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Annual Meeting Optical Society of America

Monte Carlo wave-function solution of the spectrum of Rb in a 1-D optical molasses

Paul D. Lett

Published Jan 1, 1992• 0 citations
Publisher

Abstract

A Monte Carlo wave-function approach is used to calculate the spectrum of resonance fluorescence as well as the momentum and position distributions of an 85Rb atom in a one-dimensional standing-wave optical field (“optical molasses”). The problem includes the 16 states of the F=3 to F′=4 transition and sufficient momentum states (~100) to accurately reproduce the momentum distributions expected from polarization-gradient cooling. In addition, the trapping of these cold atoms in optical potential wells formed in some polarization configurations is demonstrated. Dicke-narrowed spectra with sidebands representing transitions between bound states in the optical potentials result. The problem is in some sense “too large” to permit solving for the spectrum by using more traditional techniques. A comparison with recent 1-D experiments measuring the spectrum of 85Rb is made. Some insights into atomic dynamics can also be gained by examining the evolution of the atomic wave packet in time.

DOI
10.1364/oam.1992.mff5

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A Monte Carlo wave-function approach is used to calculate the spectrum of resonance fluorescence as well as the momentum and position distributions of an 85Rb atom in a one-dimensional standing-wave optical field (“optical molasses”). The problem includes the 16 states of the F=3 to F′=4 transition and sufficient momentum states (~100) to accurately reproduce the momentum distributions expected from polarization-gradient cooling. In addition, the trapping of these cold atoms in optical potential wells formed in some polarization configurations is demonstrated. Dicke-narrowed spectra with sidebands representing transitions between bound states in the optical potentials result. The problem is in some sense “too large” to permit solving for the spectrum by using more traditional techniques. A comparison with recent 1-D experiments measuring the spectrum of 85Rb is made. Some insights into atomic dynamics can also be gained by examining the evolution of the atomic wave packet in time.
A Monte Carlo simulation method is developed to model quantized atomic motion in one-dimensional optical molasses.1 We calculate atomic absorption and emission spectra and energy distribution functions for various J → J′ transitions coupled to linear and circularly polarized light for standing wave configurations.2 We are particularly interested in generation of sidebands in atomic spectra corresponding to quantized atomic motion.3 Our Monte Carlo method is based on propagating an atomic center-of-mass wave packet and simulating a sequence of spontaneous emissions between families of Bloch states in the periodic laser-induced potentials. Propagation of the (periodic part of the) atomic wave function in time is performed by using a variant of the split-operator FFT algorithm. Band structure calculations for these atomic configurations are presented.
Using recently developed quantum wave function techniques, we have performed a simulation of 85Rb^{85}\mathrm{Rb}85Rb atoms in a one-dimensional optical molasses, formed from counterpropagating laser beams with orthogonal linear polarizations. Both internal and external degrees of freedom are treated quantum mechanically in one dimension and the spectrum of resonance fluorescence is calculated and compared to recent experiments. Excellent agreement is obtained for the spectrum and additional insight is gained into the experimental evidence for quantized motion in the optical potentials.
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 study theoretically the spectrum of resonance fluoresence from one-dimensional molasses consisting of two-level atoms with Zeeman substructure. The center-of-mass motion of the atom is treated fully quantum mechanically. The spectrum shows sidebands due to transitions between vibrational levels in optical potentials generated by the laser light. Detailed results are presented for the Jg{\mathit{J}}_{\mathit{g}}Jg​=1/2\ensuremath{\rightarrow}Je{\mathit{J}}_{\mathit{e}}Je​=3/2 atomic transition in a laser configuration with two counterpropagating waves with orthogonal polarizations. We have solved the corresponding quantum master equation and calculated the relevant autocorrelation function of the atomic dipole using (i) a direct numerical solution of the master equation, (ii) a wave-function simulation of the master equation and correlation function employing periodic time-dependent Bloch wave functions, and (iii) a semiclassical bipotential calculation for the spectrum including a simple quantum correction.
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
A full quantum treatment of laser cooling in three dimensions is performed. It is based on a recently developed Monte Carlo wave-function technique, which reduces an otherwise unmanageable density matrix problem by at least a factor of 104{0}^{4}04 in computing requirements. For different atomic transitions and temporal phases of the laser beams, our calculated mean kinetic energies are in good agreement with experimental results. Momentum distributions are shown to develop escape lines when the optical molasses is close to disintegration. Spatial distributions are also calculated and discussed.