Dissipative quantum phase space dynamics on dynamically adapting grids

A moving grid approach to a dynamical study of dissipative systems is described. The dynamics are studied in phase space for the Caldeira-Leggett master equation. The grid movement is based on the principle of equidistribution and, by using a grid smoothing technique, the grid points trace a path that continuously adapts to reflect the dynamics of a phase-space distribution function. The technique is robust and allows accurate computations to be obtained for long propagation times. The effects of dissipation on the dynamics are studied and results are presented for systems subject to both periodic and nonperiodic multiminimum potential functions.     

Exact dynamics of dissipative electronic systems and quantum transport: Hierarchical equations of motion approach

A generalized quantum master equation theory that governs the exact, nonperturbative quantum dissipation and quantum transport is formulated in terms of hierarchically coupled equations of motion for an arbitrary electronic system in contact with electrodes under either a stationary or a nonstationary electrochemical potential bias. The theoretical construction starts with the influence functional in path integral, in which the electron creation and annihilation operators are Grassmann variables. Time derivatives on the influence functionals are then performed in a hierarchical manner. Both the multiple-frequency dispersion and the non-Markovian reservoir parametrization schemes are considered for the desired hierarchy construction. The resulting hierarchical equations of motion formalism is in principle exact and applicable to arbitrary electronic systems, including Coulomb interactions, under the influence of arbitrary time-dependent applied bias voltage and external fields. Both the conventional quantum master equation and the real-time diagrammatic formalism of Schon and co-workers can be readily obtained at well defined limits of the present theory. We also show that for a noninteracting electron system, the present hierarchical equations of motion formalism terminates at the second tier exactly, and the Landuer-Buttiker transport current expression is recovered. The present theory renders an exact and numerically tractable tool to evaluate various transient and stationary quantum transport properties of many-electron systems, together with the involving nonperturbative dissipative dynamics.     

Short time quantum phase space dynamics at a 1:2 Fermi resonance

The short time dynamics of a wavepacket initiated on the unstable classical periodic orbit of a 1:2 Fermi resonance are investigated in cases that give rise to regular and near chaotic classical motion. In both cases the spectral decomposition reveals an imprint of the periodic orbit frequency but there is a broader additional fine structure in the irregular case. A Husimi phase space representation of the wavepacket shows that two separate time components in the autocorrelation function are associated with motion on two distinct branches of the separatrix attached to the unstable orbit. The wavepacket in the irregular case is also found to be controlled by the now broken separatrix or homoclinic tangle. The autocorrelation function again shows recurrences indicative of a residual compact component to the wavepacket.     

Coherence spectroscopy in dissipative media: a Liouville space pathway approach

We address the possibility of using coherent control tools to extract useful information about the interaction of a system with a dissipative environment. To that end we extend previous work, which developed a coherence spectroscopy based on two-pathway excitation phase control, from the isolated molecule limit to dense media. Specifically, we explore the properties of the channel phase, an observable of energy-domain two-pathway excitation experiments that was shown in the isolated molecule limit to carry information about the phase properties of the material system. Our analysis is based on the combination of steady state and time-dependent analytical perturbative approaches within the density matrix formalism, complemented by nonperturbative numerical simulations. We find that the channel phase carries significantly richer information in the presence of decoherence mechanisms than in their absence. In particular, rescattering events in the structured continuum introduce new features in the channel phase spectrum, whose structure conveys information about both the molecular continuum and the system bath interaction.     

Communication: phase space approach to laser-driven electronic wavepacket propagation

We propose a phase space method to propagate a quantum wavepacket driven by a strong external field. The method employs the periodic von Neumann basis with biorthogonal exchange recently introduced for the calculation of the energy eigenstates of time-independent quantum systems [A. Shimshovitz and D. J. Tannor, Phys. Rev. Lett. (in press) [e-print arXiv:1201.2299v1]]. While the individual elements in this basis set are time-independent, a small subset is chosen in a time-dependent manner to adapt to the evolution of the wavepacket in phase space. We demonstrate the accuracy and efficiency of the present propagation method by calculating the electronic wavepacket in a one-dimensional soft-core atom interacting with a superposition of an intense, few-cycle, near-infrared laser pulse and an attosecond extreme-ultraviolet laser pulse.     

Communication: Partial linearized density matrix dynamics for dissipative, non-adiabatic quantum evolution

An approach for treating dissipative, non-adiabatic quantum dynamics in general model systems at finite temperature based on linearizing the density matrix evolution in the forward-backward path difference for the environment degrees of freedom is presented. We demonstrate that the approach can capture both short time coherent quantum dynamics and long time thermal equilibration in an application to excitation energy transfer in a model photosynthetic light harvesting complex. Results are also presented for some nonadiabatic scattering models which indicate that, even though the method is based on a "mean trajectory" like scheme, it can accurately capture electronic population branching through multiple avoided crossing regions and that the approach offers a robust and reliable way to treat quantum dynamical phenomena in a wide range of condensed phase applications.     

A reduced phase space approach to collision processes

An exact master equation for the occupation of different states during a collision is derived. For a wide dass of initial states including those typical of scattering experiments, the transition probability matrix is independent of the initial state. The stationary solution of the master equation is the “prior” distribution.     

Quantum trajectories in complex phase space: multidimensional barrier transmission

The quantum Hamilton-Jacobi equation for the action function is approximately solved by propagating individual Lagrangian quantum trajectories in complex-valued phase space. Equations of motion for these trajectories are derived through use of the derivative propagation method (DPM), which leads to a hierarchy of coupled differential equations for the action function and its spatial derivatives along each trajectory. In this study, complex-valued classical trajectories (second order DPM), along which is transported quantum phase information, are used to study low energy barrier transmission for a model two-dimensional system involving either an Eckart or Gaussian barrier along the reaction coordinate coupled to a harmonic oscillator. The arrival time for trajectories to reach the transmitted (product) region is studied. Trajectories launched from an "equal arrival time surface," defined as an isochrone, all reach the real-valued subspace in the transmitted region at the same time. The Rutherford-type diffraction of trajectories around poles in the complex extended Eckart potential energy surface is described. For thin barriers, these poles are close to the real axis and present problems for computing the transmitted density. In contrast, for the Gaussian barrier or the thick Eckart barrier where the poles are further from the real axis, smooth transmitted densities are obtained. Results obtained using higher-order quantum trajectories (third order DPM) are described for both thick and thin barriers, and some issues that arise for thin barriers are examined.     

Volterra inverse scattering series method for one‐dimensional quantum barrier scattering

The Volterra inverse scattering series method is developed to obtain the interaction potential for one‐dimensional quantum barrier scattering problems. The Lippmann–Schwinger equation describing quantum barrier scattering is renormalized from a Fredholm to a Volterra integral equation. Employing the Born–Neumann series solution of the Lippmann–Schwinger Volterra equation and a related expansion of the interaction potential in orders of the data, we derive the Volterra inverse scattering series for the reflection and transmission amplitudes. Each term of the interaction potential is computed using the scattering amplitude and the Volterra Green's function. We do not consider the separate issue of extracting scattering amplitudes from quantum cross sections. The triangular nature of the Volterra Green's function significantly reduces computational effort. The Volterra series is then applied to several one‐dimensional quantum barrier scattering problems. Computational results show that the first few terms in the Volterra series can yield accurate interaction barriers. In addition, the potential barriers are calculated using the Born inverse scattering series based on the Lippmann–Schwinger Fredholm equation with the reflection amplitude. The comparison between the Born and Volterra results demonstrates that the Volterra inverse scattering series can provide a more accurate and more efficient method for determining the interaction potential.     

A semiclassical hybrid approach to many particle quantum dynamics

We analytically derive a correlated approach for a mixed semiclassical many particle dynamics, treating a fraction of the degrees of freedom by the multitrajectory semiclassical initial value method of Herman and Kluk [Chem. Phys. 91, 27 (1984)] while approximately treating the dynamics of the remaining degrees of freedom with fixed initial phase space variables, analogously to the thawed Gaussian wave packet dynamics of Heller [J. Chem. Phys. 62, 1544 (1975)]. A first application of this hybrid approach to the well studied Secrest-Johnson [J. Chem. Phys. 45, 4556 (1966)] model of atom-diatomic collisions is promising. Results close to the quantum ones for correlation functions as well as scattering probabilities could be gained with considerably reduced numerical effort as compared to the full semiclassical Herman-Kluk approach. Furthermore, the harmonic nature of the different degrees of freedom can be determined a posteriori by comparing results with and without the additional approximation.     

量子相空间中微扰理论

在Torres-Vega和Frederick所提出来的量子相空间理论(简作TF量子相空间理论)的框架下,研究了量子相空间表象下的微扰理论,得出了量子相空间理论框架下的非简并微扰理论(perturbation theory),得到了体系存在微扰情况下的能量和波函数,并且以一维电解质在外加电场中的极化率为例,在量子相空间表象下,对量子体系的状态进行了分析.以期探索出将量子相空间理论和各种近似方法结合起来处理真实体系的问题.     

Analysis of barrier scattering with real and complex quantum trajectories

In this study, an analysis of the one-dimensional Eckart and Gaussian barrier scattering problems is undertaken using approximate quantum trajectories. Individual quantum trajectories are computed using the derivative propagation method (DPM). Both real-valued and complex-valued DPM quantum trajectories are employed. Of interest are the deep tunneling and the higher energy barrier scattering problems in cases in which the scattering barrier is "thick" by comparison to the width of the initial wave packet. For higher energy scattering problems, it is found that real-valued DPM trajectories very accurately reproduce the transmitted probability densities at low orders when compared to large fixed-grid calculations. However, higher orders must be introduced to obtain good probabilities for deep tunneling problems. Complex-valued DPM is found to accurately reproduce transmitted probability densities at low order for both the deep tunneling and the higher energy scattering problems. Of particular note, complex-classical trajectories are found to very nearly give the exact result for the deep barrier tunneling scattering problem, and the complex DPM converges well at high orders for these thick barrier scattering problems. A variety of analyses are performed to elucidate the dynamics of complex-valued DPM trajectories. The complex-extended barrier potentials are examined in detail, including an analysis of the complex force. Of particular interest are initial conditions for complex-valued DPM trajectories known as isochrones. All trajectories launched from an isochrone arrive on the real axis on the transmitted side of the barrier at the same time. The computation and properties of isochrones as well as the behavior of the initial wave packet in the complex plane are also examined.     

Complex trajectories sans isochrones: quantum barrier scattering with rectilinear constant velocity trajectories

One of the major obstacles in employing complex-valued trajectory methods for quantum barrier scattering calculations is the search for isochrones. In this study, complex-valued derivative propagation method trajectories in the arbitrary Lagrangian-Eulerian frame are employed to solve the complex Hamilton-Jacobi equation for quantum barrier scattering problems employing constant velocity trajectories moving along rectilinear paths whose initial points can be in the complex plane or even along the real axis. It is shown that this effectively removes the need for isochrones for barrier transmission problems. Model problems tested include the Eckart, Gaussian, and metastable quadratic+cubic potentials over a variety of wave packet energies. For comparison, the "exact" solution is computed from the time-dependent Schrodinger equation via pseudospectral methods.     

An approach for generating trajectory-based dynamics which conserves the canonical distribution in the phase space formulation of quantum mechanics. I. Theories

We have reformulated and generalized our recent work [J. Liu and W. H. Miller, J. Chem. Phys. 126, 234110 (2007)] into an approach for generating a family of trajectory-based dynamics methods in the phase space formulation of quantum mechanics. The approach (equilibrium Liouville dynamics) is in the spirit of Liouville's theorem in classical mechanics. The trajectory-based dynamics is able to conserve the quantum canonical distribution for the thermal equilibrium system and approaches classical dynamics in the classical (? → 0), high temperature (β → 0), and harmonic limits. Equilibrium Liouville dynamics provides the framework for the development of novel theoretical∕computational tools for studying quantum dynamical effects in large∕complex molecular systems.     

Nonlinear optical approach to multiexciton relaxation dynamics in quantum dots

Unlike the majority of molecular systems quantum dots can accommodate multiple excitations, which is a particularly important attribute for potential lasing applications. We demonstrate in this work the concept of using nth order nonlinear spectroscopies in the transient grating configuration as a means of selectively exciting (n-1)/2 excitons in a semiconductor and probing the subsequent relaxation dynamics. We report a direct observation of multiparticle dynamics on ultrashort time scales through comparison of third and fifth order experiments for CdSe colloidal quantum dots. Time constants associated with multiexciton recombination and depopulation dynamics are reported. Deviation from a Poisson model for the distribution of photoexcited excitons, biexcitons, and triexcitons is also discussed.     

Hybrid quantum/classical molecular dynamics for a proton transfer reaction coupled to a dissipative bath

A hybrid quantum/classical molecular dynamics approach is applied to a proton transfer reaction represented by a symmetric double well system coupled to a dissipative bath. In this approach, the proton is treated quantum mechanically and all bath modes are treated classically. The transition state theory rate constant is obtained from the potential of mean force, which is generated along a collective reaction coordinate with umbrella sampling techniques. The transmission coefficient, which accounts for dynamical recrossings of the dividing surface, is calculated with a reactive flux approach combined with the molecular dynamics with quantum transitions surface hopping method. The hybrid quantum/classical results agree well with numerically exact results in the spatial-diffusion-controlled regime, which is most relevant for proton transfer in proteins. This hybrid quantum/classical approach has already been shown to be computationally practical for studying proton transfer in large biological systems. These results have important implications for future applications to hydrogen transfer reactions in solution and proteins.     

Multidimensional reactive scattering with quantum trajectories: dynamics with Morse vibrational modes

The reactive scattering of a wave packet is studied by the quantum trajectory method for a model system with up to 25 Morse vibrational modes. The equations of motion are formulated in curvilinear reaction path coordinates with the restriction to a planar reaction path. Spatial derivatives are evaluated by the least squares method using contracted basis sets. Dynamical results, including trajectory evolution and time-dependent reaction probabilities, are presented and analyzed. For the case of one Morse vibrational mode, the results are in good agreement with those derived through direct numerical integration of the time-dependent Schrodinger equation.     

Quantum wave packet ab initio molecular dynamics: an approach to study quantum dynamics in large systems

A methodology to efficiently conduct simultaneous dynamics of electrons and nuclei is presented. The approach involves quantum wave packet dynamics using an accurate banded, sparse and Toeplitz representation for the discrete free propagator, in conjunction with ab initio molecular dynamics treatment of the electronic and classical nuclear degree of freedom. The latter may be achieved either by using atom-centered density-matrix propagation or by using Born-Oppenheimer dynamics. The two components of the methodology, namely, quantum dynamics and ab initio molecular dynamics, are harnessed together using a time-dependent self-consistent field-like coupling procedure. The quantum wave packet dynamics is made computationally robust by using adaptive grids to achieve optimized sampling. One notable feature of the approach is that important quantum dynamical effects including zero-point effects, tunneling, as well as over-barrier reflections are treated accurately. The electronic degrees of freedom are simultaneously handled at accurate levels of density functional theory, including hybrid or gradient corrected approximations. Benchmark calculations are provided for proton transfer systems and the dynamics results are compared with exact calculations to determine the accuracy of the approach.