Shaping interactions between polar molecules with far-off-resonant light
SShaping interactions between polar molecules with far-off-resonant light
Mikhail Lemeshko ∗ Fritz-Haber-Institut der Max-Planck-Gesellschaft, Faradayweg 4-6, D-14195 Berlin, Germany (Dated: September 5, 2018)We show that dressing polar molecules with a far-off-resonant optical field leads to new types ofintermolecular potentials, which undergo a crossover from the inverse-power to oscillating behaviordepending on the intermolecular distance, and whose parameters can be tuned by varying thelaser intensity and wavelength. We present analytic expressions for the potential energy surfaces,thereby providing direct access to the parameters of an optical field required to design intermolecularinteractions experimentally.
PACS numbers: 34.50.Cx, 34.20.-b, 34.90.+q, 32.60.+i, 33.90.+h, 33.15.Kr, 37.10.Vz, 37.10.PqKeywords: quantum gases, cold and ultracold collisions, dipole-dipole interaction, induced-dipole interaction,AC Stark effect, dynamic polarizability, far-off-resonant laser field
An intense far-off-resonant laser field induces aretarded dipole-dipole interaction between atoms ormolecules, which is of long range character and falls offas 1 /r , 1 /r , or 1 /r , depending on the separation r andthe light wavevector [1, 2]. This interaction is highlycontrollable and brings about peculiar effects in atomicBose condensates, such as “gravitational self-binding,”rotons, and density modulations leading to a supersolid-like behavior [3–5]. In contrast to atoms, molecules pos-sess anisotropic polarizability and rotational structure,which renders laser-induced interactions more complex,and thereby offers new riches in the few- and many-bodyphysics of field-dressed ultracold gases.In this contribution we study the interaction betweentwo polar and polarizable molecules in a far-off-resonantlaser field. We demonstrate that an optical field gives riseto new types of intermolecular potentials, which exhibit acrossover from the inverse-power decay at short distancesto an oscillating long-range behavior, and whose param-eters can be altered by tuning the laser intensity and fre-quency. Furthermore, for a wide range of field intensitiesthe problem can be described by an exactly solvable two-level model, which leads to simple analytic expressionsfor the effective potential energy surfaces. The theoryis exemplified by optically-induced interactions between K Rb molecules, widely employed in experiments withultracold polar gases [6].We consider two identical diatomic molecules, 1 and 2,with a dipole moment d and polarizability components, α (cid:107) and α ⊥ , parallel and perpendicular to the molecularaxis. In a far-off-resonant radiative field of intensity I ,the rotational levels of each molecule undergo a dynamicStark shift, as given by the Hamiltonian [1] H , = B J − I cε e j e ∗ l α lab , (1 , jl ( k ) , (1)with B the rotational constant, and α lab jl ( k ) the dynamicpolarizability tensor in the laboratory frame. We as- ∗ [email protected] sume a laser beam propagating along the positive Y di-rection with the wavevector k = k ˆY , and linear polar-ization along the Z axis, ˆe = ˆZ . Given that the onlynonzero polarizability components in the molecular frameare α zz = α (cid:107) and α xx = α yy = α ⊥ , and using B as a unitof energy, Hamiltonian (1) can be recast as: H , = J − ∆ η ( k ) cos θ (1 , − η ⊥ ( k ) , (2)where θ (1 , is the angle between the molecular axis andthe polarization vector of the laser field. The dimension-less interaction parameter ∆ η ( k ) = η (cid:107) ( k ) − η ⊥ ( k ) with η (cid:107) , ⊥ ( k ) = α (cid:107) , ⊥ ( k ) I/ (2 ε cB ). We note that eq. (2) wasderived in Refs. [7, 8] using the semiclassical approachand the rotating wave approximation. All rotational lev-els exhibit a constant shift of η ⊥ , given by the secondterm of eq. (2), which will be omitted hereafter.The polarization vector of an optical field defines anaxis of cylindrical symmetry, Z . The projection, M , ofthe angular momentum J on Z is then a good quantumnumber, while J is not. However, one can use the valueof J of the field-free rotational state, Y J,M ( θ, φ ), thatadiabatically correlates with the hybrid state as a label,designated by ˜ J , so that | ˜ J, M ; ∆ η (cid:105) → Y J,M for ∆ η → J by tilde, e.g.with ˜0 corresponding to ˜ J = 0.Induced-dipole interaction (2) preserves parity, hy-bridizing states with even or odd J ’s, | ˜ J, M ; ∆ η (cid:105) = (cid:88) J c ˜ J,MJM (∆ η ) Y JM , J + ˜ J even , (3)and therefore aligns molecules in the laboratory frame.Aligned molecules possess no space-fixed dipole moment,in contrast to species oriented by an electrostatic field.We note that at the far-off-resonant wavelengthsusually employed in alignment and trapping experi-ments ( ∼ α ij ( k )approaches its static limit, α ij (0), for a number ofmolecules, e.g. CO, N , and OCS. However, this is notthe case for alkali dimers having low-lying excited Σ and Π states, such as KRb and RbCs. Virtual transitions tothese states contribute to the ground-state dynamic po- a r X i v : . [ phy s i c s . a t o m - ph ] M a y -20-15-10-505 E Δ η FIG. 1. Lowest rotational energy levels of a molecule in afar-off-resonant laser field, depending on the field-strength pa-rameter ∆ η . Different colors correspond to M = 0 (black), 1(red), and 2 (blue). Energy is in units of B , two lowest tunnel-ing doublets are labeled as ˜ J, M . For ∆ η (cid:38)
15 the splitting ofthe lowest tunneling doublet can be accurately estimated as∆ E = | E | exp(3 . − √ ∆ η ), with E = 2 √ ∆ η − ∆ η − larizability, rendering it a few times larger than the staticvalue [9, 10].Figure 1 illustrates the dynamic Stark effect on rota-tional levels of a diatomic molecule. A far-off-resonantoptical field of sufficiently large intensity leads to forma-tion of “tunneling doublets” – closely lying states of op-posite parity with the same M and ∆ ˜ J = 1 [7]. The dou-blet states can be mixed by extremely weak electrostaticfields, leading to strong molecular orientation in the lab-oratory frame [11, 12]. The energy gap between neigh-boring doublets increases with the field intensity, and isproportional to 2 √ ∆ η in the strong-field limit [13]. Aswe demonstrate below, at large ∆ η interaction betweentwo ground-state molecules can be described within thelowest tunneling doublet.In the absence of fields, two polar molecules interactvia the dipole-dipole interaction: V dd ( r ) = ˆ d (1) j ˆ d (2) l r ( δ jl − r j ˆ r l ) , (4)where ˆd (1 , = d (1 , /d are unit dipole moment vectorsof the molecules, ˆ r is the unit vector defining the inter-molecular axis, and r = (cid:18) d πε B (cid:19) / (5)is introduced as a unit of length. In the field-free case,interaction between two ground-state polar molecules hasan isotropic asymptotic behavior, V dd ( r ) = − / (6 r ).Far-off-resonant laser light interacts with molecular po-larizability, thereby inducing oscillating dipole momentson each of the two molecules. Retarded interaction be-tween these instantaneous dipoles leads to an additionalterm in the intermolecular potential [1, 2, 14], V αα ( k, r ) = I πε c e ∗ i α lab , ij ( k ) V jl ( k, r ) α lab , ln ( k ) e n cos( kr ) , (6) with V jl the retarded dipole-dipole interaction tensor, V jl ( k, r ) = 1 r (cid:2) ( δ jl − r j ˆ r l )(cos kr + kr sin kr ) − ( δ jl − ˆ r j ˆ r l ) k r cos kr (cid:3) (7)For the laser light linearly polarized along the Z -axis, optically-induced dipole-dipole interaction (6) canbe rewritten in dimensionless form, V αα ( k, r ) = ∆ η ( k ) ξ ( k ) ˜ α lab , Zj ( k ) V jl ( k, r )˜ α lab , lZ ( k ) cos( kr ) , (8)where energy is measured in units of B , distance in unitsof r , and k in units of r − , and ˜ α ij = α ij / ∆ α with∆ α = α (cid:107) − α ⊥ is the reduced polarizability tensor. Thedimensionless parameter, ξ ( k ) = d α ( k ) B , (9)characterizes the relative strength of the permanent-dipole and induced-dipole interactions and is on the or-der of 10 − for polar alkali dimers. We note that ineq. (6) we neglected a “static” term due to the couplingof the dipole moment of one molecule with the hyperpo-larizability of another via the optical field [14, 15]. Thisinteraction is independent of k and only becomes com-parable to the dipole-dipole potential (4) at much largerintensities ( I (cid:38) W/cm ).In the Born-Oppenheimer approximation, effective in-teraction potentials V eff ( r ) are obtained by diagonalizingthe Hamiltonian, H = H + H + V dd + V αα , (10)for fixed intermolecular separations r = ( r, θ, φ ). In di-mensionless units, the dipole-dipole potential (4) is onthe order of unity, while the strength of the optically-induced interaction (8) is given by ∆ η/ξ . Therefore,for the field strength parameter satisfying the inequal-ity 1 (cid:28) √ ∆ η (cid:28) ξ ( k ), both interaction terms are muchsmaller than the energy gap between neighboring tunnel-ing doublets. Hence, in the basis of field-dressed states, | ˜ J M , ˜ J M (cid:105) , the interaction between two ground-statemolecules can be treated within the lowest tunneling dou-blet, | ˜00 , ˜00 (cid:105) – | ˜10 , ˜10 (cid:105) . Given that the optically-inducedpotential (8) and the dipole-dipole interaction (4) mixonly states of the same and the opposite parity, respec-tively [16, 17], the Hamiltonian matrix takes the form: H rel = (cid:18) U ˜0 αα U dd U dd U ˜1 αα + 2∆ E (cid:19) , (11)where ∆ E (∆ η ) is the splitting between the tunneling-doublet levels | ˜0 , (cid:105) and | ˜1 , (cid:105) , cf. Fig. 1, and the matrixelements are given by: U dd ( r ) = 1 − θr G (∆ η ) , (12) X Z
Ground Excited
X Z
X Z
X Z Δ = η Δ = η X Z
X Z Δ = η X Z
X Z Δ = η (a)(b)(c)(d) FIG. 2. Short-range behavior of optically-induced KRb–KRbpotentials in the XZ plane ( φ = 0), for different values of ∆ η .Left and right columns correspond, respectively, to the ground | ˜00 , ˜00 (cid:105) state, and excited | ˜10 , ˜10 (cid:105) state potentials, as given byeq. (16). The dependence of the short-range potentials on φ isnegligible. Potential energy is in units of B , with V eff ( r → ∞ )chosen as zero. The laser beam propagates along the Y axis, k (cid:107) ˆ Y , with the polarization ˆ e (cid:107) ˆ Z . See text. U ˜ Jαα ( k , r ) = ∆ η ( k ) ξ ( k ) cos ( kr ) r K ˜ J (∆ η, α/ ∆ α ) × (cid:40) − (cid:114) a ( kr ) + [ a ( kr ) − b ( kr )] (cid:114) π Y ( θ, φ ) (cid:41) , (13)with α = ( α (cid:107) + 2 α ⊥ ) / a ( x ) = x cos x , and b ( x ) = (cos x + x sin x ).The factors G (∆ η ) and K ˜ J (∆ η, α/ ∆ α ) have an ana-lytic representation in terms of the coefficients c ˜ J, J (∆ η )of eq. (3) [17]. They are on the order of unity and can be analytically estimated in the strong-field limit, ∆ η → ∞ ,with good accuracy: G (∆ η ) = (cid:104) − ∆ η − / F (cid:16) ∆ η − / (cid:17)(cid:105) , (14) K ˜ J =0 , (∆ η ) = (cid:114) (cid:104) α (cid:107) ∆ α − ∆ η − / F (cid:16) ∆ η − / (cid:17)(cid:105) , (15)where F ( x ) = exp( − x ) (cid:82) x exp( y ) dy is the Dawson in-tegral [18].In such a way, the effective potentials are given bysolutions of (11): V eff ( k , r ) = ∆ E + U ˜0 αα ( k , r ) + U ˜1 αα ( k , r )2 ± (cid:113) U ( r ) + (cid:2) U ˜1 αα ( k , r ) − U ˜0 αα ( k , r ) + 2∆ E (cid:3) , (16)with the minus sign corresponding to the ground | ˜00 , ˜00 (cid:105) state effective potential.Figure 2 shows the short-range behavior of effective po-tentials (16), induced between two K Rb molecules bya laser field of wavelength λ = 1090 nm ( k = 0 . /r ),corresponding to the following values of parameters: ξ =96 . α/ ∆ α = 0 .
62, and r = 36 . kr (cid:28)
1, the dominant contribution toeq. (16) reads: V eff ( kr (cid:28) ≈ | − θ | r (cid:34)(cid:114)
32 ∆ ηξ s ( θ ) ± G (∆ η ) (cid:35) , (17)where s ( θ ) = sgn[1 − θ ]. Eq. (17) reveals the pres-ence of a critical value, ∆ η c = ξ (cid:112) / G (∆ η ), that deter-mines the sign of the short-range potential (∆ η c ∼
65 forKRb). For ∆ η < ∆ η c , eq. (17) is governed by the secondterm in the square brackets, and the potentials are alwaysattractive in the ground state and repulsive in the excitedstate for any angle θ , except for θ = ± arccos( ± − / ),where V eff ( kr (cid:28)
1) vanishes identically, cf. Fig 2 (a), (b).This behavior is qualitatively different from the dipole-dipole interaction between two polar molecules orientedalong the Z axis, V ↑↑ dd = (1 − θ ) /r , whose signalternates in dependence on θ . On the other hand, for∆ η > ∆ η c , the sign of V eff becomes angle-dependent, dueto the interplay between the terms in the square brack-ets of eq. (17), resulting in the behavior similar to V ↑↑ dd :the potential is attractive at θ = 0 , π and repulsive at θ = π/
2, cf. Fig. 2 (c), (d). Both below and above∆ η c the inverse-power decay rate of V eff can be tuned bychanging ∆ η .At large distances, kr (cid:29)
1, the optically-induced po-tential (13) is proportional to 1 /r and hence dominates X Z -6 -10 -6 Y Z -6 -10 -6 FIG. 3. Long-range behavior of the ground | ˜00 , ˜00 (cid:105) statepotentials in the XZ ( φ = 0) and Y Z ( φ = π/
2) planes, for∆ η = 100. Long-range behavior is similar for the excited | ˜10 , ˜10 (cid:105) state, the magnitude of the oscillations scales with∆ η , as given by eq. (18). Potential energy is in units of B ,with V eff ( r → ∞ ) chosen as zero. The laser beam propagatesalong the Y axis, k (cid:107) ˆ Y , with the polarization ˆ e (cid:107) ˆ Z . Seetext. over the dipole-dipole interaction (12), resulting in anasymptotic behavior given by: V eff ( kr (cid:29) ≈ − k ∆ ηξ (cid:114)
32 cos( kr ) cos( kr ) r sin θ, (18)which manifests itself in decaying oscillations, as shownin Fig. 3 for the case of ∆ η = 100. We note that the 1 /r -like interaction (18) is of longer range than both the vander Waals and V ↑↑ dd potentials, which may lead to intrigu-ing scattering properties of optically-dressed molecules.In general, the potentials of eq. (16) depend on theazimuthal angle φ , as given by the cos( kr ) term of eqs. (13) and (18). Although in the case of kr (cid:28) φ -dependence is negligible, the long-range behavioris strongly anisotropic in φ , cf. Fig. 3. As follows fromeq. (18), the magnitude and phase of the long-range oscil-lations scale with ∆ η and k respectively, and are similarfor the ground and excited state.As one can see from Fig. 3, optically-induced po-tentials exhibit concentric minima. If these potentialwells are deep enough, they will support long-rangebound states, whose properties are completely deter-mined by the optical field, molecular dipole momentsand polarizabilities, and are independent of the detailsof the intermolecular potential, in a similar way tothe electrostatically-induced bound states predicted byAvdeenkov and Bohn [21]. Looking into the propertiesof these states represents a challenging theoretical andcomputational problem.In summary, we undertook a study of intermolecularinteractions in the presence of an intense far-off-resonantoptical field, and provided simple analytic expressionsfor the resulting potential energy surfaces. Optically-induced potentials are highly controllable and are signifi-cantly different from the dipole-dipole interaction takingplace between oriented polar molecules [22]. This couldopen a way to novel quantum phases of laser-dressed ul-tracold polar gases and new methods to control molecularcollisions in the ultracold regime.The author is indebted to Bretislav Friedrich for con-tinuous encouragement and valuable suggestions on themanuscript; to Eugene Demler, Mikhail Lukin, andBoris Sartakov for insightful discussions; to Svetlana Ko-tochigova for providing dynamic polarizabilities; and toGerard Meijer for support. [1] D. P. Craig and T. Thirunamachandran, MolecularQuantum Electrodynamics (Academic Press, 1984).[2] T. Thirunamachandran, Molecular Physics , 393(1980).[3] D. O’Dell, S. Giovanazzi, G. Kurizki, and V. Akilin,Physical Review Letters , 5687 (2000).[4] S. Giovanazzi, D. O’Dell, and G. Kurizki, Physical Re-view Letters , 130402 (2002).[5] D. O’Dell, S. Giovanazzi, and G. Kurizki, Physical Re-view Letters , 110402 (2003).[6] K.-K. Ni, S. Ospelkaus, D. Wang, G. Qu´em´ener,B. Neyenhuis, M. H. G. de Miranda, J. L. Bohn, J. Ye,and D. S. Jin, Nature , 1324 (2010).[7] B. Friedrich and D. Herschbach, Physical Review Letters , 4623 (1995).[8] B. Friedrich and D. Herschbach, Journal of PhysicalChemistry , 15686 (1995).[9] S. Kotochigova and D. DeMille, Phys. Rev. A , 063421(2010).[10] J. Deiglmayr, M. Aymar, R. Wester, M. Weidem¨uller,and O. Dulieu, Journal of Chemical Physics , 064309(2008). [11] B. Friedrich and D. Herschbach, Journal of ChemicalPhysics , 6157 (1999).[12] B. Friedrich and D. Herschbach, Journal of PhysicalChemistry A , 10280 (1999).[13] M. H¨artelt and B. Friedrich, Journal of Chemical Physics , 224313 (2008).[14] A. Salam, Physical Review A , 063402 (2007).[15] D. S. Bradshaw and D. L. Andrews, Phys. Rev. A ,033816 (2005).[16] R. J. Cross and R. G. Gordon, Journal of ChemicalPhysics , 3571 (1966).[17] M. Lemeshko, et al. , to be published (2011).[18] M. Abramowitz and I. A. Stegun, eds., Handbook ofMathematical Functions (Dover, New York, 1972).[19] J. Aldegunde, B. A. Rivington, P. S. Zuchowski, andJ. M. Hutson, Phys. Rev. A , 033434 (2008).[20] M. Aymar and O. Dulieu, Journal of Chemical Physics , 204302 (2005).[21] A. Avdeenkov and J. L. Bohn, Physical Review Letters , 043006 (2003).[22] M. A. Baranov, Physics Reports464