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Circular polarizations represent radiation states that are eigenfunctions of the operator for optical spin angular momentum SAM , whose density operator is given by;. Photons of twisted or vortex forms of structured light in principle represent quantum eigenstates of an OAM density operator, expressible as follows;. Both of the above results, equations 38 and 39 , represent gauge-dependent quantities, cast in terms of the vector potential A r.

However, the separation of angular momentum into spin and orbital parts is a simplification that applies only in the paraxial approximation.

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More generally the separation is not absolute; there are transverse components and spin—orbit coupling in any significantly structured beam [ 33 , 34 ] and there is indeed recent experimental proof of their interconversion in a cylindrically symmetric optical fibre [ 35 ]. For an objective perspective, it is therefore expedient to introduce more definitive, generalized measures of chirality for the radiation field. One suitable measure is the optical chirality density, defined as;.

We can also define a corresponding chirality flux;. When vortex modes are entertained, the key optomechanical parameter is represented by the OAM operator. In the paraxial approximation, for a mode with topological charge l this operator is expressible as;. From a different perspective this result is also consistent with the fact that fields whose mode expansions convey a phase factor , are eigenfunctions of the angular momentum operator L [ 7 ]. The wide variety of other beams conveying OAM includes several other kinds of modified-Gaussian vortex beams [ 39 ], described as having a perfect optical vortex structure [ 40 ] and propagation-invariant Bessel beams [ 41 ].

For mode structures cast in a form that necessarily involves summation over an additional parameter as is the case with perfect vortex beams, for example the associated quanta are correspondingly associated with state superposition [ 40 ]. The term 'laser cooling' refers to various methods in which an interaction of laser light is made to systematically cool atomic, molecular and condensed matter systems to lower temperatures. For general reviews see [ 42 — 46 ]. The primary processes involve the exchange of laser photons leading to momentum and hence velocity changes.

Doppler cooling is the simplest of a number of techniques leading to the systematic reduction of the temperature of atomic or molecular ensembles while Raman anti-Stokes techniques are used for cooling condensed matter systems. Besides Doppler cooling, the list of laser cooling schemes includes, among others: Sisyphus cooling [ 47 ]; Raman sideband cooling [ 48 ]; velocity-selective coherent population trapping; [ 49 , 50 ] and electromagnetically induced transparency cooling [ 51 ].

In addition to laser cooling, the laser light can be made to trap atoms in the minima of optical potential wells set up by the laser light. The physical principles underlying Doppler cooling can be summarised succinctly as follows. When the frequency of the laser light is below a strong atomic transition frequency a scenario referred to as red-detuning , then for an atom travelling in the direction of the laser source the light is blue-shifted in accordance with the Doppler effect.

The atom absorbs a photon and so is slowed down on recoil. Consider now the effects of two counter-propagating laser beams of the same wavelength on a representative atom in a dilute atomic gas. Each atom absorbs more photons belonging to the laser beam opposite to its direction of motion in each event, thereby losing a linear momentum equal to the photon momentum.

This atom is now in the excited state and so discharges its excitation by spontaneous emission in a random direction. The total effect of this basic cycle of photon absorption followed by emission is a reduction of the momentum of the atom, and so the atom loses speed. Repeated cycles then lead to a reduction of the centre of mass kinetic energy, which signifies cooling of the atom since when compared with the case of molecules they have no other centre of mass degrees of freedom.

Currently the most prominent use of laser cooling is in preparing samples of atomic ensembles with temperatures just above absolute zero, widely used for experiments that lead to a variety of effects, most notably Bose—Einstein condensation BEC. Laser cooling has primarily been applied to atoms, but recently there has been progress leading to the cooling of more complex systems such as molecules [ 52 , 53 ] and macro-scale objects [ 54 , 55 ].

Depending on the size of the molecule, the problem of dissipating the energy from internal vibrational and rotational levels can present a considerable additional challenge [ 56 ]. When laser light is employed in the context of the laser cooling techniques mentioned above, it is commonly regarded as ordinary laser light in the sense that it is not endowed with OAM unless this feature is specifically introduced. The aim in the following is to highlight what has been achieved to date as regards the modifications to processes involving cooling and trapping of atoms when the laser light is twisted.

Soon after Doppler cooling appeared to be well-explained theoretically the experimental evidence showed that the existent theory was inadequate [ 57 ], as the measured kinetic temperatures achieved were significantly lower than those predicted by the Doppler mechanism. The failure of the Doppler mechanism to account solely for the lower temperatures achieved meant that a new theory was needed. This paved the way for the development of so-called sub-Doppler cooling mechanisms, most notably the mechanism based on the Sisyphus effect [ 58 , 59 ].

As we discuss later, the Sisyphus effect is modified when the laser light is endowed with OAM, so it is helpful to first review the salient features of this effect. There are two main differences between the Sisyphus cooling mechanism and Doppler cooling.

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In Sisyphus cooling, the main process involves the interaction of atoms with a light field characterised by spatial polarisation gradients. The specific polarisation gradients which have been utilised in the Sisyphus effect can be created by the superposition of two for one-dimensional cooling counter-propagating plane wave laser beams, with either mutually orthogonal linear polarisations [ 58 ], or with opposite circular polarisations [ 59 ].

Consider now the first case where the two fields have mutually orthogonal polarisations. We assume that the two beams have mutually orthogonal polarisations, and. The total the total electric field is the vector sum:.

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This spatial variation of the wave polarisation along the common axis of the interfering beams constitutes a polarisation gradient, which—as will be explained—can lead to a spatially-dependent population differential. Figure 1. Variations of the populations spots and energy level shifts of the ground state sub-levels along the z -axis. Reproduced with permission from [ 58 ]. We, therefore, consider a transition.

The polarisation gradients created by the interfering counter-propagating beams affect the light shifts and the populations of the atomic levels which now become spatially dependent. This can be explained as follows. The optical potentials associated with the two Zeeman sub-level shifts are given by.

Early theoretical works which sought to explain the Doppler cooling argued that the damping of the atomic gross motion arises from the fact that the atomic internal state does not follow adiabatically the variations of the laser field resulting from atomic motion [ 60 ]. For multi-level atoms, we can similarly define a non-adiabaticity parameter. At low intensities, since it follows that. Thus non-adiabatic effects can appear at much lower velocities than those required by Doppler cooling, and thus can ensure the presence of damping forces even at very low velocities. Dalibard and Cohen-Tannoudji [ 58 ] explained how the damping of the atomic motion is generated.

The key point is that, as a condition, the atom must have internal states with energy sub-levels with a strong position-dependence, and which therefore experience large changes as the atom moves. The creation of polarisation gradients can ensure this condition. If the atom starts from the bottom of a valley in a given Zeeman sub-level, then it has sufficient time to reach the top of the hill.

At this position, it has a large probability to be optically pumped to the other sub-level and be shifted to the bottom of a valley, and so on: see figure 2.

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The atom is running uphill more frequently than downhill. This is called a low intensity Sisyphus effect, which arises from the correlations between the spatial modulations of light shifts and optical pumping rates. It is important to emphasise the term 'low intensity' as the Sisyphus effect we are discussing is valid in the low intensity regime. This is in contrast to another Sisyphus effect which is valid at high intensities, which we shall not discuss any further here [ 61 ].

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Figure 2. This basic set of steps is repeated in each cycle. In the process of Sisyphus cooling, an atomic sample eventually reaches an equilibrium temperature. In each optical pumping cycle, we have the emission of a fluorescence photon. Each such photon has an energy higher than the energy of the absorbed photon, by an amount in the order of the light shift. The excess energy is transferred from the atom to the light field leading to a decrease of the atomic energy by the same amount. Repeated pumping cycles, thus, lead to a stepwise decrease of the atomic energy until its total energy is so low that the atom becomes trapped in the optical potential wells associated with the spatially modulated light shifts.

The equilibrium temperature of sub-Doppler cooling is therefore expected to be given by:.

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The detailed quantitative treatment of these predictions has been given in [ 58 ]. The Sisyphus cooling leads to a damping force which for large detuning is in the form,. In typical experiments, both cooling mechanisms come into play. Although the friction force of the low intensity Sisyphus cooling acts within a much smaller velocity interval than the Doppler cooling, both mechanisms are useful. The Doppler cooling that acts over a relatively large velocity interval drags the atoms towards the velocity region where the Sisyphus cooling operates.

Thus, use of the Doppler mechanism as a first step serves to increase dramatically the number of atoms affected by the sub-Doppler mechanism. We have seen that the equilibrium temperature is proportional to the square of the Rabi frequency, which means that it is directly proportional to the laser intensity. This may incorrectly imply that lowering the intensity can lower the temperature indefinitely.

But we must take into account the fact that the scheme is based on spontaneously emitted photons in each pumping cycle. Each photon imparts a recoil momentum k to the atom which, according to its direction relative to the atomic motion, may either decrease or increase the atomic kinetic energy. The atomic motion continues to be cooled only so far as the decrease of the total atomic energy due to the Sisyphus effect, remains larger than the increase of the kinetic energy, of the order of E rec , due to recoil associated with the spontaneously emitted photon.

The qualitative description of low intensity Sisyphus cooling, as given above, is based on a classical description of the position of the atomic centre of mass. This means that the moving atom is treated as a classical point particle. This is a reasonable assumption only if the atomic wave packet, which describes quantum mechanically the centre of mass, is well localised in the laser wave.