optical manipulation of atoms and molecules using ...in the gas phase, laser cooling schemes such as...

18African Physical Review (2007) 1:0002

Optical Manipulation of Atoms and Molecules Using Structured Light

Mohamed Babiker∗ and David L. Andrews†∗ Department of Physics, University of York, Heslington, York, United Kingdom

† School of Chemical Sciences, University of East Anglia, Norwich, United Kingdom

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.

1. Introduction

Optomechanical forces have a long history. The op-eration of the eighteenth century Crookes radiome-ter (the familiar rotating propeller in an evacuatedbulb) is a popular demonstration of light produc-ing a mechanical force, through a mechanism wellknown to have its origins in thermal effects. Thevalidity of direct radiation pressure as a mechani-cal principle was also well understood in that cen-tury, and was first experimentally verified using aNichols radiometer in 1901. Following the develop-ment of the laser in 1960, the possibilities for prac-tical utilisation of optomechanical forces to manip-ulate small particles came newly to the fore, largelyowing to the pioneering work of Ashkin about tenyears later [1]. Here another principle was estab-lished, namely that motion can be produced byforces associated with spatial inhomogeneity in anoptical beam. This field evolved very rapidly andby the mid-eighties it had led to the invention ofoptical tweezers, a technique which has since be-come a mainstream tool for the optical trappingand manipulation of a diverse range of particles.

For optical manipulation, the mechanisms thatare most prominent in any specific system are pri-marily dictated by the size of the target particles.The particle size, in turn, determines the nature

of the physical system in which such effects can beobserved; the scale of size for optical tweezers andallied methods runs up to a significant fraction ofthe beam width. Microscopic particles such as cellsand polymer beads represent optically controllableparticles at this higher limit of size [2-4]. To off-set gravitational forces, such materials are mostconveniently studied in liquid suspension, and insuch cases the particle position and motion arecontrollable by various means including intensitygradients (optical tweezing of individual particlesat a laser focus, or for large numbers of particlesin holographically generated traps) and multiple-scattering (optical binding). Together, such meth-ods represent a branch of optical technology thathas already found extensive applications in thefields of medicine, sensors and micromechanical de-vices.

At the opposite extreme, the lower end of thesize scale, most such methods are clearly unus-able. Individual atoms and small molecules do notpresent sufficient cross-section to respond differen-tially, across their own dimensions, to wavelength-scale variations in intensity, nor are they so readilylocalizable. In the condensed phase, the opticalmanipulation of particles smaller than 100 nm be-comes problematic because of Brownian motion.In the gas phase, laser cooling schemes such as theconfiguration known as optical molasses, based on

African Physical Review (2007) 1:0002 19

momentum exchange and exploiting Doppler shiftto its own ultimate demise, allow the optical gen-eration of traps within which further optical ma-nipulation can be achieved. This science of coldatoms [5-9] has of course recently developed intoanother burgeoning area of study, the generationand control of Bose-Einstein condensates [10]. Insuch systems the responsive motion of individualatoms or molecules, or of the whole assembly inthe Bose-Einstein case, is determined by an opti-cally generated potential well.

There is a growing recognition of the potentialfor distinctively nanoscale mechanisms and appli-cations of optomechanical force. The range of ac-tivity in this area has recently seen a huge increase,with the latest technological advances leading tonew methods of applying optical forces, and thecreation of exquisite new beam structures for laserlight. This is a subject that now covers a broadand exciting forefront of optical technology. Itsfocus includes not only nanoscale applications ofwell developed optical tweezer techniques; it alsoaccommodates topics ranging from cold atom ma-nipulation to optically driven fluidics. In this area,theory and experiment have a particularly vigorousdynamic: theory is constantly informing and sug-gesting new experiments, while experimental re-sults challenge and invite new theory.

Against this background, the theoretical andtechnological developments that have led to theproduction of structured laser light introduce an-other tier of possibilities associated with orbital an-gular momentum (OAM) content [11-15]. For thestudies to be discussed in this review, the most de-cisive advance has been the achievement of opticalbeams with phase-imprinted wavefronts, of whichhelically twisted Laguerre-Gaussian beams are themost widely studied example (see Fig. 1). Whilespin angular momentum is one of the best knownproperties of the photon, most obviously manifestin circular polarisations, recognition has emergedthat light beams with a helical wave-front displayother kinds of angular momentum, now generallycalled orbital. This new radiation is often describedas a twisted beam or optical vortex, and the torqueit exerts an optical spanner. Specifically, if sucha beam has an integer number l twists within itswavelength, each photon conveys an orbital angu-lar momentum of ~l, distinct from any spin angu-lar momentum. Such beams, which are now read-ily producible in the laboratory, generate opticalforces and torques with no counterpart in conven-tional optics. Already it has been shown that theexploitation of such beams for atomic and molec-

ular manipulation can lead to a variety of latticestructures, clusters and rings [16-18].

In the following, we describe the general princi-ples, and we give the key equations. We also ex-hibit some of the results that have emerged fromstudies of optically trapped atoms, and moleculesin the quasi-static environment of a liquid crys-tal [19]. First, we begin with a brief overview ofthe context of the engagement of light possess-ing orbital angular momentum with atoms andmolecules.

2. Twisted light

Maxwell’s theory predicts that a light beam car-ries both energy and momentum but, until rela-tively recently, the angular momentum propertyof light was predominantly understood to refer tothe spin angular momentum associated with wavepolarisation. The orbital component was only for-mally defined in Maxwell’s theory, and was rarelydiscussed in an experimental context. The first ex-periment on optical angular momentum was car-ried out by Beth [24], but this only concerned cir-cularly polarised light and so it dealt only with thespin component. Related work on the spin opticalangular momentum was carried out by P.J. Allen[21], Simon et al. [22] and Bretenaker and Le Floch[23].

The development of lasers as sources of coherentlight beams has enabled the production of variousmodes that can collectively be described as twistedlight. This is because such light beams possess anorbital angular momentum arising from their non-uniform spatial distribution. Of particular interestin this context are the Laguerre-Gaussian modeswhich are characterised by helical wavefront struc-tures involving a number of intertwined helices, asshown schematically in Fig. 1.

FIG. 1: Schematic wavefront structure of a Laguerre-Gaussian light beam with three intertwined helices.


The production of Laguerre-Gaussian light beamsinvolves the use of an ordinary laser beam withplanar wavefronts in the form of rectangularly sym-metric Hermite-Gaussian modes, characterised bytwo integer indices m and n. A Laguerre-Gaussian(LG) mode will conveniently emerge on passingHermite-Gaussian light through a pre-designedcomputer generated hologram. Such a LG modeis characterised by two indices: an azimuthal inte-ger index l, representing the number of intertwinedhelices and a radial integer index p, representingthe number of radial nodes. We will use the no-tation LGlp to donate a Laguerre-Gaussian modeof indices (l, p). When both l and p are zero, weget a (0, 0) mode as an ordinary Gaussian distri-bution that carries no angular momentum and hasno radial nodes. For l 6= 0 and p = 0 we havethe so-called doughnut (donut) modes, which ex-hibit ring shaped intensity distributions, as shownFig.2 for the cases l = 1 and l = 3. Figure 2also shows the case of two-rings mode arising whenl = 1, p = 1. A photon of an LGlp beam carriesa well defined orbital angular momentum equal tol~. This was first shown by Allen et al. using theparaxial approximation [20]. Within the paraxialapproximation [24] we have for the electric fieldvector of a Laguerre-Gaussian beam travelling inthe +z-direction and polarised principally in the xdirection

Eklp(R) = iω

(

uklpx +i

k

∂uklp

∂xz

)

eikz (1)

The function uklp can be written in cylindrical co-ordinates R = (r, φ, z) as follows

uklp(r, φ, z) =C

(1 + z2/z2R)1/2

(

r√

2

w(z)

)l

·

Llp

(

2r2

w2(z)

)

e−(r2/w2(z))eiΘklp(r,φ,z)

(2)

where

Θklp(r, φ, z) =ikr2z

2(z2 + z2R)

+ lφ

+(2p+ l + 1) tan−1(z/zR) (3)

where zR is the Rayleigh range and w2(z) = 2(z2+z2

R)/kzR is the beam width at distance z from thebeam waist (at z = 0) and C is a normalisationfactor.

A characteristic feature of all modes possessingorbital angular momentum is the phase factor eilφ.

FIG. 2: The intensity distributions of modes, respectively,for LG1,0 (donut mode), LG3,0 (donut mode) and LG1,1

(two-ring). These radial intensity distributions are at thewaist plane z = 0. The insets exhibit graphically the corre-sponding radial intensity distributions with radial distancein units of wavelength. The functional forms of LG modesare given below.

This term will be shown here to be responsible forall rotational effects when Laguerre-Gaussian lightinteracts with atoms and molecules

3. Atoms and molecules in LG beams - brief

overview

The literature dealing with study of the engage-ment of light endowed with orbital angular momen-tum with atoms and molecules is relatively sparsein comparison with that involving optical manipu-lation of the larger particles [6] mentioned above,such as biological cells and polymer beads. Mostpublished works on atoms and molecules are con-


cerned with theoretical studies, but there are alsoa few experimental studies.

The possibility that orbital angular momentumeffects can influence matter at the atomic andmolecular level was first mentioned as a specula-tion in pioneering work by Allen et al. [24]. Thiswas followed by a number of theoretical investi-gations [25-29] that led to the prediction of thelight-induced torque [25], the azimuthal Dopplershift [26] and a number of studies on the motionof atoms and ions in Laguerre-Gaussian beams, in-cluding optical molasses in one, two and three di-mensions [27-29]. The role of photon spin whenconsidered in the same context as OAM was clar-ified [28]. This led to the identification of a spin-orbit term and a contribution involving l-s couplingin the azimuthal force due to circularly polarisedLaguerre-Gaussian light. More recent work con-centrated on trapping in dark regions of the beamprofile, indicating that under such circumstancesthe trapped atoms would experience diminishedheating effects [30]. Studies dealing with the se-lection rules governing the interaction of light withthe internal and external degrees of freedom wereundertaken by van Enk [31] and Babiker et al.[32], while Juzeliunas et al. identified novel fea-tures in the interactions of Bose-Einstein conden-sates [33,34]. As interest in the subject has grown,many other groups have also engaged with the is-sues surrounding the effects of OAM on atoms andmolecules [35-40].

The first experimental study involving atoms in-teracting with orbital angular momentum of lightwas that by Tobosa and Petrov [41] in which theydemonstrated the transfer of OAM from the beamto cold caesium atoms. Other studies have dealtwith the channelling of atoms in material struc-tures possessing cylindrical symmetry, where theoptical modes are distinguished by orbital angularmomentum features. Theoretical studies [42,43]have shown that, in such structures, the chan-nelling of atoms involves light torques similar tothose produced by free space Laguerre-Gaussianbeams - which have also been employed as atomguides [44-49]. In the molecular context particularinterest has focused on liquid crystals, despite theircomplexity, since their distinctive combination ofanisotropic local structure and relatively labile ori-entational motion is directly amenable to opticalinterrogation. The effects of OAM on liquid crys-tals have been studied recently by Piccarillo andco-workers [50,51] and a subsequent analysis em-ploying the dielectric model of the nematic liquidcrystal has been reported by Carter et al. [19].

4. Transfer of OAM to atoms and molecules

For both structured and unstructured light, mostoptical processes involve electric dipole interac-tions with the radiation fields, this type of interac-tion generally being the strongest form of coupling.Depending on the specific process, the dipolar in-teractions entailed may invoke either static or tran-sition moments. In connection with the interac-tions of structured light - twisted beams in partic-ular - distinctive issues revolve around the possibleinvolvement of other electric and magnetic multi-poles. One issue concerns the possibility that theleft- or right-handed character of a twisted beam(i.e. the handedness of its helical wavefront sur-face) might engage differentially with fluids com-prising chiral molecules, according to the molecularhandedness. This hinges on the form of the elec-tromagnetic coupling because, in such media, chi-ral discrimination requires electronic transitions tobe simultaneously allowed by multipoles of oppo-site parity: this is a problem addressed in Chapter15.

Berry [52] showed theoretically that the orbitalangular momentum is an intrinsic property of alltypes of azimuthal phase-bearing light. If so, thenit could be argued that in its interaction with anatom or a molecule, orbital angular momentumshould be exchanged in an optical transition, justas spin angular momentum is exchanged in a ra-diative transition. Another question arises in con-nection with the multipolar interactions of twistedlight. This concerns the fact that the photon or-bital angular momentum can engage in electricquadrupole and higher electric multipole interac-tions, and it is necessary to consider whether thismight introduce modified selection rules for elec-tronic transitions. These matters have been ex-plored by an explicit analysis [14], which concludedthat the exchange of OAM occurs in the electricdipole approximation and couples only the centreof mass to the light beam. The internal degrees offreedom associated with the ‘electronic’ motion arenot involved in any OAM exchange with the lightbeam to this leading order. It is only in the nextorder, namely in an electric quadrupole transition,that an exchange involving the light, the centre ofmass and the internal degrees of freedom can berealised. One unit of OAM is exchanged betweenthe light and the internal dynamics, so that thelight beam possesses (l ± 1)~ units of OAM andthe centre of mass motion gains ±1 units. Theseconclusions suggest that no experiments can detectOAM exchange between Laguerre-Gaussian light


and molecular systems through changes involvingelectric dipole transitions. The analysis confirmsthe fact that OAM effects are manifest primarilyin the centre mass motion by the imposition of ad-ditional forces and associated torques. The studyof these additional forces is best carried out by aextending the formalism of Doppler cooling andtrapping to the case of light possessing OAM, aswe discuss next.

5. Doppler forces and torques

It has long been known that the Doppler effectresponsible for broadening atomic transitions canbe exploited for laser cooling. On irradiatingan atomic gas with a laser beam detuned to thered of an absorption frequency, only a subset ofthe atoms - those that experience a compensat-ing (blue) Doppler shift due to motion towardsthe light source - can absorb the light. The de-cay of the resulting excited state releases a pho-ton in a random direction. Due to the extremelyshort lifetime usually associated with the excitedstate, this is a process that can recur with greatrapidity, and the net effect over a series of absorp-tion and emission cycles is that such atoms expe-rience a net imparted momentum against their di-rection of travel, slowing them down. For the self-selected group of atoms within the laser beam pro-file this loss of translational energy signifies cool-ing, to the extent that such a term can mean-ingfully be applied to a non-equilibrium system.With two counter-propagating beams the veloci-ties of the fastest atoms in each direction can bereduced, and as the laser frequency is graduallyincreased the breadth of the initially Maxwellianvelocity distribution becomes increasingly narrow.Transverse motions can be controlled by the ad-dition of further counter-propagating beams, witheach pair of sources in a mutually orthogonal con-figuration; this is the essence of optical molasses.

The significant features introduced by the use ofstructured light possessing orbital angular momen-tum are: (i) there is, in addition to translationaleffects, a light-induced torque which causes a ro-tational motion of the atoms about the beam axisand; (ii) there are regions of maximum and min-imum intensities in the beam cross-section. Theforces and torque are, in general, time-dependentas well as position-dependent. As we discuss be-low, the full space- and time-dependence of themotion is, in general, characterised by a transientregime, followed by a steady state regime after a

sufficiently large time has elapsed from the instantin which the beam is switched on (typically forelapsed times much larger than the characteristictimescale of the problem).

5.1. Essential formalism

Consider an atom or a molecule for which the grossmotion is that of the centre of mass and the inter-nal dynamics is modelled in terms of a two-levelatom. In the presence of a laser field, the totalHamiltonian for the whole system is

H = ~ωa†a+P2

2M+ ~ω0π

†π − i~[

π†f(R) − h.c.]

(4)where π and f(R) are given by

π = πeiωt; f(R) = (µ12 · ǫ)αFklp(R)eiΘklp(R)/~

(5)

Here π and π† are the ladder operators for the two-level system; P is the centre-of-mass momentumoperator with M the total mass and ω0 the dipoletransition frequency. The operators a and a† arethe annihilation and creation operators of the laserlight and ω is its frequency. In the classical limit,appropriate for the case of a coherent beam, the aand a† operators become c-numbers involving theparameter α such that

a(t) → αe−iωt; a†(t) → α∗eiωt. (6)

The last term in eq(4) is the interaction Hamilto-nian coupling the laser light to the two-level systemin the electric dipole and rotating wave approxi-mations, evaluated at the centre of mass positionvector R. The coupling function f(R) in Eq.(5)involves µ12, the transition dipole matrix elementof the atom interacting with a Laguerre-Gaussianlight mode characterised by ε, the mode polarisa-tion vector, the mode amplitude function, Fklp(R)and phase Θklp(R), given by

Fklp(R) = Fk00Nlp

(1 + z2/z2R)1/2

(√2r

w(z)

)|l|

·L|l|p

(

2r2

w2(z)

)

e−r2/w2(z), (7)

Θklp(R) =kr2z

2(z2 + z2R)

+ lφ+

2p+ l + 1) tan−1(z/zR) + kz,(8)


Here Fk00 may be identified as the amplitude fora plane wave propagating along the z-axis withwavevector k; the coefficient Nlp =

√

p!/(|l|+ p!)is a normalisation constant; w(z) is a characteris-tic width of the beam at axial coordinate z and isexplicitly given by w2(z) = 2(z2 + z2

R)/kzR, wherezR is the Rayleigh range. The LG mode indices land p determine the field intensity distribution andare such that l~ is the orbital angular momentumcontent carried by each quantum.

We now assume the position R and the momen-tum operator P of the atomic centre of mass shouldtake their average values r and P0 = MV, whereV is the centre of mass velocity. Thus we aretreating the atom gross motion classically, whileits internal motion continues to be treated quan-tum mechanically. This treatment is justified pro-vided that the spread in the atomic wavepacketis much smaller than the wavelength of the light,and that the recoil energy is much smaller than thelinewidth. The system density matrix can then bewritten as

ρS = δ(R − r)δ(P −MV)ρ(t), (9)

where ρ(t) is the internal density matrix, whichfollows the time evolution

dρ

dt= − i

~[H, ρ] + Rρ, (10)

and where the term Rρ represents the relaxationprocesses in the two-level system. The opticalBloch equations governing the evolution of the den-sity matrix elements can be written as follows

˙ρ21(t)˙ρ12(t)ρ22(t)

=

−(Γ2 − i∆) 0 2f(r)0 −(Γ2 + i∆) 2f∗(r)

−f∗(r) −f(r) −Γ1

·

ρ21(t)ρ12(t)ρ22(t)

+

−f(r)−f∗(r)

0

(11)

Here the relaxation processes are assumed to becharacterised by an inelastic collision rate, Γ1, andan elastic collision one, Γ2. The effective, velocity-dependent, detuning ∆ is given by ∆ = ∆0 −∇Θ.V and we have set ρ = ρ exp(−itV ·∇Θ). Wehave also made use of the relation ρ11(t)+ρ22(t) =1.

The average force due to the light acting on thecentre of mass is the expectation value of the traceof −ρ∇H ,

〈F〉 = −〈tr(ρ∇H)〉 (12)

The total force can be written as the sum of twotypes of force, a dissipative force 〈Fdiss〉 and adipole force 〈Fdipole〉, and these are related to thedensity matrix elements as follows

〈Fdiss(R, t)〉 = −~∇Θ(ρ∗21f(r) + ρ21f∗(r))(13)

〈Fdipole(R, t)〉 = i~∇Ω

Ω(ρ∗21f(r) − ρ21f

∗(r)).(14)

Here we have introduced the position dependentRabi frequency Ω(R), defined as

~Ω(R) = |(µ12 ·ǫ)αF(R)|; f(R) = Ω(R)eiΘ(R)

(15)Clearly all quantities depend on the mode type andimplicitly carry the labels klp.

The centre of mass dynamics is determined byNewton’s second law, written in the form

Md2R

dt2= 〈F(t)〉 (16)

where 〈F(t)〉 is the total average force. Since thisdifferential equation is second order in time, val-ues of the position vector components R(0) andinitial velocity vector components V(0) should bestated as initial conditions. The main outcome ofsolving Eq.(16) is a complete determination of the

trajectory function R(t), along with V(t) = R(t).Furthermore, as will become apparent, the devel-opment furnishes important information about theevolution of the light-induced torque.

5.2. Transient dynamics

The transient effects are most prominent for transi-tions with a long excited state lifetime. Rare-earthions provide such a context; we consider an Eu3+

ion which has M= 25.17 × 10−26 kg and for its5D0 →7 D1 transition λ = 614 nm and Γ = 1111Hz. We focus on the l = 1, p = 0 Laguerre-Gaussian mode and assume the laser intensity tobe I = 105 W cm−2, and the beam waist w0 = 35λ.The transient regime can be explored for three spe-cial cases, namely (a) exact resonance; (b) strongcollisions and (c) intense field. For the latter casewe shall assume the higher intensity I = 108 Wcm−2. Evaluations have been carried out for a pe-riod tmax ≈ 5Γ−1 ≈ 4.5 ms, which is sufficientlylong to exhibit effects both for the transient regimeand the steady state.

The results are shown in Fig.3 for the cases ofstrong collisions. The atom follows a characteristicpath with an axial motion superimposed on an in-plane motion. The in-plane motion is seen to be in


the form of loops in the shape of petals. It is in thischaracteristic motion that the effects of the opticaltorque are evident. A similar trajectory arises in

FIG. 3: The path in the x-y-plane of an Eu3+ ion subjectto an LG1,0 mode in the case of strong collisions. The initialposition is represented by a dot. Other parameters used forthe generation of this figure are given in the text.

the case of an intense external field, as shown inFig. 4, but here it is seen that there are manymore loops due to the atom gaining kinetic energywith a larger force and torque. Fig. 5 explores theinitial stages of the trajectory before the secondpetal is formed. In the case of exact resonance,Fig. 6, there is no dipole force acting on the atomand so no radial force, due to the zero-detuning.The atomic radial position is constant.

FIG. 4: As in Fig. 3, but for the case of a strongexternal field, as described in the text.

The time-dependent torque is defined as

T (t) = r(t) × 〈F(t)〉 (17)

The evolution of this torque can be determinedalong with the corresponding trajectories. The

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

-1 0 1 2 3 4 5

y/λ

x/λ

-1 0 1 2 3 4 5

0 0.5 1 1.5

x/λ

Γt

FIG. 5: The initial stages of the trajectories shownin Figs. 3 and 4, exhibiting the initiation of the firstpetal-like loop. Inset: variation of the x-component ofthe position with time.

-1-0.5

0 0.5

1 -1

-0.5

0

0.5

1

0

20

40

60

80

100

120

z/λ

x/λ

y/λ

z/λ

FIG. 6: The path of an Eu3+ ion subject to a LG1,0 inthe case of exact resonance. Other parameters are thesame as those in Figs. 3 and 4.

torque experienced by the Eu3+ ion for the caseof strong collisions is displayed in Fig. 7. It is ev-ident from Fig. 7 that once the beam is switchedon, there is an abrupt increase in the magnitude ofthe torque, which then oscillates and rapidly de-cays towards a steady state value. Furthermorethe evolution exhibits a collapse and revival pat-tern, with each cycle corresponding to a loop inthe trajectory. The peak of the torque correspondsto the outer tip of the loop, and the collapse cor-responds to the points near the beam axis. Thesudden jumps in the torque are real events aris-ing from the change in the direction of motion asthe atom is repelled from regions of extremum fieldintensity values.


-15

-10

-5

0

5

10

15

0 1 2 3 4 5

T/− h

kΓλ

× 1

00

Γt

-0.1

0

0.1

4 4.5 5

FIG. 7: The variation of the (axial) light-inducedtorque acting on the Eu3+ ion subject to a LG1,0 inthe case of strong collisions. The detuning ∆0 = 500Γand the initial radial position is r = 5λ

5.3. Steady state dynamics

The formal expressions for the steady state forcescan be deduced by taking the limit t → ∞, or thetime derivatives in the optical Bloch equations setequal to zero. In the steady state, where Γt ≫1 (where Γ is the de-excitation rate of the upperstate of the atomic transition), the total force ona two-level atom exhibits position-dependence andis naturally divisible into two terms. Restoring theexplicit reference to a specific Laguerre-Gaussianmode, the steady state force on a moving atom dueto a single beam propagating along the positive zaxis is written

〈F〉klp = 〈Fdiss〉klp + 〈Fdipole〉klp (18)

where 〈Fdiss〉klp is the dissipative force

〈Fdiss(R,V)〉klp = 2~ΓΩ2klp(R) ·

(

∇Θklp(R)

∆2klp(R,V) + 2Ω2

klp(R) + Γ2

)

(19)

and 〈Fdipole(R,V)〉klp is the dipole force

〈Fdipole(R,V)〉klp = −2~Ωklp(R)∇Ωklp ·(

∆klp(R,V)

∆2klp(R,V) + 2Ω2

klp(R) + Γ2

)

(20)

The effective detuning ∆klp(R,V) is now bothposition- and velocity-dependent;

∆klp(R,V) = ∆0 − V.∇Θklp(R,V) (21)

The dissipative force is due to absorption followedby spontaneous emission of light by the atom, whilethe dipole force is seen to be proportional to thegradient of the Rabi frequency. Both types of forcefeature prominently in atom cooling and trapping,with the dissipative force creating a net a frictionalforce in optical molasses, and the dipole force trap-ping the atom in regions of extremum field inten-sity.

The inference that a light induced torque is auto-matically created in this context can be confirmedby examining at the velocity-independent forces.For V = 0 and for z << zR we have

⟨

F0diss(R)

⟩

klp=

2~ΓΩ2klp(R)

∆20 + 2Ω2

klp(R) + Γ2

[

kz +l

rφ

]

(22)There are thus two components of force: an ax-ial component and an azimuthal component. Thelatter has a non-vanishing moment about the axis,i.e. a torque given by

T =2~ΓΩ2

klp(R)

∆20 + 2Ω2

klp(R) + Γ2lz (23)

In the saturation limit, where Ω >> ∆0 and Ω >>Γ we have

T ≈ ~lΓz (24)

This simple form of the light-induced torque wasfirst pointed out by Babiker et al. [25]. In general,the

torque is velocity- and position-dependent, andtherefore changes along the path of the atom.

5.4. Dipole potential

The velocity-independent dipole force can be de-rived from the dipole potential

〈U(R)〉klp =~∆0

2ln

[

1 +2Ω2

klp(R)

∆20 + Γ2

]

(25)

such that⟨

F0dipole

⟩

klp= −∇〈U(R)〉klp. This po-

tential would trap atoms in the high intensity re-gions of the beam for ∆0 < 0 (red-detuning). Forblue detuning, ∆0 > 0, the trapping would be inthe dark regions of the field. For example, considerthe LG mode for which l = 1, p = 0. On the planeof the beam waist z = 0, the potential minimumoccurs at r = r0 = w0/

√2. For a beam propa-

gating along the z axis the locus of the potential


minimum is a circle in the xy plane given by

x2 + y2 = r20 (26)

Expanding 〈U(R)〉k10 about r0 we have

〈U〉k10 ≈ U0 +1

2Λk10(r − r0)

2 (27)

where |U0| is the potential depth given by

|U0| =1

2~|∆0| ln

[

1 +2Ω2

k10(r0)

∆20 + Γ2

]

(28)

and Λk10 is an elastic constant given by

Λk10 =4~|∆0|

∆20 + 2e−1Ω2

k00 + Γ2

(

e−1Ω2k00

w20

)

(29)

An atom of mass M, trapped if its energy is lessthan |U0|, will exhibit a vibrational motion aboutr = r0 of angular frequency approximately equal

to Λk10/M1/2.

6. The Doppler shift

The force expressions contain the effective detun-ing ∆klp defined by

∆klp = ω − ω0 − ∇Θklp.V; (30)

this can be written as

∆klp = ω − ω0 − δ (31)

where δ is an effective Doppler shift associated withthe beam. On substituting for Θ(R) we obtain forδ

δ =

(

krz

z2 + z2R

)

Vr +lVφ

r

+

kr2

2(z2 + z2R)

[

1 − 2z2

z2 + z2R

]

+(2p+ l + 1)zR

z2 + z2R

+ k

Vz (32)

where Vr, Vφ and Vz are the velocity componentsin cylindrical coordinates. The Doppler shift com-prises four types of contribution: an axial term, acontribution due to the Guoy phase, a curvatureterm and an azimuthal contribution, so that

δ = δaxial + δGuoy + δcurve + δazimuth (33)

The axial term corresponds to a Doppler shift dueto a plane wave travelling along the beam axis

δaxial = kVz (34)

This would be the largest shift if the atom hasa substantial axial velocity component. The shiftassociated with the Guoy phase is

δGouy =

(

(2p+ l + 1)zR

z2 + z2R

)

Vz (35)

Since typically zR >> w0, the Guoy shift is nor-mally negligibly small. The shift arising from thebeam curvature is

δcurve =

(

krz

z2 + z2R

)

Vr

+kr2

2(z2 + z2R)

[

1 − 2z2

z2 + z2R

]

Vz (36)

This is due to beam spreading in the radial andaxial directions and arises from the curvature ofthe wavefront. It could be observable under ap-propriate conditions. The azimuthal Doppler shiftis

δazimuth =lVφ

r(37)

This shift is directly proportional to the orbitalangular momentum quantum number l of theLaguerre-Gaussian mode and is inversely propor-tional to the radial coordinate of the atom. It isjust one of a number of effects directly attributableto orbital angular momentum content of the light.The Doppler term involving the gradient of thephase is also responsible for the radiation forcesgenerating atomic trajectories, as we now discuss.

6.1. Trajectories

Newton’s second law determines the form of theatom dynamics, subject to initial conditions. Thesolutions lead to the trajectory R(t) and they alsodetermine the evolution of other variables of thesystem. Unfortunately, R(t) cannot, in general, bedetermined analytically and it is necessary to pro-ceed using numerical analysis. It is easy to checkthat the trajectories for two cases in which a Mg+

ion is subject to single separate beams differingonly in the sign of l will only display a reversal ofthe direction of rotation. This is consistent withthe existence of the light-induced torque.

6.2. Multiple Beams

Doppler cooling manifests itself in the so-called op-tical molasses configurations in one, two and three


dimensions. For beams endowed with OAM a de-scription of optical molasses requires the specifica-tion of individual field distributions referred to thelaboratory coordinate system. We therefore needto apply multiple coordinate transformations withreference to the original Cartesian axes. The to-tal force acting on the atom is the vector sum ofindividual forces in a given configuration of lightbeams.

For a light beam of frequency ω, axial wavevec-tor k and quantum numbers l and p coupled toan atom or an ion at a general position vectorR = (r, φ, z) in cylindrical coordinates, the phaseΘklp(R) and the Rabi frequency Ωklp(R) can betaken as follows

Θklp = lφ+ kz (38)

and

Ωklp(R) ≈ Ω0

(

r√

2

w0

)|l|

exp(−r2/w20)L

|l|p

(

2r2

w20

)

(39)These expressions are applicable for a Laguerre-Gaussian beam in the limit z ≪ zR, where zR isthe Rayleigh range; also setting w(z) = w0, i.e.ignoring all beam curvature effects.

The total forces acting on the atomic centreof mass moving with velocity V = R are givenabove in the steady state, but with the approxi-mate phase Θklp(R) and Rabi frequency Ωklp(R),as in Eqs.(38) and (39). These are given in cylin-drical polar coordinates, with the beams propa-gation parallel to the z-axis. However, in orderto consider multiple beams, it is convenient tobegin by expressing the position dependence inthe Rabi frequency and phase in Cartesian coor-dinates R = (x, y, z), simply by the substitutions

r =√

x2 + y2 and φ = arctan(y/x). A beam prop-agating in an arbitrary direction is determined byapplying two successive transformations. The firsttransformation is a rotation of the beam about they-axis by an angle θ and the second is a subsequentrotation about the x axis by an angle ψ. This sig-nifies to the following overall coordinate transfor-mation

x → x′ cos(θ)x + sin(θ)z (40)

y → y′ = − sin(θ) sin(ψ)x + cos(ψ)y

+ cos(θ) sin(ψ)z (41)

z → z′ − sin(θ) cos(ψ)x − sin(ψ)y

+ cos(θ) cos(ψ)z (42)

By suitable choice of the angles θ and ψ we ob-tain the force distribution due to a twisted light

beam propagating in any direction. In this man-ner, we are able to consider geometrical arrange-ments involving counter-propagating beams (espe-cially those corresponding to one-, two- and three-dimensional optical molasses configurations), forbeams possessing OAM.

We concentrate on the case of optical molassesof magnesium ions Mg+ with a transition of fre-quency ω0 corresponding to the transition wave-length λ = 280.1 nm and transition rate Γ =2.7 × 108 s−1. The Mg+ massM = 4.0×10−26 kg.We consider red-detuned light to induce trappingin areas of high intensity, ∆0 = −Γ and w0 = 35λ.The equation of motion for the Mg+ ion is now

Md2

dt2R(t) =

∑

i

〈Fi〉 (43)

where the sum is taken over individual (total) forcecontributions from each beam present. In theone-dimensional molasses configuration, a pair ofcounter-propagating beams is set up along the zaxis. The specification of the force due to the beampropagating in the negative z-direction is given byEqs.(40), (41) and (42) with θ = π and ψ = 0.Figure 8 shows the trajectory of the Mg+ ion withl1 = −l2 = 1 and p1 = p2 = 0. The initial ra-dial position is r = 10λ and the initial velocity isV(0) = 5ms−1z. The motion is for a time dura-tion equal to 2× 105Γ−1. It is clear that the atom

-60-40

-20 0

20 40

60 -80

-40

0

40

80

0

20

40

60

z/λ

x/λ

y/λ

z/λ

FIG. 8: Path of a Mg+ ion in the one-dimensionaltwisted optical molasses created by two counter-propagating Laguerre-Gaussian beans with l1 = −l2 =1 and p1 = p2 = 0 propagating along the z-axis. Theinitial velocity is v = 5z ms−1

is slowed down to a halt in the z-direction, whilein its motion in the x-y plane it is attracted tothe region of high beam intensity at approximatelyr0 > w0/

√2. The long-time motion is a uniform


circular motion, as can be deduced from Fig. 9 ,which exhibits the corresponding evolution of thevelocity components. Once the Mg+ ion is trapped

-3

-2

-1

0

1

2

3

4

5

6

0 5 10 15 20

Vel

oci

ty (

ms-1

)

Γ t ×104

vxvyvz

FIG. 9: Evolution of the three velocity components ofthe Mg+ ion in the one-dimensional optical molassesof Fig. 8. The axial component of the velocity rapidlyapproaches zero, consistent with Doppler cooling, whilethe in-plane components vx and vy show a convergencetowards uniform circular motion.

axially, it continues to rotates clockwise about theaxis, subject to a torque which, in the saturationlimit, is given by |〈T 〉| ≈ l1~Γ − l2~Γ = 2~Γ. Themotion of the ion gives rise to an electric currentequal to e/τ ≡ evs/2πr0. With vs of about 2 ms−1

and r0 ≈ w0 = 35λ we have an ionic current of theorder of a femtoAmp if a single ion is involved. Itis significant that the current scales with the num-ber of trapped ions; obviously a million or so ionscan produce a current on the nA scale.

6.3. Two- and three-dimensional molasses

We now introduce a second pair of counter-propagating beams along the x axis and one paircould be characterised by a different width w′

0.The total force is now the vector sum of individ-ual forces from the four beams. The specificationsof three of the beams is made with the help ofthe transformation equations (40)-(42). The tra-jectories of two Mg+ ions positioned at differentinitial points, each having an initial velocity ofvz = 5ms−1, are shown in Fig. 10, where eachof the four beams has an azimuthal index, l = 1,and radial index, p = 0. Since by choice of l valuesin this case, the total torque arising from eitherpair of beams is zero; each ion ends up at a spe-cific fixed point, where it remains essentially mo-tionless. To understand this, one should note that

-25

0

25-25

0

25

-25

0

25

50

z/λ

x/λ

y/λ

z/λ

FIG. 10: Trajectories of two Mg+ ions with differentinitial locations subject to a two-dimensional opticalmolasses formed by two pairs of counter-propagatingtwisted beams, with li = 1 and pi = 0 for i = 1− 4. Itis seen that each ion ends up motionless on the locus oflowest potential energy minima corresponding to twooblique orthogonal circles, as explained in the text.

the deepest potential well is four times as deep asthat of a single beam, with the potential minimasituated along the locus of spatial points definedsimultaneously by two equations x2 + y2 = w2

0/2

and y2 + z2 = w′20/2. For w′

0 = w0 these twoequations describe two orthogonal oblique circlesrepresenting the intersection curves of two cylin-ders of radii w0/

√2. Solving for x and y we have

x = ±z and y = ±√

w20/2 − z2. The locus of

spatial points where the dipole potential is min-imum can be described by the parametric equa-tions x(u) = (w′

0/√

2) cos u; y(u) = (w′0/√

2) sinu;

z(u) = ±√

w20/2 − (w′2

0/2) sin2 u . All Mg+ ions

in the two-dimensional configuration of orthogonalcounter-propagating pairs of twisted beams will betrapped at points lying on one of the two obliquecircles, as determined by the initial conditions. Anensemble of Mg+ ions with a distribution of ini-tial positions and velocities will populate the twocircles, producing two orthogonal essentially staticMg+ ion loops. Associated with this system ofcharges would be a Coulomb field whose spatialdistribution, for example, for ions uniformly dis-tributed in the ring can easily be evaluated. Whenthe values of l are such that each pair of beamsgenerates a torque, the motion becomes more com-plicated, but the ions will seek to congregate in theregion of potential minima, while responding to thecombined effects of two orthogonal torques and or-thogonal axial cooling forces.

When a third pair of counter-propagating beams


-30-15

0 15

30 -30-15

0 15

30

-20

-10

0

10

20

z/λ

x/λ

y/λ

z/λ

FIG. 11: Trajectories of eight Mg+ ions in a three-dimensional twisted optical molasses formed by threepairs of counter-propagating Laguerre-Gaussian beamswith li = 1 and pi = 0 where i = 1 − 8. The initialvelocity of each of the ions is vz = 5 ms−1. The ionsend up motionless at the corners of a cube of side w0.

is added to the two-dimensional configuration,orthogonal to the plane containing the originalbeams, we have a three-dimensional configuration.In this case the deepest potential minima are lo-cated at eight discrete points defined by the co-ordinates: x = ±w0

2 , y = ±w0

2 , z = ±w0

2 . Thesecoincide with the eight corners of a cube of sidew0, centred at the origin of coordinates, as shownin Fig.11.

7. Rotational effects on liquid crystals

As observed earlier, a liquid crystal is anotherphysical system where new physical effects shouldarise when subject to twisted light. The mostprominent effect in this case can be expected tobe an optical influence on the angular distributionof the director n(r) in the illuminated region. Tofocus on a case of direct and practical relevance, letus consider a liquid crystal film of thickness d occu-pying the region 0 ≤ z ≤ d, with the light incidentin such a manner that the beam waist coincideswith the plane z = 0.

To begin with, we first note that in the absenceof the light, the free energy density of the systemis given by [53]

F0(r) =1

2K1 [∇.n(r)]2 +

1

2K2 [n.∇ × n(r)]2

+1

2K3 [n × ∇ × n(r)]

2(44)

where K1,2,3 are the Frank elastic constants andthe terms represent splay, twist and bend contri-butions to the free energy density, respectively. Asan approximation we set K1 = K2 = K3 = K, cor-responding to elastic isotropy. Then Eq.(44) be-comes

F0(r) =1

2K

[∇.n(r)]2

+ [∇ × n(r)]2

(45)

Symmetry considerations suggest that n can bewritten in terms of Ψ(r), the local azimuthal an-gle such that n = (sin Ψ, cosΨ, 0). The free energyexpression thus reduces to

F0 =1

2K∇Ψ.∇Ψ (46)

Next consider the electric field of a twisted lightbeam, at a general position vector r = (r, φ, z) incylindrical coordinates, expressible as

Eklp(r) = fklp(r) eiΘklp(r) (47)

where fklp is the electric field amplitude functioncorresponding to the expressions given in Eq.(7).At frequencies far removed from a molecular reso-nance, the coupling of the light can be cast in termsof the dielectric properties of the liquid crystal. Ig-noring any frequency dispersion, the application ofthe twisted light thus leads to an additional field-dependent free energy term Fint given by [48]

Fint = −1

4ε0εa (n.Eklp)

(

n.E∗klp

)

(48)

where εa is is the dielectric isotropy of the liquidcrystal. Assuming that the field is plane-polarisedalong the x-axis, we find that total free energy canbe written as

F = F0 + Fint =1

2K∇Ψ.∇Ψ − Λklp sin2 Ψ(r, z)

(49)where Λl,p is given by

Λklp =1

2ε0εa |fklp|2 (50)

Using Landau’s free energy formalism, it emergesthat Ψ satisfies a second order partial differentialequation in two-dimensions (r, z);

K

∂2Ψ

∂r2+

1

r

∂Ψ

∂r+∂2Ψ

∂z2

− Λl,p sin 2Ψ(r, z) = 0

(51)The above theory has been applied to the case

of the nematic liquid crystal 5CB (pentylcyanotr-phenyl) doped with an anthraquinone derivative


dye (AD1). The system is modelled as an infinitelywide film of thickness d, sandwiched between twothin glass plates, which fix the director angle alongthe top z = d and bottom z = 0. The generalboundary conditions are then in the form

Ψ(r, 0) = Ψ0

Ψ(r, d) = Ψd (52)

and

∂Ψ(r, z)

∂r

∣

∣

∣

∣

r=∞

= 0 (53)

The relevant parameters in this case are K =0.64 × 10−12 N, and εa = 0.5832 [54]. The thick-ness of the film is taken as d = 2000λ whereλ = 600 nm is the wavelength and the intensity ofthe light is 108 Wm−2; the beam waist is taken asw0 = 50λ. We consider the cases of (l, p) = (5, 0)and (l, p) = (5, 1), but the theory can be appliedto any l, p mode and we shall choose other modesfor illustration.

Figures 12 and 13 display a vector field distri-bution and the corresponding colour-coded plot ofthe director orientations in the r − z plane withthe boundary conditions Ψ0 = 0 and Ψd = π/2corresponding to the situation in the absence ofthe twisted light. Figures 14 and 15 show the

0

500

1000

1500

2000

0 50 100 150 200

z/λ

r/λ

(a)

FIG. 12: A vector field representation of the directororientation of the twisted nematic liquid crystal beforeapplication of the twisted light. The orientation at theboundaries z = 0 (bottom) and z = d (top) are fixedas described in the text.

modified landscape once a twisted light beam withl = 5, p = 0 has been switched on. It is clear that

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

(b)

0 50 100 150 200

r/λ

0

500

1000

1500

2000

z/λ

FIG. 13: A colour-coded contour representation of thesame data as in Fig. 10. The key represents a scaleof the local director orientation angle Ψ relative to ther−axis. It spans the angular range Ψ = 0 at the bot-tom (red) to Ψ = π/2 at the top (magenta)

there is a marked re-orientation of the directors,when compared to the situation in the beam-freecase. Figures 16 and 17 concern the case where

0

500

1000

1500

2000

0 50 100 150 200

z/λ

r/λ

(a)

FIG. 14: A vector field representation of the directororientation of the twisted nematic liquid crystal afterthe application of the twisted LGl,p light where l =5, p = 0

l = 5, p = 1 for the applied twisted light mode. Itis seen that the differences in the variation of inten-sity of the light manifest themselves in the directorre-orientation.

8. Comments and conclusions

This article has been concerned primarily with atheory of two-level systems responding to the field


0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

(b)

0 50 100 150 200

r/λ

0

500

1000

1500

2000

z/λ

FIG. 15: A colour-coded contour representation of thesame data as in Fig. 14. The key represents a scaleof the local director orientation angle Ψ relative to ther−axis. It spans the angular range Ψ = 0 at the bot-tom (red) to Ψ = π/2 at the top (magenta)

0

500

1000

1500

2000

0 50 100 150 200

z/λ

r/λ

(a)

FIG. 16: A vector field representation of the directororientation of the twisted nematic liquid crystal afterthe application of the twisted LGl,p with l = 5, p = 1.Other parameters are described in the text.

of Laguerre-Gaussian light and, as a supplemen-tary topic, the influence of such light on liquidcrystals. The primary aim has been to presentan up-to-date account of work in this branch ofoptical angular momentum effects. The resultsfor atomic, ionic and molecular motion displaynovel features; not only do such particles experi-ence modified translational forces in such fields,but the radiation forces include rotational com-ponents which are solely attributable to to theOAM of the light. In the transient regime, ap-plicable within the time interval of the order ofΓ−1, where Γ is the transition rate from the up-per state, the particles experience time-dependentforces and torques leading to characteristic particle

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

(b)

0 50 100 150 200

r/λ

0

500

1000

1500

2000

z/λ

FIG. 17: A colour-coded contour representation of thesame data as in Fig. 16. The key represents a scaleof the local director orientation angle Ψ relative to ther−axis. It spans the angular range Ψ = 0 at the bot-tom (red) to Ψ = π/2 at the top (magenta).

trajectories. In the steady state regime, applicablefor times much larger than Γ−1, the particles ex-perience time-independent forces and torques andthe motion leads to cooling, trapping and novel ro-tational effects in a variety of situations. The sat-uration light-induced torque has the simple form

T ≈ ~lΓz (54)

The result has the simple interpretation that a sin-gle transition transfers angular momentum of mag-nitude ~l, and since there are Γ transitions per unittime, the product amounts to a torque of magni-tude ~Γl along the beam axis z. This supportsthe prediction that the orbital angular momentumof light is quantised in units of ~. We have alsoseen that the two-level system subject to such lightexperiences a new Doppler shift associated withthe rotational component of the interaction. Wehave seen that this additional Doppler shift is thesource of modification of the dissipative forces re-sponsible for laser cooling. The treatments of opti-cal molasses involving pairs of Laguerre-Gaussianbeams in one-, two- and three dimensions involvea variety of steady state optical forces with thepropensity to produce a novel and highly distinc-tive behaviour. The Laguerre-Gaussian light gen-erates optical potential wells associated with thetotal dipole force contributions from all the beamspresent, while the total dissipative force providesa mechanism for cooling or heating the azimuthalmotion along with the axial motion. For each pairof counter-propagating beams, the light-inducedtorque is doubled or annulled, depending on therelative sign of the OAM quantum number l.


It is clear that the subject of atomic and molecu-lar manipulation using structured light, especiallylight carrying orbital angular momentum is still atan early stage of development. Experimental workin which atoms, ions and molecules are trapped inoptical potentials generated by multiple beams isstill to be carried out. Although the theoreticalpredictions presented here involve trapping in themaximum intensity regions, similar predictions, al-beit differing in detail, can be made for trapping inthe dark regions of higher order intersecting mul-tiple beams. Moreover, for particles trapped inoptical beam arrays, it is also emerging that thelight can itself engage with the inter-particle forces.This can, for example lead to azimuthally periodicforce fields, generating necklace-like ring struc-tures. In all these potentialities, the exploratory

studies conducted so far are giving every indica-tion that there is indeed a rich scope for furthertechnical advances leading to widespread optome-chanical applications of structured light [55].

Acknowledgments

The research described in this chapter owes muchto work with a number of colleagues and researchstudents, including Les Allen, Vassilis Lembessis,Wei Lai, William Power, Luciana Davila Romero,Andrew Carter, Mohammad Al-Amri, Simon Hors-ley and Matt Probert. The authors wish to ac-knowledge their valuable contributions. Finan-cial support from the Science and Engineering Re-search Council (EPSRC) are gratefully acknowl-edged.

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Received: 17 July, 2007

Accepted: 28 July, 2007

optical manipulation of atoms and molecules using ...in the gas phase, laser cooling schemes such as...

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