Semiclassical Coherent State Propagator 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.

The semiclassical coherent state propagator in the Weyl representation

It is shown that the semiclassical coherent state propagator takes its simplest form when the quantum mechanical Hamiltonian is replaced by its Weyl symbol in defining the classical action, in that there is then no need for a Solari-Kochetov correction. It is also shown that such a correction exists if a symbol other than the Weyl symbol is chosen and that its form is different depending on the symbol chosen. The various forms of the propagator based on different symbols are shown to be equivalent provided the correspondingly correct Solari-Kochetov correction is included. All these results are shown for both particle and spin coherent state propagators. The global anomaly in the fluctuation determinant is further elucidated by a study of the connection between the discrete fluctuation determinant and the discrete Jacobi equation.

Communication: Overcoming the root search problem in complex quantum trajectory calculations

Three new developments are presented regarding the semiclassical coherent state propagator. First, we present a conceptually different derivation of Huber and Heller's method for identifying complex root trajectories and their equations of motion [D. Huber and E. J. Heller, J. Chem. Phys. 87, 5302 (1987)]. Our method proceeds directly from the time-dependent Schrödinger equation and therefore allows various generalizations of the formalism. Second, we obtain an analytic expression for the semiclassical coherent state propagator. We show that the prefactor can be expressed in a form that requires solving significantly fewer equations of motion than in alternative expressions. Third, the semiclassical coherent state propagator is used to formulate a final value representation of the time-dependent wavefunction that avoids the root search, eliminates problems with caustics and automatically includes interference. We present numerical results for the 1D Morse oscillator showing that the method may become an attractive alternative to existing semiclassical approaches.

Complex invariants in two-dimension for coupled oscillator systems

In order to extract some insight into the features of a dynamical system, we present here the possibility of its complex dynamical invariant. There are many systems which admit complex invariants. To achieve this we use Lie algebraic method to study two dimensional complex systems (coupled oscillator system) on the extended complex phase plane characterized by \(x=x_{1}+ip_{3}, y=x_{2} + ip_{4} , p_x=p_{1} + ix_{3}, p_y=p_{2}+ix_{4}\). Such invariants play an important role in the analysis of complex trajectories with regard to the calculation of semi-classical coherent state propagator in the path integral method.

Semiclassical Propagation of Coherent States for the Hartree Equation

In this paper we consider the nonlinear Hartree equation in presence of a given external potential, for an initial coherent state. Under suitable smoothness assumptions, we approximate the solution in terms of a time dependent coherent state, whose phase and amplitude can be determined by a classical flow. The error can be estimated in $L^2$ by $C \sqrt {\var}$, $\var$ being the Planck constant. Finally we present a full formal asymptotic expansion.

An initial value representation for the coherent state propagator with complex trajectories

We present an initial value representation for the semiclassical coherent state propagator based on complex trajectories. We map the complex phase space into a real one with twice as many dimensions and introduce initial valued trajectories in this double phase space. We use a procedure to eliminate non-contributing trajectories that allows for the automation of the entire calculation, rendering it simple. The resulting semiclassical formulas do not show divergences due to caustics and provide accurate results in short computational times.

Evaluation of the semiclassical coherent state propagator in the presence of phase space caustics

A uniform approximation for the coherent state propagator, valid in the vicinity of phase space caustics, was recently obtained using the Maslov method combined with a dual representation for coherent states. In this paper we review the derivation of this formula and apply it to two model systems: the one-dimensional quartic oscillator and a two-dimensional chaotic system.

Semiclassical coherent-state propagator for many spins

We obtain the semiclassical coherent-state propagator for a many-spin system with an arbitrary Hamiltonian.

Controlling phase space caustics in the semiclassical coherent state propagator

The semiclassical formula for the quantum propagator in the coherent state representation 〈 z ″ | e - i H ˆ T / ℏ | z ′ 〉 is not free from the problem of caustics. These are singular points along the complex classical trajectories specified by z′, z″ and T where the usual quadratic approximation fails, leading to divergences in the semiclassical formula. In this paper, we derive third order approximations for this propagator that remain finite in the vicinity of caustics. We use Maslov’s method and the dual representation proposed in Phys. Rev. Lett. 95, 050405 (2005) to derive uniform, regular and transitional semiclassical approximations for coherent state propagator in systems with two degrees of freedom.

Quantum mechanics on a circle: Husimi phase-space distributions and semiclassical coherent state propagators

We discuss some basic tools for an analysis of one-dimensional quantum systems defined on a cyclic coordinate space. The basic features of the generalized coherent states, the complexifier coherent states, are reviewed. These states are then used to define the corresponding (quasi)densities in phase space. The properties of these generalized Husimi distributions are discussed, in particular their zeros. Furthermore, the use of the complexifier coherent states for a semiclassical analysis is demonstrated by deriving a semiclassical coherent state propagator in phase space.

Semiclassical coherent-state propagator for many particles

We obtain the semiclassical coherent-state propagator for a many-particle system with an arbitrary Hamiltonian.

The semiclassical coherent state propagator for systems with spin

We derive the semiclassical limit of the coherent state propagator for systems with two degrees of freedom of which one degree of freedom is canonical and the other a spin. Systems in this category include those involving spin-orbit interactions and the Jaynes-Cummings model in which a single electromagnetic mode interacts with many independent two-level atoms. We construct a path integral representation for the propagator of such systems and derive its semiclassical limit. As special cases we consider separable systems, the limit of very large spins and the case of spin 1/2.

Erratum: Semiclassical coherent-state propagator via path integrals with intermediate states of variable width [Phys. Rev. A68, 062112 (2003)

Erratum: Semiclassical coherent-state propagator via path integrals with intermediate states of variable width [Phys. Rev. A<b>68</b>, 062112 (2003)

Semiclassical propagation of coherent states using complex and real trajectories

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.

Semiclassical Propagation of Coherent States with Spin-Orbit Interaction

We study semiclassical approximations to the time evolution of coherent states for general spin-orbit coupling problems in two different semiclassical scenarios: The limit \hbar to zero is first taken with fixed spin quantum number s and then with \hbar*s held constant. In these two cases different classical spin-orbit dynamics emerge. We prove that a coherent state propagated with a suitable classical dynamics approximates the quantum time evolution up to an error of size \sqrt{\hbar} and identify an Ehrenfest time scale. Subsequently an improvement of the semiclassical error to an arbitray order \hbar^{N/2} is achieved by a suitable deformation of the state that is propagated classically.

Semiclassical representations of electronic structure and dynamics.

We use a new formulation of the semiclassical coherent state propagator to derive and evaluate several different approximate representations of electron dynamics. For each representation we examine: (1) its ability to treat quantum effects and electron correlation, (2) its expected scaling with system size, and (3) the types of systems for which it can be used. We also apply two of the methods to a pair of model problems, namely the minimal basis electron dynamics in H2 and the magnetization dynamics in a cluster model of the Kagome lattice, in order to verify the feasibility of these approaches for realistic systems. Based on all these criteria, we find that the representation that takes the electron spins as the classical variables is particularly promising for the quantitative and qualitative description of large systems.

Semiclassical coherent-state propagator via path integrals with intermediate states of variable width

We derive a semiclassical approximation for the coherent state propagator $〈{z}^{\ensuremath{''}}|{e}^{\ensuremath{-}iHt/\ensuremath{\Elzxh}}|{z}^{\ensuremath{'}}〉$ using a path integral formulation in which the intermediate coherent states can have arbitrary widths. Our semiclassical formula involves complex trajectories of the smoothed Hamiltonian $\mathcal{H}(q,p,b)=〈z|\ifmmode \hat{H}\else \^{H}\fi{}|z〉$ where b, the width of the coherent state $|z〉,$ is a free function that can be chosen conveniently. The generality of this formalism enables us to derive a semiclassical approximation which contains, as particular cases, other similar approximations known in the literature, providing a natural link between them. We present numerical results showing that the semiclassical propagation can be very sensitive to the choice of b and we suggest an energy dependent value ${b=b}_{E}$ that results in considerable improvement over other choices. This value for the width will be generally different from the widths ${\ensuremath{\sigma}}^{\ensuremath{'}}$ or ${\ensuremath{\sigma}}^{\ensuremath{''}}$ of the initial and final states $|{z}^{\ensuremath{'}}〉$ and $|{z}^{\ensuremath{''}}〉.$

Similarity transformed semiclassical dynamics

In this article, we employ a recently discovered criterion for selecting important contributions to the semiclassical coherent state propagator [T. Van Voorhis and E. J. Heller, Phys. Rev. A 66, 050501 (2002)] to study the dynamics of many dimensional problems. We show that the dynamics are governed by a similarity transformed version of the standard classical Hamiltonian. In this light, our selection criterion amounts to using trajectories generated with the untransformed Hamiltonian as approximate initial conditions for the transformed boundary value problem. We apply the new selection scheme to some multidimensional Henon–Heiles problems and compare our results to those obtained with the more sophisticated Herman–Kluk approach. We find that the present technique gives near-quantitative agreement with the the standard results, but that the amount of computational effort is less than Herman–Kluk requires even when sophisticated integral smoothing techniques are employed in the latter.

Semiclassical calculation of thermal rate constants in full Cartesian space: The benchmark reaction D+H2→DH+H

Semiclassical (SC) initial-value representation (IVR) methods are used to calculate the thermal rate constant for the benchmark gas-phase reaction D+H2→DH+H. In addition to several technical improvements in the SC-IVR methodology, the most novel aspect of the present work is use of Cartesian coordinates in the full space (six degrees of freedom once the overall center-of-mass translation is removed) to carry out the calculation; i.e., we do not invoke the conservation of total angular momentum J to reduce the problem to fewer degrees of freedom and solve the problem separately for each value of J, as is customary in quantum mechanical treatments. With regard to the SC-IVR methodology, we first present a simple and straightforward derivation of the semiclassical coherent-state propagator of Herman and Kluk (HK). This is achieved by defining an interpolation operator between the Van Vleck propagators in coordinate and momentum representations in an a priori manner with the help of the modified Filinov filtering method. In light of this derivation, we examine the systematic and statistical errors of the HK propagator to fully understand the role of the coherent-state parameter γ. Second, the Boltzmannized flux operator that appears in the rate expression is generalized to a form that can be tuned continuously between the traditional half-split and Kubo forms. In particular, an intermediate form of the Boltzmannized flux operator is shown to have the desirable features of both the traditional forms; i.e., it is easy to evaluate via path integrals and at the same time it gives a numerically well-behaved flux correlation function at low temperatures. Finally, we demonstrate that the normalization integral required in evaluating the rate constant can be expressed in terms of simple constrained partition functions, which allows the use of well-established techniques of statistical mechanics.

New method for obtaining complex roots in the semiclassical coherent-state propagator formula

A semiclassical formula for the coherent-state propagator requires the determination of specific classical paths inhabiting a complex phase-space and governed by a Hamiltonian flux. Such trajectories are constrained to special boundary conditions which render their determination difficult by common methods. In this paper we present a new method based on Runge-Kutta integrator for a quick, easy and accurate determination of these trajectories. Using nonlinear one dimensional systems we show that the semiclassical formula is highly accurate as compared to its exact counterpart. Further, we clarify how the phase of the semiclassical approximation is correctly retrieved during the time evolution.