Optical binding with cold

C. E. M´aximo,1 R. Bachelard,1 and R. Kaiser2 1Instituto de F´ısica de S˜aoCarlos, Universidade de S˜aoPaulo, 13560-970 S˜aoCarlos, SP, Brazil 2Universit´eCˆoted’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 in response to the collective scattering of light. Such phenomenon has been studied mainly in the context of equilibrium stability of spheres arrays which move amid dissipative media. In this letter, we demonstrate that optically bounded states of a pair of cold atoms can exist, in the absence of non-radiative damping. We study the scaling laws for the unstable-stable phase transition at negative detuning and the unstable-metastable one for positive detuning. In addition, we show that angular momentum can lead to dynamical stabilisation with infinite range scaling.

The interaction of light with atoms, from the micro- pairs and discuss the increased range of such a dynami- scopic to the macroscopic scale, is one of the most fun- cally stabilized pair of atoms. damental mechanisms in nature. After the advent of the , new techniques were developed to manipulate pre- cisely objects of very different sizes with light, ranging from individual atoms [1] to macrosopic objects in opti- cal tweezers [2]. It is convenient to distinguish two kinds of optical forces which are of fundamental importance: the radiation pressure force, which pushes the particles in the direction of the light propagation, and the dipole force, which tends to trap them into intensity extrema, as for example in optical lattices. Beyond single-particle physics, multiple scattering of light plays an important role in modifying these forces. For instance, the radia- tion pressure force is at the origin of an increase of the size of magneto-optical traps [3] whereas dipole forces can lead to optomechanical self-structuring in a cold atomic gas [4] or to optomechanical strain [5]. For two or more scatterers, mutual exchange of light results in cooperative optical forces, which may induce FIG. 1. Two atoms evolve in the z = 0 plane, trapped in optical mutual trapping, and eventually correlations in 2D by counter-propagating plane-waves, with wave vector or- the relative positions of the particles at distances of the thogonal to that plane. The light exchange between the atoms order of the optical wavelength. This effect, called opti- induces the 2D two-body optically coupled dynamics. cal binding, has been first demonstrated by Golovchenko and coworkers [6, 7], using two dielectric microsized- We consider a system composed of two two-level atoms spheres interacting with light fields within dissipative flu- of mass m which interact with the radiation field. Using ids. Since then a number of experiments with different the dipole approximation, the -laser interaction is P2 geometries and with increasing number of scatterers have described by HAL = − j=1 Dj · E (rj), where Dj is the been reported, all using a suspension of scatterers in a dipole operator and E (rj) the electric field calculated fluid providing thus a viscous damping of the motion of at the center-of-mass rj of each atom. We will describe the scatterers [8–15]. the center of mass of each atom by its classical trajectory In this letter, we demonstrate theoretically the exis- and for simplicity we consider here the scalar linear optics tence of optically bounded motion for a pair of cold atoms regime [16]. The equations of motion for the amplitudes in the absence of non-radiative friction. We consider two of the dipoles and their positions are then given by: atoms confined in two dimensions, e.g. by counterpropa-   ˙ Γ Γ gating . After introducing the model used we first βj = i∆ − βj − iΩ − G (|rj − rl|) βl, (1) 2 2 derive the equilibrium positions of two atoms. We then Γ study the scaling laws for bound states in the case of ~  ∗  ¨rj = − Im ∇rj G (|rj − rl|) βj βl , l 6= j. (2) pairs of atoms without angular momentum, confronting m our finding to known results of optical binding [6, 7]. We Here we assume that the atoms are confined in the z = 0 then turn to the more general situation allowing for an- plane, which can e.g. be obtained by using counter- gular momentum in the initial conditions of the atomic 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 to the two-level transition frequency ωat. Γ is the decay rate of the excited state of the atom and ∆ = ωL − ωat the detuning between the laser frequency and the atomic transition frequency. For a single atom, a frictionless 2D motion emerges. Note that while the dipole ampli- tude βj responds linearly to the light field, for two atoms their coupling with the centers-of-mass rj is responsi- ble for the nonlinearity emerging from Eqs.(1–2). The light-mediated long-range interaction between the atoms is given, in the scalar light approximation, by the Green function

eik|rj −rl| G (|rj − rl|) = , (3) ik |rj − rl| which is obtained from the Markovian integration over the vacuum modes of the electromagnetic field [16]. This coupled 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, linewidth broadening and cooperative frequency shifts [21, 22]. Central force problems, as the one described by Eqs. (1-2), are best studied in the relative coordinate frame, so we define the average (and differential) posi- tions and dipoles of the atoms. One can shown that after a transient of order 1/Γ, the two atomic dipoles synchro- nize, independently of their distance, pump strength or FIG. 2. (a) Stability diagram around the equilibrium point detuning, and their relative position r = r1 − r2 gets r ≈ λ for the 1D dynamics with ` = 0. Free-particle states are coupled only to the average dipole. The center-of-mass found for low laser strength (black region), bound states on motion and the difference in dipole amplitudes thus play the red-detuned side (light blue area) and metastable states no role [23]. The angular momentum L = mrv⊥/2 of the on the blue-detuned side (color gradient). (b) Typical dynam- system is a conserved quantity, with v⊥ the magnitude ics of free-particle, bound and metastable states around the of the velocity perpendicular to the inter-particle separa- equilibrium point. Simulations realized with particles with an tion r = rˆr. This is a fundamental difference with other initial temperature T = 1µK, as throughout this work. optical binding systems where the angular momentum is quickly driven to zero by friction. Note that the stochas- tic heating due to the random atomic recoil which comes simplified condition tan kr = −1/kr which does not de- from spontaneous emission recoils is neglected here. pend on the light matter coupling Ω or ∆, but only on To study bound states we first identify the equilibrium the mutual distance between the atoms. The details of points of the system, which are given by the zeros of the the light-matter interaction will however come into play equation [23] when the stability of these equilibrium points is consid- ered. 2 2 sin kr cos kr ` Ω + 2 2 = kr k r . (4) As pointed out in Ref.[24], linear stability may not be k3r3 Γ2 sin kr 2 cos kr 2 sufficient to provide a phase diagram with bound states. 1 + kr + 2δ + kr We thus choose to study its stability by integrating nu- Note that Eq.(4) depends on the pump strength Ω/Γ, merically the dynamics, starting with a pair of atoms laser detuning δ ≡ ∆/√Γ and the dimensionless√ angular with initial velocities, and moving around the equilib- momentum ` ≡ Lk/2 ~Γm = krv⊥/ 32vDopp, where rium separation r ≈ λ. We first focus on states without 2 vDopp = ~Γ/2m corresponds to the Doppler temperature angular momentum, so the atoms are chosen with oppo- of the two-level . This equation includes both site radial velocities vk, parallel to interatomic distance. stable and unstable equilibrium points. For the partic- We compute the escape time τ at which the atoms start ular case ` = 0, the equilibrium points are given by the to evolve as free-particles by integrating the associated 3

2 one-dimensional dynamics. the two atoms (kBT = mhvki). Eq.(5) is valid for atoms Fig. 2(a) presents the stability diagram of the r ≈ λ at large distances (r  λ) since for short distances, cor- equilibrium point, for a pair of atoms with an initial tem- rections due to their coupling should be included. We perature of T = 1mK (throughout this work, the conver- note that this scaling law corresponds to the law derived sion between temperature and velocity is realized using for particles, where the interaction is given by the 87Rb atom mass m = 1.419 × 10−25 kg). For zero 1 2 2 2 2 W = − 2 α E k cos(kr)/r [6], where the α scaling of angular momentum, the initial temperature is associated the polarisability is the indication of double scattering. 2 to the radial degree of freedom T = mvk/2. In the lower The corresponding scaling law for two-level atoms yields (black) part of the diagram (low pump strength), free- a dipole potential (∝ Ω2) but with double scattering and particle states are always observed after a short transient a corresponding square dependence of the atomic polaris- τ ≈ 0.1 ms, which means that the optical forces are un- ability, which at large detuning scales as α2 ∝ 1/∆2. The able to bind the atoms (see trajectory for δ = −2.5 in difference to previous work lies in the fact that we do not Fig. 2(b)). For negative δ, a free-particle to bounded have an external friction or viscous force, which would motion phase transition occurs as the pumping strength damp the atomic motion independently from the laser Ω/Γ is increased. In the stable phase, the atoms mutual detuning. The metastable phase region is thus a novel distance r converges oscillating to r ≈ λ, as displayed feature for cold atoms compared to dielectrics embedded for δ = −1 in Fig. 2(b). We verified that these bound in a fluid. states, characterized by an infinite escape time τ, occur A fine analysis of the transition between bound states around the equilibrium points r = nλ, with n ∈ . The N to metastable states in Fig. 2(a) shows a small shift δ0 unstable equilibrium points are found around (n + 1/2)λ compared to the single atom condition. The and contains only free-particle states. We stress that the origin of this shift can partially be understood by the damping observed in the bound state region cannot be cooperative energy shift of cos(kr)/2kr for the two-atom associated to a viscous medium as in [6, 7]. In our case, state at the origin of the dynamics for these synchro- the cooling of the relative motion for negative detung- nized dipoles. We also identified an additional velocity ing can be understood as in a multiply dependence beyond this effect, with a total shift scaling scattering regime [25–27]. kv (τ) as δ ≈ − cos kr − k . For positive detuning (δ > 0), we also observe a 0 2kr Γ free-particle to bound states phase transition, symmet- An additional novel feature emerging from the friction- ric to the negative detuning case(see Fig. 2(a)). How- less nature of the cold atom system is the conservation ever we find that these bound states appear to be only of the angular momentum during the dynamics. This metastable, with the pair of atoms separating on large leads to a striking difference to optical binding with di- time scales (see Fig. 2(b)). The binding time is indeed electric particles as we can obtain rotating bound states. observed to grow exponentially with the detuning δ. We Examples of such states are shown in Fig. 3: the pair of have verified that the oscillations of an atom in the opti- atoms can reach a rotating bound states with fixed inter- cal potential of another atom kept at a fixed position also particle distance on resonance (see Figs. 3 (a) and (b)), results in an increase (decrease) of kinetic energy for posi- but it may also support stable oscillations along the two- tive (negative) detuning. These results of mutual heating atom axis far from resonance (see Figs. 3 (c) and (d)), or cooling is thus reminiscent of the asymmetry reported analogue to a molecule vibrational mode. As in the 1D in multiple-scattering based atom cooling schemes [25– case, the atoms may remain coupled for long times, before 27], but differs from previous works on optical binding eventually separating (see Figs. 3 (e) and (f) and also the of dielectric spheres, which did not study the sign of the movie in the supplemental material). As can be seen in particles refractive index [14]. Eq.(4), for ` 6= 0 the stationary points are circular orbits We have performed a systematic study of the free to in the plane z = 0 instead of fixed points. We note that bound states transition and identified the following scal- high values of angular momentum ` strongly modifies the equilibrium point landscape, suppressing the low-r equi- ing law for the critical temperature Tc of this transition: librium points [23]. 2 Tc Ω 16 = 2 2 , (5) Rotating bound states are characterized by both radial TDopp Γ + 4∆ kr vk and tangential v⊥ velocities, the latter being associ- with TDopp = ~Γ/2kB the Doppler temperature. This ated to the conserved angular momentum. For simplicity, criterion can be obtained from the balance between the we focus on initial states of atoms with purely tangential 2 2 kinetic energy Ekin = mkBvk/2 and the dipole potential and opposite velocities: T = mv⊥/2, neglecting thus any induced by the interference between the incident laser initial radial velocity, although it may appear dynami- beam and the scattered light field V (r) = 4~Ω2(Γ2 + cally. A phase diagram for the r ≈ λ and ` = 0.09 (cor- 4∆2)−1 cos(kr)/(kr). We recall that here we are consid- responding to an initial temperature T = 1µK) states is ering zero angular motion dynamics, and the tempera- presented in Fig. (4), where the escape time τ has been ture is thus associated to the parallel (radial) velocity of computed following the same procedure as before. Let us 4

FIG. 4. Stability diagram around the equilibrium point r ≈ λ for ` = 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 temperature T = 1µK, where the conversion between temperature and velocity 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 angular FIG. 3. Rotating states for the pair of atoms for (a–b) momentum implies that over a displacement of δr = λ/2 Fixed inter-particle distance, (c–d) Vibrational mode and (e– necessary to escape the radial potential well, only a por- f) Metastable state. As the orbits of both atoms coincides 2 in (a), only a single orbit appears displayed. Simulations re- tion of the kinetic energy δE = mv⊥L δr/2r is converted alized with ` = 0.09, Ω/Γ = 0.25 and T = 1µK, where the to the radial degree of freedom. As an extra consequence 87Rb atom mass was adopted. of this constraint, the diagram of the ` 6= 0 case presents lifetimes for the metastable states which are much longer than for the ` = 0. Thus, the presence of a conserved an- first remark that the purely-bounded to metastable tran- gular momentum strongly promotes the stability of the sition is not delimited by a sharp transition anymore, in system, and is particular promising for the optical stabi- contrast to the ` = 0 case. For ` 6= 0, this energy shift lization of macroscopic clouds. This dynamical stabiliza- varies nonlinearly according to the field pump strength tion of optical binding in frictionless media is a important Ω/Γ, allowing dynamically stable bound states for δ > δ0, novel feature since it opens the possibility of enhanced including the resonant line. The stable-metastable phase long range and collective effects in cold atoms and be- transition for ` 6= 0 covers a larger part of the phase yond. diagram, so the introduction of an angular momentum In conclusion, our study of optical binding in cold in the system allows to reach bound states for ranges of atoms has allowed to recover the prediction of opti- parameters where they do not exist at ` = 0. cal binding of dielectric particles for negative detuning, A more systematic investigation of the stability of where the viscosity of the embedding medium is replaced states with different angular momentum and inter– by a diffusive Doppler cooling analogue. For positive de- particle distance, leads to the following scaling law for tuning, we find a metastable region, as Doppler heating the ` 6= 0: eventually leads to an escape of the atoms from the mutu- T 8 Ω2 ally induced dipole potential. We also identified a novel c = , (6) T π Γ2 + 4∆2 dynamical stabilization with rotating bound states. We Dopp have shown that pairs of cold atoms can exhibit optically where the temperature is now related to the tangential bound states in vacuum. The absence of non-radiative 2 velocity as T kB = mhv⊥i. In excellent agreement with damping in the motion allows for a new class of dynami- simulations, Eq.(6) differs substantially from the ` = 0 cally bound states – a phenomenon not present in other case in that the distance between the atoms has disap- optical binding setups. While this demonstration of op- peared. This peculiar effect originates in the fact that tical binding for a pair of particles paves the way for the 5 study of this phenomenon on larger atomic systems, the Rev. Lett. 96, 068102 (2006). generalization of these peculiar stability properties will [12] V. Kar´asek, K. Dholakia, and P. Zem´anek,Applied be an important issue to understand the all-optical sta- Physics B 84, 149 (2006). bility of large clouds. [13] N. K. Metzger, R. F. Marchington, M. Mazilu, R. L. Smith, K. Dholakia, and E. M. Wright, Phys. Rev. Lett. One interesting generalization is the study of opti- 98, 068102 (2007). cal forces in astrophysical situations. Whereas radiation [14] K. Dholakia and P. Zem´anek, Review of Modern Physics pressure forces are well studied and participate for in- 82, 1767 (2010). stance in the determination of the size of a star, dipole [15] T. M. Grzegorczyk, J. Rohner, and J.-M. Fournier, Phys. forces are often neglected [28]. The possibility of trapping Rev. Lett. 112, 023902 (2014). a large assembly of particles in space would allow to con- [16] T. Bienaim´e,R. Bachelard, N. Piovella, and R. Kaiser, Fortschritte der Physik 61, 377 (2013). sider novel approaches in astrophysical imaging [15, 29] [17] T. Bienaim´e,S. Bux, E. Lucioni, P. W. Courteille, N. Pi- and could shed additional light on the motion of atoms ovella, and R. Kaiser, Physical Review Letters 104, such as the abondant hydrogen around high intensity re- 183602 (2010). gions of galaxies, where even small corrections to the pure [18] S. Bux, E. Lucioni, H. Bender, T. Bienaim´e,K. Lauber, gravitational attraction might be important [30]. C. Stehle, C. Zimmermann, S. Slama, P. Courteille, C.E.M. and R.B. hold Grants from So Paulo Re- N. Piovella, and R. Kaiser, Journal of Modern Optics search Foundation (FAPESP) (Grant Nos. 2014/19459- 57, 1841 (2010). 6, 2016/14324-0, 2015/50422-4 and 2014/01491-0). R.B. [19] M. O. Ara´ujo,I. Kreˇsi´c,R. Kaiser, and W. Guerin, Phys. Rev. Lett. 117, 073002 (2016). and R.K. received support from project CAPES- [20] W. Guerin, M. O. Ara´ujo, and R. Kaiser, Phys. Rev. COFECUB (Ph879-17/CAPES 88887.130197/2017-01). Lett. 116, 083601 (2016). We thank fruitful discussions with N. Piovella. [21] S. L. Bromley, B. Zhu, M. Bishof, X. Zhang, T. 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SUPPLEMENTAL MATERIAL Derivation of the equilibrium condition Eq.(4)

Effective two-dimensional dynamics and cooperative As there is no source term Eq. (S-5), the relative shift dipoles coordinate b vanishes on a time scale of the order of 1/Γ, which is very fast compared to the atomic motion. Therefore, the atoms can be considered always synchro- Eqs. (1-2) of the main text can be rewritten in a more nized and the inter-atomic dynamics r couples only to convenient form through the following variables transfor- the average dipole B. It is thus possible to write mation: iχ B = β1 = β2 ≡ βe , (S-9) with β ≥ 0 and χ ∈ R. Then, the set of dynamical b = β1 − β2, (S-1) equations reduces to: β + β B = 1 2 , (S-2) . Γ  sin kr  2 β = − 1 + β − Ω sin χ, (S-10) 2 kr r = r1 − r2, (S-3) r + r . Γ cos kr cos χ R = 1 2 . (S-4) χ = ∆ + − Ω , (S-11) 2 2 kr β 2L2 2Γ k sin kr cos kr  r¨ = − ~ + β2. (S-12) m2r3 m kr k2r2 where B and b are the average and differential dipoles, and R and r the average and differential positions. As The equilibrium points of these equations are given by the motion is restricted to two dimensions, we choose setting β˙ =χ ˙ = 0: the polar coordinates parametrization r = r (cos φ, sin φ), 1 " #− 2 which yields the following dynamical equations: Ω  sin kr 2  cos kr 2 β = 2 1 + + 2δ + (S-13), 0 Γ kr kr ! .    sin kr Γ sin kr cos kr 1 + kr b = − 1 − − i 2δ − b, (S-5) χ0 = arctan − cos kr . (S-14) 2 kr kr 2δ + kr . Γ  sin kr  cos kr  B = − 1 + − i 2δ + B − iΩ, (S-6) which, combined withr ¨ = 0, gives the equilibrium condi- 2 kr kr tion for the interatomic separations, Eq.(4) in the main 2L2 Γ k sin kr cos kr    r¨ = − ~ + 4 |B|2 − |b|2(S-7), text. If one redefines Eq.(4) as m2r3 2m kr k2r2 . 2 2 sin kr cos kr ! ` Ω + 2 2 d mr2φ F ≡ − kr k r , (S-15) L˙ = = 0, (S-8) k3r3 Γ2 sin kr 2 cos kr 2 dt 2 1 + kr + 2δ + kr the equilibrium points are obtained from the zeros of F (see Fig. 1). Fig. 1(a) illustrates the dependence of F where the center-of-mass dynamics of the two-atom sys- on the angular momentum and on the detuning, showing tem has naturally decoupled from all equations. that zeros of F tend to disappear as we get farther from The decay of the relative dipole b, given by Eq.(S-5), resonance or at larger angular momentum. is given by Γ, independently of the pump strength or of the inter-atomic distance. Once b becomes negligible, the two atoms are synchronized, and their dipole is given by r ≈ 2λ equilibrium point the average dipole B. This synchronized mode has an en- ergy − cos kr/2kr, relative to the atomic transition, and In Fig. 2 the phase diagram of the r ≈ 2λ equilib- its decay rate is 1 + sin(kr)/(kr). Nevertheless, the equi- rium point is shown: The pair of atoms presents a higher librium points which emerge from the atom dynamics, at threshold in pump strength to become stable, in order r ≈ nλ (n ∈ N), makes that this rate is approximately Γ, to compensate for their weaker interaction at a larger so no superradiant or subradiant behaviour is expected. distance. 2

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

FIG. 2. Stability diagram for the equilibrium point around r ≈ 2λ.