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Physical Review A

Sub-Doppler cooling in reduced-period optical lattice geometries

P. R. Berman, Georg A. Raithel, R. Zhang, Vladimir S. Malinovsky

Published Sep 21, 2005• 7 citations
Publisher

Abstract

It is shown that sub-Doppler cooling occurs in an atom-field geometry that can lead to reduced-period optical lattices. Four optical fields are combined to produce a ``standing wave'' Raman field that drives transitions between two ground state sublevels. In contrast to conventional Sisyphus cooling, sub-Doppler cooling to zero velocity occurs when all fields are polarized in the same direction. Solutions are obtained using both semiclassical and quantum Monte Carlo methods in the case of exact two-photon resonance. The connection of the results with conventional Sisyphus cooling is established using a dressed state basis.

DOI
10.1103/physreva.72.033415

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It is shown that sub-Doppler cooling occurs in an atom-field geometry that can lead to reduced-period optical lattices. Four optical fields are combined to produce a ``standing wave'' Raman field that drives transitions between two ground state sublevels. In contrast to conventional Sisyphus cooling, sub-Doppler cooling to zero velocity occurs when all fields are polarized in the same direction. Solutions are obtained using both semiclassical and quantum Monte Carlo methods in the case of exact two-photon resonance. The connection of the results with conventional Sisyphus cooling is established using a dressed state basis.
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 present a general theoretical description of laser cooling of three-level atoms in a Λ configuration in two standing waves. Detunings, intensity, and spatial phase difference of the two waves are arbitrary. We demonstrate that, for a zero spatial phase shift, sub-Doppler cooling requires equal detunings. The basic mechanism is then coherent population trapping. For different detunings sub-Doppler cooling occurs only when the spatial phases of the two standing waves are different. This effect can be traced back to a mechanism of the Sisyphus type. These schemes of sub-Doppler cooling require neither polarization gradients nor magnetic fields.
A theoretical analysis is given for 1-D laser cooling below the Doppler limit of J = 1/2 ground- state atoms. The laser field consists of a pair of counterpropagating, linearly polarized low- power beams, whose polarization directions differ by an angle θ (0 ≤ θ ≤ π/2). For θ ≪ 1, the effective optical pumping time is shown to increase strongly near the nodes of the standing wave and the cooling force can be much larger than that for θ ~1. Moreover, for θ ≪ 1, it can be shown that the stimulated part of the atomic diffusion is reduced considerably as compared with that for θ ~ 1. As a consequence it is possible to achieve an equilibrium atomic distribution that, for θ ≪ 1, is characterized by a mean kinetic energy lower than that predicted to occur for θ = π/2. The equilibrium velocity distribution is not necessarily Maxwellian and thus the temperature of the atomic ensemble may not be well defined. The achievable kinetic energy is so small that the cooled atoms are trapped in the vicinity of the laser field nodes. An explanation for these effects is given in terms of Sisyphus cooling.
Reduced-period optical lattices based on Raman transitions allow for sub-Doppler laser cooling. An important parameter of this Raman optical lattice is the frequency difference ${\ensuremath{\Delta}}_{\mathrm{d}}$ between two virtual energy levels involved in the atom-field interaction scheme. In this work, we use experimental time-of-flight data and quantum simulations to characterize laser cooling in the Raman lattice as a function of ${\ensuremath{\Delta}}_{\mathrm{d}}$. Two different domains of laser cooling are identified. For small ${\ensuremath{\Delta}}_{\mathrm{d}}$, atoms are cooled due to a well-known mechanism that also occurs in standard optical lattices. For large ${\ensuremath{\Delta}}_{\mathrm{d}}$, atoms are cooled based on Raman transitions. We study the transition between the two domains of laser cooling in detail.
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.
Summary form only given. Laser cooled atoms play a critical role in modern designs of frequency standards such as atomic fountains. Recently, an elegant and efficient method of two-dimensional cooling to the vibrational ground state of a far-off resonance optical lattice has been demonstrated. This method is a variant of Raman sideband cooling based on the Raman transitions between the vibrational manifolds of adjacent Zeeman substates. A static magnetic field is used to tune the Zeeman levels so that Raman resonance occurs on the red sideband and results in cooling. Two circularly polarized fields are used to recycle the atoms for repetitive Raman cooling. Unfortunately, a direct application of this scheme to frequency standards meets with several difficulties. Stimulated by these concepts and results, we propose a new variant of transverse sideband cooling, where these problems are avoided, while still maintaining most of the attractive features. The field configuration used for the optical lattice consists of three linearly polarized beams having equal amplitudes and propagating in the xy plane with angles of 2/spl pi//3 between each other. The polarization vectors of these beams are tilted through a small angle with respect to the z-axis. All the beams have the same frequency far-detuned to the red of the D/sub 2/ resonance line. Raman transitions between vibrational levels of adjacent magnetic substates are induced by the small in-plane component of the field. Two other important requirements for Raman sideband cooling are a uniform energy shift of the magnetic sublevels and optical pumping.