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Annual Meeting Optical Society of America

Monte Carlo simulation of quantized atomic motion in one-dimensional optical molasses

P. Marte, Ralph Dum, A. R. Taieb, P. Zoller

Published Jan 1, 1992• 0 citations
Publisher

Abstract

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.

DOI
10.1364/oam.1992.thuu5

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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.
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.
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.
We study theoretically the spectrum of resonance fluorescence and laser-cooling dynamics corresponding to quantized atomic motion in one-dimensional optical molasses. We consider an atom with a Jg{\mathit{J}}_{\mathit{g}}Jg​=1\ensuremath{\rightarrow}Je{\mathit{J}}_{\mathit{e}}Je​=2 transition interacting with two counterpropagating laser beams. The laser light is assumed to be linearly polarized with an angle \ensuremath{\theta} between the polarization vectors. We discuss the spectrum of resonance fluorescence, the spectroscopy of band structure in the optical potentials, and the population of the vibrational quantum levels as a function of the angle between the polarization vectors.
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.
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 Monte Carlo simulation of the atomic master equation for spontaneous emission in terms of atomic wave functions is developed. Realizations of the time evolution of atomic wave functions are constructed that correspond to an ensemble of atoms driven by laser light undergoing a sequence of spontaneous emissions. The atomic decay times are drawn according to the photon count distribution of the driven atom. Each quantum jump of the atomic electron projects the atomic wave function to the ground state of the atom. Our theory is based on a stochastic interpretation and generalization of Mollow's pure-state analysis of resonant light scattering, and the Srinivas-Davies theory of continuous measurements in photodetection. An extension of the theory to include mechanical light effects and a generalization to atomic systems with Zeeman substructure are given. We illustrate the method by simulating the solutions of the optical Bloch equations for two-level systems, and laser cooling of a two-level atom in an ion trap where the center-of-mass motion of the atom is described quantum mechanically.