Semiclassical description of chaos-assisted tunneling

Quantum dynamical tunneling breaks classical conserved quantities

We discover that quantum dynamical tunneling, occurring between phase space regions in a classically forbidden way, can break conserved quantities. We rigorously prove that a conserved quantity in a class of typical pseudointegrable systems can be broken quantum mechanically. We then numerically compute the uncertainties of this broken conserved quantity, which remain nonzero for up to 10^{5} eigenstates and exhibit universal distributions similar to energy level statistics. Furthermore, all the eigenstates with large uncertainties show the superpositions of regular orbits with different values of the conserved quantity, showing definitive manifestation of dynamical tunneling. A random matrix model is constructed to successfully reproduce the level statistics of pseudointegrable systems.

Mixed eigenstates in the Dicke model: Statistics and power-law decay of the relative proportion in the semiclassical limit

How the mixed eigenstates vary when approaching the semiclassical limit in mixed-type many-body quantum systems is an interesting but still less known question. Here, we address this question in the Dicke model, a celebrated many-body model that has a well defined semiclassical limit and undergoes a transition to chaos in both quantum and classical cases. Using the Husimi function, we show that the eigenstates of the Dicke model with mixed-type classical phase space can be classified into different types. To quantitatively characterize the types of eigenstates, we study the phase space overlap index, which is defined in terms of the Husimi function. We look at the probability distribution of the phase space overlap index and investigate how it changes with increasing system size, that is, when approaching the semiclassical limit. We show that increasing the system size gives rise to a power-law decay in the behavior of the relative proportion of mixed eigenstates. Our findings shed more light on the properties of eigenstates in mixed-type many-body systems and suggest that the principle of uniform semiclassical condensation of Husimi functions should also be valid for many-body quantum systems.

Power-law decay of the fraction of the mixed eigenstates in kicked top model with mixed-type classical phase space

The properties of mixed eigenstates in a generic quantum system with a classical counterpart that has mixed-type phase space, although important to understand several fundamental questions that arise in both theoretical and experimental studies, are still not clear. Here, following a recent work [Č. Lozej, D. Lukman, and M. Robnik, Phys. Rev. E 106, 054203 (2022)2470-004510.1103/PhysRevE.106.054203], we perform an analysis of the features of mixed eigenstates in a time-dependent Hamiltonian system, the celebrated kicked top model. As a paradigmatic model for studying quantum chaos, the kicked top model is known to exhibit both classical and quantum chaos. The types of eigenstates are identified by means of the phase-space overlap index, which is defined as the overlap of the Husimi function with regular and chaotic regions in classical phase space. We show that the mixed eigenstates appear due to various tunneling precesses between different phase-space structures, while the regular and chaotic eigenstates are, respectively, associated with invariant tori and chaotic components in phase space. We examine how the probability distribution of the phase-space overlap index evolves with increasing system size for different kicking strengths. In particular, we find that the relative fraction of mixed states exhibits a power-law decay as the system size increases, indicating that only purely regular and chaotic eigenstates are left in the strict semiclassical limit. We thus provide further verification of the principle of uniform semiclassical condensation of Husimi functions and confirm the correctness of the Berry-Robnik picture.

Chaos-assisted many-body tunnelling

We study the interplay of chaos and tunneling between two weakly coupled Bose-Josephson junctions. The classical phase space of the composite system has a mixed structure including quasi-integrable self-trapping islands for particles and excitations, separated by a chaotic sea. We show that the many-body dynamical tunneling gap between macroscopic Schrödinger cat states supported by these islands is chaos-enhanced. The many-body tunneling rate fluctuates over several orders of magnitude with small variations of the system parameters or the particle number.

Entropic comparison of Landau-Zener and Demkov interactions in the phase space of a quadrupole billiard

We investigate two types of avoided crossings in a chaotic billiard within the framework of information theory. The Shannon entropy in the phase space for the Landau-Zener interaction increases as the center of the avoided crossing is approached, whereas for the Demkov interaction, the Shannon entropy decreases as the center of avoided crossing is passed by with an increase in the deformation parameter. This feature can provide a new indicator for scar formation. In addition, it is found that the Fisher information of the Landau-Zener interaction is significantly larger than that of the Demkov interaction.

Direct observation of chaotic resonances in optical microcavities

Optical microcavities play a significant role in the study of classical and quantum chaos. To date, most experimental explorations of their internal wave dynamics have focused on the properties of their inputs and outputs, without directly interrogating the dynamics and the associated mode patterns inside. As a result, this key information is rarely retrieved with certainty, which significantly restricts the verification and understanding of the actual chaotic motion. Here we demonstrate a simple and robust approach to directly and rapidly map the internal mode patterns in chaotic microcavities. By introducing a local index perturbation through a pump laser, we report a spectral response of optical microcavities that is proportional to the internal field distribution. With this technique, chaotic modes with staggered mode spacings can be distinguished. Consequently, a complete chaos assisted tunneling (CAT) and its time-reversed process are experimentally verified in the optical domain with unprecedented certainty.

Chaos-assisted tunneling resonances in a synthetic Floquet superlattice

The field of quantum simulation, which aims at using a tunable quantum system to simulate another, has been developing fast in the past years as an alternative to the all-purpose quantum computer. So far, most efforts in this domain have been directed to either fully regular or fully chaotic systems. Here, we focus on the intermediate regime, where regular orbits are surrounded by a large sea of chaotic trajectories. We observe a quantum chaos transport mechanism, called chaos-assisted tunneling, that translates in sharp resonances of the tunneling rate and provides previously unexplored possibilities for quantum simulation. More specifically, using Bose-Einstein condensates in a driven optical lattice, we experimentally demonstrate and characterize these resonances. Our work paves the way for quantum simulations with long-range transport and quantum control through complexity.

Resonance-assisted tunneling in four-dimensional normal-form Hamiltonians

Nonlinear resonances in the classical phase space lead to a significant enhancement of tunneling. We demonstrate that the double resonance gives rise to a complicated tunneling peak structure. Such double resonances occur in Hamiltonian systems with an at least four-dimensional phase space. To explain the tunneling peak structure, we use the universal description of single and double resonances by the four-dimensional normal-form Hamiltonians. By applying perturbative methods, we reveal the underlying mechanism of enhancement and suppression of tunneling and obtain excellent quantitative agreement. Using a minimal matrix model, we obtain an intuitive understanding.

Complex-path prediction of resonance-assisted tunneling in mixed systems

We present a semiclassical prediction of regular-to-chaotic tunneling in systems with a mixed phase space, including the effect of a nonlinear resonance chain. We identify complex paths for direct and resonance-assisted tunneling in the phase space of an integrable approximation with one nonlinear resonance chain. We evaluate the resonance-assisted contribution analytically and give a prediction based on just a few properties of the classical phase space. For the standard map excellent agreement with numerically determined tunneling rates is observed. The results should similarly apply to ionization rates and quality factors.

Perturbation-free prediction of resonance-assisted tunneling in mixed regular-chaotic systems

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 nonperturbative 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 the method for the standard map.

Experimental Observation of Resonance-Assisted Tunneling

We present the first experimental observation of resonance-assisted tunneling, a wave phenomenon, where regular-to-chaotic tunneling is strongly enhanced by the presence of a classical nonlinear resonance chain. For this we use a microwave cavity made of oxygen free copper with the shape of a desymmetrized cosine billiard designed with a large nonlinear resonance chain in the regular region. It is opened in a region, where only chaotic dynamics takes place, such that the tunneling rate of a regular mode to the chaotic region increases the line width of the mode. Resonance-assisted tunneling is demonstrated by (i) a parametric variation and (ii) the characteristic plateau and peak structure towards the semiclassical limit.

Analysis of Firefly Algorithms and Automatic Parameter Tuning

Many metaheuristic algorithms are nature-inspired, and most are population-based. Particle swarm optimization is a good example as an efficient metaheuristic algorithm. Inspired by PSO, many new algorithms have been developed in recent years. For example, firefly algorithm was inspired by the flashing behaviour of fireflies. In this chapter, the authors analyze the standard firefly algorithm and study the chaos-enhanced firefly algorithm with automatic parameter tuning. They first compare the performance of these algorithms and then use them to solve a benchmark design problem in engineering. Results obtained by other methods are compared and analyzed. The authors also discuss some important topics for further research.

The impact of emission mechanisms on the long-lived states around avoided resonance crossings in chaotic microcavity

Here we demonstrate the impacts of emission mechanisms on the light confinements in open systems. Taking the oval-shaped cavities as examples, we show that the enhancements in quality (Q) factors are usually associated with the universal emissions. When the coupled resonances have similar far field patterns, the Q factor of the long-lived resonance has the possibility to be enhanced by the coherent destruction at the decay channels. Otherwise, the Q factors of long-lived resonances are usually reduced around the level crossings.

Coupling of bouncing-ball modes to the chaotic sea and their counting function

We study the coupling of bouncing-ball modes to chaotic modes in two-dimensional billiards with two parallel boundary segments. Analytically, we predict the corresponding decay rates using the fictitious integrable system approach. Agreement with numerically determined rates is found for the stadium and the cosine billiard. We use this result to predict the asymptotic behavior of the counting function N_{bb}(E)∼E^{δ}. For the stadium billiard we find agreement with the previous result δ=3/4. For the cosine billiard we derive δ=5/8, which is confirmed numerically and is well below the previously predicted upper bound δ=9/10.

Chaos-Enhanced Firefly Algorithm with Automatic Parameter Tuning

Many metaheuristic algorithms are nature-inspired, and most are population-based. Particle swarm optimization is a good example as an efficient metaheuristic algorithm. Inspired by PSO, many new algorithms have been developed in recent years. For example, firefly algorithm was inspired by the flashing behaviour of fireflies. In this paper, the author extends the standard firefly algorithm further to introduce chaos-enhanced firefly algorithm with automatic parameter tuning, which results in two more variants of FA. The author first compares the performance of these algorithms, and then uses them to solve a benchmark design problem in engineering. Results obtained by other methods will be compared and analyzed.

Fractional-power-law level statistics due to dynamical tunneling

For systems with a mixed phase space we demonstrate that dynamical tunneling universally leads to a fractional power law of the level-spacing distribution P(s) over a wide range of small spacings s. Going beyond Berry-Robnik statistics, we take into account that dynamical tunneling rates between the regular and the chaotic region vary over many orders of magnitude. This results in a prediction of P(s) which excellently describes the spectral data of the standard map. Moreover, we show that the power-law exponent is proportional to the effective Planck constant h(eff).

Direct regular-to-chaotic tunneling rates using the fictitious-integrable-system approach

We review the fictitious integrable system approach which predicts dynamical tunneling rates from regular states to the chaotic region in systems with a mixed phase space. It is based on the introduction of a fictitious integrable system that resembles the regular dynamics within the regular island. We focus on the direct regular-to-chaotic tunneling process which dominates if nonlinear resonances within the regular island are not relevant. For quantum maps, billiard systems, and optical microcavities, we find excellent agreement with numerical rates for all regular states.

Pump-induced dynamical tunneling in a deformed microcavity laser

Pump-induced dynamical tunneling has been observed in free-space resonant optical pumping of a deformed microcavity by employing excitation spectroscopy. A focused-pump beam was injected into the cavity by refraction and then coupled to a high-Q cavity mode via dynamical tunneling. Pump-coupling efficiency as high as 50% and an effective coupling constant responsible for the tunneling were obtained from the observed pumping efficiency with a mode-mode coupling model.

Regular-to-chaotic tunneling rates: from the quantum to the semiclassical regime

We derive a prediction of dynamical tunneling rates from regular to chaotic phase-space regions combining the direct regular-to-chaotic tunneling mechanism in the quantum regime with an improved resonance-assisted tunneling theory in the semiclassical regime. We give a qualitative recipe for identifying the relevance of nonlinear resonances in a given variant Planck's over 2pi regime. For systems with one or multiple dominant resonances we find excellent agreement to numerics.

Chaos-assisted tunneling and 1/falpha spectral fluctuations in the order-chaos transition

It has been shown that the spectral fluctuations of different quantum systems are characterized by 1/falpha noise, with 1< or =alpha< or =2, in the transition from integrability to chaos. This result is not well understood. We show that chaos-assisted tunneling gives rise to this power-law behavior. We develop a random matrix model for intermediate quantum systems, based on chaos-assisted tunneling, and we discuss under which conditions it displays 1/falpha noise in the transition from integrability to chaos. We conclude that the variance of the elements that connect regular with chaotic states must decay with the difference of energy between them. We compare the characteristics of the transition modeled in this way with what is obtained for the Robnik billiard.

Regular-to-chaotic tunneling rates using a fictitious integrable system

We derive a formula predicting dynamical tunneling rates from regular states to the chaotic sea in systems with a mixed phase space. Our approach is based on the introduction of a fictitious integrable system that resembles the regular dynamics within the island. For the standard map and other kicked systems we find agreement with numerical results for all regular states in a regime where resonance-assisted tunneling is not relevant.

Universality in the flooding of regular islands by chaotic states

We investigate the structure of eigenstates in systems with a mixed phase space in terms of their projection onto individual regular tori. Depending on dynamical tunneling rates and the Heisenberg time, regular states disappear and chaotic states flood the regular tori. For a quantitative understanding we introduce a random matrix model. The resulting statistical properties of eigenstates as a function of an effective coupling strength are in very good agreement with numerical results for a kicked system. We discuss the implications of these results for the applicability of the semiclassical eigenfunction hypothesis.

Influence of random bulk inhomogeneities on quasioptical cavity resonator spectrum

The statistical spectral theory of oscillations in a quasioptical cavity resonator filled with random inhomogeneities is suggested. It is shown that inhomogeneities in the resonator lead to intermode scattering which results in the shift and broadening of spectral lines. The shift and the broadening of each spectral line is strongly depended upon the frequency distance between the nearest-neighbor spectral lines. As this distance increases, the influence of inhomogeneities is sharply reduced. Solitary spectral lines that have quite a large distance to the nearest neighbors are slightly changed due to small inhomogeneities. Owing to such a selective influence of inhomogeneities on spectral lines the effective spectrum rarefaction arises. Both the shift and the broadening of spectral lines as well as spectrum rarefaction in the quasioptical cavity millimeter wave resonator were detected experimentally. We have found out that inhomogeneities result in the resonator spectrum stochastization. As a result, the spectrum becomes composite, i.e., it consists of both regular and random parts. The active self-excited system based on the inhomogeneous quasioptical cavity millimeter wave resonator with a Gunn diode was examined as well. The inhomogeneous quasioptical cavity millimeter wave resonator (passive and active) can serve as a model of a semiconductor quantum billiard. Based on our results we propose that such a billiard with the spectrum rarefied by random inhomogeneities be used as an active semiconductor laser system.

Nanowires with surface disorder: giant localization lengths and quantum-to-classical crossover

We investigate electronic quantum transport through nanowires with one-sided surface roughness. A magnetic field perpendicular to the scattering region is shown to lead to exponentially diverging localization lengths in the quantum-to-classical crossover regime. This effect can be quantitatively accounted for by tunneling between the regular and the chaotic components of the underlying mixed classical phase space.

Influence of classical resonances on chaotic tunneling

Dynamical tunneling between symmetry-related stable modes is studied in the periodically driven pendulum. We present strong evidence that the tunneling process is governed by nonlinear resonances that manifest within the regular phase-space islands on which the stable modes are localized. By means of a quantitative numerical study of the corresponding Floquet problem, we identify the trace of such resonances not only in the level splittings between near-degenerate quantum states, where they lead to prominent plateau structures, but also in overlap matrix elements of the Floquet eigenstates, which reveal characteristic sequences of avoided crossings in the Floquet spectrum. The semiclassical theory of resonance-assisted tunneling yields good overall agreement with the quantum-tunneling rates, and indicates that partial barriers within the chaos might play a prominent role.

Resonance-assisted tunneling in three degrees of freedom without discrete symmetry

We study dynamical tunneling in a near-integrable Hamiltonian with three degrees of freedom. Despite the absence of discrete symmetry we show that the mixing of near-degenerate quantum states is due to dynamical tunneling mediated by the nonlinear resonances in the classical phase space. Identifying the key resonances allows us to selectively suppress the dynamical tunneling by adding weak counter-resonant terms.

Chaos-assisted tunneling in dielectric microcavities

We demonstrate that the lifetimes and emission patterns of the optical modes in generic (asymmetric) microresonators are strongly affected by the phenomenon of chaos-assisted tunneling and develop a theory of the effect.

Flooding of chaotic eigenstates into regular phase space islands

We introduce a criterion for the existence of regular states in systems with a mixed phase space. If this condition is not fulfilled chaotic eigenstates substantially extend into a regular island. Wave packets started in the chaotic sea progressively flood the island. The extent of flooding by eigenstates and wave packets increases logarithmically with the size of the chaotic sea and the time, respectively. This new effect is observed for the example of island chains with just ten islands.

Resonance- and chaos-assisted tunneling in mixed regular-chaotic systems

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.