order to show that indeed the entropy is very sensitive to the presence of correlations among the successive values of angular variables, even when it is weak. Later on, we apply this approach to unveil strong correlations in the time evolution of the phases i…
Chaotic Hamiltonians are known to follow Random Matrix Theory (RMT) ensembles in the apparent randomness of their spectra and wavefunction statistics. Deviations form RMT also do occur, however, due to system-specific properties, or as quantum signatures of classical chaos. Scarring, for instance, is the enhancement of wavefunction intensity near classical periodic orbits, and it can be characterized by a local density of states (or local spectrum) that clearly deviates from RMT expectations, by exhibiting a peaked envelope, which has been described semiclassically. Here, the system is connect…
