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Physical Review A

Radiative cooling force in atoms with multiplet structure

Arturo Bambini, A. Agresti

Published Oct 1, 1997• 18 citations
Publisher

Abstract

This paper extends the calculation of laser cooling forces on atoms with magnetic degeneracy to the case where the excited state is composed by a multiplet of levels with arbitrary energy separation. The sub-Doppler force that arises in the ${\ensuremath{\sigma}}_{+},{\ensuremath{\sigma}}_{\ensuremath{-}}$ field configuration is found to be strongly affected if other atomic levels lie close to the excited level of the cooling transition. The paper examines in detail the case in which the excited multiplet comes from spin-orbit coupling of angular momentum $L=1$ and spin $S=1,$ forming the ${J}_{e}=0,$ ${J}_{e}=1,$ and ${J}_{e}=2$ energy levels. The cooling transition ${J}_{g}=1\ensuremath{\leftrightarrow}{J}_{e}=2$ shows the sub-Doppler structure due to magnetic degeneracy. The force is modified by the two other transitions (if sufficiently close), even if these do not provide any mechanical effect on the atom when acting alone. The paper presents a detailed scheme for the construction of the relevant optical Bloch equations for the present case, which can be generalized to treat more complex atomic structures. The solutions to these equations are then discussed; several cases of energy separation are worked out and the ensuing graphs of the radiative force are shown. The calculations were carried out in both low- and strong-field regimes.

DOI
10.1103/physreva.56.3040

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This paper extends the calculation of laser cooling forces on atoms with magnetic degeneracy to the case where the excited state is composed by a multiplet of levels with arbitrary energy separation. The sub-Doppler force that arises in the \ensuremathσ+,\ensuremathσ\ensuremath−{\ensuremath{\sigma}}_{+},{\ensuremath{\sigma}}_{\ensuremath{-}}\ensuremathσ+​,\ensuremathσ\ensuremath−​ field configuration is found to be strongly affected if other atomic levels lie close to the excited level of the cooling transition. The paper examines in detail the case in which the excited multiplet comes from spin-orbit coupling of angular momentum L=1L=1L=1 and spin S=1,S=1,S=1, forming the Je=0,{J}_{e}=0,Je​=0, Je=1,{J}_{e}=1,Je​=1, and Je=2{J}_{e}=2Je​=2 energy levels. The cooling transition Jg=1\ensuremath↔Je=2{J}_{g}=1\ensuremath{\leftrightarrow}{J}_{e}=2Jg​=1\ensuremath↔Je​=2 shows the sub-Doppler structure due to magnetic degeneracy. The force is modified by the two other transitions (if sufficiently close), even if these do not provide any mechanical effect on the atom when acting alone. The paper presents a detailed scheme for the construction of the relevant optical Bloch equations for the present case, which can be generalized to treat more complex atomic structures. The solutions to these equations are then discussed; several cases of energy separation are worked out and the ensuing graphs of the radiative force are shown. The calculations were carried out in both low- and strong-field regimes.
We give a general theoretical description of radiative forces on atoms in a monochromatic radiation field. The driven transition has a lower level consisting of several substates, and we consider the limit of low velocities and weak intensities. This situation comprises the schemes where cooling to temperatures below the Doppler limit is possible. After expressing the atomic dipole in terms of the polarizability tensor, we obtain an expression of the force as the sum of the radiation pressure, the dipole force, and a contribution from the gradient of the polarization. This latter contribution contains a part resulting from redistribution of photons between the plane waves that compose the field, and a part resulting from fluorescence. We express the average force and the momentum-diffusion tensor in terms of a closed evolution equation for the lower-state density matrix. We show by a scaling argument that in any case of cooling, the final temperature goes down linearly with the intensity, until the recoil limit is approached. The description is valid for a weak and monochromatic, but otherwise arbitrary field, and it allows for the presence of an external magnetic field, and of hyperfine splitting. It is applicable to cooling in one, two, or three dimensions. We give numerical results for the average force in a number of specific cases of two counterrunning plane waves.
We extend the kinetic theory of the laser cooling of atoms to four multilevel atomic schemes: (3+5)-, (5+7)-, (7+9)-, and (9+11)-level atom. In all four atomic schemes the atoms are considered to be excited by two counterpropagating circularly polarized laser waves. A comparison of four multilevel atomic schemes shows that the even-order multiphoton processes are responsible for the narrow structures near zero velocity in the atomic coherences, atomic populations, dipole radiation forces, and the diffusion coefficients. The even-order multiphoton processes in a multilevel atom with a big number of magnetic sublevels are shown to produce the velocity structures, which are much narrower than that produced by the two-photon processes in a (3+5)-level atom. It is found that the multiphoton processes enhance the radiation force at low velocities and the friction produced by the force at a negative detuning. The even-order multiphoton processes caused by the laser waves are identified as the basic physical mechanism responsible for the sub-Doppler laser cooling of atoms. The temperature of laser-cooled atoms is derived for all four models and found to be lower for atomic schemes with a larger number of magnetic sublevels.
We analyze the influence of an arbitrarily directed uniform magnetic field on the laser sub-Doppler cooling of atoms. The analysis is done for a (3+5)-level atom excited by a ${\ensuremath{\sigma}}^{+}\ensuremath{-}{\ensuremath{\sigma}}^{\ensuremath{-}}$ laser field configuration. Our analysis shows that the effects of the magnetic field depend strongly on the direction of the magnetic field. In an arbitrarily directed magnetic field the laser cooling configuration produces both the main resonance existing already at zero magnetic field and additional sub-Doppler resonances caused by two-photon and higher-order multiphoton processes. These sub-Doppler resonances are, however, well separated on the velocity scale if the Zeeman shift exceeds the widths of the resonances. This allows one to use the main sub-Doppler resonance for an effective laser cooling of atoms even in the presence of the magnetic field. The effective temperature of the atomic ensemble at the velocity of the main resonance is found to be almost the same as in the absence of the magnetic field. The defined structure of the multiphoton resonances may be of importance for the sub-Doppler laser cooling of atoms, atomic extraction from magneto-optical traps, and applications related to the control of atomic motion.
In the limit of asymptotically small photon momentum, a Fokker–Planck equation for the center-of-mass Wigner function suffices to describe laser cooling of an atom. We have devised a procedure to compute the light pressure force and the diffusion tensor for an atom with an arbitrary level structure, traveling with an arbitrary velocity in any given three-dimensional light field. The scheme is based on solving a set of differential equations closely related to the density matrix equations of motion encountered in conventional recoilless laser spectroscopy. Computer programs have been written that automatically construct and solve the required equations from broad specifications of the problem, such as energies and angular momenta of the atomic levels. A Langevin equation stochastic simulation has been implemented to solve the Fokker-Planck equation of laser cooling using the computed force and diffusion. Comparisons of optical-molasses experiments and simulations carried out for a model sodium atom with a J = 2→J = 3 transition show qualitative similarities such as sub-Doppler limit temperatures, but the quantitative agreement of theory and experiment is not satisfactory.
We report the first experimental study of sub-Doppler cooling in alkaline earth atoms (87Sr) enabled by the presence of nuclear spin-originated magnetic degeneracy in the atomic ground state. Sub-Doppler cooling in a sigma(+)-sigma(-) configuration is achieved despite the presence of multiple, closely spaced excited states. This surprising result is confirmed by an expanded multilevel theory of the radiative cooling force. Detailed investigations of system performance have shed new insights into (sigma(+)-sigma(-)) cooling dynamics and will likely play an important role in the future development of neutral atom-based optical frequency standards.
A realistic model of the dipole radiation forces in transverse Doppler cooling (with a s+-s- laser configuration) of an atomic beam of group 13 elements is studied within the quantum-kinetic equation framework. The full energy level sub-structure for such an atom with I = 0 (such as 66Ga) is analysed. Two cooling strategies are investigated; the first involving the 2P3/2 ? 2D5/2 transition and the second a dual laser cooling experiment involving transitions 2P1/2 and 2P3/2 ? 2S1/2. The latter scheme creates a velocity-independent dark-state resonance that inhibits a steady-state dipole cooling force. However, time-dependent calculations show that transient cooling forces are present that could be exploited for laser cooling purposes in pulsed laser fields.