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AIP conference proceedings

Hole burning: A discrete kinetic approach

Alexander Prüll

Published Jan 1, 2001• 0 citations
Publisher

Abstract

Investigations in the field of quantum optics very often requires a molecular simulation of transport phenomena. This paper presents a discrete kinetic model to deal with the quantum optical problem of hole burning. Based on a full discretization of the velocity space, Boltzmann-like equations are established to govern the temporal evolution of heavily disturbed Maxwell distributions of optically excited particles diffusing in a buffer gas. These model equations involve elastic collisions and the interaction of photons with three-level atoms by taking into account the Doppler effect. Important features of the developed model are the realistic simulation of the relaxation process and the existence of a stable and unique equilibrium solution.

DOI
10.1063/1.1407543

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Investigations in the field of quantum optics very often requires a molecular simulation of transport phenomena. This paper presents a discrete kinetic model to deal with the quantum optical problem of hole burning. Based on a full discretization of the velocity space, Boltzmann-like equations are established to govern the temporal evolution of heavily disturbed Maxwell distributions of optically excited particles diffusing in a buffer gas. These model equations involve elastic collisions and the interaction of photons with three-level atoms by taking into account the Doppler effect. Important features of the developed model are the realistic simulation of the relaxation process and the existence of a stable and unique equilibrium solution.
The authors consider a system of two-level atoms coupled to a radiation field and immersed in a buffer gas. The corresponding Bloch-like equations are augmented by collision operators stemming from the linear Boltzmann equation. Collision operators describe velocity-changing collisions that give rise to various kinetic effects of light. They study the general properties of collision operators, the main one consisting of the fact that such operators are Hermitian in a suitably defined space of velocity distribution functions They introduce a set of functions that serve as approximations to the eigenfunctions. The eigenvalues are connected with transport coefficients (they give an example of such a relation), but they keep eigenvalues as free parameters of the approach. In this way the authors can describe the whole class of collision operators, the Keilson-Storer model being a particular example. Employing the introduced eigenfunctions they derive and formally solve the equations of motion for the expansion coefficients of the elements of the density matrix of active atoms.
The interaction of a plane monochromatic traveling wave with two-level particles suffering collisions with buffer-gas particles is considered. Collision rates are assumed to be velocity dependent. The collision integral is obtained on the basis of the strong-collision model, generalized to the case of velocity-dependent collision rates (the so-called ``kangaroo'' model). We obtained the exact analytical solution of the problem for arbitrary intensity of radiation, arbitrary ratio of homogeneous and Doppler widths of the absorption line, and arbitrary mass ratio between absorbing- and buffer-gas particles. The obtained analytical solutions of the quantum kinetic equations allowed us to analyze the spectral shape of the strong-field absorption line as well as the probe-field absorption line (the nonlinear part of the work done by the probe field) and the frequency dependence of the light-induced drift (LID) velocity. A comprehensive comparative analysis for the three- and one-dimensional versions of the model is given. On the basis of this analysis, we reach the conclusion that the one-dimensional quantum kinetic equation has quite a wide range of application. We also reveal the conditions for the strongest manifestation of the velocity dependence of the collision rates, which affects most strongly the anomalous LID.
Multiphoton excitation and dissociation of large molecules under collisionless conditions is discussed in terms of an intercontinuum random coupling model. The mathematical approach described in a previous paper is used to obtain the general solution for a system of consecutively coupled discrete states, quasicontinuous manifolds, and continuous (dissociative) manifolds of molecular levels (eigenstates of the total molecular Hamiltonian), where the radiative coupling matrix elements are assumed to be given as a linear combination of smoothly varying and randomly varying (over level indices in the molecular manifolds) components. In the range of discrete molecular levels the time evolution is coherent and described in terms of the optical Bloch equation. In the quasicontinuous and continuous ranges the time evolution may be described in terms of Markoffian kinetic equations for the number of photons absorbed by the molecule, provided that the intramolecular vibrational relaxation widths associated with the optically active molecular modes is much larger than the Rabi frequency associated with the excitation of these modes. The kinetic evolution itself consists of direct multiphoton excitation processes (simultaneous transitions from the upper discrete levels to all higher energy molecular manifolds) resulting from the smooth component in the radiative coupling, and a consecutive excitation process described by the Pauli master equation with rates given by the golden rule expression. The interaction which enters into the golden rule expression is the variance in the radiative coupling. The direct excitation component contributes a negligible part of the overall excitation even if the random and smooth radiative coupling components are comparable. The resulting incoherent time evolution of the multiphoton excitation process is consistent with available experimental results. Coherent effects in the time evolution are expected for higher radiation field intensities, where the Rabi frequency becomes comparable to the intramolecular vibrational relaxation rate.
Based on the momentum method the Fokker-Planck equation is derived for a velocity distribution perturbed by monomode laser excitation (hole burning). If the used collision kernel applies to marked forward scattering and if the collision partner is assumed to be infinitely heavy, the FPE degenerates into the Legendre equation. While a proportional term accounts for the decay of atoms, an inhomogeneity accounts for their creation. The considered species is metastable argon ions. These ions are probably created by electronic ionization, while both electronic deexcitation and metastability exchange interfere in their destruction. The laser keeps the solution of the FPE at zero level for any velocity whose component along the laser beam upsilon z,0 follows from the laser frequency by the Doppler condition. We treat this constraint as a boundary condition for the solution. The latter is put up as a superposition of the Legendre function which solves the homogeneous part, with a special solution of the inhomogeneous equation. Free parameters are adjusted to the boundary condition. The thus obtained solution is finally integrated over all the velocity groups of atoms which under our assumptions are not coupled collisionally, to yield the expected shape of the signal. The velocity groups were assumed to be distributed according to Maxwell. We devised two types of experiments to measure the shape of the hole. Both are suited to measure holes on the wings as well as on the centre of the line. Remarkable similarity of the measured and the calculated signals is found.
Kinetic equations have played a significant role in optics since the introduction of the radiative transport equation (RTE) by Schuster in the early 20th century. Now they are applied in many scientific fields such as atmospheric physics and biomedical imaging to understand how light interacts with a complex medium. In this thesis we investigate how these equations arise in: 1. models of classical optics in an inhomogeneous disordered scattering medium 2. in fully quantum real space models with a random distribution of atoms In certain spatial and temporal regimes it has been empirically observed that light interacting with a highly disordered scattering medium may be described by a radiative transport equation with a constant scattering coefficient. One way to derive such equations from microscopic principles is to model the medium as a random field which is statistically homogeneous. We show that in a quasi-homogeneous media, where statistical correlations vary rapidly on small length scales and slowly on large length scales, light obeys a radiative transport equation with variable scattering coefficient. In the second part of this thesis we introduce a new model in quantum optics which allows us to treat a quantized electromagnetic field coupled to a medium of two level atoms, both in real space. The rest of the chapter focuses on the interaction of atoms with a field consisting of at most one photon. First we recover a classical result due to Wigner and Weisskopf on the rate of single atom spontaneous emission. Next we show that in a statistically homogeneous distribution of two level atoms the field and atomic amplitude satisfy linear kinetic equations. Finally we show that at sufficiently long times and large distances these amplitudes are asymptotically governed by diffusion equations which may be solved analytically. In the final chapter we use the previous model to analyze a system in which there can be at most two photons. This allows us to study the time evolution of states that initially contain two entangled photons, a phenomenon which has no classical analogue. We show that the various probability amplitudes satisfy linear two particle kinetic equations which again may be asymptotically evaluated at large distances and long times.
We present results of simulations of a quantum Boltzmann master equation (QBME) describing the kinetics of a dilute Bose gas confined in a trapping potential in the regime of Bose condensation. The QBME is the simplest version of a quantum kinetic master equation derived in previous work. We consider two cases of trapping potentials: a three-dimensional square-well potential with periodic boundary conditions and an isotropic harmonic oscillator. We discuss the stationary solutions and relaxation to equilibrium. In particular, we calculate particle distribution functions, fluctuations in the occupation numbers, the time between collisions, and the mean occupation numbers of the one-particle states in the regime of onset of Bose condensation.