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Physical Review Letters

Monte Carlo Wave-Function Analysis of 3D Optical Molasses

Yvan Castin, Klaus Mølmer

Published May 8, 1995• 97 citations
Publisher

Abstract

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 1${0}^{4}$ 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.

DOI
10.1103/physrevlett.74.3772

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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.
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.
The purpose of our master thesis is to present an original synthesis of various works dealing with the Monte-Carlo wavefunction (MCWF) method in quantum optics and its applications to laser cooling of atoms with a particular focus on Doppler and Sisyphus cooling mechanisms. We introduce the MCWF method which is used to solve master equations describing the dynamics of open quantum systems in quantum optics. This method turns out to be much more efficient than a method based on the full integration of the density matrix. We perform laser cooling simulations in 1D both with a classical and a quantum treatment of the atomic centre-of-mass motion and investigate Doppler and Sisyphus cooling. Our numerical results turn out to be in very good agreement with those predicted by semiclassical theories, except for large detuning and/or low intensity laser where semiclassical theories fail. For such cases, our method provides a good alternative and predicts temperatures much larger than the recoil temperature. Finally, we also consider a classical treatment of the centre-of-mass motion and we compare the subsequent results with those obtained with a full quantum approach.
In the limit of asymptotically small photon momentum, a Fokker–Planck equation for the center-of-mass Wigner function suffices to describe laser cooling of an atom. We have devised a procedure to compute the light pressure force and the diffusion tensor for an atom with an arbitrary level structure, traveling with an arbitrary velocity in any given three-dimensional light field. The scheme is based on solving a set of differential equations closely related to the density matrix equations of motion encountered in conventional recoilless laser spectroscopy. Computer programs have been written that automatically construct and solve the required equations from broad specifications of the problem, such as energies and angular momenta of the atomic levels. A Langevin equation stochastic simulation has been implemented to solve the Fokker-Planck equation of laser cooling using the computed force and diffusion. Comparisons of optical-molasses experiments and simulations carried out for a model sodium atom with a J = 2→J = 3 transition show qualitative similarities such as sub-Doppler limit temperatures, but the quantitative agreement of theory and experiment is not satisfactory.
The technique of laser cooling, whose basic idea was proposed in 1975 [1], provided the development of many important areas like high-resolution spectroscopy [2], atomic clocks [3] and quantum gases confined in magnetic or optical traps [4, 5]. Among the techniques applied to cooling neutral atoms is the optical molasses [6] that will be explored in this work. Here we study the standard molasses and a new alternated molasses configuration, which consists of turning on just one pair of counter-propagating laser beams per cycle, and alternating this pair between the three spatial directions. We analyse the two techniques under the lens of the radiation force, by exploring a semiclassical treatment, in order to obtain the minimum energy attained by each of them at the end of the cooling process. We compare, then, these two cooling techniques, pointing out their advantages.
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.