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Physical review. E

Brillouin propagation modes of cold atoms undergoing Sisyphus cooling

David Cubero

Published Mar 2, 2023• 1 citation
Publisher

Abstract

An exact expression for the average velocity of cold atoms in a driven, dissipative optical lattice in terms of the amplitudes of atomic density waves is derived from semiclassical equations for the phase space densities of the Zeeman ground-state sublevels. The calculations are for a J_{g}=1/2→J_{e}=3/2 transition, as it is customary in theoretical studies of Sisyphus cooling. While the driver, an additional beam of small amplitude, sets the atoms into directed motion, the new expression permits the quantification of the contribution to the atomic motion of a specific atomic wave, revealing unexpected counterpropagating contributions from many modes. Additionally, the method is shown to provide the generic threshold for the transition into the regime of infinite density, regardless of the details, or even the presence, of driving.

DOI
10.1103/physreve.107.034102

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An exact expression for the average velocity of cold atoms in a driven, dissipative optical lattice in terms of the amplitudes of atomic density waves is derived from semiclassical equations for the phase space densities of the Zeeman ground-state sublevels. The calculations are for a J_{g}=1/2→J_{e}=3/2 transition, as it is customary in theoretical studies of Sisyphus cooling. While the driver, an additional beam of small amplitude, sets the atoms into directed motion, the new expression permits the quantification of the contribution to the atomic motion of a specific atomic wave, revealing unexpected counterpropagating contributions from many modes. Additionally, the method is shown to provide the generic threshold for the transition into the regime of infinite density, regardless of the details, or even the presence, of driving.
An exact series expansion of the average velocity of cold atoms in dissipative optical lattices under probe driving, based on the amplitudes of the excited atomic density waves, is derived from the semiclassical equations for the phase space densities of the Zeeman ground-state sublevels. This expansion permits the identification of the precise contribution to the current of a propagating atomic wave for the specific driving, as well as providing the general threshold for the transition into the regime of infinite density.
Atoms confined in a three-dimensional dissipative optical lattice oscillate inside potential wells, occasionally hopping to adjacent wells, thereby diffusing in all directions. Illumination by a weak probe beam modulates the lattice, yielding propagating atomic density waves, referred to as Brillouin modes which travel perpendicular to the direction of travel of the probe. The probe is made incident at a small angle relative to a lattice symmetry axis, yielding a driving potential perturbation whose spatial period is not a multiple of the period of the underlying optical potential, thus enabling exploration of the regime of space quasiperiodic drive. A theory, based on the Fourier decomposition of the current into its atomic density wave contributions, reveals that unlike the previously studied time quasiperiodic case, wherein a lattice driven by two incommensurate frequencies may exhibit abrupt suppression in directed current as the driving transitions from quasiperiodic to periodic, a spatial-quasiperiodically driven lattice exhibits no such abrupt response. Further, detailed modeling of spatial-quasiperiodically driven lattices reveals that directed propagation occurs not only as a consequence of velocity-matching between the propagating modulation and the average velocity of the atom oscillating inside a well as was previously reported in the literature, but also as a distinct consequence of a new mechanism, namely, frequency-matching between the modulation frequency and the oscillation frequencies. A systematic measurement of the transmitted probe spectra as a function of off-axis probe angle is presented, which is consistent with the velocity- and frequency-matching predictions from the detailed model.
Using the basis of Wannier states, we study the local dynamics of polarization gradient cooling for an atom driven on a ${J}_{g}=2\ensuremath{\rightarrow}{J}_{e}=3$ 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 ${J}_{g}=1/2\ensuremath{\rightarrow}{J}_{e}=3/2$ 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.
Atoms confined in a three-dimensional dissipative optical lattice oscillate inside potential wells, occasionally hopping to adjacent wells, thereby diffusing in all directions. Illumination by a weak probe beam modulates the lattice, yielding propagating atomic density waves, referred to as Brillouin modes, which travel perpendicular to the direction of travel of the probe. The probe is made incident at a small angle relative to a lattice symmetry axis, yielding a driving potential perturbation whose spatial period is not a multiple of the period of the underlying optical potential, thus enabling exploration of the regime of space quasiperiodic drive. A theory, based on the Fourier decomposition of the current into its atomic density wave contributions, reveals that, unlike the previously studied time quasiperiodic case, wherein a lattice driven by two incommensurate frequencies may exhibit abrupt suppression in directed current as the driving transitions from quasiperiodic to periodic, a spatial-quasiperiodically driven lattice exhibits no such abrupt response. Further, detailed modeling of spatial-quasiperiodically driven lattices reveals that directed propagation occurs not only as a consequence of velocity matching between the propagating modulation and the average velocity of the atom oscillating inside a well, as was previously reported in the literature, but also as a distinct consequence of another mechanism, namely, frequency matching between the modulation frequency and the oscillation frequencies. A systematic measurement of the transmitted probe spectra as a function of off-axis probe angle is presented, which is consistent with the velocity- and frequency-matching predictions from the detailed model.
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.
We present theoretical descriptions of Sisyphus cooling in a two-dimensional laser configuration, based on semiclassical as well as quantum approaches. A detailed comparison of these various approaches and a discussion of their range of validity are given. Features of the cooled atomic distribution such as channeling of the atoms along specific lines in momentum space and the observability of tunneling induced resonances in the population of the ground state of the atomic motion in the optical potential wells are discussed.