Artificial abelian gauge potentials induced by dipole-dipole interactions between Rydberg atoms
aa r X i v : . [ qu a n t - ph ] D ec Artificial abelian gauge potentials induced by dipole-dipole interactions betweenRydberg atoms
A. Cesa and J. Martin ∗ Institut de Physique Nucléaire, Atomique et de Spectroscopie,Université de Liège, Bât. B15, B - 4000 Liège, Belgium (Dated: October 16, 2018)We analyze the influence of dipole-dipole interactions between Rydberg atoms on the generationof Abelian artificial gauge potentials and fields. When two Rydberg atoms are driven by a uniformlaser field, we show that the combined atom-atom and atom-field interactions give rise to new,nonuniform, artificial gauge potentials. We identify the mechanism responsible for the emergenceof these gauge potentials. Analytical expressions for the latter indicate that the strongest artificialmagnetic fields are reached in the regime intermediate between the dipole blockade regime and theregime in which the atoms are sufficiently far apart such that atom-light interaction dominatesover atom-atom interactions. We discuss the differences and similarities of artificial gauge fieldsoriginating from resonant dipole-dipole and van der Waals interactions. We also give an estimationof experimentally attainable artificial magnetic fields resulting from this mechanism and we discusstheir detection through the deflection of the atomic motion.
PACS numbers: 11.15.-q, 34.20.Cf, 32.80.Rm, 37.10.Vz
I. INTRODUCTION
In 1996, Lloyd [1] showed that the dynamics of com-plex many-body quantum systems can be efficiently sim-ulated by quantum computers, an idea first put forwardby Manin [2] and further developed by Feynman [3]. Al-though the first quantum computers of a few qubits havebeen realized experimentally [4, 5], the advent of scalablequantum computers might take another few decades. Analternative tool in the context of simulation is a highlycontrollable quantum system able to mimic the dynamicsof other complex quantum systems, known as an analogquantum simulator. Cold neutral atoms and trapped ionshave been shown to be versatile quantum simulators [6, 7]thanks to their high flexibility, controllability, and scala-bility. They permit one to study a wide range of problemsarising from atomic physics, relativistic quantum physics,or cosmology [8]. Since neutral atoms do not carry anynet charge, the simulation of electric and magnetic con-densed matter phenomena, such as the spin Hall effect,seems out of reach. To overcome this apparent difficulty,the idea has been proposed to create artificial electro-magnetic potentials for neutral atoms based on atom-light interaction [9–12]. These artificial potentials act onneutral atoms as real electromagnetic potentials act oncharged particles. Soon after, proposals for the gener-ation of non-Abelian gauge potentials came out [13–15],inspired by the work of Wilczek and Zee [16] on adiabaticevolution in the presence of degenerate eigenstates.Many works on artificial gauge potentials induced byatom-light interaction adopt a single-particle approach.The predicted potentials are then supposed to be valid fora system of many weakly interacting atoms, like, e.g., in a ∗ [email protected] Bose-Einstein condensate [17–20]. However, new physicsarise in ultracold dipolar gases where long-range inter-actions between atoms are known to play an importantrole [21–23]. So far, the consequences of atom-atom in-teractions on the generation of artificial gauge fields haslittle been studied. In particular, dipole-dipole interac-tions could significantly modify artificial gauge fields, oreven give rise to new artificial gauge fields otherwise notpresent. The aim of this paper is to address this problemanalytically by studying a system of two long-lived in-teracting Rydberg atoms driven by a common laser field.In a recent work [24], another configuration was studiedin which two interacting Rydberg atoms are subjectedto a static electric field. Abelian and non-Abelian artifi-cial gauge fields were computed numerically for differentamounts of Stark shifts asymmetry of the Zeeman sub-levels, and were shown to realize a velocity-dependentbeam splitter [25].The paper is organized as follows. In Sec. II, we presentsome established results on artificial gauge fields for a sin-gle two-level atom evolving adiabatically [12] as these willbe used throughout the paper. We give at the same timea different formulation of the single-atom artificial gaugepotentials. We then generalize the analysis to two nonin-teracting two-level atoms. In Sec. III, we study a systemof two interacting Rydberg atoms driven by a uniformlaser field and calculate the associated artificial gaugepotentials when the system evolves adiabatically. We dis-cuss the general expressions and explain the mechanismresponsible for these potentials and fields. We estimatethe attainable field strengths and consider their detec-tion in view of recent experiments. A brief conclusionis drawn in Sec. IV. Sections V and VI are appendixesdedicated to the calculation of the artificial gauge po-tentials in the center-of-mass coordinate system and tothe derivation of an effective Hamiltonian in the dipoleblockade regime.
II. ARTIFICIAL GAUGE POTENTIALSWITHOUT ATOM-ATOM INTERACTIONSA. A single two-level atom
We consider a single two-level atom interacting witha classical electric field E ( r , t ) = E ǫ cos( k L · r − ω L t ) of amplitude E , polarization ǫ , and wavevector k L . Inthe electric dipole and rotating wave approximations, theHamiltonian accounting for the internal dynamics of theatom is given, in the interaction picture, by ˆ H l = ~ (cid:0) Ω e i k L · r | e ih g | + H . c . (cid:1) − ~ δ σ z (1)where | e i , | g i are the atomic excited and ground statesrespectively, separated in energy by ~ ω , Ω = − d ∗ · ǫ E / ~ is the Rabi frequency with d = h g | ˆ d | e i the dipole ma-trix element for the transition | e i ↔ | g i , δ = ω L − ω is the detuning between the laser and atomic transi-tion frequencies, H . c . stands for Hermitian conjugate and σ z = | e ih e | − | g ih g | . Hamiltonian (1) admits two nonde-generate eigenvectors: | χ ± ( r ) i = ( − δ ± Λ) e i k L · r p ∓ δ Λ) | e i + Ω ∗ p ∓ δ Λ) | g i (2)of energy E ± = ± ~ Λ / where Λ ≡ Λ( r ) = p | Ω( r ) | + δ ( r ) (3)is the generalized Rabi frequency. The latter can varywith the atomic position both through the Rabi fre-quency Ω via the electric field amplitude and throughthe detuning which can be made position dependent bymeans of the Zeeman effect [18, 19].When the atomic motion is treated quantum-mechanically, the Hamiltonian governing the system isgiven in the position representation by ˆ H = (cid:18) ˆ p m + U (cid:19) ⊗ ˆ int + ˆ ext ⊗ ˆ H l (4)where ˆ p = ~ ∇ r /i is the atomic momentum operator, m is the atomic mass, U is a potential energy insensitiveto the atomic internal state, ˆ H l , given by Eq. (1), isthe atomic internal Hamiltonian, and ˆ int ( ˆ ext ) denotesthe identity operator in the atomic internal (external)Hilbert space.The internal state of the atom can always be decom-posed onto the basis states (2) which depend parametri-cally on the atomic position r . With such a decomposi-tion, the global wave function of the atom in the positionrepresentation reads [12] h r | ψ ( t ) i = X j = ± ψ j ( r , t ) | χ j ( r ) i . (5)When the atom is initially in the internal state | χ + ( r ) i and moves sufficiently slowly to ensure adiabatic evolu-tion, it stays over time in the internal state | χ + ( r ) i . In this case, the population of the state | χ − ( r ) i remains neg-ligible such that h r | ψ ( t ) i ≈ ψ + ( r , t ) | χ + ( r ) i at any time t . Plugging this expression of the wave function into thetime-dependent Schrödinger equation for Hamiltonian(4) and projecting onto | χ + ( r ) i , we obtain a Schrödinger-like equation for ψ + ( r , t ) , i ~ ∂∂t ψ + ( r , t ) = (cid:20) (cid:0) ˆ p − q A + (cid:1) m + qφ + + U + ~ Λ2 (cid:21) ψ + ( r , t ) , (6)with q A + ( r ) = i ~ h χ + | ∇ r χ + i , (7)and qφ + ( r ) = ~ m |h χ − | ∇ r χ + i| . (8)Equation (6) is formally equivalent to Schrödinger’sequation for a particle of electric charge q immersedin an electromagnetic field described by the potentials A + ( r ) and φ + ( r ) and experiencing an additionnal poten-tial U + ~ Λ / . The potentials A + ( r ) and φ + ( r ) are there-fore referred to as Abelian artificial gauge potentials. Thecompleteness relation | χ + ih χ + | + | χ − ih χ − | = ˆ int allowsus to rewrite Eqs. (7) and (8) in terms of the expecta-tion value h ˆ p i χ + and variance (∆ˆ p ) χ + of the momentumoperator ˆ p = ~ ∇ r /i in the position-dependent internalstate | χ + ( r ) i , q A + ( r ) = −h ˆ p i χ + , qφ + ( r ) = (∆ˆ p ) χ + m . (9)This formulation of the artificial gauge potentials makesexplicit the interpretation of the term (cid:0) ˆ p + h ˆ p i χ + (cid:1) / m appearing in (6) as the kinetic energy associated withthe slow center-of-mass motion of the atom during itsadiabatic evolution. It also makes clear that the originof the scalar potential φ + lies in the quantum fluctua-tions of momentum as measured by the variance, in fullagreement with the interpretation of this term as addi-tional kinetic energy associated with the micro-motionof the atom resulting from its interaction with the laserfield [26].In the remainder of this paper, we set the artificialcharge q equal to one unless otherwise stated. When theRabi frequency has a constant phase ϕ , Ω( r ) = | Ω( r ) | e iϕ ,we directly obtain from Eqs. (2), (7), and (8), A + ( r ) = (cid:18) − δ Λ (cid:19) ~ k L ,φ + ( r ) = (cid:20) ( δ | ∇ r Ω | + | Ω | | ∇ r δ | ) k L Λ + | Ω | Λ (cid:21) ~ k L m . (10)If the system had adiabatically followed the state | χ − i (instead of | χ + i ), the resulting scalar potential would beidentical to Eq. (10) whereas the vector potential wouldappear with the opposite sign in front of δ/ Λ .The artificial magnetic and electric fields associatedwith these potentials are B ± = ∇ r × A ± and E ± = −∇ r φ ± . For instance, the artificial magnetic field is givenby B ± ( r ) = ± ~ | Ω | (cid:0) | Ω | ∇ r δ − δ ∇ r | Ω | (cid:1) × k L . (11)Whenever δ and Ω are uniform, i.e., do not vary in space,the artificial vector potential is uniform and the magneticfield vanishes everywhere. B. Two noninteracting two-level atoms
For comparison with latter results and to introduce no-tations, we briefly consider the case of two noninteractingtwo-level atoms driven by a common laser field. Let usdenote by r α the position of atom α = a, b . The Hamil-tonian describing the internal dynamics of the atoms in-teracting with the laser field is given by ˆ H non − int = ˆ H l,a ⊗ ˆ int b + ˆ int a ⊗ ˆ H l,b , (12)where ˆ int α denotes the identity operator in the internalHilbert space of atom α and ˆ H l,α is the single-atom in-teraction Hamiltonian (1) for atom α ( α = a, b ). Theeigenvectors of Hamiltonian (12) follow directly from theeigenvectors | χ ± ( r ) i [Eq. (2)] of (1), | χ ij ( r a , r b ) i = | χ i ( r a ) i a ⊗ | χ j ( r b ) i b , (13)with i, j = ± .The full Hamiltonian, including quantization of theatomic motion, is given by ˆ H = (cid:18) ˆ p a m a + ˆ p b m b + U (cid:19) ⊗ ˆ int ab + ˆ ext ab ⊗ ˆ H non − int , (14)where ˆ · ab = ˆ · a ⊗ ˆ · b , ˆ p α = ~ ∇ r α /i is the momentumoperator for atom α , and U is a potential energy insensi-tive to the atomic internal state. The global state of thetwo-atom system is given in the position representationby h r a , r b | ψ ( t ) i = X i,j = ± ψ ij ( r a , r b , t ) | χ ij ( r a , r b ) i , (15)where the wave functions ψ ij ( r a , r b , t ) describe theatomic motion. When the atoms are initially inthe (separable) internal eigenstate | χ ij ( r a , r b ) i , theiradiabatic evolution ensure them to follow the sameseparable internal state such that h r a , r b | ψ ( t ) i ≈ ψ ij ( r a , r b , t ) | χ ij ( r a , r b ) i at any time t . Similar devel-opments as before then lead us to the Schrödinger-likeequation i ~ ∂∂t ψ ij ( r a , r b , t ) = " X α = a,b [ˆ p α − A ijα ( r α )] m α + φ ijα ( r α )+ U + E i ( r a ) + E j ( r b ) ψ ij ( r a , r b , t ) , (16) with the artificial gauge potentials, A ijα ( r α ) = i ~ h χ ij | ∇ r α χ ij i , (17) φ ijα ( r α ) = ~ m α X kl = ij |h χ kl | ∇ r α χ ij i| , (18)where the sum runs over all eigenstates of the two-atomHamiltonian except the initial state. For independentatoms, the eigenstates | χ ij i are separable [Eq. (13)] andEqs. (17) and (18) reduce to the single-atom potentials(7) and (8). As could be expected when the electromag-netic field is treated classically, e.g., as an external field,the noninteracting atoms experience the same artificialgauge potentials as those calculated for a single atom,with the slight difference that they can experience dif-ferent potentials depending on their respective internalstate. Again, when δ and Ω are constant over space, theartificial vector potentials are constant and the artificialmagnetic fields vanish everywhere.The generalization of these results to a system of N noninteracting atoms in a classical laser field proceedsalong the same lines. III. ARTIFICIAL GAUGE POTENTIALS FORTWO INTERACTING RYDBERG ATOMS
In order to highlight the contribution of atom-atominteractions on the generation of artificial gauge fields,we consider uniform Rabi frequency and detuning. Inthis case, the single-atom artificial gauge potentials areconstant and do not give rise to any magnetic or electricfields, as recalled in the previous section.
A. Hamiltonian
We consider a system of two atoms interacting witheach other when they are both in an excited Rydbergstate. The interaction energy between a ground-stateatom and the other atom is assumed to be negligible [27].To account for the energy shift of the doubly excited state | ee i ≡ | e i a ⊗ | e i b caused by dipole-dipole interactions,the term ~ V | ee ih ee | is added to the Hamiltonian (12)describing two independent atoms driven by a commonlaser field. This leads us to the Hamiltonian, ˆ H d − d = ˆ H non − int + ~ V | ee ih ee | . (19)This simple but realistic model provides the core founda-tion for several theoretical works on Rydberg gases [28].Gillet et. al. [29] showed that it successfully reproducesexperimental observations on the dipole blockade ef-fect [30]. Very recently, it was applied by Béguin andcoworkers [27] to deduce from experimentally measuredexcitations probabilities the /r ab dependence of the vander Waals interaction between two Rydberg atoms (here r ab = | r a − r b | is the interatomic distance).For resonant dipole-dipole (RDD) interactions, the en-ergy shift takes the form ~ V = ~ C /r ab . In the ab-sence of an external field, C has no angular depen-dence because of rotational invariance. However, whenthe atoms are excited by a laser field, C may dependon the angle between the interatomic axis and light po-larization. Here, we shall consider the case of s -stateatoms for which the energy shift is almost sphericallysymmetric [31], even though our analysis can be general-ized directly to account for an angular dependence. Forvan der Waals (vdW) interactions, when the atoms arefar apart or in the absence of Förster resonance, the en-ergy shift displays the characteristic /r ab dependence, ~ V = ~ C /r ab . Let us define a crossover distance r c atwhich the atom-atom interaction energy equals the atom-field interaction energy. The distance r c is implicitelydetermined by the equality | V ( r c ) | = Λ , (20)where Λ is given by Eq. (3). For RDD interactions, wehave r c = p | C | / Λ , whereas for vdW interactions, r c = p | C | / Λ . B. Eigenstates
It is convenient to introduce the symmetric and anti-symmetric one-excitation states | ψ ± i = 1 √ e i k L · r a | eg i ± e i k L · r b | ge i ) , (21)because | ψ − i is a trivial eigenstate of ˆ H d − d with eigen-value . In the basis {| ψ − i , | ee i , | ψ + i , | gg i} , Hamiltonian ˆ H d − d reads ˆ H d − d = ~ ( V − δ ) | ee ih ee | + ~ δ | gg ih gg | + (cid:20) ~ Ω √ (cid:16) e i k L · ( r a + r b ) | ee ih ψ + | + | ψ + ih gg | (cid:17) + h . c . (cid:21) (22)It has non-degenerate eigenvalues E = 0 , E = ~ (cid:20) s + + s − + 23 V (cid:21) E ± = ~ " −
12 ( s + + s − ) + 23 V ± i √
32 ( s + − s − ) (23)where s ± = q γ ± p η + γ (24)and η = 43 (cid:18) δ ( V − δ ) − | Ω | − V (cid:19) γ = V (cid:18) V − δ ( V − δ ) − | Ω | (cid:19) (25) The associated eigenvectors are given by ( i = 1 , ± ) | χ i i = N i h ~ Ω e i k L · ( r a + r b ) E i | ee i + √ E i F i | ψ + i + ~ Ω ∗ F i | gg i i (26)where N i ≡ N i ( r a , r b ) is a normalization constant and E i ( r a , r b ) = E i − ~ δ, F i ( r a , r b ) = E i + ~ ( δ − V ) (27) C. Artificial gauge potentials and fields
1. General expressions
When the system is initially in the internal state | χ i i ( i = 1 , ± ) [see Eq. (26)], the general expressions of theartificial gauge potentials are still given by Eqs. (17) and(18) but with | χ ij i replaced by the two-atom eigenstates(26). A direct calculation yields A iα ( r ab ) = A iα ( r ab ) e k L , (28)with r ab = | r a − r b | , e k L = k L /k L and A iα ( r ab ) = − N i E i (cid:0) ~ | Ω | + F i (cid:1) ~ k L . (29)The dependence of A iα ( r ab ) on the position r α of atom α appears only through the interatomic distance r ab via N i , E i , and V . Since the artificial potentials (28) areidentical for both atoms and depend only on r ab , themagnetic fields for atoms a and b have opposite signs. Toreduce the amount of notation, we only give the magneticfield experienced by atom a , which reads B ia ( r ab ) = ∇ r a A ia × e k L = dA ia dr ab e r ab × e k L , (30)with e r ab = r ab /r ab . In a reference frame in which atom b is at the origin, and equipped with spherical coordinates { r ab , θ, ϕ } where the z -axis points in the same directionas the laser wave vector, the artificial magnetic field takesthe form, B ia ( r ab ) = dA ia dr ab sin θ e ϕ = B ia,ϕ e ϕ (31)where θ is the angle between the z axis and r ab . Thestructure of this vector field is illustrated in Fig. 1.From Eqs. (18) and (26), we obtain for the artificialscalar potentials φ iα ( r ab ) = N i (cid:26) E i F i X j = i N j (cid:2) ( E ′ i E j C F ij + C E ij F ′ i F j ) /k L + E i E j ( ~ | Ω | + F i F j ) (cid:3) (cid:27) ~ k L m α , (32) z ab /r c y ab /r c x ab /r c FIG. 1. (Color online) Vector plot of B + a for δ/ | Ω | = 0 and k L = (0 , , k L ) . Color indicates the magnetic field strength,from black (strong magnetic field) to white (zero magneticfield). The blue dot shows the position of the fixed atom withwhich atom a interacts. with C E ij = ~ | Ω | + 2 E i E j , C F ij = ~ | Ω | + 2 F i F j , i, j = 1 , ± and where the prime denotes a derivative withrespect to r ab .Artificial scalar potentials are of the order of the recoilenergy ~ k L / m , which for Rubidium atoms and an opti-cal transition is about 1 µ K. This is usually much weakerthan the trapping potential U and can be compensatedby additional light shifts. We shall therefore concentrateour attention on artificial vector potentials and magneticfields. For the sake of completeness, we also provide inAppendix A the derivation and a brief discussion of theartificial gauge potentials in the center-of-mass coordi-nate system.
2. Dipole blockade regime
In the blockade regime, dipole-dipole interactions be-tween excited atoms dominate over atom-light interac-tions. This prevents the system from populating the dou-bly excited state | ee i , which can be eliminated from theequations of motion, thus leading to an effective Hamil-tonian that captures the dynamics as long as the inter-atomic distance is much smaller than the crossover dis-tance r c defined by Eq. (20). The derivation of the effec-tive Hamiltonian and the determination of its eigenvaluesand eigenvectors are exposed in Appendix B. There wealso show that the artificial gauge potentials take thesimple form, A eff , ± α ( r α ) = ∓ (Γ + δ ) − √ Ξ2 √ Ξ ~ k L ,φ eff ± α ( r α ) = (cid:20) ± Γ + δ √ Ξ + | Ω | Ξ + 4 | Ω | | ∇ r α Γ | k L Ξ (cid:21) ~ k L m α , (33) where Γ = | Ω | V − δ/ , Ξ = (Γ + δ ) + 2 | Ω | , (34)with the correspondence A eff , + α ↔ A + α , A eff , − α ↔ A − α in the case of repulsive interactions and A eff , + α ↔ A − α , A eff , − α ↔ A α in the case of attractive interactions. Thesame correspondence holds for the scalar potentials. Inthis regime, the third vector potential is constant andequal to − ~ k L because the corresponding eigenstate ofenergy ~ ( V − δ ) reduces to exp[ i k L · ( r a + r b )] | ee i . Forconsistency, we checked numerically that these potentialsare close to the general expressions (28) and (32) for in-teratomic distances much smaller than r c .Expression (33) for the artificial vector potentialsshows a crucial feature: The artificial magnetic fields ap-pear only because of the combined atom-atom and atom-field interactions. Indeed, in the absence of field, Ω = 0 ,the vector potentials are constant and the magnetic fieldsvanish. On the other hand, when there is no interaction, V = 0 and the vector potentials are constant which againleads to zero magnetic fields.For a vanishing detuning ( δ = 0 ), the vector potentialsbecome A eff , ± α ( r α ) = − ∓ | Ω | p | Ω | + 8 V ( r ab ) ! ~ k L . (35)Comparison of Eq. (35) with Eq. (10) shows that theartifical vector potentials have the same form, up to amultiplicative factor / , as those felt by a single two-level atom irradiated by a laser field where | Ω | plays therole of the detuning and V ( r ab ) the role of the Rabifrequency. This similarity breaks down for the scalarpotentials and for both potentials in the presence of adetuning in the interacting two-atom system.
3. Weak interaction regime
For large interatomic distances, r ≫ r c , atom-light in-teraction dominates over atom-atom interactions ( ~ V ≪ ~ Λ ). In this limit, a series expansion of the general ex-pression (29) yields, after some algebra, A α = (cid:20)(cid:18) − δ Λ (cid:19) + (cid:18) δ + 3 | Ω | ( δ − Λ)6Λ (cid:19) V (cid:21) ~ k L , A + α = (cid:20)(cid:18) − − δ Λ (cid:19) + (cid:18) δ + 3 | Ω | ( δ + Λ)6Λ (cid:19) V (cid:21) ~ k L , A − α = (cid:20) − − (cid:18) δ | Ω | (cid:19) V (cid:21) ~ k L . (36)In the absence of atom-atom interactions, the vector po-tentials A α and A + α and the scalar potentials reduce tothe single-atom potentials (10) as required. We note thatfor weak interactions, the artificial vector potentials arelinear in the interatomic potential. D. Discussion
Resonant dipole-dipole interactions give rise to an at-tractive or repulsive interaction potential between ex-cited atoms of the form ~ V = ~ C /r ab , whereas vander Waals interactions are usually attractive with an in-teraction potential of the form ~ V = ~ C /r ab [31, 32].When the laser frequency matches the atomic transi-tion frequency ( δ = ω L − ω = 0 ), the sign of the in-teraction potential does not affect the artificial magneticfields [33]. More generally, it follows from Eq. (23) that E ( V, δ ) = − E + ( − V, − δ ) and E − ( V, δ ) = − E − ( − V, − δ ) where E i ( V, δ ) denotes the energy eigenvalue for an in-teratomic potential V and a detuning δ . These rela-tions, together with Eqs. (27), (29) and (31), imply that B α ( V, δ ) = B + α ( − V, − δ ) and B − α ( V, δ ) = B − α ( − V, − δ ) .In the remainder, we choose to focus on attractive poten-tials ( C , C < ) both for RDD and vdW interactions,given that the artificial magnetic fields for repulsive inter-actions can directly be deduced from those for attractiveinteractions.Before we discuss the features of the artificial magneticfields, it is useful to construct from the characteristiclength r c [see Eq. (20)], the laser wave number k L andthe elementary charge e , a characteristic magnetic fieldstrength, B = ~ k L er c . (37)As we shall see, B gives the typical strength of the arti-ficial magnetic fields induced by the joint atom-laser andatom-atom interactions for a particle with electric charge q = e .We show in Fig. 2 the only non-vanishing componentof the artificial vector potential as a function of the di-mensionless interatomic distance r ab /r c for a vanishingdetuning. Firstly, we note that the artificial vector po-tentials for RDD and vdW interactions display the samequalitative behavior. For small interatomic distances, r ab /r c ≪ , the system is dipole blockaded and the com-ponent along the laser propagation axis of the vectorpotential tends to a non-zero value ( − ~ k L / or − ~ k L )which only depends on the atomic internal state, andnot on the type of interaction. The largest variations ofthe vector potentials occur around r ab = r c , and are morepronounced in the case of vdW interactions (see bottompanel). For large r ab /r c , atom-atom interactions becomenegligible with respect to atom-field interactions, and A iα tends in both cases to the value obtained for noninteract-ing atoms, i.e. − ~ k L / [see Eq. (10)]. Figure 3 displaysthe corresponding artificial magnetic fields, which are siz-able over a distance interval of the order of r c . A max-imum of intensity appears around r ab = r c , where theatom-atom interaction energy ~ V equals the atom-fieldinteraction energy ~ Λ [see Eq. (20)]. The magnetic fieldprofiles depend markedly on the atomic internal state,and are more squeezed in the case of vdW interactions.The behavior of the artificial magnetic field is strongly − . − . − . − . − r ab /r c A iα ~ k L − . − . − . − . − A iα ~ k L FIG. 2. (Color online) Only nonvanishing component of thedimensionless artificial vector potentials A iα / ~ k L (green dot-ted curve, i = 1 ; blue dashed curve, i = − ; orange solidcurve, i = + ) as a function of the interatomic distance r ab /r c for δ/ | Ω | = 0 , (top panel) resonant dipole-dipole interactions,and (bottom panel) van der Waals interactions. dependent on the sign of the detuning, as shown inFigs. 4, 5 and 6 for RDD interactions. The curves forvdW interactions are not shown as they display the samequalitative features. For positive detunings (sign oppo-site to that of the energy shift ~ V ), the overall mag-netic field amplitude decreases whereas for negative de-tunings it increases as compared to the zero detuningcase. A series expansion of Eq. (29) inserted into Eq. (31)shows that for large negative detunings ( δ < with | δ/ Ω | ≫ ), the peak height of the dimensionless mag-netic field B a /B scales linearly with the dimensionlessdetuning δ/ | Ω | according to B a, min /B ≈ β δ/ | Ω | with β = 3 / (4 √ for RDD interactions and β = 3 / (2 √ for vdW interactions. Given Eqs. (20) and (37), it followsthat | B min | scales like | δ | / / | Ω | for RDD interactions andlike | δ | / / | Ω | for vdW interactions. This behavior is il-lustrated in the inset of Fig. 4. The position of the mag-netic field peak appears around r ab = r c as in the zerodetuning case, which corresponds to the distance whereatom-atom and atom-field interaction energies are equal.The magnetic field B + a displays a single peak around . . . − . − r ab /r c B ia,ϕ B . . . − . − B ia,ϕ B FIG. 3. (Color online) The ϕ -component of the dimensionlessmagnetic fields B ia /B (green dotted curve, i = 1 ; blue dashedcurve, i = − ; orange solid curve, i = + ) as a function ofthe interatomic distance r ab /r c for δ/ | Ω | = 0 , (top panel)resonant dipole-dipole interactions, and (bottom panel) vander Waals interactions. r ab = γ + r c , as Fig. 5 shows. For large negative detun-ings, its intensity scales quadratically with δ/ | Ω | accord-ing to B + a, max /B ≈ β + ( δ/ | Ω | ) with γ + = 2 − / ≈ . , β + = 3 √ for RDD interactions (see inset of Fig. 5), and γ + = 2 − / ≈ . , β + = 6 √ for vdW interactions. Asregards B − a , it displays both a maximum and a minimum(see Fig. 6). For large negative detunings, the minimumoccurs around r ab = γ − min r c and scales quadraticallywith the detuning, i.e., B − a, min /B ≈ − β − min ( δ/ | Ω | ) with γ − min = 2 − / ≈ . , β − min = 3 √ for RDD interactions(see inset of Fig. 6), and γ − min = 2 − / ≈ . , β − min =6 √ for vdW interactions. In the same limit, the max-imum occurs around r ab ≈ r c , and scales linearly withthe detuning according to B − a, max /B ≈ − β − max δ/ | Ω | for δ/ | Ω | ≫ with β − max = 3 / (4 √ for RDD interactionsand β − max = 3 / (2 √ for vdW interactions.The location of the intensity peaks displayed by themagnetic field in the regime of large detunings can berelated to transitions between bare states. For negativedetunings large compared to the Rabi frequency, transi- − − − − δ/ | Ω | B a, min B − . − − . − r ab /r c B a,ϕ B FIG. 4. (Color online) The ϕ -component of the dimension-less magnetic field B a /B as a function of the interatomicdistance r ab /r c for different values of the detuning and RDDinteractions. (From bottom to top) δ/ | Ω | = − , − , − , , .(Inset) Largest value of the artificial magnetic field given bythe minimum value of its ϕ -component as a function of thedimensionless detuning δ/ | Ω | . − − − δ/ | Ω | B + a, max B r ab /r c B + a,ϕ B FIG. 5. (Color online) The ϕ -component of the dimension-less magnetic field B + a /B as a function of the interatomicdistance r ab /r c for different values of the detuning and RDDinteractions. (From top to bottom) δ/ | Ω | = − , − , − , , .(Inset) Largest value of the artificial magnetic field given bythe maximum value of its ϕ -component as a function of thedimensionless detuning δ/ | Ω | . tions between bare states are highly inhibited. However,at small interatomic distances, dipole-dipole interactionsgive rise to an energy shift of the doubly excited statewhich can compensate the energy mismatch stemmingfrom the detuning. When the two-photon antiblockadecondition ~ ω + ~ V = 2 ~ ω L is met, the | gg i ↔ | ee i − − − − − δ/ | Ω | B − a, min B − − − − r ab /r c B − a,ϕ B FIG. 6. (Color online) The ϕ -component of the dimension-less magnetic field B − a /B as a function of the interatomicdistance r ab /r c for different values of the detuning and RDDinteractions. (From top to bottom) δ/ | Ω | = − , − , − , , .(Inset) Largest value of the artificial magnetic field given bythe minimum value of its ϕ -component as a function of thedimensionless detuning δ/ | Ω | . transition is on resonance. Similarly, the | ψ + i ↔ | ee i transition becomes resonant when the single-photon an-tiblockade condition ~ ω + ~ V = ~ ω L holds [34, 35]. Interms of interatomic distances, the two-photon conditionreads r ab = p | C | / (2 δ ) = r c p Λ / (2 δ ) for RDD interac-tions and r ab = p | C | / (2 δ ) = r c p Λ / (2 δ ) for vdW in-teractions. For large values of the detuning, Λ ≈ | δ | suchthat the condition becomes r ab ≈ r c / √ ≈ . r c (RDD)and r ab ≈ r c / √ ≈ . r c (vdW). Likewise, the single-photon antiblockade condition reads r ab ≈ r c for bothtypes of interactions. The distances corresponding tothese antiblockade conditions coincide with the locationswhere the magnetic fields are found to be the most in-tense. This can be understood as follows. Equations (17)and (30) show that large artificial magnetic fields ap-pear where the eigenstates present strong nonuniformspatial variations. This is not the case in the pres-ence of large detunings because transitions between thebare states | gg i , | ψ + i , and | ee i are then highly inhib-ited. However, some of these transitions are enabled atinteratomic distances where the antiblockade conditionsare met. This is only possible when the detuning andthe energy shift ~ V have the same sign. In this case(negative detunings), the eigenstates display importantspatial variations which lead to large artificial magneticfields, whereas in the absence of antiblockade (positivedetunings) the spatial variation of the eigenstates, andthus the magnetic fields, are small (see insets of Figs. 4,5 and 6). Indeed, in the case of B a , the location of theintensity peak ( r ab ≈ r c ) satisfies to the single-photonantiblockade condition. As regards the corresponding eigenstate | χ i , it coincides nearly with | ψ + i at smallinteratomic distances, turns into a superposition of | ψ + i and | ee i around r ab ≈ r c , and coincides nearly with | ee i at larger distances. The same observation holds for theartificial magnetic field B + a , which displays a maximumof intensity around r ab ≈ r c / √ (RDD) or r ab ≈ r c / √ (vdW) satisfying the two-photon antiblockade condition.In this case, | χ + i reduces to | ee i for r ab ≪ r c and to | gg i for r ab ≫ r c . When the two-photon antiblockadecondition is met, | χ + i becomes an equally weighted su-perposition of | ee i and | gg i . As for the artificial magneticfield B − a , the two antiblockade conditions are successivelymet as the interatomic distance grows because the asso-ciated eigenstate coincides with | gg i for r ab ≪ r c . Thefirst (second) maximum of intensity corresponds to thetwo-photon (single-photon) antiblockade condition. Atinteratomic distances in-between the two maxima, | χ − i coincides nearly with | ee i , and at large distances with | ψ + i . At the locations of the maxima, | χ − i is a super-position of the two states involved in the antiblockademechanism.This mechanism also allows one to understand qual-itatively the width of the intensity peaks in the artifi-cial magnetic fields. Indeed, the larger the detuning, thesmaller the crossover distance r c and the more impor-tant the spatial variations of the dipole shift ~ V around r ab = r c . Therefore, the interval of distances where thedipole shift and the detuning counterbalance to allowsignificant transitions between bare states becomes nar-rower as the detuning increases.A similar line of reasoning can be pursued to explainthe cause of the magnetic field peaks and their location( r ab ≈ r c ) when the system is at resonance ( δ = 0 ). In-deed, at small interatomic distances, the dipole blockadeeffect prevents the system from populating the doublyexcited state, which as a matter of fact do not contributeto the artificial magnetic field. When the dipole shift is ofthe order of the atom-light interaction energy, transitionsbetween | ee i and lower excitations states are enabled, re-sulting in spatial variations of the eigenstates leading toartificial magnetic fields. At large distances, the atomscan be considered as independent and the magnetic fieldsvanish. E. Experimental considerations
In this section, we give an estimation of the attainableartificial magnetic field strengths in the RDD and vdWregimes in view of recent experiments. Moreover, weshow that the artificial gauge potentials could be detectedthrough the deflection of the atomic motion caused by theartificial Lorentz force. To verify our theoretical predic-tions, similar experimental setups as those designed byGaëtan et. al. [30] (RDD regime) or Béguin et. al. [27](vdW regime) could be considered. In those experiments,two Rb atoms with residual temperature T ≈ µ K( v rms ≈ cm/s) are trapped in two optical tweezers witha beam waist w ≈ µ m. The atoms are laser excitedto Rydberg states with high principal quantum number( n = 53 , , in [27] and n = 58 in [30]) character-ized by a radiative lifetime τ ranging from to µ s.Depending on the principal quantum number, either theRDD or the vdW regime can be reached.Let us now consider that one atom is kept at a fixedposition in space and a second atom is sent towards thefirst one, e.g., by means of an optical conveyor belt [36].The results of the preceding sections show that the mov-ing atom will experience artificial gauge fields as a resultof its joint interaction with the trapped atom and thelaser field. As a consequence, its trajectory will be mod-ified by the action of the artificial Lorentz force. In theRDD regime, we consider | Ω | / π = 6 . MHz, λ L = 296 nm [37], and C / π = 3200 MHz. µ m as in [30]. In thiscase, the crossover distance at zero detuning is r c ≈ µ mand the characteristic magnetic field strength B ≈ mTfor a particle with electric charge equal to the elementarycharge. In the vdW regime, we base our estimation onRef. [27] in which the single-atom Rabi frequency | Ω | / π can be varied in the range from 500 kHz to 5 MHz, and | C | can be varied from to GHz. µ m by changingthe principal quantum number n of the atomic Rydbergstate. In this case, the crossover distance at zero detun-ing can be tuned from r c ≈ . µ m to r c ≈ µ m and thecharacteristic magnetic field strength from B ≈ mTto B ≈ . mT. Note that in both regimes the crossoverdistance is much larger than the waist of the tweezers.For an initial velocity of cm/s in the xy plane andan impact parameter equal to r c ≈ µ m, a semiclassicalcalculation predicts a deflection of the atomic trajectoryin the z direction (laser propagation direction) of the or-der of 1 µ m for a traveled distance equal to r c in the xy plane. For such a velocity, the adiabatic approximationis still valid to about [12] and it takes a time equal to µ s < τ to travel a distance r c , during which sponta-neous emission can be neglected to a good approximation.This approximation holds even better if we consider thatthe system follows adiabatically the internal state | χ + i corresponding to the two atoms in their ground state atlarge interatomic distances with respect to r c . In this sit-uation, the Rydberg states are populated only during asmall time in comparison with their radiative lifetime τ .Larger (smaller) initial velocities would lead to smaller(larger) deflections. The main experimental challenge isthus to control the atomic velocity with sufficient pre-cision to avoid a drift due to an initial velocity in the z direction that would mask the deflection due to theartificial Lorentz force. IV. CONCLUSION
We have shown that dipole-dipole interactions betweenRydberg atoms submitted to a uniform laser field giverise to nonuniform artificial Abelian gauge potentials.We have obtained general analytical expressions for the latter, as well as approximate expressions in the dipoleblockade and weak interaction regimes. We have iden-tified the mechanism responsible for the artificial gaugefields and have shown that they are the strongest whenatom-atom and atom-field interaction energies are of thesame order of magnitude. Note that a similar featurehas been observed experimentally in the population dy-namics of a pair of interacting Rydberg atoms [27]. Wehave discussed the differences and similarities of artificialgauge fields originating from resonant dipole-dipole andvan der Waals interactions. We have estimated on thebasis of recent experiments the attainable artificial mag-netic field to a few mT extending over a distance rangeof a few micrometers. Finally, we have shown that thesefields lead under realistic conditions to a deflection of theatomic motion of the order of 1 µ m, measurable withcurrent imaging techniques [38]. ACKNOWLEDGMENTS
J.M. is grateful to the University of Liège (SEGI facil-ity) for the use of the NIC3 supercomputer.
V. APPENDIX A : ARTIFICIAL GAUGEFIELDS IN THE CENTER-OF-MASSREFERENCE FRAME
In this appendix, we give the expressions of the artifi-cial electromagnetic potentials induced by dipole-dipoleinteractions between two Rydberg atoms in the centerof mass reference frame. This is the most natural ref-erence frame in view of the interaction potential whichonly depends on the relative coordinate. We first recallthe center-of-mass coordinates, R = m a r a + m b r b m a + m b , r = r a − r b (38)and their conjugate momenta, P = p a + p b , p = m b p a − m a p b m a + m b (39)where m α , r α and, p α are, respectively, the mass, theposition and the momentum of atom α = a, b . The totalmass of the system is M = m a + m b and the reduced massis µ = m a m b / ( m a + m b ) . In the center-of-mass referenceframe, the full Hamiltonian takes the form, ˆ H = ˆ P M + ˆ p µ + U ! ⊗ ˆ int ab + ˆ ext ab ⊗ ˆ H d − d ( R , r ) , (40)where ˆ P = ~ ∇ R /i and ˆ p = ~ ∇ r /i in the position rep-resentation, and with ˆ H d − d ( R , r ) given by Eq. (19) but0now expressed in terms of the center-of-mass coordinates(38). The eigenstates of ˆ H d − d ( R , r ) are still given byEq. (26) and, following the same procedure as in Sec. II,we obtain A i R ( r ) = A ia + A ib , A i r ( r ) = m b A ia − m a A ib m a + m b . (41)The resulting artificial vector potentials have the sameform as the relations (39) between the center of massand relative momenta, and the momenta of atoms a and b . This fully agrees with the reformulation of the artifi-cial vector potentials as the expectation values of the mo-mentum operator evaluated in the atomic internal statesas presented in Sec. II. In the center-of-mass referenceframe, the scalar potentials are given by φ i R ( r ) = X j = i N i N j (cid:2) E i E j (cid:0) ~ | Ω | + F i F j ) ] ~ k L M ,φ i r ( r ) = N i (cid:26) E i F i X j = i N j (cid:2) ( E ′ i E j C F ij + C E ij F ′ i F j ) /k L + (cid:18) m b − m a M (cid:19) E i E j ( ~ | Ω | + F i F j ) ~ k L µ (42)where C E ij = ~ | Ω | + 2 E i E j , C F ij = ~ | Ω | + 2 F i F j , i, j = 1 , ± and where the prime denotes a derivativewith respect to r ab . They correspond to (∆ ˆ P ) χ i / M and (∆ˆ p ) χ i / µ , respectively, and are thus determinedby the variance of center of mass and relative momentain the two-atom internal state | χ i i [see Eq. (26)].Expressions (41) and (42) show that the potentials inthe center-of-mass reference frame are simply connectedto those in the laboratory frame. Moreover, when oneatom is kept at a fixed position in space and another atomis traveling around it, the vector potential for the relativecoordinate reduces to the one for the moving atom in thelaboratory frame. VI. APPENDIX B : DERIVATION OF ANEFFECTIVE HAMILTONIAN IN THE DIPOLEBLOCKADE REGIME
In this appendix, we derive an effective Hamiltoniandescribing the internal dynamics of two interacting Ry-dberg atoms in the dipole blockade regime. We also de-termine its eigenvalues and eigenvectors. For this pur-pose, we eliminate the doubly excited state | ee i fromHamiltonian (22) following a method recently proposedby Paulisch et al. [39]. For the effective Hamiltonian to bevalid in the largest possible range, we add, beforehand,a constant term C ˆ int ab to ˆ H d − d . This term does obvi-ously not affect the dynamics of the system but leadsto a different effective Hamiltonian. Following [39], C ischosen so as to satisfy the condition Tr( ˆ H ′ + C ˆ ) = 0 where ˆ H ′ is the restriction of ˆ H d − d to the subspacespanned by {| ψ − i , | ψ + i , | gg i} and ˆ the identity opera-tor in this subspace. A straightforward calculation showsthat C = − ~ δ/ . We now eliminate the state | ee i fromHamiltonian ˆ H ′ d − d = ˆ H d − d − ( ~ δ/ int ab by first writingthe internal states of the two-atom system in the form | ψ ( t ) i = c e ( t ) | ee i + c + ( t ) | ψ + i + c − ( t ) | ψ − i + c g ( t ) | gg i . (43)Inserting this expression into the time-dependentSchrödinger’s equation, we obtain the set of equations: i ˙ c e = ∆ c e + Ω √ e i k L · ( r a + r b ) c + , (44) i ˙ c + = − δ c + + Ω √ c g + Ω ∗ √ e − i k L · ( r a + r b ) c e , (45) i ˙ c − = − δ c − , (46) i ˙ c g = 2 δ c g + Ω ∗ √ c + , (47)where a dot denotes a time derivative and ∆ = V − δ/ .Solving Eq. (44) for c e ( t ) , we get c e ( t ) = − i √ Z t e − i ∆( t − t ′ ) Ω e i k L · ( r a + r b ) c + ( t ′ ) dt ′ . (48)In the Markov approximation, memory effects are ne-glected, which amounts to taking the coefficient c + ( t ′ ) out of the integral. This approximation is valid as longas Λ = p δ + | Ω | ≪ | V | and implies that c i ( t ) ( i = e )oscillates slowly in comparison to exp ( − iV t ) . In thiscase, c e ( t ) takes the simple form, c e ( t ) = − Ω √ e i k L · ( r a + r b ) c + ( t ) . (49)Inserting this expression into Eq. (45), we readily de-duce from the equations of motion (45)–(47) the effectiveHamiltonian, ˆ H effd − d = − ~ δ ) | ψ + ih ψ + | + ~ δ | gg ih gg | − | ψ − ih ψ − | )+ (cid:18) ~ Ω √ | ψ + ih gg | + h . c . (cid:19) (50)where Γ = | Ω | / . The state | ψ − i ≡ | χ i remainseigenstate of the effective Hamiltonian, but with energy E = − ~ δ/ . The two other eigenstates are | χ eff ± i = N ± (cid:20) (cid:16) − (Γ + δ ) ∓ p (Γ + δ ) + 2 | Ω | (cid:17) | ψ + i + √ ∗ | gg i (cid:21) (51)with eigenvalues E eff ± = ~ h δ − ∓ p (Γ + δ ) + 2 | Ω | i . (52)1Similarly to Eqs. (17) and (18) of Sec. II, the artificialgauge potentials are given by A eff , ± α ( r α ) = i ~ h χ eff ± | ∇ r α χ eff ± i ,φ eff , ± α ( r α ) = ~ m α (cid:0) |h χ eff ∓ | ∇ r α χ eff ± i| + |h χ | ∇ r α χ eff ± i| (cid:1) . (53) After some algebra, we arrive at Eq. (33). When the signsof the detuning ( δ ) and the interatomic potential ( V ) arechanged simultaneously, E eff ± → − E eff ∓ , and A eff , ± α → A eff , ∓ α . [1] S. Lloyd, Science , 1073 (1996).[2] Yu. I. Manin, Computable and uncomputable , SovetskoyeRadio, Moscow, 1980.[3] R. P. Feynman, Int. J. Theor. Phys. , 467 (1982).[4] L. DiCarlo, J. M. Chow, J. M. Gambetta, Lev S. Bishop,B. R. Johnson, D. I. Schuster, J. Majer, A. Blais, L. Frun-zio, S. M. Girvin, and R. J. Schoelkopf, Nature , 240(2009).[5] N. Xu, J. Zhu, D. Lu, X. Zhou, X. Peng, and J. Du,Phys. Rev. Lett. , 130501 (2012).[6] I. Buluta and F. Nori, Science , 108 (2009).[7] I. Bloch, J. Dalibard and S. Nascimbéne, Nature Physics , 267 (2012).[8] R. Blatt and C. F. Roos, Nature Physics 8, (2012).[9] G. Juzeli¯unas and P. Öhberg, Phys. Rev. Lett. , 033602(2004).[10] G. Juzeli¯unas, P. Öhberg, J. Ruseckas, and A. Klein,Phys. Rev. A , 053614 (2005).[11] G. Juzeli¯unas, J. Ruseckas, P. Öhberg, and M. Fleis-chhauer, Phys. Rev. A , 025602 (2006).[12] J. Dalibard, F. Gerbier, G. Juzeli¯unas, and P. Öhberg,Rev. Mod. Phys. , 1523 (2011).[13] R. G. Unanyan, B. W. Shore, and K. Bergmann,Phys. Rev. A , 2910 (1999).[14] K. Osterloh, M. Baig, L. Santos, P. Zoller, and M. Lewen-stein, Phys. Rev. Lett. , 010403 (2005).[15] J. Ruseckas, G. Juzeli¯unas, P. Öhberg, and M. Fleis-chhauer, Phys. Rev. Lett. , 010404 (2005).[16] F. Wilczek and A. Zee, Phys. Rev. Lett. , 2111 (1984).[17] Y. J. Lin, R. L. Compton, A. R. Perry, W. D. Phillips,J. V. Porto, and I. B. Spielman, Phys. Rev. Lett. ,130401 (2009).[18] Y-J. Lin, R. L. Compton, K. Jiménez-García, J. V. Porto,and I. B. Spielman, Nature , 628 (2009).[19] Y-J. Lin, R. L. Compton, K. Jiménez-García,W. D. Phillips, J. V. Porto, and I. B. Spielman, NaturePhysics , 531 (2011).[20] M. C. Beeler, R. A. Williams, K. Jiménez-García,L. J. LeBlanc, A. R. Perry, and I. B. Spielman, Nature , 201 (2013).[21] J. Stuhler, A. Griesmaier, T. Koch, M. Fattori, T. Pfau,S. Giovanazzi, P. Pedri, and L. Santos, Phys. Rev. Lett. , 150406 (2005).[22] A. Dauphin, M. Müller, and M. A. Martin-Delgado,Phys. Rev. A , 053618 (2012).[23] F. Grusdt and M. Fleischhauer, Phys. Rev. A , 043628(2013).[24] M. Kiffner, W. Li, and D. Jaksch, J. Phys. B:At. Mol. Opt. Phys. , 134008 (2013).[25] M. Kiffner, W. Li, and D. Jaksch, Phys. Rev. Lett. ,170402 (2013).[26] M. Cheneau, S. P. Rath, T. Yefsah, K. J. Günter,G. Juzeli¯unas and J. Dalibard, Europhysics Letters ,60001 (2008).[27] L. Béguin, A. Vernier, R. Chicireanu, T. Lahaye, andA. Browaeys, Phys. Rev. Lett. , 263201 (2013).[28] See C. Ates, T. Pohl, T. Pattard, and J. M. Rost, Phys.Rev. A , 013413 (2007) and citing articles.[29] J. Gillet, G. S. Agarwal, and T. Bastin, Phys. Rev. A ,013837 (2010).[30] A. Gaëtan, Y. Miroshnychenko, T. Wilk, A. Chotia,M. Viteau, D. Comparat, P. Pillet, A. Browaeys, andP. Grangier, Nature Physics , 115 (2009).[31] M. Saffman, T. G. Walker, and K. Mølmer, Rev. Mod.Phys. , 2313 (2010).[32] Note that in our model, the sign of the interatomic po-tential is fixed.[33] However, a change of sign of the interaction potential atzero detuning turns B α into B + α and vice versa, whereas B − α remains unchanged.[34] C. Ates, T. Pohl, T. Pattard, and J. M. Rost, Phys. Rev.Lett. , 023002 (2007).[35] T. Amthor, C. Giese, C. S. Hofmann, and M. Wei-demüller, Phys. Rev. Lett. , 013001 (2010).[36] D. Schrader, S. Kuhr, W. Alt, M. Müller, V. Gomer, andD. Meschede, Appl. Phys. B , 819 (2001).[37] It should be mentioned that the experiment exploits atwo-photon transition driven by lasers with wavevectors k and k . The resulting wavevector k L = k + k leadsto an effective wavelength λ L = 2 π/k L .[38] J. F. Sherson, C. Weitenberg, M. Endres, M. Cheneau,I. Bloch, and S. Kuhr, Nature467