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Chinese Physics B

Exact results on cavity cooling in a system of a two-level atom and a cavity field

Yuqing Zhang, Lei Tan, Zhu Zhong-Hua, Liwei Liu

Published Mar 1, 2010• 1 citation
Publisher

Abstract

Using the algebraic dynamical method, this paper investigates the laser cooling of a moving two-level atom coupled to a cavity field. Analytical solutions of optical forces and the cooling temperatures are obtained. Considering Rb atoms as an example, it finds that the numerical results are relevant to the recent experimental laser cooling investigations.

DOI
10.1088/1674-1056/19/3/033202

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Using the algebraic dynamical method, this paper investigates the laser cooling of a moving two-level atom coupled to a cavity field. Analytical solutions of optical forces and the cooling temperatures are obtained. Considering Rb atoms as an example, it finds that the numerical results are relevant to the recent experimental laser cooling investigations.
Using quantum wavefunction simulations, the dynamics of an inverted two-level atom strongly coupled to a mode of an optical high-Q resonator is investigated. It is found that the generated laser light attracts the atom to field antinodes and cools its motion if the cavity mode eigenfrequency is larger than the atomic transition frequency. The system is treated via the Heisenberg-Langevin equations (HLE) and derive analytic expressions for the photon number, the force acting on the atom, and the atomic equilibrium temperature.
We compare the efficiencies of two optical cooling schemes, where a single particle is either inside or outside an optical cavity, under experimentally-realisable conditions. We evaluate the cooling forces using the general solution of a transfer matrix method for a moving scatterer inside a general one-dimensional system composed of immobile optical elements. Assuming the same atomic saturation parameter, we find that the two cooling schemes provide cooling forces and equilibrium temperatures of comparable magnitude.
We present a microscopic laser model for many atoms coupled to a single-cavity mode, including the light forces, which result from atom-field momentum exchange. Within a semiclassical description, we solve the equations for the atomic motion and the internal dynamics to obtain analytic expressions for the optical potential and the friction force seen by each atom. When optical gain is maximum at frequencies where the light field extracts kinetic energy from the atomic motion, the dynamics combines optical lasing and motional cooling. From the corresponding momentum-diffusion coefficient we predict sub-Doppler temperatures in the stationary state. This generalizes the theory of cavity-enhanced laser cooling to active cavity systems. We identify the gain-induced reduction of the effective resonator linewidth as the key origin for the faster cooling and the lower temperatures, which implies that a bad cavity with a gain medium can replace a high-QQQ cavity. In addition, this shows the importance of light forces for gas lasers in the low-temperature limit, where atoms can arrange in a periodic pattern maximizing gain and counteracting spatial hole burning. Ultimately, in the low-temperature limit, such a setup should allow us to combine optical lasing and atom lasing in a single device.
We study a single incoherently pumped atom moving within an optical high-QQQ resonator in the strong-coupling regime. Using a semiclassical description for the atom and field dynamics, we derive a closed system of differential equations to describe this coupled atom-field dynamics. For sufficiently strong pumping, the system starts lasing when the atom gets close to a field antinode, and the associated light forces provide for self-trapping of the atom. For a cavity mode blue detuned with respect to the atomic transition frequency, this is combined with cavity-induced motional cooling, allowing for long-term steady-state operation of such a laser. The analytical results for temperature and field statistics agree well with our earlier predictions based on quantum Monte Carlo simulations. We find sub-Doppler temperatures that decrease with gain and coupling strength, and can even go beyond the limit of passive cavity cooling. Besides demonstrating the importance of light forces in single-atom lasers, this result also gives strong evidence to enhance laser cooling through stimulated emission in resonators.
We investigate the external and internal dynamics of a two-level atom in a standing wave cavity. In the strong coupling regime, where the atom field coupling ggg dominates the atomic and cavity decay rates (\ensuremath{\Gamma},\ensuremath{\kappa}), a cooling mechanism entirely different from free-space Doppler cooling appears. Under suitable operating conditions, the cavity dynamics induces a Sisyphus type cooling, which is the dominant contribution to the total friction force acting on a moving atom. Simple equations describing the key properties of this effect are derived from a completely classical picture and confirmed by a semiclassical approach. The model is investigated in the bad and good cavity limits, and analytic expressions for the friction coefficient and the momentum diffusion for slow atoms are derived. Using a continued fractions expansion the cooling force for arbitrary velocities is evaluated numerically. The result is used to calculate the equilibrium temperatures of the atom of the order of $\mathrm{kT}\ensuremath{\approx}\ensuremath{\Elzxh}\ensuremath{\kappa},$ which can be much lower than the free-space Doppler limit and agree well to those obtained by quantum wave-function simulations.