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Physical Review A

Quantum theory of a magneto-optical trap

Прудников Олег Николаевич, Valeriy I. Yudin, A. V. Taǐchenachev, Lin Zhou, Mingsheng Zhan

Published Sep 17, 2025• 2 citations
Publisher

Abstract

Despite their prevalence in cold-atom physics, magneto-optical traps are still modeled semiclassically, with the exception of some numerical works on quantum models performed decades ago. Here, the authors present a quantum theory of the magneto-optical trap from first principles, providing a significant step towards a complete theoretical understanding of laser cooling and trapping of atoms.

DOI
10.1103/5v11-cwv2

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•Jan 1, 1996•1 citation
Despite their prevalence in cold-atom physics, magneto-optical traps are still modeled semiclassically, with the exception of some numerical works on quantum models performed decades ago. Here, the authors present a quantum theory of the magneto-optical trap from first principles, providing a significant step towards a complete theoretical understanding of laser cooling and trapping of atoms.
Since the first proposal and the first experiment on magneto-optical trapping of atoms, magneto-optical traps (MOT) became the most popular traps for cold atoms. Despite the numerous successful applications of MOT it is still unclear what ultimate parameters could be expected for atoms trapped in MOT. We develop the theory of MOT for cold atoms under the multilevel dipole interaction schemes. Theory is based on consideration of one-dimensional motion of atoms which have three ground-state sublevels (F/sub g/=1) and five upper-state sublevels (F/sub e/=2) and are in inhomogeneous magnetic field and in the field of two counter-propagating circular polarized laser waves. It is shown that in multilevel interaction schemes two-photon processes are responsible for formation of a narrow velocity and spatial structure in the radiation force. It is found that two-photon coherent process enhances considerably the friction force and the oscillation frequency at negative detuning. Single-photon and two-photon processes are shown to produce the two-component structure of the MOT potential and accordingly the two-component coordinate distribution.
We present a quantum theory of a magneto-optical trap (MOT) from first principles based on the quantum kinetic equation for the atomic density matrix with taking into account the recoil effects caused by the interaction of atoms with the laser field. An efficient method for solving the quantum kinetic equation is proposed. It is shown that the steady-state solution describing the atoms in the MOT has a significantly non-equilibrium nature and can be described within the framework of a two-temperature distribution. The momentum distribution of cold atoms in the MOT depends on the magnetic field gradient and, in general, significantly differs from the momentum distribution of atoms in the optical molasses, which is usually used as an approximation to describe the MOT. We have also shown that with an increase in the magnetic field gradient, a spatial two-component distribution of atoms in the trap is formed even for a single particle approximation when interatomic interactions are neglected.
The advent of the magneto-optical trap (MOT) introduced to the field of atomic physics an unprecedented surge in research opportunities and the potential for widespread application in relating fields, such as those of quantum computation, quantum optics, laser physics, and plasma physics. The MOT makes use of a process called laser cooling, whereby atoms can be manipulated through the absorption and emission of photons, in conjunction with inhomogeneous magnetic fields, to cool atoms to kinetic temperatures as low as 200 µK and contain them in a region of space. This process has given researchers a remarkable level of control and has provided the means for extensive
Laser cooled and trapped atoms have been the workhorse of atomic physics for the past four decades. The predominant method has been the highly versatile Magneto-Optical Trap. We describe an alternative laser trap involving a simple geometry of collimated laser beams that provides both a velocity and position dependent restoring force such that a dense cloud of cold atoms is formed. This technique produces similar atom number (>106>10^6>106) and density (101010^{10}1010\,atoms/cm3^{3}3) to the Magneto-Optical Trap, albeit with \emph{no magnetic field}. The beam geometry is compatible with conventional sub-Doppler cooling techniques, allowing the trapped cloud to be cooled to <10 μ< 10~μ<10 μK. We demonstrate the validity and robustness of the trap by capturing 87^{87}87Rb atoms directly from the background vapor and provide a theoretical discussion of the underlying principles. This trap has many unique properties that make it highly suitable for quantum sensing, timing, and computing applications as well as a new tool in fundamental science and metrology.
We report on a novel mechanism that allows for strong laser cooling of atoms that do not have a closed cycling transition. This mechanism is observed in a magneto-optical trap (MOT) for erbium, an atom with a very complex energy level structure with multiple pathways for optical-pumping losses. We observe surprisingly high trap populations of over 10(6) atoms and densities of over 10(11) atoms cm(-3), despite the many potential loss channels. A model based on recycling of metastable and ground state atoms held in the quadrupole magnetic field of the trap explains the high trap population, and agrees well with time-dependent measurements of MOT fluorescence. The demonstration of trapping of a rare-earth atom such as erbium opens a wide range of new possibilities for practical applications and fundamental studies with cold atoms.
This thesis describes theoretical and experimental work concerning radiation forces on atoms, with particular reference to rubidium atoms confined in a magneto-optical trap. After a short history of the field of laser cooling, a review of the semiclassical theory of mechanical interactions between two-level atoms and electromagnetic radiation is given. Different formulations of the semiclassical theory are discussed, including a new formulation in terms of momentum transfer amongst the plane wave modes of the electromagnetic field. Two important applications of light forces on atoms, namely 'optical molasses' and the 'magneto-optical trap', are then described with emphasis on experimental parameters. Three sub-Doppler cooling mechanisms, 'sisyphus cooling', 'motion-induced orientation cooling' and the 'magnetically-assisted sisyphus effect', are described and their role in optical molasses and the magneto-optical trap is discussed. A new study of the polarisation gradients which occur in 3-D monochromatic light fields is presented and quantifies their relative presence in different light field configurations. Polarisation gradient parameters are developed and shown to be directly related to the relativistic spin tensor of the light field. Implications of this polarisation gradient study for laser cooling work are discussed. The design, construction from scratch, operation and testing of a magneto-optical trap for rubidium are described, including novel designs for two vacuum cells. Preliminary experiments to characterise the trap are described and results are presented; they primarily concern the number and distribution of atoms in the trap. Finally. the theory of time domain spectroscopy is reviewed. The construction and testing of a pulsed dye laser for study of coherent transients in samples of laser-cooled atoms and a proposed experiment to measure the temperature of cold atoms using coherent transients are described. Factors expected to influence the shape of coherent transients in cold atoms are discussed.