A practical search technique for finding the complex saddle points used in wave packet and coherent state propagation is developed which works for a large class of Hamiltonian dynamical systems with many degrees of freedom. The method can be applied to problems in atomic, molecular, and optical physics and other domains. A Bose-Hubbard model is used to illustrate the application to a many-body system where discrete symmetries play an important and fascinating role. For multidimensional wave packet propagation, locating the necessary saddles involves the seemingly insurmountable difficulty of solving a boundary value problem in a high-dimensional complex space, followed by determining whether each particular saddle found actually contributes. In principle, this must be done for each propagation time considered. The method derived here identifies a real search space of minimal dimension, which leads to a complete set of contributing saddles up to intermediate times much longer than the Ehrenfest timescale for the system. The analysis also gives a powerful tool for rapidly identifying the various dynamical regimes of the system.
A practical search technique for finding the complex saddle points used in wave packet and coherent state propagation is developed which works for a large class of Hamiltonian dynamical systems with many degrees of freedom. The method can be applied to problems in atomic, molecular, and optical physics and other domains. A Bose-Hubbard model is used to illustrate the application to a many-body system where discrete symmetries play an important and fascinating role. For multidimensional wave packet propagation, locating the necessary saddles involves the seemingly insurmountable difficulty of solving a boundary value problem in a high-dimensional complex space, followed by determining whether each particular saddle found actually contributes. In principle, this must be done for each propagation time considered. The method derived here identifies a real search space of minimal dimension, which leads to a complete set of contributing saddles up to intermediate times much longer than the Ehrenfest timescale for the system. The analysis also gives a powerful tool for rapidly identifying the various dynamical regimes of the system.
In a previous publication [J. Chem. Phys. 118, 9911 (2003)], the derivative propagation method (DPM) was introduced as a novel numerical scheme for solving the quantum hydrodynamic equations of motion (QHEM) and computing the time evolution of quantum mechanical wave packets. These equations are a set of coupled, nonlinear partial differential equations governing the time evolution of the real-valued functions C and S in the complex action, S=C(r,t) + iS(r,t)/Planck's over 2pi, where Psi(r,t)=exp(S). Past numerical solutions to the QHEM were obtained via ensemble trajectory propagation, where the required first- and second-order spatial derivatives were evaluated using fitting techniques such as moving least squares. In the DPM, however, equations of motion are developed for the derivatives themselves, and a truncated set of these are integrated along quantum trajectories concurrently with the original QHEM equations for C and S. Using the DPM quantum effects can be included at various orders of approximation; no spatial fitting is involved; there is no basis set expansion; and single, uncoupled quantum trajectories can be propagated (in parallel) rather than in correlated ensembles. In this study, the DPM is extended from previous one-dimensional (1D) results to calculate transmission probabilities for 2D and 3D wave packet evolution on coupled Eckart barrier/harmonic oscillator surfaces. In the 2D problem, the DPM results are compared to standard numerical integration of the time-dependent Schrodinger equation. Also in this study, the practicality of implementing the DPM for systems with many more degrees of freedom is discussed.
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.
A new method is proposed for computing the time evolution of quantum mechanical wave packets. Equations of motion for the real-valued functions C and S in the complex action S=C(r,t)+iS(r,t)/ℏ, with ψ(r,t)=exp(S), involve gradients and curvatures of C and S. In previous implementations of the hydrodynamic formulation, various time-consuming fitting techniques of limited accuracy were used to evaluate these derivatives around each fluid element in an evolving ensemble. In this study, equations of motion are developed for the spatial derivatives themselves and a small set of these are integrated along quantum trajectories concurrently with the equations for C and S. Significantly, quantum effects can be included at various orders of approximation, no spatial fitting is involved, there are no basis set expansions, and single quantum trajectories (rather than correlated ensembles) may be propagated, one at a time. Excellent results are obtained when the derivative propagation method is applied to anharmonic potentials involving barrier transmission.
A semiclassical approximation to the time evolution of coherent states may be derived from a saddle point approximation to the exact quantum propagator, and in general it involves complex classical dynamics. We generalize previous one-dimensional results to d dimensions, and for the case d=2 we present several applications. We also consider other simple approximations that depend only on real classical trajectories, but are not initial value representations. These approximations are able to reproduce interference and tunneling effects and involve propagating a few classical initial conditions compatible with the quantum uncertainties.
In two previous studies, the time-dependent scattering of a wave packet from a Gaussian barrier was investigated computationally in the complex z-plane. One of these involved the 'direct' propagation of the wave packet in the complex space, and the other used numerical analytic continuation techniques to generate the dynamics in the complex plane from the wave function computed on the real-axis. In the current study, the dynamics of exact quantum trajectories are analyzed for the same barrier scattering problem. Thousands of quantum trajectories were launched from positions near the center of the initial wave packet. These trajectories were computed by integrating equations-of-motion involving the quantum momentum function, which was obtained from the time-dependent wave function and its derivative. In order to analyze the dynamics, many trajectories were plotted on space-time diagrams. Particular emphasis was placed upon trajectories undergoing reflection in the barrier region. Some groups of strongly correlated trajectories form long-lived highly organized patterns, including helical wrappings around a series of stagnation filaments. These curves alternate with quasi-nodes where the amplitude of the wave function reaches low values. In addition, other trajectories for short times follow hyperbolic paths as they propagate near vorticity tubes surrounding these quasi-nodes.
The quantum trajectory method was recently developed to solve the hydrodynamic equations of motion in the Lagrangian, moving-with-the-fluid, picture. In this approach, trajectories are integrated for fluid elements (“particles”) moving under the influence of the combined force from the potential surface and the quantum potential. To accurately compute the quantum potential and the quantum force, it is necessary to obtain the derivatives of a function given only the values on the unstructured mesh defined by the particle locations. However, in some regions of space–time, the particle mesh shows compression and inflation associated with regions of large and small density, respectively. Inflation is especially severe near nodes in the wave function. In order to circumvent problems associated with highly nonuniform grids defined by the particle locations, adaptation of moving grids is introduced in this study. By changing the representation of the wave function in these local regions (which can be identified by diagnostic tools), propagation is possible to much longer times. These grid adaptation techniques are applied to the reflected portion of a wave packet scattering from an Eckart potential.