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Physical Review A

Exact solution of the one- and three-dimensional quantum kinetic equations with velocity-dependent collision rates: Comparative analysis

Timofei Privalov, Anatolii M. Shalagin

Published Jun 1, 1999• 20 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.59.4331

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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.
The solution of many problems in light-induced gas kinetics can be simplified significantly using quantum kinetic equations in the context of the so-called one-dimensional approximation, in which the initial equations are averaged over transverse (relative to the direction of radiation) velocities. The errors introduced in such an approach are usually assumed to be small; however, this has been confirmed quantitatively only on the basis of the simplest (two- and three-level) particle models. We analyze the accuracy of the one-dimensional approximation for multilevel particles quantitatively for the light-induced drift (LID) effect in cesium atoms in the atmosphere of inert buffer gases. It is shown that in the case of the so-called “normal” LID, one-dimensional kinetic equations can always be used instead of three-dimensional equations without a risk of losing some important fine details in the dependence of the drift velocity on the radiation frequency. In the case of anomalous LID, the error of the one-dimensional approximation is also insignificant, but it can be disregarded only in the case of light buffer particles. For comparable masses of resonant and buffer particles, the one-dimensional approximation may give a noticeable error in determination of drift velocity amplitudes; however, the positions of drift velocity zeros and extrema depending on radiation-frequency detuning can be described successfully. Results show that the error introduced by using the one-dimensional approximation for multilevel particles turns out to be more significant than for the simplest particle models.
The effect of the velocity ( v ) dependence of the transport collision frequency ν tr v on the Dicke line narrowing is analyzed in terms of the strong-collision model generalized to velocity-dependent collision frequencies (the so-called kangaroo model). This effect has been found to depend on the mass ratio of the resonance ( M ) and buffer ( M b ) particles, β = M b / M : it is at a minimum for β ≪ 1 and reaches a maximum for β ≳ 3. A power-law particle interaction potential, U ( r ) ∝ r − n , is used as an example to show that, compared to ν tr v ( v ) = const ( n = 4), the line narrows if ν tr v ( v ) decreases with increasing v ( n < 4) and broadens if ν tr v ( v ) increases with v ( n > 4). At β ≳ 3, the line width can increase [compared to ν tr v ( v ) = const] by 5 and 12% for the potentials with n = 6 and n ≳ 10, respectively; for the potentials with n = 1 (Coulomb potential) and n = 3, it can decrease by more than half and 6%, respectively. The line profile I (Ω) has been found to be weakly sensitive to ν tr v ( v ) at some detuning Ω c of the radiation frequency Ω. Dicke line narrowing is used as an example to analyze the collisional transport of nonequilibrium in the resonance-particle velocity distribution in a laser field. The transport effect is numerically shown to be weak. This allows simpler approximate one-dimensional quantum kinetic equations to be used instead of the three-dimensional ones to solve spectroscopic problems in which it is important to take into account the velocity dependence of the collision frequency when the phase memory is preserved during collisions.
The authors consider a system of optically active two-state atoms immersed in a buffer gas and irradiated by a light beam. The corresponding Bloch equations are augmented by Keilson-Storer (KS) collision kernels (1952) which account for velocity-changing collisions. Expansion of the density operator elements in the eigenfunctions of KS kernels allows one to obtain exact solutions for the Laplace transforms of the density operator elements which may be expressed as rational approximants with any prescribed degree of accuracy. As an example they apply this method to discussion of the light-induced drift effect. In particular they give analytic expressions for the drift velocity, which are illustrated by numerical results. Finally, they briefly discuss possible generalisations and applications of the presented method.
The spectrum of absorption (amplification) of a weak probe field by two-level atoms colliding with atoms of a buffer gas in the field of a strong electromagnetic wave is studied theoretically. A universal, free of any collision model, solution is obtained for systems with small (compared to collision rates) Doppler broadening with no restrictions on the strong-field intensity and with the possible absence of dephasing of the light-induced dipole moment in elastic collisions of the gas particles. The relation of the collisional parameters of the problem with the characteristics of an elementary scattering event and with the macroscopic characteristics connected with the transport phenomena is established. It is found that the nontrivial features of the probe-field spectrum discovered by us previously [Zh. Éksp. Teor. Fiz. 127 , 320 (2005)] in the framework of the strong-collision model are not connected with a particular model of collisions and have a universal nature.
The spectrum of weak probe field absorption (amplification) by two-level atoms experiencing collisions with buffer gas atoms in a strong resonance laser field is studied theoretically. Analysis is carried out for systems with a weak Doppler broadening under relatively mild constraints on the strong field intensity for the general case of an arbitrary change in the phase of the radiation-induced dipole moment in elastic collisions of gas particles. It is shown that, in spite of uniform broadening of the absorption line, the probe field spectrum exhibits a clearly manifested anisotropy to mutual orientation of the wavevectors of strong and probe radiation. It is found that the width of resonances in the probe field spectrum under definite conditions (that can easily be created in experiments) is proportional to the diffusion coefficient for atoms interacting with radiation. This fact can form the basis of the spectroscopic method for measuring the transport frequencies of collisions between particles absorbing radiation and buffer particles. It is shown that phase memory effects in collisions strongly modify the probe field spectrum both qualitatively and quantitatively. Simple operative formulas proposed for the probe field spectrum are convenient for experimental data processing.
We study theoretically how the dependences of transport collision frequencies n i , collision broadening g and collision shift D of the levels on the velocity u of resonant particles influence lightinduced diffusive pulling (pushing) (LDP) effects in the framework of a generalised model of strong collisions in the case of velocitydependent collision rates (so-called kangaroo model).It is found that allowance for the dependences n i (u), g(u) and D(u) does not change the spectral shape of an LDP signal.In particular, in the case of low-intensity radiation, the spectral dependence of the LDP signal coincides with the absorption line shape.It is shown that the magnitude of the LDP effect is proportional to the difference between the diffusion coefficients of particles in the excited and ground states.It is found that the spectral anomalies previously predicted in the LDP effect [Gel'mukhanov F.Kh. JETP Lett., 55, 214 (1992)] for an idealised model of the Lorentz gas (the limiting case of heavy buffer particles), which arise due to the dependences n i (u), g(u) and D(u), are typical only for this gas.At a realistic ratio of the masses of absorbing and buffer particles, spectral anomalies do not occur in the LDP effect.