Optical binding with cold atoms

C. E. Maximo,1 R. Bachelard,1 and R. Kaiser2

1Instituto de Fısica de Sao Carlos, Universidade de Sao Paulo, 13560-970 Sao Carlos, SP, Brazil2Universite Cote d’Azur, CNRS, INPHYNI, 06560 Valbonne, France

(Dated: November 16, 2017)

Optical binding is a form of light-mediated forces between elements of matter which emerge inresponse to the collective scattering of light. Such phenomenon has been studied mainly in thecontext of equilibrium stability of dielectric spheres arrays which move amid dissipative media. Inthis letter, we demonstrate that optically bounded states of a pair of cold atoms can exist, in theabsence of non-radiative damping. We study the scaling laws for the unstable-stable phase transitionat negative detuning and the unstable-metastable one for positive detuning. In addition, we showthat angular momentum can lead to dynamical stabilisation with infinite range scaling.

The interaction of light with atoms, from the micro-scopic to the macroscopic scale, is one of the most fun-damental mechanisms in nature. After the advent of thelaser, new techniques were developed to manipulate pre-cisely objects of very different sizes with light, rangingfrom individual atoms [1] to macrosopic objects in opti-cal tweezers [2]. It is convenient to distinguish two kindsof optical forces which are of fundamental importance:the radiation pressure force, which pushes the particlesin the direction of the light propagation, and the dipoleforce, which tends to trap them into intensity extrema,as for example in optical lattices. Beyond single-particlephysics, multiple scattering of light plays an importantrole in modifying these forces. For instance, the radia-tion pressure force is at the origin of an increase of thesize of magneto-optical traps [3] whereas dipole forces canlead to optomechanical self-structuring in a cold atomicgas [4] or to optomechanical strain [5].

For two or more scatterers, mutual exchange of lightresults in cooperative optical forces, which may induceoptical mutual trapping, and eventually correlations inthe relative positions of the particles at distances of theorder of the optical wavelength. This effect, called opti-cal binding, has been first demonstrated by Golovchenkoand coworkers [6, 7], using two dielectric microsized-spheres interacting with light fields within dissipative flu-ids. Since then a number of experiments with differentgeometries and with increasing number of scatterers havebeen reported, all using a suspension of scatterers in afluid providing thus a viscous damping of the motion ofthe scatterers [8–15].

In this letter, we demonstrate theoretically the exis-tence of optically bounded motion for a pair of cold atomsin the absence of non-radiative friction. We consider twoatoms confined in two dimensions, e.g. by counterpropa-gating lasers. After introducing the model used we firstderive the equilibrium positions of two atoms. We thenstudy the scaling laws for bound states in the case ofpairs of atoms without angular momentum, confrontingour finding to known results of optical binding [6, 7]. Wethen turn to the more general situation allowing for an-gular momentum in the initial conditions of the atomic

pairs and discuss the increased range of such a dynami-cally stabilized pair of atoms.

FIG. 1. Two atoms evolve in the z = 0 plane, trapped in2D by counter-propagating plane-waves, with wave vector or-thogonal to that plane. The light exchange between the atomsinduces the 2D two-body optically coupled dynamics.

We consider a system composed of two two-level atomsof mass m which interact with the radiation field. Usingthe dipole approximation, the atom-laser interaction isdescribed by HAL = −

∑2j=1 Dj ·E (rj), where Dj is the

dipole operator and E (rj) the electric field calculatedat the center-of-mass rj of each atom. We will describethe center of mass of each atom by its classical trajectoryand for simplicity we consider here the scalar linear opticsregime [16]. The equations of motion for the amplitudesof the dipoles and their positions are then given by:

βj =

(i∆− Γ

2

)βj − iΩ−

Γ

2G (|rj − rl|)βl, (1)

rj = −~Γ

mIm[∇rjG (|rj − rl|)β∗j βl

], l 6= j. (2)

Here we assume that the atoms are confined in the z = 0plane, which can e.g. be obtained by using counter-propagating waves. The strength of the atom-laser cou-

2

pling is given for a homogeneous laser by the Rabi fre-quency Ω and laser frequency ωL = ck tuned close tothe two-level transition frequency ωat. Γ is the decayrate of the excited state of the atom and ∆ = ωL − ωatthe detuning between the laser frequency and the atomictransition frequency. For a single atom, a frictionless2D motion emerges. Note that while the dipole ampli-tude βj responds linearly to the light field, for two atomstheir coupling with the centers-of-mass rj is responsi-ble for the nonlinearity emerging from Eqs.(1–2). Thelight-mediated long-range interaction between the atomsis given, in the scalar light approximation, by the Greenfunction

G (|rj − rl|) =eik|rj−rl|

ik |rj − rl|, (3)

which is obtained from the Markovian integration overthe vacuum modes of the electromagnetic field [16]. Thiscoupled dipole model, has been shown to provide an ac-curate description of many phenomena based on cooper-ative light scattering, such as the observation of a cooper-ative radiation pressure force [17, 18], superradiance [19]and subradiance [20] in dilute atomic clouds, linewidthbroadening and cooperative frequency shifts [21, 22].

Central force problems, as the one described byEqs. (1-2), are best studied in the relative coordinateframe, so we define the average (and differential) posi-tions and dipoles of the atoms. One can shown that aftera transient of order 1/Γ, the two atomic dipoles synchro-nize, independently of their distance, pump strength ordetuning, and their relative position r = r1 − r2 getscoupled only to the average dipole. The center-of-massmotion and the difference in dipole amplitudes thus playno role [23]. The angular momentum L = mrv⊥/2 of thesystem is a conserved quantity, with v⊥ the magnitudeof the velocity perpendicular to the inter-particle separa-tion r = rr. This is a fundamental difference with otheroptical binding systems where the angular momentum isquickly driven to zero by friction. Note that the stochas-tic heating due to the random atomic recoil which comesfrom spontaneous emission recoils is neglected here.

To study bound states we first identify the equilibriumpoints of the system, which are given by the zeros of theequation [23]

`2

k3r3=

Ω2

Γ2

sin krkr + cos kr

k2r2(1 + sin kr

kr

)2+(2δ + cos kr

kr

)2 . (4)

Note that Eq.(4) depends on the pump strength Ω/Γ,laser detuning δ ≡ ∆/Γ and the dimensionless angularmomentum ` ≡ Lk/2

√~Γm = krv⊥/

√32vDopp, where

v2Dopp = ~Γ/2m corresponds to the Doppler temperature

of the two-level laser cooling. This equation includes bothstable and unstable equilibrium points. For the partic-ular case ` = 0, the equilibrium points are given by the

FIG. 2. (a) Stability diagram around the equilibrium pointr ≈ λ for the 1D dynamics with ` = 0. Free-particle states arefound for low laser strength (black region), bound states onthe red-detuned side (light blue area) and metastable stateson the blue-detuned side (color gradient). (b) Typical dynam-ics of free-particle, bound and metastable states around theequilibrium point. Simulations realized with particles with aninitial temperature T = 1µK, as throughout this work.

simplified condition tan kr = −1/kr which does not de-pend on the light matter coupling Ω or ∆, but only onthe mutual distance between the atoms. The details ofthe light-matter interaction will however come into playwhen the stability of these equilibrium points is consid-ered.

As pointed out in Ref.[24], linear stability may not besufficient to provide a phase diagram with bound states.We thus choose to study its stability by integrating nu-merically the dynamics, starting with a pair of atomswith initial velocities, and moving around the equilib-rium separation r ≈ λ. We first focus on states withoutangular momentum, so the atoms are chosen with oppo-site radial velocities v‖, parallel to interatomic distance.We compute the escape time τ at which the atoms startto evolve as free-particles by integrating the associated

3

one-dimensional dynamics.Fig. 2(a) presents the stability diagram of the r ≈ λ

equilibrium point, for a pair of atoms with an initial tem-perature of T = 1mK (throughout this work, the conver-sion between temperature and velocity is realized usingthe 87Rb atom mass m = 1.419 × 10−25 kg). For zeroangular momentum, the initial temperature is associatedto the radial degree of freedom T = mv2

‖/2. In the lower

(black) part of the diagram (low pump strength), free-particle states are always observed after a short transientτ ≈ 0.1 ms, which means that the optical forces are un-able to bind the atoms (see trajectory for δ = −2.5 inFig. 2(b)). For negative δ, a free-particle to boundedmotion phase transition occurs as the pumping strengthΩ/Γ is increased. In the stable phase, the atoms mutualdistance r converges oscillating to r ≈ λ, as displayedfor δ = −1 in Fig. 2(b). We verified that these boundstates, characterized by an infinite escape time τ , occuraround the equilibrium points r = nλ, with n ∈ N. Theunstable equilibrium points are found around (n+ 1/2)λand contains only free-particle states. We stress that thedamping observed in the bound state region cannot beassociated to a viscous medium as in [6, 7]. In our case,the cooling of the relative motion for negative detung-ing can be understood as Doppler cooling in a multiplyscattering regime [25–27].

For positive detuning (δ > 0), we also observe afree-particle to bound states phase transition, symmet-ric to the negative detuning case(see Fig. 2(a)). How-ever we find that these bound states appear to be onlymetastable, with the pair of atoms separating on largetime scales (see Fig. 2(b)). The binding time is indeedobserved to grow exponentially with the detuning δ. Wehave verified that the oscillations of an atom in the opti-cal potential of another atom kept at a fixed position alsoresults in an increase (decrease) of kinetic energy for posi-tive (negative) detuning. These results of mutual heatingor cooling is thus reminiscent of the asymmetry reportedin multiple-scattering based atom cooling schemes [25–27], but differs from previous works on optical bindingof dielectric spheres, which did not study the sign of theparticles refractive index [14].

We have performed a systematic study of the free tobound states transition and identified the following scal-ing law for the critical temperature Tc of this transition:

TcTDopp

=Ω2

Γ2 + 4∆2

16

kr, (5)

with TDopp = ~Γ/2kB the Doppler temperature. Thiscriterion can be obtained from the balance between thekinetic energy Ekin = mkBv

2‖/2 and the dipole potential

induced by the interference between the incident laserbeam and the scattered light field V (r) = 4~Ω2(Γ2 +4∆2)−1 cos(kr)/(kr). We recall that here we are consid-ering zero angular motion dynamics, and the tempera-ture is thus associated to the parallel (radial) velocity of

the two atoms (kBT = m〈v2‖〉). Eq.(5) is valid for atoms

at large distances (r λ) since for short distances, cor-rections due to their coupling should be included. Wenote that this scaling law corresponds to the law derivedfor dielectrics particles, where the interaction is given byW = − 1

2α2E2k2cos(kr)/r [6], where the α2 scaling of

the polarisability is the indication of double scattering.The corresponding scaling law for two-level atoms yieldsa dipole potential (∝ Ω2) but with double scattering anda corresponding square dependence of the atomic polaris-ability, which at large detuning scales as α2 ∝ 1/∆2. Thedifference to previous work lies in the fact that we do nothave an external friction or viscous force, which woulddamp the atomic motion independently from the laserdetuning. The metastable phase region is thus a novelfeature for cold atoms compared to dielectrics embeddedin a fluid.

A fine analysis of the transition between bound statesto metastable states in Fig. 2(a) shows a small shift δ0compared to the single atom resonance condition. Theorigin of this shift can partially be understood by thecooperative energy shift of cos(kr)/2kr for the two-atomstate at the origin of the dynamics for these synchro-nized dipoles. We also identified an additional velocitydependence beyond this effect, with a total shift scaling

as δ0 ≈ − cos kr2kr −

kv‖(τ)

Γ .

An additional novel feature emerging from the friction-less nature of the cold atom system is the conservationof the angular momentum during the dynamics. Thisleads to a striking difference to optical binding with di-electric particles as we can obtain rotating bound states.Examples of such states are shown in Fig. 3: the pair ofatoms can reach a rotating bound states with fixed inter-particle distance on resonance (see Figs. 3 (a) and (b)),but it may also support stable oscillations along the two-atom axis far from resonance (see Figs. 3 (c) and (d)),analogue to a molecule vibrational mode. As in the 1Dcase, the atoms may remain coupled for long times, beforeeventually separating (see Figs. 3 (e) and (f) and also themovie in the supplemental material). As can be seen inEq.(4), for ` 6= 0 the stationary points are circular orbitsin the plane z = 0 instead of fixed points. We note thathigh values of angular momentum ` strongly modifies theequilibrium point landscape, suppressing the low-r equi-librium points [23].

Rotating bound states are characterized by both radialv‖ and tangential v⊥ velocities, the latter being associ-ated to the conserved angular momentum. For simplicity,we focus on initial states of atoms with purely tangentialand opposite velocities: T = mv2

⊥/2, neglecting thus anyinitial radial velocity, although it may appear dynami-cally. A phase diagram for the r ≈ λ and ` = 0.09 (cor-responding to an initial temperature T = 1µK) states ispresented in Fig. (4), where the escape time τ has beencomputed following the same procedure as before. Let us

4

FIG. 3. Rotating states for the pair of atoms for (a–b)Fixed inter-particle distance, (c–d) Vibrational mode and (e–f) Metastable state. As the orbits of both atoms coincidesin (a), only a single orbit appears displayed. Simulations re-alized with ` = 0.09, Ω/Γ = 0.25 and T = 1µK, where the87Rb atom mass was adopted.

first remark that the purely-bounded to metastable tran-sition is not delimited by a sharp transition anymore, incontrast to the ` = 0 case. For ` 6= 0, this energy shiftvaries nonlinearly according to the field pump strengthΩ/Γ, allowing dynamically stable bound states for δ > δ0,including the resonant line. The stable-metastable phasetransition for ` 6= 0 covers a larger part of the phasediagram, so the introduction of an angular momentumin the system allows to reach bound states for ranges ofparameters where they do not exist at ` = 0.

A more systematic investigation of the stability ofstates with different angular momentum and inter–particle distance, leads to the following scaling law forthe ` 6= 0:

TcTDopp

=8

π

Ω2

Γ2 + 4∆2, (6)

where the temperature is now related to the tangentialvelocity as TkB = m〈v2

⊥〉. In excellent agreement withsimulations, Eq.(6) differs substantially from the ` = 0case in that the distance between the atoms has disap-peared. This peculiar effect originates in the fact that

FIG. 4. Stability diagram around the equilibrium point r ≈ λfor ` = 0.09. Free-particle states are found for low pump in-tensity (black region), bound states on the red-detuned side(light blue) and metastable states in the blue-detuned side,where the color gradient denotes the state lifetime. Sim-ulations realized with particles with an initial temperatureT = 1µK, where the conversion between temperature andvelocity was realized using the Rb atom mass.

while the kinetic energy is initially associated to the rota-tional degree of freedom, the conservation of the angularmomentum implies that over a displacement of δr = λ/2necessary to escape the radial potential well, only a por-tion of the kinetic energy δE = mv⊥L

2δr/2r is convertedto the radial degree of freedom. As an extra consequenceof this constraint, the diagram of the ` 6= 0 case presentslifetimes for the metastable states which are much longerthan for the ` = 0. Thus, the presence of a conserved an-gular momentum strongly promotes the stability of thesystem, and is particular promising for the optical stabi-lization of macroscopic clouds. This dynamical stabiliza-tion of optical binding in frictionless media is a importantnovel feature since it opens the possibility of enhancedlong range and collective effects in cold atoms and be-yond.

In conclusion, our study of optical binding in coldatoms has allowed to recover the prediction of opti-cal binding of dielectric particles for negative detuning,where the viscosity of the embedding medium is replacedby a diffusive Doppler cooling analogue. For positive de-tuning, we find a metastable region, as Doppler heatingeventually leads to an escape of the atoms from the mutu-ally induced dipole potential. We also identified a noveldynamical stabilization with rotating bound states. Wehave shown that pairs of cold atoms can exhibit opticallybound states in vacuum. The absence of non-radiativedamping in the motion allows for a new class of dynami-cally bound states – a phenomenon not present in otheroptical binding setups. While this demonstration of op-tical binding for a pair of particles paves the way for the

5

study of this phenomenon on larger atomic systems, thegeneralization of these peculiar stability properties willbe an important issue to understand the all-optical sta-bility of large clouds.

One interesting generalization is the study of opti-cal forces in astrophysical situations. Whereas radiationpressure forces are well studied and participate for in-stance in the determination of the size of a star, dipoleforces are often neglected [28]. The possibility of trappinga large assembly of particles in space would allow to con-sider novel approaches in astrophysical imaging [15, 29]and could shed additional light on the motion of atomssuch as the abondant hydrogen around high intensity re-gions of galaxies, where even small corrections to the puregravitational attraction might be important [30].

C.E.M. and R.B. hold Grants from So Paulo Re-search Foundation (FAPESP) (Grant Nos. 2014/19459-6, 2016/14324-0, 2015/50422-4 and 2014/01491-0). R.B.and R.K. received support from project CAPES-COFECUB (Ph879-17/CAPES 88887.130197/2017-01).We thank fruitful discussions with N. Piovella.

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[16] T. Bienaime, R. Bachelard, N. Piovella, and R. Kaiser,Fortschritte der Physik 61, 377 (2013).

[17] T. Bienaime, S. Bux, E. Lucioni, P. W. Courteille, N. Pi-ovella, and R. Kaiser, Physical Review Letters 104,183602 (2010).

[18] S. Bux, E. Lucioni, H. Bender, T. Bienaime, K. Lauber,C. Stehle, C. Zimmermann, S. Slama, P. Courteille,

N. Piovella, and R. Kaiser, Journal of Modern Optics57, 1841 (2010).

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[20] W. Guerin, M. O. Araujo, and R. Kaiser, Phys. Rev.Lett. 116, 083601 (2016).

[21] S. L. Bromley, B. Zhu, M. Bishof, X. Zhang, T. Both-well, J. Schachenmayer, T. L. Nicholson, R. Kaiser, S. F.Yelin, M. D. Lukin, A. M. Rey, and J. Ye, Nature Com-munications 7 (2016).

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1

SUPPLEMENTAL MATERIAL

Effective two-dimensional dynamics and cooperativeshift

Eqs. (1-2) of the main text can be rewritten in a moreconvenient form through the following variables transfor-mation:

b = β1 − β2, (S-1)

B =β1 + β2

2, (S-2)

r = r1 − r2, (S-3)

R =r1 + r2

2. (S-4)

where B and b are the average and differential dipoles,and R and r the average and differential positions. Asthe motion is restricted to two dimensions, we choosethe polar coordinates parametrization r = r (cosφ, sinφ),which yields the following dynamical equations:

.

b = −Γ

2

[1− sin kr

kr− i(

2δ − cos kr

kr

)]b, (S-5)

.

B = −Γ

2

[1 +

sin kr

kr− i(

2δ +cos kr

kr

)]B − iΩ, (S-6)

r =2L2

m2r3− Γ~k

2m

(sin kr

kr+

cos kr

k2r2

)(4 |B|2 − |b|2

),(S-7)

L =d

dt

(mr2

.

φ

2

)= 0, (S-8)

where the center-of-mass dynamics of the two-atom sys-tem has naturally decoupled from all equations.

The decay of the relative dipole b, given by Eq.(S-5),is given by Γ, independently of the pump strength or ofthe inter-atomic distance. Once b becomes negligible, thetwo atoms are synchronized, and their dipole is given bythe average dipole B. This synchronized mode has an en-ergy − cos kr/2kr, relative to the atomic transition, andits decay rate is 1 + sin(kr)/(kr). Nevertheless, the equi-librium points which emerge from the atom dynamics, atr ≈ nλ (n ∈ N), makes that this rate is approximately Γ,so no superradiant or subradiant behaviour is expected.

Derivation of the equilibrium condition Eq.(4)

As there is no source term Eq. (S-5), the relativedipoles coordinate b vanishes on a time scale of the orderof 1/Γ, which is very fast compared to the atomic motion.Therefore, the atoms can be considered always synchro-nized and the inter-atomic dynamics r couples only tothe average dipole B. It is thus possible to write

B = β1 = β2 ≡ βeiχ, (S-9)with β ≥ 0 and χ ∈ R. Then, the set of dynamicalequations reduces to:

.

β = −Γ

2

(1 +

sin kr

kr

)β − Ω sinχ, (S-10)

.χ = ∆ +

Γ

2

cos kr

kr− Ω

cosχ

β, (S-11)

r =2L2

m2r3− 2Γ~k

m

(sin kr

kr+

cos kr

k2r2

)β2. (S-12)

The equilibrium points of these equations are given bysetting β = χ = 0:

β0 = 2Ω

Γ

[(1 +

sin kr

kr

)2

+

(2δ +

cos kr

kr

)2]− 1

2

,(S-13)

χ0 = arctan

(−

1 + sin krkr

2δ + cos krkr

). (S-14)

which, combined with r = 0, gives the equilibrium condi-tion for the interatomic separations, Eq.(4) in the maintext. If one redefines Eq.(4) as

F ≡ `2

k3r3− Ω2

Γ2

sin krkr + cos kr

k2r2(1 + sin kr

kr

)2+(2δ + cos kr

kr

)2 , (S-15)

the equilibrium points are obtained from the zeros of F(see Fig. 1). Fig. 1(a) illustrates the dependence of Fon the angular momentum and on the detuning, showingthat zeros of F tend to disappear as we get farther fromresonance or at larger angular momentum.

r ≈ 2λ equilibrium point

In Fig. 2 the phase diagram of the r ≈ 2λ equilib-rium point is shown: The pair of atoms presents a higherthreshold in pump strength to become stable, in orderto compensate for their weaker interaction at a largerdistance.

2

FIG. 1. Two-atoms equilibrium distances pattern for different` (a) and different detunings (b).

FIG. 2. Stability diagram for the equilibrium point aroundr ≈ 2λ.