Dynamical Tunneling in More than Two Degrees of Freedom.

Recent progress towards understanding the mechanism of dynamical tunneling in Hamiltonian systems with three or more degrees of freedom (DoF) is reviewed. In contrast to systems with two degrees of freedom, the three or more degrees of freedom case presents several challenges. Specifically, in higher-dimensional phase spaces, multiple mechanisms for classical transport have significant implications for the evolution of initial quantum states. In this review, the importance of features on the Arnold web, a signature of systems with three or more DoF, to the mechanism of resonance-assisted tunneling is illustrated using select examples. These examples represent relevant models for phenomena such as intramolecular vibrational energy redistribution in isolated molecules and the dynamics of Bose-Einstein condensates trapped in optical lattices.

Phase space perspective on a model for isomerization in an optical cavity.

Explanation for the modification of rates and mechanism of reactions carried out in optical cavities still eludes us. Several studies indicate that the cavity-mediated changes in the nature of vibrational energy flow within a molecule may play a significant role. Here, we study a model polaritonic system, proposed and analyzed earlier by Fischer et al., J. Chem. Phys. 156, 154305 (2022), comprising a one-dimensional isomerization mode coupled to a single photon mode in a lossless cavity. We show that the isomerization probability in the presence of virtual photons, for specific cavity-system coupling strengths and cavity frequencies, can exhibit suppression or enhancement for different choices of the initial reactant vibropolariton wavepacket. We observe a qualitative agreement between the classical and quantum average isomerization probabilities in the virtual photon case. A significant part of the effects due to coupling to the cavity can be rationalized in terms of a "chaos-order-chaos" transition of the classical phase space and the phase space localization nature of the polariton states that dominantly participate in the quantum isomerization dynamics. On the other hand, for initial states with zero photons (i.e., a "dark cavity"), the isomerization probability is suppressed when the cavity frequency is tuned near to the fundamental frequency of the reactive mode. The classical-quantum correspondence in the zero photon case is unsatisfactory. In this simple model, we find that the suppression or enhancement of isomerization arises due to the interplay between cavity-system energy flow dynamics and quantum tunneling.

Dissociation dynamics of a diatomic molecule in an optical cavity.

We study the dissociation dynamics of a diatomic molecule, modeled as a Morse oscillator, coupled to an optical cavity. A marked suppression of the dissociation probability, both classical and quantum, is observed for cavity frequencies significantly below the fundamental transition frequency of the molecule. We show that the suppression in the probability is due to the nonlinearity of the dipole function. The effect can be rationalized entirely in terms of the structures in the classical phase space of the model system.

Dynamically Hidden Reaction Paths in the Reaction of CF3 + + CO.

Reaction paths on a potential energy surface are widely used in quantum chemical studies of chemical reactions. The recently developed global reaction route mapping (GRRM) strategy automatically constructs a reaction route map, which provides a complete picture of the reaction. Here, we thoroughly investigate the correspondence between the reaction route map and the actual chemical reaction dynamics for the CF3 + + CO reaction studied by guided ion beam tandem mass spectrometry (GIBMS). In our experiments, FCO+, CF2 +, and CF+ product ions were observed, whereas if the collision partner is N2, only CF2 + is observed. Interestingly, for reaction with CO, GRRM-predicted reaction paths leading to the CF+ + F2CO product channel are found at a barrier height of about 2.5 eV, whereas the experimentally obtained threshold for CF+ formation was 7.48 ± 0.15 eV. In other words, the ion was not obviously observed in the GIBMS experiment, unless a much higher collision energy than the requisite energy threshold was provided. On-the-fly molecular dynamics simulations revealed a mechanism that hides these reaction paths, in which a non-statistical energy distribution at the first collisionally reached transition state prevents the reaction from proceeding along some reaction paths. Our results highlight the existence of dynamically hidden reaction paths that may be inaccessible in experiments at specific energies and hence the importance of reaction dynamics in controlling the destinations of chemical reactions.

Intramolecular vibrational energy redistribution and the quantum ergodicity transition: a phase space perspective.

Intramolecular vibrational energy redistribution (IVR) impacts the dynamics of reactions in a profound way. Theoretical and experimental studies are increasingly indicating that accounting for the finite rate of energy flow is critical for uncovering the correct reaction mechanisms and calculating accurate rates. This requires an explicit understanding of the influence and interplay of the various anharmonic (Fermi) resonances that lead to the coupling of the vibrational modes. In this regard, the local random matrix theory (LRMT) and the related Bose-statistics triangle rule (BSTR) model have emerged as powerful and predictive quantum theories for IVR. In this Perspective we highlight the close correspondence between LRMT and the classical phase space perspective on IVR, primarily using model Hamiltonians with three degrees of freedom. Our purpose for this is threefold. First, this clearly brings out the extent to which IVR pathways are essentially classical, and hence crucial towards attempts to control IVR. Second, given that LRMT and BSTR are designed to be applicable for large molecules, the exquisite correspondence observed even for small molecules allows for insights into the quantum ergodicity transition. Third, we showcase the power of modern nonlinear dynamics methods in analysing high dimensional phase spaces, thereby extending the deep insights into IVR that were earlier gained for systems with effectively two degrees of freedom. We begin with a brief overview of recent examples where IVR plays an important role and conclude by mentioning the outstanding problems and the potential connections to issues of interest in other fields.

Stable chaos and delayed onset of statisticality in unimolecular dissociation reactions.

Statistical models provide a powerful and useful class of approximations for calculating reaction rates by bypassing the need for detailed, and often difficult, dynamical considerations. Such approaches invariably invoke specific assumptions about the extent of intramolecular vibrational energy flow in the system. However, the nature of the transition to the statistical regime as a function of the molecular parameters is far from being completely understood. Here, we use tools from nonlinear dynamics to study the transition to statisticality in a model unimolecular reaction by explicitly visualizing the high dimensional classical phase space. We identify generic features in the phase space involving the intersection of two or more independent anharmonic resonances and show that the presence of correlated, but chaotic, intramolecular dynamics near such junctions leads to nonstatisticality. Interestingly, akin to the stability of asteroids in the Solar System, molecules can stay protected from dissociation at the junctions for several picoseconds due to the phenomenon of stable chaos.

Relevance of the Resonance Junctions on the Arnold Web to Dynamical Tunneling and Eigenstate Delocalization.

We study the competition and correspondence between the classical and quantum routes to intramolecular vibrational energy redistribution (IVR) in a three degrees of freedom model effective Hamiltonian. Specifically, we focus on the classical and the quantum dynamics near the resonance junctions on the Arnold web that are formed by an intersection of independent resonances. The regime of interest models the IVR dynamics from highly excited initial states near dissociation thresholds of molecular systems wherein both classical and purely quantum, involving dynamical tunneling, routes to IVR coexist. In the vicinity of a resonance junction, classical chaos is inevitably present, and hence one expects the quantum IVR pathways to have a strong classical component as well. We show that with increasing resonant coupling strengths the classical component of IVR leads to a transition from coherent dynamical tunneling to incoherent dynamical tunneling. Furthermore, we establish that the quantum IVR dynamics can be predicted based on the structures on the classical Arnold web. In addition, we investigate the nature of the highly excited eigenstates to identify the quantum signatures of the multiplicity-2 junctions. For the parameter regimes studies herein, by projecting the eigenstates onto the Arnold web, we find that eigenstates in the vicinity of the junctions are primarily delocalized due to dynamical tunneling.

Detecting reactive islands using Lagrangian descriptors and the relevance to transition path sampling.

It has been known for sometime now that isomerization reactions, classically, are mediated by phase space structures called reactive islands (RI). RIs provide one possible route to correct for the nonstatistical effects in the reaction dynamics. In this work, we map out the reactive islands for the two dimensional Müller-Brown model potential and show that the reactive islands are intimately linked to the issue of rare event sampling. In particular, we establish the sensitivity of the so called committor probabilities, useful quantities in the transition path sampling technique, to the hierarchical RI structures. Mapping out the RI structure for high dimensional systems, however, is a challenging task. Here, we show that the technique of Lagrangian descriptors is able to effectively identify the RI hierarchy in the model system. Based on our results, we suggest that the Lagrangian descriptors can be useful for detecting RIs in high dimensional systems.

One Versus Two Photon Control of Dynamical Tunneling: Influence of the Irregular Floquet States.

A useful approach to control quantum processes involves driving systems with two colored laser fields and varying the relative phase between the fields to control the quantum interferences. A particularly interesting class of bichromatic control schemes involves the so-called M versus N-photon control that results in laser-induced symmetry breaking and leads to directed transport; however, recent studies have shown that the mechanism of laser-induced symmetry breaking has a common classical and quantum origin. In this context, a relevant question is the extent to which such a detailed classical-quantum correspondence holds if the process to be controlled involves quantum tunneling. In this work, we address this issue in terms of controlling dynamical tunneling between field-induced islands of stability in the classical phase space of a model system, a periodically driven pendulum. This is also a paradigmatic model for Hamiltonian ratchets wherein the islands of stability, that is, nonlinear resonances, play a crucial role in the observed directed transport. We compute an appropriate control landscape for the process and show that despite breaking the relevant symmetries, there exist regions in the control landscape where the control fails. The lack of control can be understood in terms of the phase-space nature of the quantum Floquet states that participate in the dynamics of the initial wavepacket. We argue that robust regions of no control arise due to the phenomenon of chaos-assisted tunneling and comment on the possible influence of such regions on the directed transport in the model system.

Breaking of a bond: when is it statistical?

Unimolecular dissociation dynamics of a model three degree of freedom triatomic molecule is studied in order to understand the mechanisms for deviations from statisticality. Performing a wavelet based time-frequency analysis of the dynamics allows for the dynamics to be followed on the network of nonlinear resonances, also called as the Arnold web. The results indicate that the long lifetime trajectories spend a considerable amount of time trapped near junctions in the web. It is argued that characterizing the dynamics near such junctions might lead to deeper insights into the origins of nonstatistical dynamics.

Dynamical traps lead to the slowing down of intramolecular vibrational energy flow.

The phenomenon of intramolecular vibrational energy redistribution (IVR) is at the heart of chemical reaction dynamics. Statistical rate theories, assuming instantaneous IVR, predict exponential decay of the population with the properties of the transition state essentially determining the mechanism. However, there is growing evidence that IVR competes with the reaction timescales, resulting in deviations from the exponential rate law. Dynamics cannot be ignored in such cases for understanding the reaction mechanisms. Significant insights in this context have come from the state space model of IVR, which predicts power law behavior for the rates with the power law exponent, an effective state space dimensionality, being a measure of the nature and extent of the IVR dynamics. However, whether the effective IVR dimensionality can vary with time and whether the mechanism for the variation is of purely quantum or classical origins are issues that remain unresolved. Such multiple power law scalings can lead to surprising mode specificity in the system, even above the threshold for facile IVR. In this work, choosing the well-studied thiophosgene molecule as an example, we establish the anisotropic and anomalous nature of the quantum IVR dynamics and show that multiple power law scalings do manifest in the system. More importantly, we show that the mechanism of the observed multiple power law scaling has classical origins due to a combination of trapping near resonance junctions in the network of classical nonlinear resonances at short to intermediate times and the influence of weak higher-order resonances at relatively longer times.

Eigenstates of thiophosgene near the dissociation threshold: deviations from ergodicity.

A subset of the highly excited eigenstates of thiophosgene (SCCl2) near the dissociation threshold are analyzed using sensitive measures of quantum ergodicity. We find several localized eigenstates, suggesting that the intramolecular vibrational energy flow dynamics is nonstatistical even at such high levels of excitations. The results are consistent with recent observations of sharp spectral features in the stimulated emission spectra of SCCl2.

Decoding the dynamical information embedded in highly excited vibrational eigenstates: state space and phase space viewpoints.

We study the nature of highly excited eigenstates of strongly coupled multimode systems with three degrees of freedom. Attempts to dynamically assign the eigenstates using classical-quantum correspondence techniques poses a considerable challenge, due to both the number of degrees of freedom and the extensive chaos in the underlying classical phase space. Nevertheless, we show that sequences of localized states interspersed between delocalized states can be identified readily by using the parametric variation technique proposed earlier by us. In addition, we introduce a novel method to lift the quantum eigenstates onto the classical resonance web using a wavelet-based local time-frequency approach. It is shown that the lifting procedure provides clear information on the various dominant nonlinear resonances that influence the eigenstates. Analyzing the spectroscopic Hamiltonians for CDBrClF and CF(3)CHFI as examples, we illustrate our approach and demonstrate the consistency between state space and phase space perspectives of the eigenstates. Two exemplary cases of highly mixed quantum states are discussed to highlight the difficulties associated with their assignment. In particular, we provide arguments to distinguish between the states in terms of their predominantly classical or quantum nature of the mixing.

Bichromatically driven double well: parametric perspective of the strong field control landscape reveals the influence of chaotic states.

The aim of this work is to understand the influence of chaotic states in control problems involving strong fields. Towards this end, we numerically construct and study the strong field control landscape of a bichromatically driven double well. A novel measure based on correlating the overlap intensities between Floquet states and an initial phase space coherent state with the parametric motion of the quasienergies is used to construct and interpret the landscape features. "Walls" of no control, which are robust under variations of the relative phase between the fields, are seen on the control landscape and associated with multilevel interactions involving chaotic Floquet states.

Intramolecular vibrational energy redistribution from a high frequency mode in the presence of an internal rotor: classical thick-layer diffusion and quantum localization.

We study the effect of an internal rotor on the classical and quantum intramolecular vibrational energy redistribution (IVR) dynamics of a model system with three degrees of freedom. The system is based on a Hamiltonian proposed by Martens and Reinhardt [J. Chem. Phys. 93, 5621 (1990)] to study IVR in the excited electronic state of para-fluorotoluene. We explicitly construct the state space and show, confirming the mechanism proposed by Martens and Reinhardt, that an excited high frequency mode relaxes via diffusion along a thick layer of chaos created by the low frequency-rotor interactions. However, the corresponding quantum dynamics exhibits no appreciable relaxation of the high frequency mode. We attribute the quantum suppression of the classical thick-layer diffusion to the rotor selection rules and, possibly, dynamical localization effects.

Intramolecular vibrational energy redistribution as state space diffusion: classical-quantum correspondence.

We study the intramolecular vibrational energy redistribution (IVR) dynamics of an effective spectroscopic Hamiltonian describing the four coupled high frequency modes of CDBrClF. The IVR dynamics ensuing from nearly isoenergetic zeroth-order states, an edge (overtone) and an interior (combination) state, is studied from a state space diffusion perspective. A wavelet based time-frequency analysis reveals an inhomogeneous phase space due to the trapping of classical trajectories. Consequently the interior state has a smaller effective IVR dimension as compared to the edge state.

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.

On dynamical tunneling and classical resonances.

This work establishes a firm relationship between classical nonlinear resonances and the phenomenon of dynamical tunneling. It is shown that the classical phase space with its hierarchy of resonance islands completely characterizes dynamical tunneling and explicit forms of the dynamical barriers can be obtained only by identifying the key resonances. Relationship between the phase space viewpoint and the quantum mechanical superexchange approach is discussed in near-integrable and mixed regular-chaotic situations. For near-integrable systems with sufficient anharmonicity the effect of multiple resonances, i.e., resonance-assisted tunneling, can be incorporated approximately. It is also argued that the presumed relation of avoided crossings to nonlinear resonances does not have to be invoked in order to understand dynamical tunneling. For molecules with low density of states the resonance-assisted mechanism is expected to be dominant.