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Physical Review A

Sub-Doppler laser cooling of atoms: Comparison of four multilevel atomic schemes

Soo Chang, Taeg Yong Kwon, Ho Seong Lee, Vladimir G. Minogin

Published Jun 1, 2001• 12 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.64.013404

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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 extend the kinetic theory of deep laser cooling of multilevel atoms by counterpropagating laser waves to a case of linearly polarized waves and compare the results with those for circularly polarized waves. Considering the excitation of a $(3+5)\ensuremath{-}\mathrm{level}$ atom by two counterpropagating waves with orthogonal linear polarizations $(\mathrm{lin}\ensuremath{\perp}\mathrm{lin}$ configuration), we show that even-order stepwise processes composed of the two-photon optical processes are responsible for narrow structures near zero velocity in the ground-state atomic coherence, atomic populations, and optical coherences. The stepwise processes composed of the two-photon structures are shown to produce velocity structures much narrower than the two-photon structures produced by a single ${\ensuremath{\sigma}}^{+}\ensuremath{-}{\ensuremath{\sigma}}^{\ensuremath{-}}$ configuration [Phys. Rev. A 60, 3148 (1999)]. A Fokker-Planck type kinetic equation is derived for a model of a $(3+5)\ensuremath{-}\mathrm{level}$ atom interacting with two linearly polarized waves in a $\mathrm{lin}\ensuremath{\perp}\mathrm{lin}$ configuration. It is found that stepwise multiphoton coherent processes enhance considerably the radiation force at low velocities and the friction produced by the force at a negative detuning. Contrary to a case of a ${\ensuremath{\sigma}}^{+}\ensuremath{-}{\ensuremath{\sigma}}^{\ensuremath{-}}$ configuration the stepwise processes are found to increase the diffusion coefficient. The stepwise even-order processes caused by two linearly polarized waves are identified as the basic physical mechanism responsible for deep laser cooling of atoms in a $\mathrm{lin}\ensuremath{\perp}\mathrm{lin}$ configuration. The temperature of laser-cooled atoms is derived for a model of a $(3+5)\ensuremath{-}\mathrm{level}$ atom excited by a $\mathrm{lin}\ensuremath{\perp}\mathrm{lin}$ configuration and found to be asymptotically close to that for a case of ${\ensuremath{\sigma}}^{+}\ensuremath{-}{\ensuremath{\sigma}}^{\ensuremath{-}}$ configuration at large detuning.
A theory of deep laser cooling of atoms by counterpropagating circular polarized laser waves in multilevel interaction schemes is developed. Using a (3+5)-level atom as an example, we show that in multilevel interaction schemes two-photon processes are responsible for creating a narrow structure near zero velocity in the ground-state atomic coherence. This structure is in turn responsible for narrow velocity structures in atomic populations and optical coherences. A Fokker-Planck-type kinetic equation is derived for a model of a (3+5)-level interaction scheme. It is found that two-photon coherent processes considerably enhance the radiation force at low velocities and the friction produced by the force at a negative detuning, and considerably decrease the diffusion coefficient. Two-photon processes in multilevel interaction schemes are identified as the basic physical mechanism responsible for deep laser cooling of atoms. The temperature of laser-cooled atoms is derived for a model (3+5)-level atom, and found to be in good agreement with the experimentally observed temperature for laser-cooled cesium atoms.
We present two cooling mechanisms that lead to temperatures well below the Doppler limit. These mechanisms are based on laser polarization gradients and work at low laser power when the optical-pumping time between different ground-state sublevels becomes long. There is then a large time lag between the internal atomic response and the atomic motion, which leads to a large cooling force. In the simple case of one-dimensional molasses, we identify two types of polarization gradient that occur when the two counterpropagating waves have either orthogonal linear polarizations or orthogonal circular polarizations. In the first case, the light shifts of the ground-state Zeeman sublevels are spatially modulated, and optical pumping among them leads to dipole forces and to a Sisyphus effect analogous to the one that occurs in stimulated molasses. In the second case (σ+−σ− configuration), the cooling mechanism is radically different. Even at very low velocity, atomic motion produces a population difference among ground-state sublevels, which gives rise to unbalanced radiation pressures. From semiclassical optical Bloch equations, we derive for the two cases quantitative expressions for friction coefficients and velocity capture ranges. The friction coefficients are shown in both cases to be independent of the laser power, which produces an equilibrium temperature proportional to the laser power. The lowest achievable temperatures then approach the one-photon recoil energy. We briefly outline a full quantum treatment of such a limit.
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.
A theoretical discussion of laser cooling of multilevel atomic systems using intense, blue detuned laser beams is given. A method based on matrix continued fractions is presented, which enables the efficient calculation of the semi-classical force on a multilevel system. The method is applied to a three-level system driven by one standing wave and one travelling wave. The force curves obtained exhibit strong cooling features. Cooling of the three-level system is simulated using the fully quantum mechanical Monte Carlo wave function technique. The simulation predicts efficient cooling to sub-Doppler temperatures. The three-level model is related to a multilevel cooling experiment performed on sp7\\sp7sp7Li. The experimental results are found to be in reasonable agreement with the three-level model. However, a comparison of the simulation and experiment for a two-level atomic system reveals significant discrepancies, which raises questions about the model and experiment.
In sub-Doppler laser cooling of atoms, the lowest temperatures are known to occur at low (but not too low) excitation, and the temperature increases roughly linearly with laser intensity. However, under conditions for which the lowest temperatures are obtained, the small velocity capture range and low optical pumping rate limit the number of atoms that can be collected into the cold sample. In this study, we present measurements for laser cooling of Rb and metastable He atoms for two counterpropagating laser beams with orthogonal linear polarization (lin\ensuremath{\perp}lin) over a wide range of saturation parameters and laser detunings, together with results of semiclassical (Fokker-Planck) and quantum density matrix calculations. We find that at higher laser intensity, a larger number of atoms can be collected into a final velocity distribution that is significantly narrower than that given by a linear extrapolation vs the square root of laser intensity. Two cooling mechanisms are at work: the sub-Doppler Sisyphus mechanism and also Doppler cooling. Under certain conditions that we discuss, Doppler cooling aids Sisyphus cooling by collecting high-velocity atoms beyond the Sisyphus capture range. He∗{\mathrm{He}}^{*}He∗ presents a rather unusual situation in that the minimum average kinetic energy (Eav){(E}_{\mathrm{av}})(Eav​) for purely Sisyphus cooling is comparable to the minimum value of Eav{E}_{\mathrm{av}}Eav​ with Doppler cooling alone.