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Physical Review A

Four-wave mixing in one-dimensional optical molasses

Jingzhong Guo

Published Mar 1, 1995• 5 citations
Publisher

Abstract

We present a detailed calculation of the four-wave-mixing signal from one-dimensional optical molasses, in which atoms are cooled by a pair of counterpropagating, cross-polarized fields. The contributions from both Raman and Rayleigh resonances between the discrete motional bound states and the recoil-induced resonances between the motional continuum states are analyzed. It is shown that the recoil-induced resonances can lead to a narrow central resonance peak in the four-wave-mixing signal with a width given by the transverse residual Doppler width. The calculated Raman sidebands also exhibit an asymmetry in the signal strength, a feature observed in recent experiments.

DOI
10.1103/physreva.51.2338

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Bulletin of the American Physical Society•May 1, 1993•0 citations
We present a detailed calculation of the four-wave-mixing signal from one-dimensional optical molasses, in which atoms are cooled by a pair of counterpropagating, cross-polarized fields. The contributions from both Raman and Rayleigh resonances between the discrete motional bound states and the recoil-induced resonances between the motional continuum states are analyzed. It is shown that the recoil-induced resonances can lead to a narrow central resonance peak in the four-wave-mixing signal with a width given by the transverse residual Doppler width. The calculated Raman sidebands also exhibit an asymmetry in the signal strength, a feature observed in recent experiments.
We present a calculation of the probe absorption spectrum of atoms in a one-dimensional optical molasses formed by a pair of counterpropagating, cross-polarized fields. When the finite angle \ensuremath{\theta} between the propagation direction of a probe field and that of one of the cooling fields is taken into account, we find that, apart from the contributions to the probe absorption signal that are related to the Raman and Rayleigh resonances between the discretized atomic motional states in the molasses direction, there is also a signal arising from the recoil-induced resonances between atomic states with different center-of-mass momenta in the directions transverse to the molasses beams. The magnitudes of these different contributions are compared.
We present a detailed theoretical investigation of the absorption spectrum of a probe beam after transmission through a one-dimensional optical molasses obtained with circularly cross-polarized counterpropagating pump beams (corkscrew molasses). The study is performed for a Jg{\mathit{J}}_{\mathit{g}}Jg​=1\ensuremath{\rightarrow}Je{\mathit{J}}_{\mathit{e}}Je​=2 atomic transition and the external degrees of freedom are described by means of a Fokker-Planck equation in the regime where the stationary momentum distribution lies within the range of linearity of the cooling force. We analyze the stimulated Raman transitions occurring between differently populated and light-shifted ground-state Zeeman sublevels. In the case where the probe beam has the same polarization as the copropagating cooling beam, an additional narrow Rayleigh resonance appears for a probe frequency close to the pump frequency. Its width is equal to 2\ensuremath{\alpha}/M, where \ensuremath{\alpha} is the friction coefficient of the cooling force and M is the atomic mass. In the case where the probe beam has the same polarization as the counterpropagating cooling beam, a central recoil-induced resonance is expected. Its width is related to the Doppler width of the optical molasses.
We have measured the spectrum of fluorescence from 85Rb in 1-D optical molasses. The spectrum consists of a central peak and one set of sidebands characteristic of quantized atomic motion. In a 1-D molasses consisting of two counterpropagating, linear, and orthogonally polarized laser beams, optical potential wells are spaced λ/4 apart where the polarization is circular. Polarization gradient cooling reduces the temperature to a point where the atoms are trapped in these wells. The atomic motion is approximated by a quantum mechanical harmonic oscillator. We interpret the central peak as transitions beginning and ending in the same bound level and the sidebands as Raman transitions changing the vibrational quantum number by ±1. The observed level spacing in the well is proportional to the square root of the light shift. The relative strength of the blue and red sideband is given by a Boltzman factor, allowing us to determine the temperature. As expected, the temperature increases linearly with the light shift.
We present a detailed theoretical investigation of the transmission spectrum of a probe beam interacting with atoms in a one-dimensional optical molasses obtained with linearly cross-polarized counterpropagating pump beams. The study is performed for a Jg{\mathit{J}}_{\mathit{g}}Jg​=1/2\ensuremath{\rightarrow}Je{\mathit{J}}_{\mathit{e}}Je​=3/2 atomic transition in the limit where the Hamiltonian part of the atom-field coupling is predominant over the relaxation part. We analyze the stimulated Raman transitions occurring between different vibrational levels of the atoms in the periodic potential created by the light shifts, and we show a dramatic lengthening of the damping time of coherences between such levels due to the Lamb-Dicke effect. Very narrow Rayleigh resonances with a shape sensitive to the probe polarization appear for a probe frequency close to the pump frequency. We interpret these resonances in terms of scattering of the pump waves on density and magnetization gratings, and show that they provide important information about the dynamics and localization of atoms at the bottom of the potential wells. Such information should also be accessed by phase-conjugation experiments. Finally, indications on the treatment of other atomic transitions are given.
We present a calculation of the probe-transmission spectrum of a two-dimensional optical lattice, in which atoms are cooled by a pair of standing-wave fields with perpendicular polarizations within the plane of incidence. A probe laser is directed along the axis of one of the cooling fields, with a polarization parallel to the incident plane of the cooling fields. The calculated probe-transmission spectrum exhibits both sidebands, which originate from stimulated Raman transitions between different vibrational states of the atoms in the molasses, and a central resonance structure, which originates from Rayleigh resonances among these vibrational states. We also include in our calculation a possible misalignment of the probe field relative to the axis of the cooling field. It is shown that the effrects of the probe misalignment can lead to significant modifications to the central resonance signal.
In two recent experiments Verkerk et al. and our group have observed quantization of the center-of-mass motion of laser cooled atoms in sub-wavelength sized potential wells formed by the light field of a 1-dimensional optical molasses. Our 1-dimensional optical molasses consists of a pair of counterpropagating laser beams of orthogonal, linear polarization, creating a series of potential wells separated by {lambda}/4. Polarization gradient laser cooling reduces the thermal energy of the atoms until they are trapped in well resolved vibrational levels near the bottom of the wells. Transitions between these vibrational levels are seen in a high resolution spectrum of fluorescence, where they give rise to a Rayleigh peak centered at the laser frequency, and well resolved Raman sidebands shifted by a frequency {plus_minus}{omega}, typically on the order of 100 kHz. We find quantitative agreement between the measured level separation h{omega} and the results of a calculation of the bandstructure in the periodic optical potential. Based on the observed ratio of the strength of the blue to the red sideband we determine the temperature of the atoms, and find agreement with theoretical predictions. Based on the level spacing and temperature we find the atoms to be localized to {approx}/15, withmore » up to 50-60% of the population of the trapped atoms in the vibrational ground state. We observe intriguing differences between fluorescence radiated by localized atoms, and the fluorescence radiated by non-localized atoms in a 1-dimensional optical molasses composed of a pair of counterpropagating laser beams of opposite circular polarization. The prospect of observing quantum motion and spatial order in 2 and 3 dimensions, and the consequences for the spectrum of fluorescence, will be discussed.« less