Semiclassical Wave Packet Propagation Research Articles

Semiclassical propagation of coherent states and wave packets: Hidden saddles.

Semiclassical methods are extremely important in the subjects of wave-packet and coherent-state dynamics. Unfortunately, these essentially saddle-point approximations are considered nearly impossible to carry out in detail for systems with multiple degrees of freedom due to the difficulties of solving the resulting two-point boundary-value problems. However, recent developments have extended the applicability to a broader range of systems and circumstances. The most important advances are first to generate a set of real reference trajectories using appropriately reduced dimensional spaces of initial conditions, and second to feed that set into a Newton-Raphson search scheme to locate the exposed complex saddle trajectories. The arguments for this approach were based mostly on intuition and numerical verification. In this paper, the methods are put on a firmer theoretical foundation and then extended to incorporate saddles hidden from Newton-Raphson searches initiated with real trajectories. This hidden class of saddles is relevant to tunneling-type processes, but a hidden saddle can sometimes contribute just as much as or more than an exposed one. The distinctions between hidden and exposed saddles clarifies the interpretation of what constitutes tunneling for wave packets and coherent states in the time domain.

Dualities and non-Abelian mechanics

Dualities are mathematical mappings that reveal links between apparently unrelated systems in virtually every branch of physics1-8. Systems mapped onto themselves by a duality transformation are called self-dual and exhibit remarkable properties, as exemplified by the scale invariance of an Ising magnet at the critical point. Here we show how dualities can enhance the symmetries of a dynamical matrix (or Hamiltonian), enabling the design of metamaterials with emergent properties that escape a standard group theory analysis. As an illustration, we consider twisted kagome lattices9-15, reconfigurable mechanical structures that change shape by means of a collapse mechanism9. We observe that pairs of distinct configurations along the mechanism exhibit the same vibrational spectrum and related elastic moduli. We show that these puzzling properties arise from a duality between pairs of configurations on either side of a mechanical critical point. The critical point corresponds to a self-dual structure with isotropic elasticity even in the absence of spatial symmetries and a twofold-degenerate spectrum over the entire Brillouin zone. The spectral degeneracy originates from a version of Kramers' theorem16,17 in which fermionic time-reversal invariance is replaced by a hidden symmetry emerging at the self-dual point. The normal modes of the self-dual systems exhibit non-Abelian geometric phases18,19 that affect the semiclassical propagation of wavepackets20, leading to non-commuting mechanical responses. Our results hold promise for holonomic computation21 and mechanical spintronics by allowing on-the-fly manipulation of synthetic spins carried by phonons.

Dynamical consequences of time-reversal symmetry for systems with odd number of electrons: Conical intersections, semiclassical dynamics, and topology

In this manuscript we identify the main differences between the effects of Kramers symmetry on the systems with even and odd number of electrons, the ways how the aforementioned symmetry affects the structure of the Conical Seams (CSs), and how it shows up in semiclassical propagation of nuclear wavepackets, crossing the CSs. We identify the topological invariants, associated with CSs, in three cases: even and odd number of electrons with time-reversal symmetry, as well as absence of the latter. We obtain asymptotically exact semiclassical analytical solutions for wavepackets scattered on a CS for all three cases, identify topological features in a non-trivial shape of the scattered wavepacket, and connect them to the topological invariants, associated with CSs. We argue that, due to robustness of topology, the non-trivial wavepacket structure is a topologically protected evidence of a wavepacket having passed through a CS, rather than a feature of a semiclassical approximation.

Validating and implementing modified Filinov phase filtration in semiclassical dynamics.

The Mixed Quantum-Classical Initial Value Representation (MQC-IVR) is a recently introduced approximate semiclassical (SC) method for the calculation of real-time quantum correlation functions. MQC-IVR employs a modified Filinov filtration (MFF) scheme to control the overall phase of the SC integrand, extending the applicability of SC methods to complex systems while retaining their ability to accurately describe quantum coherence effects. Here, we address questions regarding the effectiveness of the MFF scheme in combination with SC dynamics. Previous work showed that this filtering scheme is of limited utility in the context of semiclassical wavepacket propagation, but we find that the MFF is extraordinarily powerful in the context of correlation functions. By examining trajectory phase and amplitude contributions to the real-time SC correlation function in a model system, we clearly demonstrate that the MFF serves to reduce noise by damping amplitude only in regions of highly oscillatory phase leading to a reduction in computational effort while retaining accuracy. Further, we introduce a novel and efficient MQC-IVR formulation that allows for linear scaling in computational cost with the total simulation length, a significant improvement over the more-than quadratic scaling exhibited by the original method.

Generalized Gaussian wave packet dynamics: Integrable and chaotic systems.

The ultimate semiclassical wave packet propagation technique is a complex, time-dependent Wentzel-Kramers-Brillouin method known as generalized Gaussian wave packet dynamics (GGWPD). It requires overcoming many technical difficulties in order to be carried out fully in practice. In its place roughly twenty years ago, linearized wave packet dynamics was generalized to methods that include sets of off-center, real trajectories for both classically integrable and chaotic dynamical systems that completely capture the dynamical transport. The connections between those methods and GGWPD are developed in a way that enables a far more practical implementation of GGWPD. The generally complex saddle-point trajectories at its foundation are found using a multidimensional Newton-Raphson root search method that begins with the set of off-center, real trajectories. This is possible because there is a one-to-one correspondence. The neighboring trajectories associated with each off-center, real trajectory form a path that crosses a unique saddle; there are exceptions that are straightforward to identify. The method is applied to the kicked rotor to demonstrate the accuracy improvement as a function of ℏ that comes with using the saddle-point trajectories.

Complex time paths for semiclassical wave packet propagation with complex trajectories.

The use of complex-valued trajectories in semiclassical wave packet methods can lead to problems that prevent calculation of the wave function in certain regions of the configuration space. We investigate this so-called bald spot problem in the context of generalized Gaussian wave packet dynamics. The analysis shows that the bald spot phenomenon is essentially due to the complex nature of the initial conditions for the trajectories. It is, therefore, expected to be a general feature of several semiclassical methods that rely on trajectories with such initial conditions. A bald region is created when a trajectory, needed to calculate the wave function at a given time, reaches a singularity of the potential energy function in the complex plane at an earlier, real time. This corresponds to passage of a branch point singularity across the real axis of the complex time plane. The missing portions of the wave function can be obtained by deforming the time path for the integration of the equations of motion into the complex plane so that the singularity is circumvented. We present examples of bald spots, singularity times, and suitable complex time paths for one-dimensional barrier transmission in the Eckart and Gaussian systems. Although the bald regions for the Eckart system are often localized, they are found to be semi-infinite for the Gaussian system. For the case of deep tunneling, the bald regions for both systems may encompass the entire portion of space occupied by the transmitted wave packet. Thus, the use of complex time paths becomes essential for a treatment of barrier tunneling.

Evaluation of new spin foam vertex amplitudes

The Christensen–Egan algorithm is extended and generalized to efficiently evaluate new spin foam vertex amplitudes proposed by Engle, Pereira and Rovelli and Freidel and Krasnov, with or without (factored) boundary states. A concrete pragmatic proposal is made for comparing the different models using uniform methodologies, applicable to the behavior of large spin asymptotics and of expectation values of specific semiclassical observables. The asymptotics of the new models exhibit non-oscillatory, power-law decay similar to that of the Barrett–Crane model, though with different exponents. Also, an analysis of the semiclassical wave packet propagation problem indicates that the Magliaro, Rovelli and Perini's conjecture of good semiclassical behavior of the new models does not hold for generic factored states, which neglect spin–spin correlations.

Semiclassical Propagation of Gaussian Wave Packets

We analyze the semiclassical evolution of Gaussian wave packets in chaotic systems. We show that after some short time a Gaussian wave packet becomes a primitive WKB state. From then on, the state can be propagated using the standard time-dependent WKB scheme. Complex trajectories are not necessary to account for the long-time propagation. The Wigner function of the evolving state develops the structure of a classical filament plus quantum oscillations, with phase and amplitude being determined by geometric properties of a classical manifold.

Semi-classical propagation of wavepackets for the phase space Schrödinger equation: interpretation in terms of the Feichtinger algebra

The nearby orbit method is a powerful tool for constructing semi-classical solutions of Schrödinger's equation when the initial datum is a coherent state. In this paper, we first extend this method to arbitrary squeezed states and thereafter apply our results to the Schrödinger equation in phase space. This adaptation requires the phase-space Weyl calculus developed in previous work of ours. We also study the regularity of the semi-classical solutions from the point of view of the Feichtinger algebra familiar from the theory of modulation spaces.

Semiclassical wave packet study of anomalous isotope effect in ozone formation

We applied the semiclassical initial value representation method to calculate energies, lifetimes, and wave functions of scattering resonances in a two-dimensional potential for O+O2 collision. Such scattering states represent the metastable O3* species and play a central role in the process of ozone formation. Autocorrelation functions for scattering states were computed and then analyzed using the Prony method, which permits one to extract accurate energies and widths of the resonances. We found that the results of the semiclassical wave packet propagation agree well with fully quantum results. The focus was on the 16O16O18O isotopomer and the anomalous isotope effect associated with formation of this molecule, either through the 16O16O+18O or the 16O+16O18O channels. An interesting correlation between the local vibration mode character of the metastable states and their lifetimes was observed and explained. New insight is obtained into the mechanism by which the long-lived resonances in the delta zero-point energy part of spectrum produce the anomalously large isotope effect.

Semiclassical Dynamics of Electron Transfer at Metal Surfaces

In this Letter, we demonstrate that nonadiabatic dynamics of molecular scattering from metal surfaces can be efficiently simulated by semiclassical Gaussian wave packet propagation on a local complex potential. The method relies on the wideband limit decoupling of the nuclear equations of motion on different electronic states. If the continuum diabatic potential surfaces are assumed to be parallel, the number of Gaussian wave packets spawned scales at most linearly with propagation time, allowing efficient propagation of nuclear dynamics.

Dynamics of Nonadiabatic Chemical Reactions

New methods are proposed to treat nonadiabatic chemical dynamics in realistic large molecular systems by using the Zhu-Nakamura (ZN) theory of curve-crossing problems. They include the incorporation of the ZN formulas into the Herman-Kluk type semiclassical wave packet propagation method and the trajectory surface hopping (TSH) method, formulation of the nonadiabatic transition state theory, and its application to the electron transfer problem. Because the nonadiabatic coupling is a vector in multidimensional space, the one-dimensional ZN theory works all right. Even the classically forbidden transitions can be correctly treated by the ZN formulas. In the case of electron transfer, a new formula that can improve the celebrated Marcus theory in the case of normal regime is obtained so that it can work nicely in the intermediate and strong electronic coupling regimes. All these formulations mentioned above are demonstrated to work well in comparison with the exact quantum mechanical numerical solutions and are expected to be applicable to large systems that cannot be treated quantum mechanically numerically exactly. To take into account another quantum mechanical effect, namely, the tunneling effect, an efficient method to detect caustics from which tunneling trajectories emanate is proposed. All the works reported here are the results of recent activities carried out in the author's research group. Finally, the whole set of ZN formulas is presented in Appendix.

A regular semiclassical approximation for the propagation of wave packets with complex trajectories

The semiclassical propagation of Gaussian wave packets by complex classical trajectories involves multiple contributing and noncontributing solutions interspersed by phase space caustics. Although the phase space caustics do not generally lie exactly on the relevant trajectories, they might strongly affect the semiclassical evolution depending on their proximity to them. In this paper, we derive a third-order regular semiclassical approximation which correctly accounts for the caustics and which is finite everywhere. We test the regular formula for the potential V( x)= 1/x 2 , where the complex classical trajectories and phase space caustics can be computed analytically. We make a detailed analysis of the structure of the complex functions involved in the saddle point approximations and show how the changes in the steepest descent integration contour control both the contributing and noncontributing trajectories and the type of Airy function that appears in the regular approximation.

Semiclassical propagation of wavepackets with complex and real trajectories

We consider a semiclassical approximation for the time evolution of an originally gaussian wave packet in terms of complex trajectories. We also derive additional approximations replacing the complex trajectories by real ones. These yield three different semiclassical formulae involving different real trajectories. One of these formulae is Heller's thawed gaussian approximation. The other approximations are non-gaussian and may involve several trajectories determined by mixed initial-final conditions. These different formulae are tested for the cases of scattering by a hard wall, scattering by an attractive gaussian potential, and bound motion in a quartic oscillator. The formula with complex trajectories gives good results in all cases. The non-gaussian approximations with real trajectories work well in some cases, whereas the thawed gaussian works only in very simple situations.

Semiclassical Wave Packet Dynamics with Electronic Structure Computed on the Fly: Application to Photophysics of Electronic Excited States in Condensed Phase

Many semiclassical wave packet propagation methods require only local potential energy surface information in order to update a Gaussian wave packet over a short time interval. These data, which include the evaluation of the potential energy at the instantaneous configuration space center of the wave packet, plus the gradient vector and Hessian (second derivative) matrix at the same configuration, can be generated efficiently by extant electronic structure packages. This leads to an algorithm for propagating semiclassical Gaussian wave packets using electronic structure data computed “on the fly” in the course of the propagation. The feasibility of such a strategy for condensed phase systems is demonstrated by using it (with an appropriate approximate level of electronic structure theory) to calculate Franck−Condon absorption and emission spectra of all-trans 1,3,5,7-octatetraene in the gas phase, and in both chloroform and methanol solvents. Good agreement with the corresponding experimentally measured s...

Semiclassical wave-packet propagation on potential surfaces coupled by ultrashort laser pulses

We demonstrate the applicability of the time-dependent semiclassical initial-value formalism to the problem of population transfer between adiabatic potential surfaces which are coupled through a pulsed laser field. Despite the fact that solutions of highly nonlinear classical equations are entering the semiclassical expression, our results for ultrashort pulse excitation are in good agreement with full quantum solutions. Even in the case where the Rosen-Zener approach to population transfer breaks down, the semiclassical results are still close to the quantum ones.

Complex trajectory method in semiclassical propagation of wave packets

We propose a semiclassical wave packet propagation method relying on classical trajectories in a complex phase space. It is based on the Schrödinger wave equation and the usual expansion with respect to ℏ, except that the amplitude of the wave packet is taken into account at the very zeroth order, unlike in the usual WKB method where it is treated as a corrective or first order term. Formally, it amounts to making both the wavelength and the width of the wave packet tend to zero with ℏ. The action and consequently the classical trajectories derived are complex. This method is tested successfully in many cases, analytically or numerically, including the bounce and even the splitting of the wave packet. Our method appears to be much more accurate than the WKB method while less computationally demanding than the Van-Vleck formula. Moreover, it has a particularly interesting property: the singularities (caustics) of the usual semiclassical theories do not appear in this formalism in all cases tested.

Improving the accuracy of semiclassical wavepacket propagation using integral conditioning techniques

An initial value representation semiclassical wavepacket propagation procedure is tested numerically for the two dimensional Henon-Heiles system. It has been shown in previous calculations on this system that the accuracy of semiclassical wavepacket propagation significantly degrades with time, due to the presence of chaotic trajectories. We find that the accuracy of the semiclassical propagation is greatly improved by the use of the integral conditioning technique of Filinov for preaveraging the phase space integral over frozen Gaussians in the semiclassical method.

The Semi-Classical Trace Formula and Propagation of Wave Packets

We study spectral and propagation properties of operators of the form Sh = ∑Nj=0hjPj where ∀ j P j is a differential operator of order j on a manifold M, asymptotically as h → 0. The estimates are in terms of the flow {φt} of the classical Hamiltonian H(x, p) = ∑Nj=0 σPj(x, p) on T*M, where σPj, is the principal symbol of P j. We present two sets of results. (I) The "semiclassical trace formula", on the asymptotic behavior of eigenvalues and eigenfunctions of Sh in terms of periodic trajectors of H. (II) Associated to certain isotropic submanifolds Λ ⊂ T*M we define families of functions {ψh} and prove that ∀t {exp(− ithSħ)(ψħ)} is a family of the same kind associated to φt(Λ).

Uniform semiclassical wave-packet propagation and eigenstate extraction in a smooth chaotic system.

A uniform semiclassical propagator is used to time evolve a wave packet in a smooth Hamiltonian system at energies for which the underlying classical motion is chaotic. The propagated wave packet is Fourier transformed to yield a scarred eigenstate.