Liouville-von Neumann Equation Research Articles

Simultaneous integration of mixed quantum‐classical systems by density matrix evolution equations using interaction representation and adaptive time step integrator

A density matrix evolution method [H. J. C. Berendsen and J. Mavri, J. Phys. Chem., 97, 13464 (1993)] to simulate the dynamics of quantum systems embedded in a classical environment is applied to study the inelastic collisions of a classical particle with a five-level quantum harmonic oscillator. We improved the numerical performance by rewriting the Liouville–von Neumann equation in the interaction representation and so eliminated the frequencies of the unperturbed oscillator. Furthermore, replacement of the fixed time step fourth-order Runge–Kutta integrator with an adaptive step size control fourth-order Runge–Kutta resulted in significantly lower computational effort at the same desired accuracy. © 1996 by John Wiley & Sons, Inc.

Simultaneous integration of mixed quantum-classical systems by density matrix evolution equations using interaction representation and adaptive time step integrator.

A density matrix evolution method [H. J. C. Berendsen and J. Mavri, J. Phys. Chem., 97, 13464 (1993)] to simulate the dynamics of quantum systems embedded in a classical environment is applied to study the inelastic collisions of a classical particle with a five-level quantum harmonic oscillator. We improved the numerical performance by rewriting the Liouville-von Neumann equation in the interaction representation and so eliminated the frequencies of the unperturbed oscillator. Furthermore, replacement of the fixed time step fourth-order Runge-Kutta integrator with an adaptive step size control fourth-order Runge-Kutta resulted in significantly lower computational effort at the same desired accuracy. © 1996 by John Wiley & Sons, Inc.

Hydrogen transfer in vibrationally relaxing benzoic acid dimers: Time-dependent density matrix dynamics and infrared spectra

We employ time-dependent density matrix theory to characterize the concerted double-hydrogen transfer in benzoic acid dimers—the ‘‘system’’—embedded in their crystalline environment—the ‘‘bath.’’ The Liouville–von Neumann equation for the time evolution of the reduced nuclear density matrix is solved numerically, employing one- and two-dimensional models [R. Meyer and R. R. Ernst, J. Chem. Phys. 93, 5528 (1990)], the state representation for all operators and a matrix propagator based on Newton’s polynomials [M. Berman, R. Kosloff, and H. Tal-Ezer, J. Phys. A 25, 1283 (1992)]. Dissipative processes such as environment-induced vibrational energy and phase relaxation, are accounted for within the Lindblad dynamical semigroup approach. The calculation of temperature-dependent relaxation matrix elements is based on a microscopic, perturbative theory proposed earlier [R. Meyer and R. R. Ernst, J. Chem. Phys. 93, 5528 (1990)]. For the evaluation of the dissipative system dynamics, we compute (i) time-dependent state populations, (ii) energy and entropy flow between system and bath, (iii) expectation values for the hydrogen transfer coordinate, (iv) characteristic dephasing times and (v) temperature-dependent infrared spectra, determined with a recently proposed method by Neugebauer et al. Various ‘‘pure’’ and ‘‘thermal’’ nonequilibrium initial states are considered, and their equilibration with the bath followed in time.

Quantum generalized Vlasov equation

An electronic system is considered as a quantum plasma characterized by a plasma parameter γ 0 ⪡ 1, over short times. Using the Dynamics of Correlations (a many-body theory based on the Liouville-von Neumann equation in the Fourier transform of the Wigner-Weyl representation), in first and second order, a generalized Vlasov equation (GVE) has been derived. GVE describes as a particular situation the STLS (Singwi, Tosi, Land and Sjölander) ansatz. Discarding correlation terms in GVE, a non-linear Vlasov equation with exchange terms is derived. The classical limit and the linearization of GVE are also studied.

Density matrix model for hydrogen transfer in the benzoic acid dimer

The double hydrogen transfer in benzoic acid dimers (‘the system’) embedded in a crystal (‘the bath’) is studied with time-dependent density matrix theory. The Liouville-von Neumann equation is solved for a two-dimensional model, using a state representation for the operators, and a polynomial expansion to treat the time evolution of the reduced nuclear density matrix. The bath-induced vibrational relaxation and dephasing processes are treated within the Lindblad formalism, with interlevel transition probabilities obtained from a microscopic perturbative theory due to Meyer and Ernst. The approach of various initial non-equilibrium states towards thermal equilibrium, is monitored with the help of: (i) state populations, (ii) system-bath energy exchange, (iii) the von Neumann and relative entropies and (iv) the decay of ‘coherences’. Simulations are carried out at low and at high bath temperatures.

WHY IRREVERSIBILITY? THE FORMULATION OF CLASSICAL AND QUANTUM MECHANICS FOR NONINTEGRABLE SYSTEMS

Nonintegrable Poincaré systems with continuous spectrum (so-called Large Poincaré Systems, LPS) lead to the appearance of diffusive terms in the framework of dynamics. These terms break time symmetry. They lead, therefore, to limitations to classical trajectory dynamics and of wave functions. These diffusive terms correspond to well-defined classes of dynamical processes (i.e., so-called “vacuum-vacuum” transitions). The diffusive effects are amplified in situations corresponding to persistent interactions. As a result, we have to include already in the fundamental dynamical description the two aspects, probability and irreversibility, which are so conspicuous on the macroscopic level. We have to formulate both classical and quantum mechanics on the Liouville level of probability distributions (or density matrices). For integrable systems, we recover the usual formulations of classical or quantum mechanics. Instead of being irreducible concepts, which cannot be further analyzed, trajectories and wave functions appear as special solutions of the Liouville-von Neumann equations. This extension of classical and quantum dynamics permits us to unify the two concepts of nature we inherited from the 19th century, based on the one hand on dynamical time-reversible laws and on the other on an evolutionary view associated to entropy. It leads also to a unified formulation of quantum theory avoiding the conventional dual structure based on Schrödinger’s equation on the one hand, and on the “collapse” of the wave function on the other. A dynamical interpretation is given to processes such as decoherence or approach to equilibrium without any appeal to extra dynamic considerations (such as the many-world theory, coarse graining or averaging over the environment). There is a striking parallelism between classical and quantum theory. For LPS we have, in general, both a “collapse” of trajectories and of wave functions for LPS. In both cases, we need a generalized formulation of dynamics in terms of probability distributions or density matrices. Since the beginning of this century, we know that classical mechanics had to be generalized to take into account the existence of universal constants. We now see that classical as well as quantum mechanics also have to be extended to include unstable dynamical systems such as LPS. As a result, we achieve a new formulation of "laws of physics" dealing no more with certitudes but with probabilities. The formulation is appropriate to describe an open, evolving universe.

Why irreversibility? The formulation of classical and quantum mechanics for nonintegrable systems

AbstractNonintegrable Poincaré systems with a continuous spectrum (so‐called large Poincaré systems, LPS) lead to the appearance of diffusive terms in the frame of classical or quantum dynamics. These terms break time symmetry. They lead, therefore, to limitations to classical trajectory theory and of wave‐function formalism (Schrödinger's equation). These diffusive terms correspond to well‐defined classes of dynamical processes (i.e., so‐called vacuum‐vacuum transitions). The diffusive effects are amplified in situations corresponding to persistent interactions. As a result, we have to include, already, in the fundamental dynamical description the two basic aspects, probability and irreversibility, which are so conspicuous on the macroscopic level. We have to formulate both classical and quantum mechanics on the Liouville level of probability distributions (or density matrices). For integrable systems, we recover the usual formulation of classical or quantum mechanics. Instead of being primitive concepts, which cannot be further analyzed, trajectories and wave functions appear as special solutions of the Liouville–von Neumann equations. This extension of classical and quantum dynamics permits us to unify the two concepts of nature that we inherited from the nineteenth century, based, on the one hand, on dynamical time‐reversible laws and, on the other, on an evolutionary view associated to entropy. It leads also to a unified formulation of quantum theory, avoiding the conventional dual structure based on Schrödinger's equation, on the one hand, and on the “collapse” of the wave function, on the other. A dynamical interpretation is given to processes such as decoherence or approach to equilibrium without any appeal to extra dynamic considerations (such as many‐world theory, coarse graining, or averaging over the environment). There is a striking parallelism between classical and quantum theory. For large Poincaré systems (LPS), we have, in general, both a “collapse” of trajectories and of wave functions. In both cases, we need a generalized formulation of dynamics in terms of probability distributions or density matrices. Since the beginning of this century, we have known that classical mechanics had to be generalized to take into account the existence of universal constants. We now see that classical as well as quantum mechanics also have to be extended to include unstable dynamical systems such as LPS. As a result, we achieve a new formulation of “laws of physics” dealing no more with certitudes but with probabilities. This formulation is appropriate to describe an open, evolving universe. © 1995 John Wiley & Sons, Inc.

Density matrix description of laser-induced hot electron mediated photodesorption of NO from Pt(111)

Based on the numerical solution of the Liouville-von Neumann equation for dissipative systems, the photodesorption dynamics of NO/Pt(111) are studied. Dissipative terms are used to describe the quenching of electronically excited states on the metal, electronic dephasing and the indirect (hot-electron mediated) excitation processes in the DIMET and DIET limits. Norm and energy flow, desorption probabilities and density time-of-flight spectra are computed.

Derivation of quantum kinetic equation from the Liouville-von Neumann equation

A quantum Markov kinetic equation is derived from the Liouville—von Neumann equation in the framework of second-order kinetic perturbation theory. The proposed method can be used to obtain an equation that describes the quantum diffusion of a particle in a crystal.

Holonomy on Lie groups and energy uncertainty

First, we give a general group-theoretical interpretation of quantum holonomy independent of a particular Hilbert space representation. Then we apply this to the Liouville-von Neumann equation for density matrices and obtain the appropriate generalization of the time-energy uncertainty relation (recently obtained by Anandan and Aharonov) from pure to mixed states. Different Riemannian structures on the space of density matrices are discussed and the relationship of the holonomy to its canonical symplectic structure is clarified.

Far from equilibrium evolution in highly excited semiconductors with a consistent treatment of the photon field

Evolution equations of a laser-excited direct gap two-band semiconductor in the transient stage have been derived by using the first principle Liouville-von Neumann equation and the statistical projection operator method. To be more general, the photon field is described in a quantum mechanical manner. Quantum effects of the electron-photon coupoling as well as charge carrier Coulomb screening and dissipation are involved. The evolution of the time dependent coherent photon state is investigated. It results in additionally evolution equations. Numerical solutions of these systems are cast out for realistic parameters and are compared with previous results.

Non-linearity, dissipation and transport in a molecular trimer

This study investigates the effects of dissipation on the quasi-particle transport in a degenerate molecular trimer in which the time evolution is described by the discrete non-linear Schrodinger equation. The standard procedure of introducing dissipation as a phase-destroying stochastic parameter through the corresponding Liouville-von Neumann equation is employed. A closed second-order integrodifferential equation describing the time evolution of the site occupation probability difference is derived in the simultaneous presence of non-linearity and dissipation for trimers in compact geometry subjected to specific initial conditions. The Painleve property of the evolution equation is examined and it is found that the system is not of p type. A conflict arising from the two opposing views existing in the literature on the effects of dissipation on non-linear dynamics in a closely related system is focused on. Both the resolution of this conflict and a description of trimer dynamics are provided by elucidating the effects of the interplay between dissipation and non-linearity through numerical experiments.

COHERENT MOTION AND TRANSFER OF EXCITATION ENERGY IN THE PRIMARY PROCESSES OF PHOTOSYNTHESIS

AbstractFluorescence decay curves are calculated within the framework of the Liouville von Neumann equation for simple model systems of a photosynthetic unit consisting of a reaction center coupled to either a linear chain or a hexagonal array of identical antenna pigments. A comparison of the results with those obtained by solving the Pauli Master equations reveals that in the case of coherent excitation energy transfer the fluorescence decay kinetics exhibit a strong dependence on the topological array of the antenna pigments. For systems with a different pigment array, markedly different kinetics can be expected upon decreasing the pure dephasing time.

From classical chaos to quantum chaos

The role of “chaos” in the fundamental dynamical description, both classical and quantum, is discussed. It is shown that it leads to a new formulation of laws of nature which is irreducible ( it can only be applied to probability distributions, or in quantum mechanics, to the density matrices ) and includes time symmetry breaking. This formulation is first applied to chaotic maps and then to Hamiltonian systems, classical or quantum, which belong to the class of non-integrable systems in the sense of Poincaré. Those are the Large Poincaré Systems ( LPS ) which include all systems studied in non-equilibrium statistical mechanics as well as interacting fields. This formulation requires the giving up of the traditional Hilbert space description and going to generalized spaces ( such as “rigged” Hilbert spaces ). In these spaces we obtain a complex spectral representation which includes irreversible processes. Moreover the solutions of the Liouville-von Neumann equation are not factorisable ( in quantum mechanics ). This leads to a purely dynamical interpretation of the “collapse” of the wave function. In this paper emphasis is put on simple physical examples in preference to general theory to introduce the reader to the ideas at the basics of the extension of the traditional formulation of classical or quantum mechanics.

Boltzmann kinetic equation with correction term for intracollisional field effect

The author has included a higher order term in the approximation of the electron density matrix used in deriving the Boltzmann kinetic equation from the Liouville-von Neumann equation by means of Kubo's formalism. This higher term has been added to the Boltzmann kinetic equation as the correction term. In a simplified case the field effect collision rate is introduced, representing the intracollisional field effect. The correction term gives a significant contribution to the Boltzmann kinetic equation for an electric field strength of 2.5 MV m-1, which was determined by numerical calculation.

Computer-simulation of the multiple-quantum dynamics of one-, two- and three-dimensional spin distributions

Multiple-quantum NMR has previously been used to characterize the distribution of spins, whether cluster or homogenously distributed, in solids. In this work, through computer simulation of the multiple-quantum coherence dynamics of nine spin networks, we examine the potential of this spectroscopy to differentiate between one-, two- and three-dimensional distributions of spins. In order to compute the multiple-quantum dynamics of networks of spins large enough to vary in dimensionality, the solution of the Liouville-von Neumann equation was written as an explicit, single time-step equation for the density matrix with pairwise, spatially averaged dipole interactions. In these nine spin simulations, differentiation of the multiple-quantum dynamics of a linear system in comparison with two- and three-dimensional networks was observed indicating the influence of the dimensionality of the network on spin correlation dynamics. While no differentiation between higher dimensional networks was observed, this can be attributed to the small number of spins in the simulation. Based on the occupancy of various lattice types, differences between spin correlation rates for surface and bulk, homogeneous distributions are expected when the number of correlated spins exceeds 20.

Minimal extensions of the Liouville-von Neumann equation accounting for measurement and state preparation in quantum mechanics

We examine the question, what are the minimal extensions of the Liouville-von Neumann equation for a density matrix, describing a quantum system interacting with an environment, which can account for the essential aspects of the measurement and state preparation processes. For the case of a system of spin-(1/2) particles, it is found that linear extensions are sufficient to describe the transitions of a system from a pure state to a mixed state with equal polarization components. It is further suggested that the saturation of the polarization amplitudeP, which characterizes the preparation of a fully polarized state, can be described by a dynamical system with a limit cycle. The minimal nonlinear extension of the Liouville-von Neumann equation presenting this feature is also discussed.

Transient quantum transport simulation based on the statistical density matrix

A new quantum device simulation technology has been developed based on the statistical density matrix theory. By introducing the Hartree self-consistent-field model for electron-electron interactions and the relaxation time approximation for scattering processes, the one-dimensional time-dependent Liouville-von Neumann equation for the electron density matrix has been solved. The density matrix is a wavefunction-wavefunction correlation function with its off-diagonal elements directly measuring the phase coherence of wavefunctions in quantum mechanical devices. The authors report a numerical model for the transient density matrix, calculated initially in thermal equilibrium by self-consistently solving the Schrodinger equation and Poisson's equation. The present simulation technology is applied to the simple AlGaAs/GaAs resonant tunnelling diode and the femtosecond time evolution of the density matrix under an applied field is demonstrated. Remarkable oscillatory behaviour observed in the off-diagonal elements reveals the existence of long. range phase correlations of the electron wave in the resonance state. Intrinsic bistability and switching characteristics of the device are investigated from the viewpoint of charge accumulation in the quantum well, and the peak-to-valley current ratio is discussed in terms of phase coherence degradation of electron waves due to scattering.

Two Types of Schrodinger Time Evolution in the Formalism of Thermo Field Dynamics and Their Equivalence to the Liouville-von Neumann Equation

It is proved that two types of the time-dependent Schrodinger equation in the formulation of thermo field dynamics, which may be regarded as equivalent to the Liouville von Neumann equation, are connected with each other under a certain unitary transformation in the tilde-space

Two Types of Schrödinger Time Evolution in the Formalism of Thermo Field Dynamics and Their Equivalence to the Liouville-von Neumann Equation

Two Types of Schrödinger Time Evolution in the Formalism of Thermo Field Dynamics and Their Equivalence to the Liouville-von Neumann Equation