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Journal of the Optical Society of America B

Temporal behavior of the velocity distribution function and the influence of collisions in laser cooling

В. А. Алексеев, D. D. Krylova

Published Sep 1, 1993• 5 citations
Publisher

Abstract

We present exact time-dependent solutions of the Fokker–Planck equation including the collisional term, which describes the influence of the laser field and collisions with the surrounding gas on the temporal behavior of the distribution function of cooled particles in the last stage of the cooling process. These solutions allow us to estimate the influence of collisions on the limiting temperature of free particles and trapped ions.

DOI
10.1364/josab.10.001545

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We present exact time-dependent solutions of the Fokker–Planck equation including the collisional term, which describes the influence of the laser field and collisions with the surrounding gas on the temporal behavior of the distribution function of cooled particles in the last stage of the cooling process. These solutions allow us to estimate the influence of collisions on the limiting temperature of free particles and trapped ions.
In this paper we derive the Fokker-Planck equation that describes the motion of a charged particle, trapped in a static harmonic potential and cooled by radiation pressure forces exerted by a laser beam. This equation is then discussed in both the classical and quantum regimes. The model adopted here is a simple one, in which one-dimensional motion and a two-level atom are assumed, and the time evolution of the system is described in the quasiclassical limit, in which the quantum nature of the photon exchanged between the particle and the field plays no role. We report relevant graphs for the loss of kinetic energy suffered by the particle in the cooling process and estimates of the lowest kinetic temperature that can be reached in these experiments.
We used the concept of quasienergy of the quantum state of a particle in a time-oscillating parabolic potential to find the particle’s distribution function thermalized by collisions with surrounding gas molecules. When the collision frequency is much smaller than the inverse period of the potential this distribution has a Boltzmannian form with respect to quasienergy. Using quasienergy distribution and exact wave functions we found the Wigner function of the particle and its classical time-dependent space-velocity distribution [eq. (18)]. To illustrate the applicability of the quasienergy approach we compare this result with the exact solution of the Fokker–Planck equation, which gives a model description at arbitrary collision frequency.
We present a master equation that describes the dynamics of laser cooling of a trapped ion. It is valid in the Lamb–Dicke limit and rests on an adiabatic elimination combined with a degenerate perturbation treatment. It describes relaxation of probabilities and coherences in the harmonic-trap degrees of freedom. The eigenvalue spectrum the of the time-evolution operator is derived, and it follows that only one zero eigenvalue exists, giving the unique steady-state probability distribution. The coherences all decay to zero with time. The ultimate steady state is distribution, which can be characterized by a temperature. We also report a numerical calculation that a Planck supports our analytical work. The final energy of the cooling is given and discussed. Finally there is a comparison between the present results and our earlier, approximate, treatments.
Applying the momentum method, the Fokker-Planck equation is derived from the Boltzmann equation for the r -4 potential between atoms and ions. Two earlier attempts are revealed to be limiting cases of our more general equation, namely for very large and very small mass ratio of the collision partners. The Fokker-Planck equation, after being completed with a decay term and a source term, is solved by the Green's function method. The solution turns out to be composed of modes in the velocity space which decay independently towards a stationary spectrum. The amplitudes of the latter depend on the source term which we assume to be shaped like a delta -function if laser pumping is accounted for. Only for very long relaxation times does the lowest mode prevail in the spectrum which then merges into the Maxwell distribution. Several new lines for ion-atom pairs as well as atom-atom pairs have been measured. They are presented along with the fits using the aforementioned formula. As expected, the agreement is satisfactory only in the first case. In the second case the Fokker-Planck approximation no longer applies.
Atomic beam deceleration and cooling by frequency sweep techniques is analysed by means of a Fokker-Planck equation (FPE) with time dependent drift and diffusion coefficients. The effect of compression of the velocity distribution is discussed with respect to theoretical and experimental limitations of atomic beam cooling imposed by the photon recoil. The cooling limit derived from the linear approximation of the FPE fails, if the broadening of the cooled velocity distribution peak by diffusion causes the loss of atoms from the resonant velocity range. This phenomenon is studied by numerical solutions of the FPE. The occurrence of higher final temperatures in previous Monte-Carlo calculations and in experiments is discussed. The anomalous diffusion coefficient caused by the antibunching of the scattered photons is taken into account. The possibility of transverse compression of the atomic beam is investigated by means of a reduced FPE for the transverse degrees of freedom.