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Physica A Statistical Mechanics and its Applications

Phase variable and phase relaxation processes in the Liouville space

Masashi Ban

Published Nov 1, 1991• 11 citations
Publisher
DOI
10.1016/0378-4371(91)90217-z

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Abstract A generalized master equation is derived to describe intramolecular rearrangement processes. It is an inhomogeneous equation, including memory effects. The derivation is based on the Liouville space formalism. Because chemically relevant information is contained in the off-diagonal elements of the density matrix, a non-diagonal coarse-graining projector is used. All necessary assumptions are stated explicitly. By making further approximations, the master equation can be reduced to an inhomogeneous von Neumann equation with an effective Liouville operator the imaginary part of which is responsible for relaxation-like coarse-grained solutions. All neglected terms are given in closed form. The character of the solutions of the master equation is discussed in "coordinate-free" manner, i.e. without referring to the underlying Hilbert space.
A time-dependent functional approach is developed for calculating the dynamical optical response and ground-state properties of nonrelativistic fermion systems. The functional (effective action) uses the particle-hole components of the reduced single-electron density matrix as variables and is constructed using the natural orbitals as a reference. It is defined on pairs of trajectories that constitute a Liouville space path in the phase space of the classical analog of the quantum system. A perturbative ``semiclassical'' calculation of the effective Liouville-space action is used to incorporate correlation effects in the linear optical response.
We develop closed expressions for a time-resolved photon counting signal induced by an entangled photon pair in an interferometric spectroscopy setup. Superoperator expressions in Liouville-space are derived that can account for relaxation and dephasing induced by coupling to a bath. Interferometric setups mix matter and light variables non-trivially, which complicates their interpretation. We provide an intuitive modular framework for this setup that simplifies its description. Based on the separation between the detection stage and the light-matter interaction processes, we show that the pair entanglement time and the interferometric time-variables control the observed physics time scale. Only a few processes contribute in the limiting case of small entanglement time with respect to the sample response, and specific contributions can be singled out.