Quantum and classical dynamics in the stadium billiard.

By computer simulation, using both quantum and classical dynamics, we determined the rate constant and the kinetic isotope effect of the rate-determining step in the neutral hydrolysis of p-methoxyphenyl dichloroacetate in aqueous solution. This step involves a proton transfer concerted with the formation of a CO bond. A method of biased sampling was used; the Gibbs free energy of the biased configuration from which proton transfer is likely to occur was determined by a combination of semiempirical quantum calculations and thermodynamic integration. The proton dynamics was modeled with the quantum-dynamical density matrix evolution method that includes nonadiabatic pathways. The proton dynamics is driven by a fluctuating proton potential that was derived from a classical molecular dynamics simulation of the system including solvent. The calculated rate constant of 3×10-2 s-1 agrees within the error of the calculation with the experimentally observed value of 2.78×10-3 . The calculated pseudo-first-order kinetic isotope effect of 3.9 is in good agreement with the experimentally observed value of 3.2. The results show the feasibility of computational approaches to slow reactions in complex environments, where proton transfer with an essential quantum-dynamical nature is the rate-limiting step. ©1999 John Wiley & Sons, Inc. J Comput Chem 20: 886-895, 1999.

Using semiclassical methods, it is possible to construct very accurate approximations in the short-wavelength limit of quantum dynamics that rely exclusively on classical dynamical input. For systems whose classical realization is strongly chaotic, there is an exceedingly short logarithmic Ehrenfest time scale, beyond which the quantum and classical dynamics of a system necessarily diverge, and yet the semiclassical construction remains valid far beyond that time. This fact leads to a paradox if one ponders the reversibility and predictability properties of quantum and classical mechanics. They behave very differently relative to each other, with classical dynamics being essentially irreversible/unpredictable, whereas quantum dynamics is reversible/stable. This begs the question: 'How can an accurate approximation to a reversible/stable dynamics be constructed from an irreversible/unpredictable one?' The resolution of this incongruity depends on a couple of key ingredients: a well-known, inherent, one-way structural stability of chaotic systems; and an overlap integral not being amenable to the saddle point method.

Classical and Quantum Chaos in Atomic Systems

Quantum corrections to Newton’s equations are obtained and used to illustrate that classical dynamics is embedded explicitly in quantum dynamics. Originally, the resultant set of dynamical equations has been used to shed light on quantum chaos. We show that the method can provide insight into the dynamics of free particles and the harmonic oscillator. We then use it to determine whether Stern–Gerlach magnets can be constructed for free electrons.

Evidences of classical and quantum chaos in the time evolution of nonequilibrium ensembles

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.

The study of optomechanical systems has attracted much attention, most of which are concentrated in the physics in the small-amplitude regime. While in this article, we focus on optomechanics in the extremely-large-amplitude regime and consider both classical and quantum dynamics. Firstly, we study classical dynamics in a membrane-in-the-middle optomechanical system in which a partially reflecting and flexible membrane is suspended inside an optical cavity. We show that the membrane can present self-sustained oscillations with limit cycles in the shape of sawtooth-edged ellipses and exhibit dynamical multistability. Then, we study the dynamics of the quantum fluctuations around the classical orbits. By using the logarithmic negativity, we calculate the evolution of the quantum entanglement between the optical cavity mode and the membrane during the mechanical oscillation. We show that there is some synchronism between the classical dynamical process and the evolution of the quantum entanglement.

We present the results of a comprehensive study of the classical and quantum dynamics of spin Dirac fermion particles with dipole moments under the simultaneous action of arbitrary external fields, including the gravitational, inertial and electromagnetic ones. We demonstrate the complete consistency of the quantum and classical dynamics. Physical applications are discussed.

We introduce an efficient configuration interaction (CI) method for the calculation of mixed quantum and classical nonadiabatic molecular dynamics for multiple electrons. For any given realization of the classical degrees of freedom (e.g., a solvent), the method uses a novel real-space quadrature to efficiently compute the Coulomb and exchange interactions between electrons. We also introduce an approximation whereby the classical molecular dynamics is propagated for several time steps on electronic potential energy surfaces generated using only a particularly important subset of the CI basis states. By only updating the important-states subset periodically, we achieve significant reductions in the computational cost of solving the multielectron quantum problem. We test the real-space quadrature for the cases of two electrons confined in a cubic box with infinitely repulsive walls and two electrons dissolved in liquid water that occupy a single cavity, so-called hydrated dielectrons. We then demonstrate how to perform mixed quantum and classical nonadiabatic dynamics by combining these computational techniques with the mean-field with surface hopping algorithm of Prezhdo and Rossky [J. Chem. Phys. 107, 825 (1997)]. Finally, we illustrate the practicality of the approach to multielectron nonadiabatic dynamics by examining the nonadiabatic relaxation dynamics of both spin singlet and spin triplet hydrated dielectrons following excitation from the ground to the first excited state.

We study the localization of energy in a nonlinear coupled system, exhibiting so-called breather modes, using quantized Hamilton dynamics (QHD). Already at the lowest order, which is only twice as complex as classical mechanics, this simple semiclassical method incorporates quantum-mechanical effects. The transition between the localized and delocalized regimes is instantaneous in classical mechanics, while it is gradual due to tunneling in both quantum mechanics and QHD. In contrast to classical mechanics, which predicts an abrupt appearance of breathers, quantum mechanics and QHD show an alternation of localized and delocalized behavior in the transient region. QHD includes zero-point energy that is reflected in a shifted energy asymptote for the localized states, providing another improvement on the classical perspective. By detailed analysis of the distribution and transfer of energy within classical mechanics, QHD, and quantum dynamics, we conclude that QHD is an efficient approach that accounts for moderate quantum effects and can be used to identify quantum breathers in large nonlinear systems.

Graduate students who want to become familiar with advanced computational strategies in classical and quantum dynamics will find here both the fundamentals of a standard course and a detailed treatmen

We present a consistent framework of coupled classical and quantum dynamics. Our result allows us to overcome severe limitations of previous phenomenological approaches, like evolutions that do not preserve the positivity of quantum states or that allow to activate quantum nonlocality for superluminal signaling. A `hybrid' quantum-classical density is introduced and its evolution equation derived. The implications and applications of our result are numerous: it incorporates the back-reaction of quantum on classical variables, it resolves fundamental problems encountered in standard mean field theories, and remarkably, also the quantum measurement process, i.e. the most controversial example of quantum-classical interaction is consistently described within our approach, leading to a theory of dynamical collapse.

Can Bohmian trajectories account for quantum recurrences having classical periodicities?

The basic experimental and theoretical concepts underlying molecular spectroscopy are presented. The equations and relations needed in practical applications of high-resolution spectroscopy are provided. Group theoretical tools and their use in high-resolution spectroscopy are introduced and illustrated by simple examples. Time-dependent processes and quantum dynamics are presented and discussed in relation to spectroscopic observations. Coherent and incoherent radiative excitation and radiationless transitions are treated in relation to applications in molecular spectroscopy. Keywords: Quantum mechanics; time-dependent and time-independent Schrodinger equation; selection rules; stationary states; time-dependent perturbation theory; Heisenberg uncertainty principle; Fourier transformations; quantum dynamics; time-resolved spectroscopy; light scattering; Raman spectroscopy; absorption; spontaneous emission; stimulated emission; Born–Oppenheimer approximation; group theory; character tables; reducible and irreducible representations; normal modes; electronic configurations and states; lineshapes; line broadening; radiationless transitions; molecular symmetry; coherent excitation; classical mechanics

An efficient method, the self-consistent hybrid method, is proposed for accurately simulating time-dependent quantum dynamics in complex systems. The method is based on an iterative convergence procedure for a dynamical hybrid approach. In this approach, the overall system is first partitioned into a “core” and a “reservoir” (an initial guess). The former is treated via an accurate quantum mechanical method, namely, the time-dependent multiconfiguration self-consistent field or multiconfiguration time-dependent Hartree approach, and the latter is treated via a more approximate method, e.g., classical mechanics, semiclassical initial value representations, quantum perturbation theories, etc. Next, the number of “core” degrees of freedom, as well as other variational parameters, is systematically increased to achieve numerical convergence for the overall quantum dynamics. The method is applied to two examples of quantum dissipative dynamics in the condensed phase: the spin-boson problem and the electronic resonance decay in the presence of a vibrational bath. It is demonstrated that the method provides a practical way of obtaining accurate quantum dynamical results for complex systems.