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Physical Review A

Cooling an atom in a weakly driven high- Q cavity

Gerald Hechenblaikner, Markus Gangl, Péter Horák, Helmut Ritsch

Published Oct 1, 1998• 138 citations
Publisher

Abstract

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 $g$ 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.

DOI
10.1103/physreva.58.3030

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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 kT\ensuremath≈\ensuremath\Elzxh\ensuremathκ,\mathrm{kT}\ensuremath{\approx}\ensuremath{\Elzxh}\ensuremath{\kappa},kT\ensuremath≈\ensuremath\Elzxh\ensuremathκ, which can be much lower than the free-space Doppler limit and agree well to those obtained by quantum wave-function simulations.
A theoretical study is carried out for the cavity cooling of a ΛΛΛ-type three level atom in a high-finesse optical cavity with a weakly driven field. Analytical expressions for the friction, diffusion coefficients and the equilibrium temperatures are obtained by using the Heisenberg equations, then they are calculated numerically and shown graphically as a function of controlling parameters. For a suitable choice of these parameters, the dynamics of the cavity field interaction with the ΛΛΛ-type three-level atom introduces a sisyphus cooling mechanism yielding lower temperatures below the Doppler limit and allowing larger cooling rate, avoiding the problems induced by spontaneous emission.
We extend an earlier semiclassical model to describe the dissipative motion of N atoms coupled to M modes inside a coherently driven high-finesse cavity. The description includes momentum diffusion via spontaneous emission and cavity decay. Simple analytical formulas for the steady-state temperature and the cooling-time for a single atom are derived and show surprisingly good agreement with direct stochastic simulations of the semiclassical equations for N atoms with properly scaled parameters. A thorough comparison with standard free-space Doppler cooling is performed and yields a lower temperature and a cooling-time enhancement by a factor of M times the square of the ratio of the atom-field coupling constant to the cavity decay rate. Finally, it is shown that laser cooling with negligible spontaneous emission should indeed be possible, especially for relatively light particles in a strongly coupled field configuration.
We investigate the coupled dynamics of N two-level atoms in the field of a weakly pumped multimode high-Q cavity. While the field mode amplitudes are coupled by the atoms depending on their positions, the motion of the atoms is influenced by the radiation pressure and dipole force induced by the total intracavity field. In the strong coupling regime, where the atom field coupling g dominates the atomic and cavity decay rates (\ensuremath{\Gamma}, \ensuremath{\kappa}) strong correlations between the atoms themselves and between the atoms and the light field build up. For suitable operating conditions an efficient motional cooling of the atoms occurs. At the example of two atoms coupled to two counterpropagating modes in a ring cavity, we show that this can lead to a strong binding force between atoms trapped at a large distance at different antinodes of the cavity field.
A semiclassical model of the force and momentum diffusion on a point particle is used to describe the motion of a two-level atom strongly coupled to a single Gaussian cavity mode. The effects of the momentum diffusion on the motion of an atom in a cavity are investigated in a regime similar to that of the experiment performed by Mabuchi et al. [Opt. Lett. 21, 1393 (1996)]. It is found that a slow atom quickly develops significant velocities along the cavity axis. The limited bandwidth in the experiment of Mabuchi et al. means that the full intensity signal due to atomic motion in the standing wave is filtered leading to the apparently smaller velocities observed. It is shown that a negative detuning of the laser and cavity from the atomic resonance would lead to nonzero dipole forces and significantly reduced velocities along the standing wave. An analysis of the intensity signal with a larger bandwidth is proposed, which would track the velocity of the atom along the cavity axis. These results are compared with a Monte Carlo wave-function simulation similar to that used to treat Doppler cooling.
Cavity-mediated cooling of the center--of--mass motion of a transversally, coherently pumped atom along the axis of a high--Q cavity is studied. The internal dynamics of the atomic dipole strongly coupled to the cavity field is treated by a non-perturbative quantum mechanical model, while the effect of the cavity on the external motion is described classically in terms of the analytically obtained linear friction and diffusion coefficients. Efficient cavity-induced damping is found which leads to steady-state temperatures well-below the Doppler limit. We reveal a mathematical symmetry between the results here and for a similar system where, instead of the atom, the cavity field is pumped. The cooling process is strongly enhanced in a degenerate multimode cavity. Both the temperature and the number of scattered photons during the characteristic cooling time exhibits a significant reduction with increasing number of modes involved in the dynamics. The residual number of spontaneous emissions in a cooling time for large mode degeneracy can reach and even drop below the limit of a single photon.
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.