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The Journal of Chemical Physics

Reduced equations of motion and reduced S matrix for scattering processes

Shaul Mukamel

Published Jul 1, 1981• 4 citations
Publisher

Abstract

Reduced equations of motion (REM) are derived for the reduced density matrix of some of the degrees of freedom taking part in a scattering process. The derivation is made making use of the Zwanzig–Mori projection operator technique and utilizing a partial time ordering prescription (POP) which results in simple REM which contain no convolution in time. The present formulation is convenient for the direct calculation of reduced information on scattering events since it avoids the necessity of calculating the complete S matrix. The entire dynamical information relevant for the present description is expressed in terms of a hierarchy of correlation functions which in turn depend on the potential surface only in the region of interest. We further develop an exponential approximation for the reduced tetradic scattering S matrix in Liouville space, which forms a convenient framework for systematic approximations.

DOI
10.1063/1.441817

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Reduced equations of motion (REM) are derived for the reduced density matrix of some of the degrees of freedom taking part in a scattering process. The derivation is made making use of the Zwanzig–Mori projection operator technique and utilizing a partial time ordering prescription (POP) which results in simple REM which contain no convolution in time. The present formulation is convenient for the direct calculation of reduced information on scattering events since it avoids the necessity of calculating the complete S matrix. The entire dynamical information relevant for the present description is expressed in terms of a hierarchy of correlation functions which in turn depend on the potential surface only in the region of interest. We further develop an exponential approximation for the reduced tetradic scattering S matrix in Liouville space, which forms a convenient framework for systematic approximations.
Reduced equations of motion (REM) are derived for molecular multiphoton processes by constructing a small set of relevant operators corresponding to populations and coherences of the molecule ’’dressed’’ by the radiation field and adopting the Mori projection operator technique. The resulting REM contain time-dependent dephasing operators which are intimately related to our reduction scheme and whose definition and evaluation do not rely on any perturbative expansion in intramolecular interactions. These operators are expressed in terms of intramolecular dipole correlation functions which are the key quantities in the present formulation and are then evaluated using a simple model. In the limit where our physical system may be separated into a ’’system’’ and a ’’bath’’ with a weak interaction, we recover the ordinary expressions of line broadening. The present formulation provides us with a hiearchy of REM which interpolate smoothly all the way from the coherent to the incoherent driving limits and are in good quantitative agreement with the existing experimental data.
Reduced density matrix descriptions are developed for linear and non-linear electromagnetic interactions of moving atomic systems, taking into account applied magnetic fields. Atomic collision processes are treated as environmental interactions. Applications of interest include electromagnetically induced transparency and related pump-probe optical phenomena in atomic vapors. Time-domain (equation-of-motion) and frequency-domain (resolvent-operator) formulations are developed in a unified manner. The standard Born (lowest-order perturbation-theory) and Markov (short-memory-time) approximations are systematically introduced within the framework of the general nonperturbative and non-Markovian formulations. A preliminary semiclassical treatment of the electromagnetic interaction is introduced. However, the need for a fully quantum mechanical approach is emphasized. Compact Liouville-space operator expressions are derived for the linear and the general (n'th order) non-linear electromagnetic-response tensors occurring in a perturbation-theory treatment of the electromagnetic interaction. These expressions can be evaluated for coherent initial atomic excitations and for the full tetradic-matrix form of the Liouville-space self-energy operator representing the environmental interactions in the Markov approximation. Intense-field electromagnetic interactions can be treated by means of an alternative method, which is based on a Liouville-space Floquet representation of the reduced density operator. Collisional interactions between atoms in a vapor can be treated in various approximations for the self-energy operator and the influence of Zeeman coherences on the electromagnetic response can be incorporated.
The quantum dynamics of many-electron spin systems is investigated using a reduced-density-matrix description. Applications of interest include trapped atomic systems in optical lattices, semiconductor quantum dots, and vacancy defect centers in solids. Complimentary time-domain (equation-of-motion) and frequency-domain (resolvent-operator) formulations are self-consistently developed. The general non-perturbative and non-Markovian formulations provide a fundamental framework for systematic evaluations of corrections to the standard Born (lowest-order-perturbation) and Markov (short-memory-time) approximations. Particular attention is given to decoherence and relaxation processes, together with spectral-line broadening phenomena, that are induced by interactions of many-electron spin systems with photons, phonons, nuclear spins, external electric, magnetic, and electromagnetic fields. Within the framework of a quantum-open-systems (reduced-density-operator) approach, these processes are treated either as coherent interactions or as environmental interactions. The environmental interactions are incorporated by means of the general expressions that are derived for the time-domain and frequency-domain Liouville-space self-energy operators, for which the tetradic-matrix elements are explicitly evaluated in the diagonal-resolvent, lowest-order, and Markov (short-memory time) approximations.
A reduced-density-matrix description is developed for linear and non-linear electromagnetic interactions of quantized electronic systems in the presence of environmental decoherence and relaxation phenomena. Applications of interest include many-electron atomic systems (in electron-ion beam interactions, gases, and high-temperature plasmas) and semiconductor materials (bulk crystals and nanostructures). Time-domain (equation-of-motion) and frequency-domain (resolvent-operator) formulations are developed in a unified manner. The standard Born (lowest-order perturbation-theory) and Markov (short-memory-time) approximations are systematically introduced within the framework of the general non-perturbative and non-Markovian formulations. A preliminary semiclassical treatment of the electromagnetic interaction is introduced. Compact Liouville-space operator expressions are derived for the linear and the general (n’th order) non-linear electromagnetic-response tensors, allowing for coherent initial electronic excitations and for the full tetradic-matrix form of the Liouville-space self-energy operator representing the environmental interactions. It is emphasized that quantum-coherent many-body interactions cannot be adequately treated as environmentally induced phenomena.
Evolution of a system, coupled to its environment and influenced by external driving fields, is an old problem that remains of interest. In this paper, we derive an equation of motion for the reduced system density matrix, which is time convolutionless and free of the partial trace with respect to the environment states. This new approach uses an extension of the projection-operator technique, which incorporates an isomorphism between the system's Liouville space and the unit eigenspace of the projection operator induced by the uniform environment density matrix. Numerical application of the present approach is particularly useful in large externally driven systems, as the partial-trace-free equation is given in terms of submatrices significantly smaller than the matrices in the conventional time-convolutionless approaches, which alleviates the computational burden. We also show that all time-convolutionless approaches, conventional or partial-trace-free, are based upon a hidden underlying assumption of time reversibility of the system's evolution. This feature puts significant constraints on applicability of time-convolutionless approaches when employing approximations that yield time irreversibility. Also, we investigate the application of the approach in the description of far-from-equilibrium systems.
Reduced density matrix descriptions are developed for linear and non-linear electromagnetic interactions of moving atomic systems, taking into account applied magnetic fields as well as atomic collisions together with other environmental decoherence and relaxation processes. Applications of interest include electromagnetically induced transparency and related pump-probe optical phenomena in warm atomic vapors. Time-domain (equation-of-motion) and frequency-domain (resolvent-operator) formulations are developed in a unified manner. The standard Born (lowestorder perturbation-theory) and Markov (short-memory-time) approximations are systematically introduced within the framework of the general non-perturbative and non-Markovian formulations. A preliminary semiclassical treatment of the electromagnetic interaction is adopted. However, the need for a fully quantum mechanical approach is emphasized. Compact Liouville-space operator expressions are derived for the linear and the general (n'th order) non-linear macroscopic electromagnetic-response tensors occurring in a perturbation-theory treatment of the semiclassical electromagnetic interaction. These expressions can be evaluated for coherent initial atomic excitations and for the full tetradic-matrix form of the Liouville-space self-energy operator representing the environmental interactions in the Markov approximation. Collisional interactions between atoms can be treated in various approximations for the selfenergy operator, and the influence of Zeeman coherences on the macroscopic electromagnetic response can be investigated.