Van Vleck Propagator Research Articles

The semiclassical propagator in Fock space: dynamical echo and many-body interference.

We present a semiclassical approach to many-body quantum propagation in terms of coherent sums over quantum amplitudes associated with the solutions of corresponding classical nonlinear wave equations. This approach adequately describes interference effects in the many-body space of interacting bosonic systems. The main quantity of interest, the transition amplitude between Fock states when the dynamics is driven by both single-particle contributions and many-body interactions of similar magnitude, is non-perturbatively constructed in the spirit of Gutzwiller's derivation of the van Vleck propagator from the path integral representation of the time evolution operator, but lifted to the space of symmetrized many-body states. Effects beyond mean-field, here representing the classical limit of the theory, are semiclassically described by means of interfering amplitudes where the action and stability of the classical solutions enter. In this way, a genuinely many-body echo phenomenon, coherent backscattering in Fock space, is presented arising due to coherent quantum interference between classical solutions related by time reversal.

Mixed semiclassical-classical propagators for the Wigner phase space representation.

We formulate mixed semiclassical-classical (SC-Cl) propagators by adding a further approximation to the phase-space SC propagators, which have been formulated in our previous paper [S. Koda, J. Chem. Phys. 143, 244110 (2015)]. We first show that the stationary phase approximation over the operation of the phase-space van Vleck propagator on initial distribution functions results in the classical mechanical time propagation. Then, after dividing the degrees of freedom (DOFs) of the total system into the semiclassical DOFs and the classical DOFs, the SC-Cl van Vleck propagator and the SC-Cl Herman-Kluk (HK) propagator are derived by performing the stationary phase approximation only with respect to the classical DOFs. These SC-Cl propagators are naturally decomposed to products of the phase-space SC propagators and the classical mechanical propagators when the system does not have any interaction between the semiclassical and the classical DOFs. In addition, we also numerically compare the original phase-space HK (full HK) propagator and the SC-Cl HK propagator in terms of accuracy and efficiency to find that the accuracy of the SC-Cl HK propagator can be comparable to that of the full HK propagator although the latter is more accurate than the former in general. On the other hand, we confirm that the convergence speed of the SC-Cl HK propagator is faster than that of the full HK propagator. The present numerical tests indicate that the SC-Cl HK propagator can be more accurate than the full HK propagator when they use a same and finite number of classical trajectories due to the balance of the accuracy and the efficiency.

Initial-value semiclassical propagators for the Wigner phase space representation: Formulation based on the interpretation of the Moyal equation as a Schrödinger equation.

We formulate various semiclassical propagators for the Wigner phase space representation from a unified point of view. As is shown in several studies, the Moyal equation, which is an equation of motion for the Wigner distribution function, can be regarded as the Schrödinger equation of an extended Hamiltonian system where its "position" and "momentum" correspond to the middle point of two points of the original phase space and the difference between them, respectively. Then we show that various phase-space semiclassical propagators can be formulated just by applying existing semiclassical propagators to the extended system. As a result, a phase space version of the Van Vleck propagator, the initial-value Van Vleck propagator, the Herman-Kluk propagator, and the thawed Gaussian approximation are obtained. In addition, we numerically compare the initial-value phase-space Van Vleck propagator, the phase-space Herman-Kluk propagator, and the classical mechanical propagation as approximation methods for the time propagation of the Wigner distribution function in terms of both accuracy and convergence speed. As a result, we find that the convergence speed of the Van Vleck propagator is far slower than others as is the case of the Hilbert space, and the Herman-Kluk propagator keeps its accuracy for a long period compared with the classical mechanical propagation while the convergence speed of the latter is faster than the former.

Semiclassical propagator to evaluate off-diagonal matrix elements of the evolution operator between quantum states.

We present a powerful semiclassical expression to evaluate off-diagonal matrix elements of the evolution operator between quantum states constructed in the neighborhood of unstable short periodic orbits, which is valid up to the Heisenberg time. The expression is much easier to evaluate than the Van Vleck propagator and consists of a sum over the set of heteroclinic orbits, where each term of the series is computed by canonical invariants. Here we introduce relevant canonical invariants of heteroclinic orbits and with them at hand, the semiclassical expression is derived. Finally, our formula is successfully verified in the hyperbola billiard.

Comment on the Breakdown of Van Vleck's Propagator and Its Implications for the Energy Domain Green's Function

Van Vleck's propagator is perhaps one of the most fundamental objects of semiclassical theory. Its Laplace transform gives the energy Green's function G(x, x′, E), which is the starting point for the derivation of Gutzwiller trace formula. The validity of Van Vleck's formula for chaotic and mixed systems is therefore of crucial importance, especially since during the derivation of the trace formula it is assumed to be valid for infinite time. In the last two years there has been numerical evidence for the time-scale validity of the semiclassical propagator when the underlying dynamics is chaotic. The time-scales were longer that previously predicted by Tabor et al. An explanation for this long time validity based on the multidimensional stationary phase approximation was recently proposed by Sepúlveda-Tomsovic-Heller. In this contribution we would like to review these recent results and comment on their implications in the energy domain and particularly in the validity of the trace formula.

Phase quantization of chaos in the semiclassical regime

Since the early stage of the study of Hamilton chaos, semiclassical quantization based on the low-order Wentzel-Kramers-Brillouin theory, the primitive semiclassical approximation to the Feynman path integrals (or the so-called Van Vleck propagator), and their variants have been suffering from difficulties such as divergence in the correlation function, nonconvergence in the trace formula, and so on. These difficulties have been hampering the progress of quantum chaos, and it is widely recognized that the essential drawback of these semiclassical theories commonly originates from the erroneous feature of the amplitude factors in their applications to classically chaotic systems. This forms a clear contrast to the success of the Einstein-Brillouin-Keller quantization condition for regular (integrable) systems. We show here that energy quantization of chaos in semiclassical regime is, in principle, possible in terms of constructive and destructive interference of phases alone, and the role of the semiclassical amplitude factor is indeed negligibly small, as long as it is not highly oscillatory. To do so, we first sketch the mechanism of semiclassical quantization of energy spectrum with the Fourier analysis of phase interference in a time correlation function, from which the amplitude factor is practically factored out due to its slowly varying nature. In this argument there is no distinction between integrability and nonintegrability of classical dynamics. Then we present numerical evidence that chaos can be indeed quantized by means of amplitude-free quasicorrelation functions and Heller's frozen Gaussian method. This is called phase quantization. Finally, we revisit the work of Yamashita and Takatsuka [Prog. Theor. Phys. Suppl. 161, 56 (2007)] who have shown explicitly that the semiclassical spectrum is quite insensitive to smooth modification (rescaling) of the amplitude factor. At the same time, we note that the phase quantization naturally breaks down when the oscillatory nature of the amplitude factor is comparable to that of the phases. Such a case generally appears when the Planck constant of a large magnitude pushes the dynamics out of the semiclassical regime.

A justification for a nonadiabatic surface hopping Herman-Kluk semiclassical initial value representation of the time evolution operator

A justification is given for the validity of a nonadiabatic surface hopping Herman-Kluk (HK) semiclassical initial value representation (SC-IVR) method. The method is based on a propagator that combines the single surface HK SC-IVR method [J. Chem. Phys. 84, 326 (1986)] and Herman's nonadiabatic semiclassical surface hopping theory [J. Chem. Phys. 103, 8081 (1995)], which was originally developed using the primitive semiclassical Van Vleck propagator. We show that the nonadiabatic HK SC-IVR propagator satisfies the time-dependent Schrodinger equation to the first order of variant Planck's over 2pi and the error is O(variant Planck's over 2pi(2)). As a required lemma, we show that the stationary phase approximation, under current assumptions, has an error term variant Planck's over 2pi(1) order higher than the leading term. Our derivation suggests some changes to the previous development, and it is shown that the numerical accuracy in applications to Tully's three model systems in low energies is improved.

Semiclassical Propagator of the Wigner Function

Propagation of the Wigner function is studied on two levels of semiclassical propagation: one based on the Van Vleck propagator, the other on phase-space path integration. Leading quantum corrections to the classical Liouville propagator take the form of a time-dependent quantum spot. Its oscillatory structure depends on whether the underlying classical flow is elliptic or hyperbolic. It can be interpreted as the result of interference of a pair of classical trajectories, indicating how quantum coherences are to be propagated semiclassically in phase space. The phase-space path-integral approach allows for a finer resolution of the quantum spot in terms of Airy functions.

Guided Gaussian Wave Packets

We discuss several semiclassical Gaussian wave packet approaches with emphasis on one that is not very well-known, that is, the off-center guiding approach. Off-center guiding of (thawed) Gaussian wave packets uses the same information as other Gaussian propagation schemes, that is, the full van Vleck determinant and multiple trajectories. It retains the well-known caustic smoothing property of Gaussians. The off-center guiding of Gaussians can handle hard chaos and other highly nonlinear situations, where the van Vleck propagator, for example, has over 30,000 separate branches.

WKB for a damped spin

The master equation for a damped spin well known from the theory of superradiance, is written as a finite-difference equation and solved by a WKB-like method. The propagator thus obtained looks like the van Vleck propagator of a certain classical Hamiltonian system with one degree of freedom. A new interpretation is provided of the temporal broadening of initially sharp probability distributions as the analogue of the spreading of the quantum mechanical wave packet.

WKB for a damped spin

The master equation for a damped spin well known from the theory of superradiance, is written as a finite-difference equation and solved by a WKB-like method. The propagator thus obtained looks like the van Vleck propagator of a certain classical Hamiltonian system with one degree of freedom. A new interpretation is provided of the temporal broadening of initially sharp probability distributions as the analogue of the spreading of the quantum mechanical wave packet.

Uniform regularized semiclassical propagator for the x-2 potential.

We apply recent methods for semiclassical time propagation involving non-Cartesian variables to the repulsive one-dimensional potential V(x)=${\mathit{x}}^{\mathrm{\ensuremath{-}}2}$,x\ensuremath{\ge}0. In order to properly treat non-Cartesian variables, a quantum regularization is first performed which leads to a Langer-type potential correction term in the Gutzwiller--Van Vleck propagator. A nonuniform semiclassical treatment of V(x)=${\mathit{x}}^{\mathrm{\ensuremath{-}}2}$ using this regularization improves earlier unregularized results, and a uniform regularized propagator is very nearly exact for all times. \textcopyright{} 1996 The American Physical Society.

Semiclassical theory of molecular nonlinear optical polarization

The calculation of linear and nonlinear optical response as well as nonadiabatic curve crossing processes depends on the time evolution of the electronic coherences (off-diagonal elements of the density matrix). Unlike their diagonal counterparts, the off-diagonal elements do not have an obvious classical limit. A semiclassical approximation for the nonlinear optical response function, which reveals the classical orbit structure underlying the electronic coherences between Born–Oppenheimer surfaces, is developed. The resulting numerical propagation, which applies to arbitrary anharmonic potentials, is based on integrating the time-dependent Schrödinger equation using the semiclassical time evolution operator, Van Vleck propagator. Using the present formalism it is possible to describe semiclassically multiphoton processes involving several Born–Oppenheimer surfaces.

Integral expressions for the semiclassical time-dependent propagator

Rather general expressions are derived which represent the semiclassical time-dependent propagator as an integral over initial conditions for classical trajectories. These allow one to propagate time-dependent wave functions without searching for special trajectories that satisfy two-time boundary conditions. In many circumstances, the integral expressions are free of singularities and provide globally valid uniform asymptotic approximations. In special cases, the expressions for the propagators are related to existing semiclassical wave function propagation techniques. More generally, the present expressions suggest a large class of other, potentially useful methods. The behavior of the integral expressions in certain limiting cases is analyzed to obtain simple formulas for the Maslov index that may be used to compute the Van Vleck propagator in a variety of representations.

Comment on the Breakdown of Van Vleck's Propagator and Its Implications for the Energy Domain Green's Function

Van Vleck's propagator is perhaps one of the most fundamental objects of semiclassical theory. Its Laplace transform gives the energy Green's function G(x, x', E), which is the starting point for the derivation of Gutzwiller trace formula. The validity of Van Vleck's formula for chaotic and mixed systems is therefore of crucial importance, especially since during the derivation of the trace formula it is assumed to be valid for infinite time. In the last two years there has been numerical evidence for the time-scale validity of the semiclas­ sical propagator when the underlying dynamics is chaotic. The time-scales were longer that previously predicted by Tabor et a!. An explanation for this long time validity based on the multidimensional stationary phase approximation was recently proposed by Sepiilveda­ Tomsovic-Heller. In this contribution we would like to review these recent results and comment on their implications in the energy domain and particularly in the validity of the trace formula.