Local dynamics of laser cooling in an optical lattice

Abstract

Using the basis of Wannier states, we study the local dynamics of polarization gradient cooling for an atom driven on a Jg52!Je53 transition by a one-dimensional optical lattice. This analysis allows us to formulate a physical picture of the cooling mechanism, analogous to Sisyphus cooling for a J g51/2!Je53/2 atom, which depends strongly on coherences in the Zeeman sublevels of the ground state. In addition, we are able to explain the steady-state properties of the laser-cooled atoms in a regime where the standard semiclassical analysis breaks down. @S1050-2947~97!50309-X# PACS number~s!: 32.80.Pj, 71.23.An The paradigm for sub-Doppler laser cooling is the Sisyphus effect as formulated by Dalibard and Cohen-Tannoudji @1#. The beauty of this model is that it gives a simple physical picture of the cooling mechanism. An atom driven on a Jg51/2!J e53/2 transition moves in a one-dimensional ~1D! light field formed by two counterpropagating plane waves with orthogonal linear polarizations—the 1D lin ’lin configuration. Cooling occurs as atoms are preferentially optically pumped from the peaks of the light shift potential associated with one ground state to the valleys of the other. Indeed, the simplicity of this model led to the realization of this 1D model in the laboratory, signifying a shift away from the image of atoms moving in a disordered ‘‘optical molasses’’ to that of an ordered ‘‘optical lattice’’ @2#. These experiments showed that in the steady state most of the atoms are trapped in the microscopic potential wells of the lattice, oscillating many periods before inelastic photon scattering occurs, and localized to a small fraction of the optical wavelength ~the ‘‘oscillating regime’’ !@ 3 # . Despite the considerable success of the simple Sisyphus model in explaining the basic steady-state properties of the laser-cooled atoms, there has been little detailed study of the dynamics. Recent experiments have shown some discrepancies with the dynamical model proposed in the simple Sisyphus picture @4#. The goal of this article is to develop a physical picture of the cooling mechanism applicable to the experiments: atoms with larger angular momentum trapped in the oscillating regime. Consider, for example, an atom with a ground-state angular momentum Jg52 and excited state Je53, moving in a 1D lin’lin lattice. In this more complex system there is a set of optical potentials associated with the five ground states, coupled by stimulated Raman transitions. For such an atom, two possible semiclassical pictures of the cooling mechanism arise. In one @Fig. 1~a!#, cooling occurs locally at a given lattice site as atoms preferentially climb steep potentials and descend shallow ones. Because of the difference in curvature between these two potentials, atoms expend more of their kinetic energy climbing wells than they regain on the descent, the difference being dissipated in optical pumping. Alternatively, atoms may cool when hopping between lattice sites by making nonadiabatic transitions between the coupled set of optical potentials and preferentially pumping to the potential with the largest light shift @Fig. 1~b!#. The difference between these mechanisms has implications for the diffusion of atoms within the lattice, as studied in recent experiments @5#, and theories @6#. Though fully quantum-mechanical numerical simulations for large-angular-momentum atoms in optical lattices @7,8# have yielded fantastic agreement with experiments, it is difficult to extract the local cooling mechanism. In these simulations the center-of-mass motion was expanded in a basis of free-particle plane waves or Bloch states of the periodic potential @9#, which were delocalized over the entire lattice. In order to gain physical insight into the local cooling dynamics, we reanalyze the system using the basis of Wannier states for the periodic potential. These represent a complete orthonormal set of functions localized at a particular lattice site @10#. As discussed in @2,7#, after identifying the complete set of symmetries of the Hamiltonian, the energy eigenstates are Bloch spinors,