Introduction The theoretical foundations for describing the influence of twisted light on atoms, ions and molecules are essentially the same as those leading to their cooling and trapping by ordinary plane wave laser light [1–4]. A third of a century has passed since the Doppler mechanism was first put forward by Wineland and Dehmelt as a mechanism for cooling ions [5], and then by Hänsch and Schawlaw for neutral atoms [6]. The key role played by the Doppler mechanism is best illustrated by the simplest case of one-dimensional optical molasses, where a two-level atom is subject to two identical counter-propagating plane waves of a frequency near to resonance with the atomic transition frequency. In terms of a classical description, at small velocities the atom experiences a friction force F = −α v , where v is the velocity vector of the atomic centre of mass parallel to the common axis of the light beams, and α is a friction coefficient. The friction force gradually cools the atoms to progressively smaller, albeit limited, velocities and can be made to operate in all three dimensions by use of multiple beams. The cooling technique based on the Doppler mechanism has been superseded by more effective cooling mechanisms, especially evaporative and Sisyphus cooling mechanisms, culminating, with the achievement of super-cooling, in the realisation of Bose-Einstein condensates (BECs) [7]. Nevertheless, the Doppler mechanism remains important as a mechanism for the early stages of cooling and still plays a key role in the interaction of atomic systems with laser light.
Introduction The theoretical foundations for describing the influence of twisted light on atoms, ions and molecules are essentially the same as those leading to their cooling and trapping by ordinary plane wave laser light [1–4]. A third of a century has passed since the Doppler mechanism was first put forward by Wineland and Dehmelt as a mechanism for cooling ions [5], and then by Hänsch and Schawlaw for neutral atoms [6]. The key role played by the Doppler mechanism is best illustrated by the simplest case of one-dimensional optical molasses, where a two-level atom is subject to two identical counter-propagating plane waves of a frequency near to resonance with the atomic transition frequency. In terms of a classical description, at small velocities the atom experiences a friction force F = −α v , where v is the velocity vector of the atomic centre of mass parallel to the common axis of the light beams, and α is a friction coefficient. The friction force gradually cools the atoms to progressively smaller, albeit limited, velocities and can be made to operate in all three dimensions by use of multiple beams. The cooling technique based on the Doppler mechanism has been superseded by more effective cooling mechanisms, especially evaporative and Sisyphus cooling mechanisms, culminating, with the achievement of super-cooling, in the realisation of Bose-Einstein condensates (BECs) [7]. Nevertheless, the Doppler mechanism remains important as a mechanism for the early stages of cooling and still plays a key role in the interaction of atomic systems with laser light.
Theoretical work has already established the existence of a light-induced torque acting on the centre of mass of an atom, ion or molecule immersed in twisted light, where the transition frequency is suitably detuned from that of the twisted light beam. The twisted beam carries l units of orbital angular momentum per photon, and the steady-state saturation form of the torque is also determined by the width of the upper state in the atomic transition. It has been shown that, to leading order, the transfer of orbital angular momentum can only occur between the twisted light and the centre of mass motion. We argue here that, for small linewidth, the full time-dependence of the torque is needed to account correctly for the dynamics of atoms in a twisted light beam. We outline the theoretical framework needed to derive this full time-dependence, applying the theory to the motion in a twisted light beam of Eu3+ ions, which possess a particularly narrow linewidth state. For relatively large linewidth, the steady-state forces and torque are appropriate, but the processes of cooling and trapping require the application of several suitably oriented twisted beams. The description of atomic motion in multiple twisted beams demands the application of special coordinate transformations. We show how to construct the appropriate transformation matrices to represent a twisted light beam propagating in an arbitrary direction, and we proceed to investigate the cooling and trapping of Mg+ ions in sets of pairs of counter-propagating twisted beams in two-dimensional and three-dimensional molasses configurations.
It has been known for some time that the interaction of laser light with a free atom can give rise to electromagnetic pressure forces that act on its centre of mass. These forces have been extensively examined both theoretically and experimentally: see, for example Metcalf and van der Straten (5.1) or Adams and Riis (5.2). A simple model of a two-level atom subject to a plane electromagnetic wave may be shown to give rise to two kinds of force acting on the atom centre of mass. These are, a dissipative force arising from the absorption of the light by the atom and its subsequent spontaneous emission in a random direction and a dipole force arising from the non-uniformity of the field distribution which can attract the atom to the regions of intense field. These forces underpin the manipulation of atoms by lasers in a variety of beam configurations. The dissipative force has been exploited in the Doppler cooling of the atomic motion (5.3, 5.4) and the dipole force used for trapping (5.5), while a combination of these effects has led to the realisation of Bose-Einstein condensation (5.6). These forces are, of course, closely related to the ones discussed in Section 4.
The physics of optical vortices, also known as twisted light, is now a well-established and a growing branch of optical physics with a number of important applications and significant interdisciplinary connections.Optical vortex fields of widely varying forms and degrees of complexity can be realised in the laboratory by a host of different means.The interference between such beams with designated orbital angular momenta and optical spins (the latter is associated with wave polarisations) can be structured to conform to various geometrical arrangements.The focus of this review is on how such tailored forms of light can exert a controllable influence on atoms with which they interact.The main physical effects involve atoms in motion due to application of optical forces.The now mature area of atom optics has had notable successes both of fundamental nature and in applications such as atom lasers, atom guides and Bose-Einstein condensates.The concepts in atom optics encompass not only atomic beams interacting with light, but atomic motion in general as influenced by optical and other fields.Our primary concern in this review is on atoms in structured light where, in particular, the twisted nature of the light is made highly complex with additional features due to wave polarisation.These features bring to the fore a variety of physical phenomena not realisable in the context of atomic motion in more conventional forms of laser light.Atoms near resonance with such structured light fields become subject to electromagnetic fields with complex polarisation and phase distributions, as well as intricately structured intensity gradients and radiative forces.From the combined effect of optical spin and orbital angular momenta, atoms may also experience forces and torques involving an interplay between the internal and centre of mass degrees of freedom.Such interactions lead to new forms of processes including scattering, trapping and rotation and, as a result, they exhibit characteristic new features at the micro-scale and below.A number of distinctive properties involving angular momentum exchange between the light and the atoms are highlighted, and prospective applications are discussed.Comparison is made between the theoretical predictions in this area and the corresponding experiments that have been reported to date.
The advent of the laser opened the way to the development of powerful new methods for manipulating and cooling atoms which were exploited in the realization of Bose–Einstein condensation in alkali atom vapours. To set the stage we describe a typical experiment, which is shown schematically in Fig. 4.1. A beam of sodium atoms emerges from an oven at a temperature of about 600 K, corresponding to a speed of about 800 m s –1 , and is then passed through a so-called Zeeman slower, in which the velocity of the atoms is reduced to about 30 m s –1 , corresponding to a temperature of about 1 K. In the Zeeman slower, a laser beam propagates in the direction opposite that of the atomic beam, and the radiation force produced by absorption of photons retards the atoms. Due to the Doppler effect, the frequency of the atomic transition in the laboratory frame is not generally constant, since the atomic velocity varies. However, by applying an inhomogeneous magnetic field designed so that the Doppler and Zeeman effects cancel, the frequency of the transition in the rest frame of the atom may be held fixed. On emerging from the Zeeman slower the atoms are slow enough to be captured by a magneto-optical trap (MOT), where they are further cooled by interactions with laser light to temperatures of order 100 μK. Another way of compensating for the changing Doppler shift is to increase the laser frequency in time, which is referred to as ‘chirping’. In other experiments the MOT is filled by transferring atoms from a second MOT where atoms are captured directly from the vapour.
The force acting on a two-level atom in a low-intensity standing laser wave is calculated. On account of this force, atoms can be trapped at the loops (antinodes) of the field. By quantizing the atomic motion near the bottom of the sinusoidal potential, we have studied the atomic behavior when they are trapped in bound states as well as when they are in the free state. Laser cooling of neutral atoms was first suggested by T. Hansh and A. Schawlow in 1975. The principle involves the Doppler effect in a detuned laser beam. The absorption rate of a photon depends on the apparent frequency of the photon that is shifted ωv/c from ω, where ω is the frequency of the laser, v is the velocity of the atom, and c is the velocity of light. For a single laser beam detuned slightly below an atomic resonance frequency, the apparent frequency for atoms moving toward the laser is closer to resonance, which causes a higher absorption rate. And these atoms lose momentum when they absorb a photon from the beam. The velocity of atoms can be decreased from 10$\sp5$ cm/sec, the velocity they have when they escape from an oven, to 10$\sp2$ cm/sec (about 1mK) by a laser beam. For further cooling, counterpropagating beams are necessary. In this study we propose a new mechanism for supercooling, that is, cooling well below the Doppler cooling limit. Within a standing laser wave there exists a force called a "dipole force", which has a sinusoidal spatial dependence. Once atoms are trapped by the potential, their motion around the minimum of the potential well is that of a simple harmonic oscillator. Restricting our attention to the two lowest quantized states, we calculate transition rates between free states and bound states, from which the average residence times in the bound states can be obtained. This first quantum treatment of an atom trapped in the sinusoidal potential is restricted to the one-dimensional case. Generalization to higher dimensions is left for future research.
The interaction of atoms and molecules with structured light, specifically laser light endowed with the property of orbital angular momentum, such as Laguerre-Gaussian light, is discussed. The primary effects of interest here are the influence of the light on the gross motion of atoms and molecules and the possibilities this motion provides for particle manipulation in cooling, heating and trapping experiments. It turns out that, in addition to the possibility of modifying translational motion, suitably structured light can facilitate the manipulation of rotational motion. The latter possibility arises from a light-induced torque that is directly attributable to the orbital angular momentum property of the light. We outline the physics responsible for these effects and consider applications to typical cases in which atoms and ions are subject to near resonant Laguerre-Gaussian beams, leading to characteristic trajectories and eventual trapping in specific regions. Details are given for optical molasses configurations based on twisted light beams arranged in one-, two- and three-dimensional counter-propagating pairs. We extend consideration to the case of liquid crystals, subject to Laguerre-Gaussian light tuned far off-resonance, and show how this leads to the twisting of the directors in the liquid crystal, coinciding with the intensity distribution of the light.