Mixed Weyl symbol calculus and spectral line shape theory

Abstract

A new and computationally viable full quantum version of line shape theory is obtained in terms of a mixed Weyl symbol calculus. The basic ingredient in the collision-broadened line shape theory is the time dependent dipole autocorrelation function of the radiator-perturber system. The observed spectral intensity is the Fourier transform of this correlation function. A modified form of the Wigner-Weyl isomorphism between quantum operators and phase space functions (Weyl symbols) is introduced in order to describe the quantum structure of this system. This modification uses a partial Wigner transform in which the radiator-perturber relative motion degrees of freedom are transformed into a phase space dependence, while operators associated with the internal molecular degrees of freedom are kept in their original Hilbert space form. The result of this partial Wigner transform is called a mixed Weyl symbol. The star product, Moyal bracket and asymptotic expansions native to the mixed Weyl symbol calculus are determined. The correlation function is represented as the phase space integral of the product of two mixed symbols: one corresponding to the initial configuration of the system, the other being its time evolving dynamical value. There are, in this approach, two semiclassical expansionsone associated with the perturber scattering process, the other with the mixed symbol star product. These approximations are used in combination to obtain representations of the autocorrelation that are sufficiently simple to allow numerical calculation. The leading O(h 0 ) approximation recovers the standard classical path approximation for line shapes. The higher order O(h 1 ) corrections arise from the noncommutative nature of the star product.

Computer simulation of quantum dynamics in a classical spin environment

“…(2) and in the following the sum over repeated indices is implied. Equation (2), which is just the quantum-classical Liouville equation [21][22][23][24][25][26][27] written in matrix form, defines what is known as quantum-classical bracket [16][17][18][19][20] or non-Hamiltonian commutator [30]. The bracket couples the dynamics of the phase space degrees of freedom with that of the quantum operators; it takes into account both the conservation of the energy and the quantum back-reaction.…”

Quasi-Lie Brackets and the Breaking of Time-Translation Symmetry for Quantum Systems Embedded in Classical Baths

“…It has been known for a long time, that the dynamics and statistical mechanics of a quantum subsystem coupled to classical-like DOF can be formulated in terms of operator-valued quasiprobability functions in phase space [27][28][29][30][31][32]. For example, the dynamics of nano-mechanical oscillators has been previously described by one of the authors in terms of operator-valued quasiprobability functions [33].…”

Communication: Quantum dynamics in classical spin baths

“…When such coordinates are canonically conjugate momenta and positions, one approach to treat these situations is provided by the quantum-classical Liouville equation. [14][15][16][17][18][19][20][21][22][23][24][25][26][27][28] Nevertheless, there are also many interesting cases when the bath is described in terms of classical spins, e.g., in order to model complex molecules where magnetic effects are important. 29,30 Such models can be studied by means of Monte Carlo methods or by molecular dynamics simulations.…”

“…29,30 Such models can be studied by means of Monte Carlo methods or by molecular dynamics simulations. 31,32 The scope of this Communication is to generalize the quantum-classical Liouville equation [14][15][16][17][18][19][20][21][22][23][24][25][26][27][28] in order to study the dynamics of quantum systems embedded in baths of classical spins. This is naturally achieved within a quantumclassical theory formulated by means of generalized antisymmetric brackets.…”

“…In practice, fast bath decoherence alleviates this problem and, indeed, the canonical version of the quantum-classical bracket, or of the quantum-classical Liouville equation, is used for many applications in chemistry and physics. [14][15][16][17][18][19][20][21][22][23][24][25][26][27][28] Moreover, it is worth reminding that the non-Lie (or, as they are also called, non-Hamiltonian) brackets, with their lack of time translation invariance, are also used to impose thermodynamical (such as constant temperature and/or pressure) [36][37][38] and holonomic constraints 39,40 in classical molecular dynamics simulations. 31,32 In order to represent the abstract equation (6) the quantum-classical Hamiltonian of Eq.…”