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Journal of the Optical Society of America B

Sisyphus cooling of a bound atom

David J. Wineland, Jean Dalibard, Claude N. Cohen-Tannoudji

Published Jan 1, 1992• 79 citations
Publisher

Abstract

Cooling that results from optical dipole forces is considered for a bound atom. Through optical pumping, the atom can be made to feel decelerating optical dipole forces more strongly than accelerating optical dipole forces. This effect, which has previously been realized for free atoms, is called Sisyphus cooling. A simple model for a bound atom is examined in order to reveal the basic aspects of cooling and heating when the atom is confined in the Lamb–Dicke regime. Results of semiclassical and quantum treatments show that the minimum energy achieved is near the zero-point energy and can be much lower than the Doppler cooling limit. Two practical examples that approximate the model are briefly examined.

DOI
10.1364/josab.9.000032

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Cooling that results from optical dipole forces is considered for a bound atom. Through optical pumping, the atom can be made to feel decelerating optical dipole forces more strongly than accelerating optical dipole forces. This effect, which has previously been realized for free atoms, is called Sisyphus cooling. A simple model for a bound atom is examined in order to reveal the basic aspects of cooling and heating when the atom is confined in the Lamb–Dicke regime. Results of semiclassical and quantum treatments show that the minimum energy achieved is near the zero-point energy and can be much lower than the Doppler cooling limit. Two practical examples that approximate the model are briefly examined.
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.
Using a laser polarization gradient, we realize 3D Sisyphus cooling of ^{171}Yb^{+} ions confined in and near the Lamb-Dicke regime in a linear Paul trap. The cooling rate and final mean motional energy of a single ion are characterized as a function of laser intensity and compared to semiclassical and quantum simulations. Sisyphus cooling is also applied to a linear string of four ions to obtain a mean energy of 1-3 quanta for all vibrational modes, an approximately order of magnitude reduction below Doppler cooled energies. This is used to enable subsequent, efficient sideband laser cooling.
The semiclassical theory of laser cooling is applied for the analysis of cooling of unbound atoms with the values of the ground and exited state angular moments 1/2 in a one-dimensional nondissipative optical lattice. We show that in the low-saturation limit with respect to the pumping field a qualitative interpretation of the cooling mechanisms can be made by the consideration of effective two-level system of the ground-state sublevels. It is clarified that in the limit of weak Raman transitions the cooling mechanism is similar to the Doppler mechanism, which is known in the theory of two-level atom. In the limit of strong Raman transitions the cooling mechanism is similar to the known Sisyphus mechanism. In the slow atom approximation the analytical expressions for the coefficients of friction, spontaneous and induced diffusion are given, and the kinetic temperature is estimated.
We propose a laser-driven Sisyphus-cooling scheme for atoms confined in a far-off resonance optical dipole trap. Utilizing the differential trap-induced ac Stark shift, two electronic levels of the atom are resonantly coupled by a cooling laser preferentially near the trap bottom. After absorption of a cooling photon, the atom loses energy by climbing the steeper potential, and then spontaneously decays preferentially away from the trap bottom. The proposed method is particularly suited to cooling alkaline-earth-metal-like atoms where two-level systems with narrow electronic transitions are present. Numerical simulations for the cases of 88{}^{88}88Sr and 174{}^{174}174Yb demonstrate the expected recoil and Doppler temperature limits. The method requires a relatively small number of scattered photons and can potentially lead to phase-space densities approaching quantum degeneracy in subsecond time scales.
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.
A cooling process for optical dipole traps, rf-induced Sisyphus cooling in a circularly polarized far-off-resonance trap, is demonstrated. Cooling rates of up to $\ensuremath{\approx}300 \ensuremath{\mu}\mathrm{K}/\mathrm{s}$ are observed without significant atom loss. The final temperature achieved was (a somewhat disappointing) $17 \ensuremath{\mu}\mathrm{K}.$ Experimental studies and theoretical analysis of this cooling process were carried out, which explain that the temperature limit is due to an equilibrium between the small fraction of total thermal energy that is removed per Sisyphus cycle and the multiple photon recoils that are required to complete each cycle. Light-induced collisional loss made it difficult to operate under conditions where a larger fraction of the energy would be removed. A method for storing atoms in an optical trap with only a single confined spin state is also demonstrated.