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Physical Review A

Relaxation processes in coherent-population trapping

E. Arimondo

Published Sep 1, 1996• 82 citations
Publisher

Abstract

The role played by a buffer gas in the ground-state superposition created in coherent-population-trapping experiments is investigated theoretically. The density matrix equations are solved on the basis of the coupled-noncoupled states. The velocity-changing collisions are examined in the limit of strong collisions. The relaxation rate for transfer of the ground-state superposition between atoms with different velocities is derived from a fit of available experimental results. \textcopyright{} 1996 The American Physical Society.

DOI
10.1103/physreva.54.2216

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The role played by a buffer gas in the ground-state superposition created in coherent-population-trapping experiments is investigated theoretically. The density matrix equations are solved on the basis of the coupled-noncoupled states. The velocity-changing collisions are examined in the limit of strong collisions. The relaxation rate for transfer of the ground-state superposition between atoms with different velocities is derived from a fit of available experimental results. \textcopyright{} 1996 The American Physical Society.
We examine the role of ground-state coherence on the fluorescence from the excited state in a three-level $\ensuremath{\Lambda}$ system under continuous-wave excitation. When resonantly pumped, the fluorescence is strongly affected by any perturbation in the ground-state coherent population trapping. From our analytical results, we suggest how to use this strong dependence of fluorescence on the ground-state decays as a new method for the measurement of the ground-state decoherence rates. Our calculations should work well for a wide range of systems such as atomic vapors with a buffer gas, active dopants in solids, and quantum dots, etc. We present a proof of principle experiment using 87Rb^{87}\mathrm{Rb}87Rb vapor.
We discuss optical pumping when (a) the collision rates of optically pumped atoms with atoms or molecules of the background gas are small enough that individual velocity groups can be preferentially excited by a monochromatic light beam, (b) the collision rates are still fast enough to partially transfer the spin polarization to other velocity groups, and (c) there are nonnegligible losses of polarization due to collisional spin relaxation and Larmor precession. These conditions lead to a strong correlation between the velocity and the spin polarization of the atoms---that is, to ``spin-tagging'' of the different velocity groups. This regime is similar to that of optically pumped 23Na^{23}\mathrm{Na}23Na atoms of the Earth's upper atmosphere, but it is seldom encountered in laboratory experiments. For cooling and trapping experiments, the collision rates with background gas are negligible. For gas-cell experiments the velocity-changing rates are normally so fast compared to spin relaxation or Larmor precession rates that the atoms have a Maxwellian velocity distribution with negligible correlation between the spin polarization and the velocity. We analyze the limiting cases of strong and weak collisions, which change the velocity by a large or small fraction, respectively, of the mean thermal velocity. The Keilson-Storer model [J. Keilson and A. E. Storer, Q. Appl. Math. 10, 243 (1952)] is used to discuss strong collisions, with memory parameter $\ensuremath{\alpha}=0$ and weak collisions with $\ensuremath{\alpha}\ensuremath{\rightarrow}1$. For weak collisions, the physics can be modeled by coupled Fokker-Planck equations, identical to those for forced diffusion in a harmonic-oscillator potential well. In this limit there are solutions analogous to the quantum mechanical coherent states of a harmonic oscillator.
An analytical theory of optical pumping of velocity-selective coherent population trapping states in three-level Λ-atoms is presented. The theory is based on an atomic density matrix in the superpositional state representation. The representation introduced for the atomic density matrix directly describes coherent population trapping (CPT) states in a three-level Λ-atom and yields an approximate analytical description for the shape of the two-peak momentum (velocity) distribution of atoms. Optical pumping of velocity-selective CPT states is described in the superpositional state representation as a process of redistribution of atoms in the region of small velocities due to the photon recoil. Typical times of formation of velocity-selective CPT states are found to be inversely proportional to the square of the recoil energy.
The velocity changing collisions (vcc) play an important role in the description of velocity redistribution of particles caused by nearly resonant interaction with the laser light and collisions (1). Collision relaxation rates are conveniently described by collision kernels which, in turn, depend on interatomic potential and scattering amplitudes and is proportional to the collision cross section. The population kernel gives the probability density per unit time that an active particle in state 1 i> undergoes collisions with perturbers (usually a noble gas atom or impurity), which changes the active particle velocity from v′ to v. Due to the fact that perturber-active particle interaction is state dependent the collisions cause different changes in the ground and excited states of active particle. Although the shape of the populations distributions resembles those from well-known Bennett hole theory, the distribution of the total population has two peaks and obviously is asymmetrical (2). Vcc together with velocity selective laser excitation lead to the asymmetry of the ground and excited state population. The flux of excited particles has properties which are different from that of ground state because of interaction collision potentials for both states. The intensity of the laser light strongly influences the velocity distributions of the ground and excited state populations.
The shape of the coherent population trapping resonance in optically thin cells with antirelaxation coating of the walls has been determined by solving the quantum kinetic equation for the single-atom density matrix. It has been found that motion and collisions with walls in the case of nondegenerate ground and excited states result in a significant change in the shape of the spectrum, in the appearance of a comb of additional resonances, and in shifts of the main resonance of coherent population trapping, which nonmonotonically depend on the size of the cell.