Quantum evolution represented by Brownian motion.

We propose a stochastic Schrödinger equation in which the momentum is coupled to a white Gaussian noise. In the stochastic representation, the kinetic energy representing the self-interaction of momentum is reduced to a linear term of momentum. As such, the quantum evolution operator factorizes into two contributions due to the momentum and the potential, respectively. The exact quantum propagator thereby becomes an expectation of the stochastic one in which the amplitude results from the potential with a fluctuating position: The particle moves with a constant velocity, subjected to a complex Brownian motion. We demonstrate that the stochastic Schrödinger equation can be feasibly used to derive the quantum propagators for the linear potential and the harmonic oscillator system. Novel semiclassical and other approximations may be developed from the new representation of quantum mechanics.

A semiclassical initial-value representation for quantum propagator and boltzmann operator.

Starting from the position-momentum integral representation, we apply the correction operator method to the derivation of a uniform semiclassical approximation for the quantum propagator and then extend it to approximate the Boltzmann operator. In this approach, the involved classical dynamics is determined by the method itself instead of given beforehand. For the approximate Boltzmann operator, the corresponding classical dynamics is governed by a complex Hamiltonian, which can be described as a pair of real Hamiltonian systems. It is demonstrated that the semiclassical Boltzmann operator is exact for linear systems. A quantum propagator in the complex time is thus proposed and preliminary numerical results show that it is a reasonable approximation for calculating thermal correlation functions of general systems. © 2018 Wiley Periodicals, Inc.

Quantum Phase Transition in the Spin-Boson Model: A Multilayer Multiconfiguration Time-Dependent Hartree Study.

The multilayer improved relaxation is applied to study the delocalization-localization transition in the spin-boson model at zero temperature-a well-known example of quantum phase transition. Calculations of energy eigenstates are obtained by iteratively diagonalizing the matrix of the Boltzmann operator in the top layer representation, using a Lanczos/Arnoldi method while relaxing the single particle functions of all layers using the multilayer multiconfiguration time-dependent Hartree imaginary time propagation. Two properties are used to examine the quantum phase transition: the energy splitting for the lowest pair of eigenstates and the magnetic susceptibility. Consistent findings are obtained with appropriate scaling parameters.

A unified view of hierarchy approach and formula of differentiation.

The stochastic differential equation is a powerful tool for describing the dynamics of a dissipative system in which noise characterizes the influence of the environment. For the Ornstein-Uhlenbeck noise, both the formula of differentiation and the hierarchy approach provide efficient numerical simulations, with the stochastic differential equation transformed into a set of coupled, linear ordinary differential equations. We show that while these two deterministic schemes result in different sets of equations, they can be regarded as two representations of an underlying linear-dynamics. Moreover, by manipulating the involved Ornstein-Uhlenbeck noise, we propose a unified algorithm that may reduce to the hierarchy approach or the formula of differentiation in different limits. We further analyze the numerical performance of this algorithm and find that the hierarchy approach appears to be more efficient for our numerical model studies.

Stationary state distribution and efficiency analysis of the Langevin equation via real or virtual dynamics.

Langevin dynamics has become a popular tool to simulate the Boltzmann equilibrium distribution. When the repartition of the Langevin equation involves the exact realization of the Ornstein-Uhlenbeck noise, in addition to the conventional density evolution, there exists another type of discrete evolution that may not correspond to a continuous, real dynamical counterpart. This virtual dynamics case is also able to produce the desired stationary distribution. Different types of repartition lead to different numerical schemes, of which the accuracy and efficiency are investigated through studying the harmonic oscillator potential, an analytical solvable model. By analyzing the asymptotic distribution and characteristic correlation time that are derived by either directly solving the discrete equations of motion or using the related phase space propagators, it is shown that the optimal friction coefficient resulting in the minimum characteristic correlation time depends on the time interval chosen in the numerical implementation. When the recommended "middle" scheme is employed, both analytical and numerical results demonstrate that, for good numerical performance in efficiency as well as accuracy, one may choose a friction coefficient in a wide range from around the optimal value to the high friction limit.

Theoretical Modeling of Plasmon-Enhanced Raman Images of a Single Molecule with Subnanometer Resolution.

Under local plasmonic excitation, Raman images of single molecules can now surprisingly reach subnanometer resolution. However, its physical origin has not been fully understood. Here we report a quantum-mechanical description of the interaction between a molecule and a highly confined plasmonic field. We show that when the spatial distribution of the plasmonic field is comparable to the size of the molecule, the optical transition matrix of the molecule becomes dependent on the position and distribution of the plasmonic field, resulting in a spatially resolved high-resolution Raman image of the molecule. The resonant Raman image reflects the electronic transition density of the molecule. In combination with first-principles calculations, the simulated Raman image of a porphyrin derivative adsorbed on a silver surface nicely reproduces its experimental counterpart. The present theory provides the basic framework for describing linear and nonlinear responses of molecules under highly confined plasmonic fields.

Quantum Markovian master equation for scattering from surfaces.

We propose a semi-phenomenological Markovian Master equation for describing the quantum dynamics of atom-surface scattering. It embodies the Lindblad-like structure and can describe both damping and pumping of energy between the system and the bath. It preserves positivity and correctly accounts for the vanishing of the interaction of the particle with the surface when the particle is distant from the surface. As a numerical test, we apply it to a model of an Ar atom scattered from a LiF surface, allowing for interaction only in the vertical direction. At low temperatures, we find that the quantum mechanical average energy loss is smaller than the classical energy loss. The numerical results obtained from the space dependent friction master equation are compared with numerical simulations for a discretized bath, using the multi-configurational time dependent Hartree methodology. The agreement between the two simulations is quantitative.

Dynamics of a two-level system coupled to a bath of spins.

The dynamics of a two-level system coupled to a spin bath is investigated via the numerically exact multilayer multiconfiguration time-dependent Hartree (ML-MCTDH) theory. Consistent with the previous work on linear response approximation [N. Makri, J. Phys. Chem. B 103, 2823 (1999)], it is demonstrated numerically that this spin-spin-bath model can be mapped onto the well-known spin-boson model if the system-bath coupling strength obeys an appropriate scaling behavior. This linear response mapping, however, may require many bath spin degrees of freedom to represent the practical continuum limit. To clarify the discrepancies resulted from different approximate treatments of this model, the population dynamics of the central two-level system has been investigated near the transition boundary between the coherent and incoherent motions via the ML-MCTDH method. It is found that increasing temperature favors quantum coherence in the nonadiabatic limit of this model, which corroborates the prediction in the previous work [J. Shao and P. Hanggi, Phys. Rev. Lett. 81, 5710 (1998)] based on the non-interacting blip approximation (NIBA). However, the coherent-incoherent boundary obtained by the exact ML-MCTDH simulation is slightly different from the approximate NIBA results. Quantum dynamics in other physical regimes are also discussed.

Frozen Gaussian series representation of the imaginary time propagator theory and numerical tests.

Thawed Gaussian wavepackets have been used in recent years to compute approximations to the thermal density matrix. From a numerical point of view, it is cheaper to employ frozen Gaussian wavepackets. In this paper, we provide the formalism for the computation of thermal densities using frozen Gaussian wavepackets. We show that the exact density may be given in terms of a series, in which the zeroth order term is the frozen Gaussian. A numerical test of the methodology is presented for deep tunneling in the quartic double well potential. In all cases, the series is observed to converge. The convergence of the diagonal density matrix element is much faster than that of the antidiagonal one, suggesting that the methodology should be especially useful for the computation of partition functions. As a by product of this study, we find that the density matrix in configuration space can have more than two saddle points at low temperatures. This has implications for the use of the quantum instanton theory.

Effects of initial correlations on the dynamics of dissipative systems.

The time correlation functions for a Gaussian wave-packet preparation of the dissipative harmonic oscillator evolving from three initial conditions for the heat bath are calculated and compared with each other for Ohmic heat baths. The three initial distributions for the bath are the factorized, partially factorized, and unfactorized distributions. Explicit analytical formulas are derived and then used to study the effect of the three initial distributions on the subsequent dynamics. We find that the transient behavior does not depend sensitively on the initial condition as long as the initial Gaussian wave function of the system is centered at the equilibrium point. Differences become noticeable as the center of the wave packet is significantly shifted from the equilibrium point. These observations justify to some extent the prevalent use of factorized initial conditions for studying real time quantum dynamics in dissipative systems. The total energy in the system is also calculated for the three initial states and its relation to features in the decay is pointed out.

Derivation of exact master equation with stochastic description: dissipative harmonic oscillator.

A systematic procedure for deriving the master equation of a dissipative system is reported in the framework of stochastic description. For the Caldeira-Leggett model of the harmonic-oscillator bath, a detailed and elementary derivation of the bath-induced stochastic field is presented. The dynamics of the system is thereby fully described by a stochastic differential equation, and the desired master equation would be acquired with statistical averaging. It is shown that the existence of a closed-form master equation depends on the specificity of the system as well as the feature of the dissipation characterized by the spectral density function. For a dissipative harmonic oscillator it is observed that the correlation between the stochastic field due to the bath and the system can be decoupled, and the master equation naturally results. Such an equation possesses the Lindblad form in which time-dependent coefficients are determined by a set of integral equations. It is proved that the obtained master equation is equivalent to the well-known Hu-Paz-Zhang equation based on the path-integral technique. The procedure is also used to obtain the master equation of a dissipative harmonic oscillator in time-dependent fields.

Solving the spin-boson model of strong dissipation with flexible random-deterministic scheme.

The zero-temperature dynamics of the spin-boson model with strong dissipation has been a challenging problem for more than 20 years. To solve this and quantum dynamics of dissipative systems at large, we recently proposed a mixed random-deterministic method. This scheme has been successfully used to simulate the time evolution of the spin-boson model at zero temperature for weak to moderate dissipation. For a better numerical performance, the approach is further modified so that it is flexible to convert a certain part of the random treatment to a deterministic one a la hierarchical equations. Applying the new method to the strong dissipated spin-boson model at zero temperature, we observe that the population in the localized state obeys a simple decay dynamics and the time scale is proportional to the reciprocal of the cutoff frequency.

Stochastic fluctuations and chiral symmetry breaking: exact solution of Lente model.

The stochastic description for the autocatalytic process has been proposed by Lente (J. Phys. Chem. A 2004, 108, 9475) to demonstrate chiral symmetry breaking. He assumed that the number of reacting molecules is macroscopic and that no products are present initially. The Lente model consisting of a finite number of molecules that may include the product molecules as chiral seeds is explored and the characteristics of stochastic distributions of the product are examined. It is shown that the presence of racemic product in the substrate reduces the possibility of chiral symmetry breaking while a few more molecules of a specific enantiomer added can yield chiral dominance for strong autocatalysis. Besides, small reactive volumes or dense reactant concentrations have a preference for chiral symmetric breaking.

Toward quantitative prediction of molecular fluorescence quantum efficiency: role of duschinsky rotation.

It is a highly desirable but difficult task to predict the molecular fluorescence quantum efficiency from first principles. The molecule in the excited state can undergo spontaneous radiation, conversion of electronic energy to nuclear motion, or chemical reaction. For relatively large molecules, it is impossible to obtain the full potential energy surfaces for the ground state and the excited states to study the excited-state dynamics. We show that, under harmonic approximation by considering the Duschinsky rotation effect, the molecular fluorescence properties can be quantitatively calculated from first principles coupled with our correlation function formalism for the internal conversion. In particular, we have explained the peculiar fluorescence behaviors of two isomeric compounds, cis,cis-1,2,3,4-tetraphenyl-1,3-butadiene and 1,1,4,4-tetraphenyl-butadiene, the former being nonemissive in solution and strongly emissive in aggregation or at low temperature, and the latter being strongly emissive in solution. The roles of low-frequency phenyl ring twist motions and their Duschinsky mode mixings are found to be crucial, especially to reveal the temperature dependence. As an independent check, we take a look at the well-established photophysics of 1,4-diphenylbutadiene for its three different conformers. Both the calculated radiative and nonradiative rates are in excellent agreement with the available experimental measurements.

Excited state radiationless decay process with Duschinsky rotation effect: formalism and implementation.

Duschinsky rotation effect is a simple and effective way to characterize the difference between the ground state and excited state potential energy surfaces. For complex molecules, harmonic oscillator model is still the practical way to describe the dynamics of excited states. Based on the first-order perturbation theory a la Fermi golden rule, the authors have applied the path integral of Gaussian type for the correlation function to derive an analytic formalism to calculate the internal conversion rate process with Duschinsky rotation effect being taken into account. The validity of their formalism is verified through comparison with previous work, both analytically for the case of neglecting Duschinsky rotation and numerically for the ethylene molecules with two-mode mixing. Their expression is derived for multimode mixing.

A new time evolving Gaussian series representation of the imaginary time propagator.

Frantsuzov and Mandelshtam [J. Chem. Phys. 121, 9247 (2004)] have recently demonstrated that a time evolving Gaussian approximation (TEGA) to the imaginary time propagator exp(-betaH) is useful for numerical computations of anharmonically coupled systems with many degrees of freedom. In this paper we derive a new exact series representation for the imaginary time propagator whose leading order term is the TEGA. One can thus use the TEGA not only as an approximation but also to obtain the exact imaginary time propagator. We also show how the TEGA may be generalized to provide a family of TEGA's. Finally, we find that the equations of motion governing the evolution of the center and width of the Gaussian may be thought of as introducing a quantum friction term to the classical evolution equations.

Decoupling quantum dissipation interaction via stochastic fields.

Based on the Hubbard-Stratonovich transformation, the dissipative interaction between the system of interest and the heat bath is decoupled and the separated system and bath thus evolve in common classical random fields. This manipulation allows us to establish a novel theoretical methodology by which the reduced density matrix is formulated as an ensemble average of its random realizations in the auxiliary white noise fields. Within the stochastic description, the interaction between the system and the bath is reflected in the mutually induced mean fields. The relationship between the bath-induced field and the influence functional in the path integral framework is revealed. As a demonstration of this approach, we derive the exact master equations for two model systems.

Nonperturbative vibrational energy relaxation effects on vibrational line shapes.

A general formulation of nonperturbative quantum dynamics of solutes in a condensed phase is proposed to calculate linear and nonlinear vibrational line shapes. In the weak solute-solvent interaction limit, the temporal absorption profile can be approximately factorized into the population relaxation profile from the off-diagonal coupling and the pure-dephasing profile from the diagonal coupling. The strength of dissipation and the anharmonicity-induced dephasing rate are derived in Appendix A. The vibrational energy relaxation (VER) rate is negligible for slow solvent fluctuations, yet it does not justify the Markovian treatment of off-diagonal contributions to vibrational line shapes. Non-Markovian VER effects are manifested as asymmetric envelops in the temporal absorption profile, or equivalently as side bands in the frequency domain absorption spectrum. The side bands are solvent-induced multiple-photon effects which are absent in the Markovian VER treatment. Exact path integral calculations yield non-Lorentzian central peaks in absorption spectrum resulting from couplings between population relaxations of different vibrational states. These predictions cannot be reproduced by the perturbative or the Markovian approximations. For anharmonic potentials, the absorption spectrum shows asymmetric central peaks and the asymmetry increases with anharmonicity. At large anharmonicities, all the approximation schemes break down and a full nonperturbative path integral calculation that explicitly accounts for the exact VER effects is needed. A numerical analysis of the O-H stretch of HOD in D(2)O solvent reveals that the non-Markovian VER effects generate a small recurrence of the echo peak shift around 200 fs, which cannot be reproduced with a Markovian VER rate. In general, the nonperturbative and non-Markovian VER contributions have a stronger effect on nonlinear vibrational line shapes than on linear absorption.