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Ultralong-range polyatomic Rydberg molecules formed by a polar perturber Seth T. Rittenhouse,1 M. Mayle,2,∗ P. Schmelcher,2 and H. R. Sadeghpour1 1ITAMP, Harvard-Smithsonian Center for Astrophysics, Cambridge, MA 02138 2Zentrum fu¨r Optische Quantentechnologien, Universita¨t Hamburg, Luruper Chaussee 149, D-22761 Hamburg, Germany. (Dated: January 28, 2011) The internal electric field of a Rydberg atom electron can bind a polar molecule to form a giant ultralong-range stable polyatomic molecule. Such molecules not only share their properties with Rydberg atoms, they possess huge permanent electric dipole moments and in addition allow for coherent control of the polar molecule orientation. In this work, we include additional Rydberg manifoldswhichcoupletothenearlydegeneratesetofRydbergstatesemployedin[S.T.Rittenhouse 1 and H. R. Sadeghpour, Phys. Rev. Lett. 104, 243002 (2010)]. The coupling of a set of (n+3)s 1 Rydberg states with the n(l>2) nearly degenerate Rydberg manifolds in alkali metal atoms leads 0 to pronounced avoided crossings in the Born-Oppenheimer potentials. Ultimately, these avoided 2 crossingsenabletheformationofthegiantpolyatomicRydbergmoleculeswithstandardtwo-photon n laser photoassociation techniques. a J PACSnumbers: 33.80.Rv,31.50.Df 7 2 I. INTRODUCTION a polar molecule. This system was first studied in [15] ] where it was shown that the Rydberg electron can be h p used to “drag” the polar molecule into a preferred orien- ThephysicsofRydbergatomsandmoleculeshasdevel- - tation and that this process can be controlled by a Ra- m oped into a quasi-sustainable “ecosystem” over the last man microwave pulse scheme. Ultralong range Rydberg decade, mainly due to a) the exaggerated properties of o molecules in general came in vogue after a proposal that Rydbergatomsandmolecules(longlifetimes,largesizes, t suchmoleculescouldformfromzero-rangeinteractionof a and scalability), and b) exquisite experimental control . electronswithgroundstateatoms[16]. Therecentexper- s over these properties through the advent of ultracold imentalrealizationofaclassofsuchmolecules[8]hasled c atomic Rydberg samples [1, 2]. Some landmark devel- i to revived theoretical and experimental activity [17, 18]. s opments over the last few years in the field of Rydberg y physics have been the creation of a frozen Rydberg gas h [3],thedemonstrationofaRydbergblockadeschemeand p The interaction of the polar perturber with the Ryd- [ the subsequent realization of a Rydberg qubit gate [4– 7], the formation of ultralong-range isotropic Rydberg bergatomstronglycouplesthefield-freeatomicRydberg 1 states by means of the dipole’s electric field. In [15], the molecule [8], the creation ultracold neutral plasmas [9], v mixing of states was restricted to the nearly degenerate as well as the formation of highly-magnetized Rydberg 3 (negligible quantum defects) manifold of Rydberg states antihydrogen atoms [10]. These rapid advances now al- 5 inRb(nl(cid:38)3). Inthepresentwork, weextendourprevi- 3 low for few-body and many-body effects to be realized ous study by going beyond this degenerate perturbation 5 in laboratory settings and be prototyped for simulation theory approach. In this manner, the effect of additional . of strongly interacting spin chains [11–13]. The other in- 1 Rydberg orbitals on the molecular Born-Oppenheimer 0 gredient of our proposed polyatomic Rydberg complex, (BO) potentials is probed. In particular, the proxim- 1 namely, ultracold polar molecules, have also been touted ity of the n(l>2) Rydberg manifold to a single (n+3)s 1 astoymodelsforthesimulationofmany-bodycondensed : systems and for the realization of quantum gates [14]. A stateintroducesastrongcoupling,allowingfortheexper- v imentalpreparationoftheproposedpolyatomicRydberg i main appeal of polar molecules is that they may possess X molecules via a standard two-photon laser excitation. sizeable permanent dipole moments. r a In this work, we harness the long-range interaction of a Rydberg atom with a polar molecule (Rb is used as an The paper is outlined as follows. In section II the ubiquitous example, but other alkali metal atoms would model Hamiltonian describing the polar molecule per- serve the purpose), to demonstrate that ultralong-range turber is introduced. The adiabatic Hamiltonian for the polyatomic molecules with enormous permanent dipole full Rydberg atom plus polar molecule complex is subse- moments can form from combining a Rydberg atom and quently provided in section III. The resulting adiabatic potential surfaces are discussed in section IV. In section V we discuss the admixture of s-wave electron character in the giant Rydberg molecule. We conclude by a brief ∗Presentaddress: JILA,UniversityofColoradoandNationalInsti- summary and an outlook on further research directions tuteofStandardsandTechnology, Boulder, Colorado80309-0440, USA. in section VI. 2 II. POLAR MOLECULE MODEL dipole (d > d ) is sensitive to the detailed, short-range 0 c HAMILTONIAN structure of the polar molecule. To avoid the complica- tionofsuper-criticaldipolescatteringandelectrontrans- Before we discuss the properties of the proposed poly- ferfromtheRydbergatomtothepolarmolecule,wewill atomic Rydberg molecules, we must introduce an effec- deal exclusively with polar molecules whose dipole mo- tive Hamiltonian for the molecular perturber. In this ments are subcritical, i.e., d0 <1.63 D. context, we model the polar molecule as a two-level molecule in which the opposite parity states are mixed in the presence of an external field, i.e., III. THE ADIABATIC HAMILTONIAN (cid:18) (cid:19) H = 0 d0Fext , (1) For simplicity, we assume that the external electric mol d0Fext ∆ field is along the intermolecular axis, such that (2) be- comes where∆isthezerofieldsplittingbetweenthetwomolec- ularstates(asinaΛ-doubletmolecule),d0 istheperma- F (cid:16)R(cid:126),(cid:126)r(cid:17)= e + ecosθ(cid:126)r−R(cid:126) (4) nent dipole moment of the molecule in the body fixed ext R2 (cid:12) (cid:12)2 (cid:12)(cid:126)r−R(cid:126)(cid:12) frame, and Fext is an electric field external to the polar (cid:12) (cid:12) molecule. Specifically, the external field stems from the (cid:16) (cid:17) (cid:12) (cid:12) Rydberg electron and the Rydberg core, i.e., where θ = (cid:126)r−R(cid:126) ·R(cid:126)/R(cid:12)(cid:126)r−R(cid:126)(cid:12). This means that (cid:126)r−R(cid:126) (cid:12) (cid:12) (cid:12) (cid:16) (cid:17)(cid:12) the projection m of the Rydberg electron angular mo- (cid:16) (cid:17) (cid:12)(cid:12) Rˆ (cid:126)r−R(cid:126) (cid:12)(cid:12) mentumalongR(cid:126) isconserved. TofindtheBOpotentials, F R(cid:126),(cid:126)r =e(cid:12) + (cid:12), (2) ext (cid:12)R2 (cid:12) (cid:12)3(cid:12) we solve the adiabatic Schrödinger equation at fixed po- (cid:12)(cid:12) (cid:12)(cid:12)(cid:126)r−R(cid:126)(cid:12)(cid:12) (cid:12)(cid:12) lar molecule location R(cid:126) =Rzˆ, where e is the electron charge, R(cid:126) is the core-polar H ψ(R;(cid:126)r,σ)=U(R)ψ(R;(cid:126)r,σ), (5) ad moleculeseparationvectorand(cid:126)risthepositionoftheRy- dberg electron with respect to the core. In other words, with (1) describes the coupling between the internal state of H =H +H . (6) the polar molecule and the Rydberg atom. ad A mol Itshouldbenotedthat(1)onlydependsonthemagni- The first term, H = − (cid:126)2 ∇2 + V (r), describes the tudeoftheexternalfield, whichisappropriateforstatic, A 2me r l unperturbed Rydberg atom; the core penetration, scat- homogeneous fields. For a rigid rotor molecule, such tering, and polarization effects of its valence electron are as KRb, the rotation of the molecule happens on much accounted for by the l-dependent model potential V (r) slowertimescalesthantheRydbergelectronorbitaltime. l [20], giving rise to the quantum defects of the low angu- In this case, the coupling between the Rydberg atom lar momentum Rydberg states. m is the electron mass, and the polar molecule accounts for the rotation of the e ψ is the electron wave function, and σ is a coordinate molecule, signifying the internal states of the polar molecule. The V (cid:16)R(cid:126)(cid:17)=−d(cid:126)·F(cid:126) , (3) eigenvaluesU(R)serveasthesought-afterBOpotentials mol ext for the polar perturber. To solve (5), the total wave function is expanded in the basis {ψ ((cid:126)r)|±(cid:105)} where where d(cid:126)is the rigid rotor dipole moment. This interac- nlm ψ ((cid:126)r) is an unperturbed Rydberg orbital and |±(cid:105) are nlm tionleadstoBOpotentialsurfaceswhichdependonboth the polar molecule parity states. The atomic and molec- R(cid:126) and the orientation of d(cid:126)with respect to R(cid:126); this more ular degrees of freedom are coupled by the electric field complete treatment is the subject of an ongoing study. (4) that mixes both the Rydberg orbitals as well as the Incontrast,aΛdoubletmolecule,suchasOH,hasaper- parity states of the polar molecule, cf. (1). manent dipole moment that arises from the interaction In [15], the BO potentials were found using degener- oftwooppositeparityelectronic(e,f)states. Rotational ate perturbation theory in the (nearly) degenerate set of transitionenergiesinΛ-doubletmoleculesareusuallyor- Rydbergorbitals{ψ ((cid:126)r)}. ThemolecularHamilto- n(l>2)0 ders of magnitude larger than the typical doublet split- nian (1) can be prediagonalized to yield the eigenvalues ting energies. For the fields provided by highly excited Rydberg atoms, of order F ∼ 10−6 a.u., our model (cid:18) (cid:113) (cid:19) ext ε(F )=d F ± F2 +F2 , (7) Hamiltonian(1)thusprovidesanexcellentdescriptionof ext 0 c ext c such molecules. The electron-dipole interaction in (3) and in the off- where F = ∆/2d is the critical external field strength c 0 diagonalelementsof(1)hasacriticalvalue. Ifthedipole atwhichthemoleculebecomescompletelypolarized. In- moment,d ,islargerthantheFermi-Tellercriticalvalue, sertingin(7)forF theR-dependenteigenstatesofthe 0 ext d =1.63D,aninfinitenumberofboundstatesform[19]. Rydberg atom exposed to the field (4) results in the po- c Furthermore, an electron scattering off a super critical tentials found in [15]. A more complete treatment of the 3 system requires that the total adiabatic Hamiltonian is Theoff-diagonalmatrixelementsof(6)areduetoH mol diagonalized in a complete set of Rydberg orbitals. Be- whichcouplestheRydbergelectronstatestotheinternal cause in the latter case, the orbitals are in general not paritystatesofthepolarmolecule. Theyaredetermined degenerate, the prediagonalization scheme cannot be di- by evaluating the integrals rectly used beyond the degenerate perturbation theory. Inthepresentwork, wegobeyondthedegenerateper- (cid:90) cosθ turbationtheoryframeworkadaptedin[15]. Tothisend, (cid:104)ψ |F |ψ (cid:105)=−e d3rψ∗ ((cid:126)r) (cid:126)r−R(cid:126)ψ ((cid:126)r). n(cid:48)l;0 ext nl0 n(cid:48)l(cid:48)0 (cid:12) (cid:12)2 nl0 the Hamiltonian (6) is diagonalized in an extended basis (cid:12)(cid:126)r−R(cid:126)(cid:12) (cid:12) (cid:12) set comprising several Rydberg n manifolds. By con- (8) struction, the atomic Hamiltonian H is diagonal in the A An intuitive closed form of (8) is obtained if we expand Rydberg basis {ψ (r)}, nl0 the electron electric field contribution in spherical har- (cid:28) (cid:12) (cid:12) (cid:29) monics. Using the multipole expansion of [21] yields ψn(cid:48)l(cid:48)0,σ(cid:48)(cid:12)(cid:12)(cid:12)−21∇2r+Vl(r)(cid:12)(cid:12)(cid:12)ψnl0,σ =∆δσ,−δn(cid:48)nδl(cid:48)lδσ(cid:48)σ √  √ Rl−1l 2l+1 −δn(cid:48)nδl(cid:48)lδσ(cid:48)σ2(n−1 µ )2,cosθ(cid:126)r−R(cid:126) = − 4π(cid:80)l rl+1 (2l√+1) Yl0(θ,φ), r >R . l (cid:12) (cid:12)2 √ rl (l+1) 2l+1 where µ is the quantum defect for the l-th partial wave (cid:12)(cid:12)(cid:126)r−R(cid:126)(cid:12)(cid:12)  4π(cid:80)Rl+2 (2l+1) Yl0(θ,φ), r <R l l Rydbergstateandσ denotestheparitystateofthepolar (9) molecule. The quantum defects used in this work are We note that, with some generalization, this procedure those for the rubidium atom [2]: µ = 3.13, µ = 2.65, can also be used to evaluate the interaction matrix ele- s p µ =1.35, µ =0.016, and µ ≈0. While our focus is mentsbetweenapermanentdipoleandtheRydbergelec- d f l>3 onRb,themethodcanbeextendedtoanyhighlyexcited tron, cf. (3). Using (9), the off diagonal matrix elements Rydbergatombyusingtheappropriatequantumdefects. of (5) are given by (cid:12) (cid:12) (cid:42)ψ (cid:12)(cid:12)(cid:12)cosθ(cid:126)r−R(cid:126)(cid:12)(cid:12)(cid:12)ψ (cid:43)=(cid:112)(2l(cid:48)+1)(2l+1) (cid:88)l+l(cid:48) (cid:18) l l(cid:48) l(cid:48)(cid:48) (cid:19)2 (10) n(cid:48)l(cid:48)0(cid:12)(cid:12) (cid:12)2(cid:12) nl0 0 0 0 (cid:12)(cid:12)(cid:12)(cid:12)(cid:126)r−R(cid:126)(cid:12)(cid:12) (cid:12)(cid:12) l(cid:48)(cid:48)=|l−l(cid:48)| (cid:34) (cid:35) 1 (cid:90) R (cid:90) ∞ 1 × (l(cid:48)(cid:48)+1) rl(cid:48)(cid:48)+2R∗ (r)R (r)dr−Rl(cid:48)(cid:48)−1l(cid:48)(cid:48) R∗ (r)R (r)dr , Rl(cid:48)(cid:48)+2 nl n(cid:48)l(cid:48) rl(cid:48)(cid:48)−1 nl n(cid:48)l(cid:48) 0 R where R (r) is the radial wave function for an electron IV. BORN-OPPENHEIMER POTENTIALS nl in the nl Rydberg state. TheadiabaticHamiltonianmatrixassumesthefollow- In [15], it was assumed that for l>2, the Rydberg or- ing form, bitalsofRbaredegenerateandthatonlythisdegenerate set of states is required to converge the BO potentials. Here we extend this treatment to including additional Rydberg orbitals and the small, but finite, f-wave quan- (cid:18)E¯ d F¯ (cid:19) H¯ = Ryd 0 , (11) tum defect. Figure 1 compares the BO potentials using ad d0F¯ ∆I¯+E¯Ryd degenerate perturbation theory (solid curves, as in [15]) to those found by including the f-wave quantum defect (dotted curves) for the n=25 state of rubidium; a polar where I¯is the unity matrix, E¯ is the diagonal matrix molecule dipole moment of d0 = 0.40 a.u., and a zero- representation of H , and F¯ iRsytdhe electric field matrix field splitting of ∆=1.85×10−7 a.u. is considered. A whoseelementsaregivenby(10)withadiagonaloffsetof We reiterate some of the most salient features of the 1/R2 duetothecontributionoftheelectricfieldfromthe proposed molecules. The modulations that form a se- positiveioniccoretothetotalelectricfield, asdefinedin ries of wells reflect the oscillatory nature of the Ryd- (4). The BO potentials are found by diagonalizing the berg electron wave function. The outer most wells in matrix (11) at each R. Special attention must thereby the lowest two potentials are deep enough to support be drawn to the actual size of the basis set in order to many vibrational levels. The resulting giant polyatomic ensure convergence of the potentials. Rydberg molecules share several features with previ- 4 0 0 -5 25f -10 -10 -15 -20 z)-20 z) H H G G-30 R) (-25 R) ( U(-30 U(-40 -35 (a) (b) -50 -40 D -45 L R -60 28s 600 900 1200 600 900 1200 600 900 1200 R (a.u.) R (a.u.) R (a.u.) FIG.1: TheBOpotentialsfortheRb(n=25)Rydbergatom FIG. 2: (a) The BO potentials for the Rydberg-polar and a polar molecule are shown. The potentials are calcu- molecule system are shown, calculated including lower angu- latedatthelevelofdegenerateperturbationtheoryincluding larmomentumstatesusingapolarmoleculeΛ-doubletparity (dashed red curves) and ignoring (solid black curves) the f- splitting of ∆ = 1.85×10−7 a.u. and a dipole moment of wave quantum defect of the Rydberg electron. The polar d =0.40 a.u. (subcritical dipole). (b) The same as (a) with 0 perturberinthisexampleisamoleculewithadipolemoment adipolemomentofd =0.57a.u.(thepermanentdipolemo- 0 d =0.40 a.u. and a zero field splitting ∆=1.85×10−7 a.u. mentofCD).ThezeroinenergyistheRb(n=25)threshold. 0 The28slevelandzerofieldmolecularsplittinghavealsobeen labelled. ously predicted homonuclear Rydberg molecules, the so called “trilobite” molecules [16, 22, 23]. The size of izationrequiresaninordinatelylargenumberofRydberg these molecules scales as R ∝ n2 and the well depths ryd orbitals. Furthermore, for larger values of d , the elec- scale as V ∝ d /n3. Unlike for homonuclear Rydberg 0 D 0 tron wave function penetrates into the short-range re- molecules, the anisotropic nature of the electron-dipole gion of the perturber molecule and probes the detailed interaction creates two different internal configurations. structure of the molecule. We therefore restrict here to Corresponding to the well labelled R in figure 1, in one sub-critical dipole moments, d ≤ 0.6 a.u. Because the 0 configuration the dipole of the polar molecule is oriented Rydberg spacing scales as n−3, larger sets of orbitals are towards the positive core. In the other configuration – required to converge the potentials attached to higher labelled L – the dipole of the polar molecule is oriented Rydberg thresholds. The calculations reported here are away from the positive core. forRb(n=25)manifoldofstates,thoughthequalitative Asanticipated, inclusionofthesmallf-wavequantum behaviourswediscusswillpersistforhighern. Toacquire defect only modifies the potentials close to the dissocia- convergedpotentialsforn=25atthelargestdipolemo- tion threshold limit, i.e., near the Rb(n = 25) limit. In ments considered here, d =0.60 a.u., our basis includes 0 the lowest wells, where the Rydberg molecules form, the all electron angular momenta for n = 19,20,...,30,31 as two sets of potentials are slightly shifted with respect to well as the 32(s,p,d), 33(s,p) and 34s Rydberg orbits, each other while being otherwise almost identical, indi- yielding a total of 331 electron basis states. cating that including the f-wave quantum defect has al- Figure 2 shows the converged potentials attached to mostnoinfluenceonthebehaviouroftheresultinggiant the n = 25 Rydberg level. As in figure 1, the polar molecule, nor on the polyatomic molecular dipole mo- molecule in figure 2(a) has a dipole moment d = 0.40 0 ment. a.u. and a zero field splitting ∆ = 1.85×10−7 a.u. (∆ By including more Rydberg orbitals in our basis set, has been chosen to be the Λ-doublet splitting of CD). we can explore the range of validity of the degenerate Comparing the two figures shows that the molecular po- perturbation theory approach. For smaller dipole mo- tentialsareunaffectedbythepresenceofthenon-binding ments, d (cid:46) 0.4 a.u., relatively few basis states are re- s-wave states for cases when the molecular dipole is less 0 quired to achieve convergence. In fact, the converged than∼0.4a.u. ThetwocurvesconvergingtotheRb(28s) potentials are only slightly different from those shown thresholdcorrespondtothetwooppositeparitystatesof in figure 1. As the dipole moment increases, more and the polar molecule and are correspondingly split by the more Rydberg orbitals are required. This is due to the zero field splitting ∆. localizing effects of the dipole-electron potential. For a For larger dipole moments, it is essential to include nearly critical dipole moment, d ≈ 0.63 a.u., the elec- the lower angular momentum states in our treatment. 0 tron is almost entirely localized at the location of the While for an unperturbed Rydberg atom, the latter are dipole. To accurately describe this (small angle) local- isolated from the degenerate l > 2 manifold, increasing 5 the dipole moment has the interesting effect of forcing -45 thepotentialwellsofthen=25tocrossthroughthe28s potentials. ThisispossiblebecausetheRb(ns)quantum z)-50 (a) H defect (µ =3.13) has a small non-integer fraction, plac- G ing the Rsb(28s) close to the n=25 degenerate manifold R) (-55 of states. In figure 2(b), a corresponding example with U(-60 d = 0.57 a.u.= 1.46 D – the permanent dipole moment 0 of CD – is provided. In this case, the interaction be- n tweenthen=25molecularstatesandtheRb(28s)state atio0.8 (b) is significant. This is generally true of all Rb(n(l > 2)) pul0.6 o degenerate manifolds and (n+3)s interacting states. As e p0.4 a result and due to strong avoided crossings, potential av w0.2 wells capable of supporting bound vibrational levels ap- s- 0 pearintheBOcurvesattachedtotheRb(28s)threshold. 600 900 1200 R (a.u.) Figure3(a)providesaclose-upofthefourBOpotentials directly involved in the binding of the polyatomic Ryd- berg molecule. The vibrational wave functions for two FIG. 3: (a) The four BO potentials, directly involved in such bound states in the outermost wells are also shown. molecular binding, are shown for the same parameters as in figure2(b). Twovibrationalwavefunctionsarealsoshownat theirbindingenergiesineachoftheoutermostwells. (b)The V. S-WAVE ADMIXTURE s-wave character of the molecular wave function is shown for each of the four potentials. The BO potential curves in figure 1 (also in [15]) include a nearly degenerate superposition of atomic or- such states are shown at their corresponding binding en- bitals. As such, they solely contain contributions from ergies in the outer two wells. These molecular levels are high angular momentum states (l > 2) and hence are well isolated and have readily accessible vibrational fre- only accessible in experiments which ”photoassociate” quencies. these molecules, if and when low-lying angular momenta Theaddeds-wavecharacterintheelectronwavefunc- are admixed into the degenerate manifold. The strong tiondecreasesthechargelocalizationexhibitedintheRy- interactionofthe(n+3)sRydbergstatewiththelowest dberg molecule slightly. Even with this decrease, these BO potential curves belonging to the n(l>2) manifolds Rydberg molecules exhibit massive dipole moments. Us- in figure 3(b) admixes large amount of s-wave character ing the scaling behaviour of [15], the Rydberg molecule into the wave functions. dipole moment d can be found, Ryd Atthepotentialminimum,theadiabaticchannelfunc- tion is well approximated by d ≈1.3[a(d )]2n2. Ryd 0 ψ(r,χ)=a(d )ψ ((cid:126)r)|χ (cid:105)+b(d )ψ (r)|χ (cid:105), (12) For the case shown in figure 3(a) this yields d ≈ 0 d d 0 s s Ryd 1400D, a truly large dipole moment. The presence of where ψ and |χ (cid:105) are the electronic wave function and these enormous dipole moments indicates that the poly- d d internal polar molecule state, respectively, that include atomicRydbergmoleculescouldbesensitivelycontrolled higher electron angular momentum with l > 2. ψ and through the use of small external electric fields. Due to s |χ (cid:105) are the s-wave Rydberg electron wave function and the extreme sensitivity of Rydberg electron to external s the corresponding molecular state, respectively. The ex- fields, the behaviour of the BO potentials under the in- pansioncoefficientsa(d )andb(d )dependonthedipole fluence of such fields is not immediately obvious and is 0 0 moment of the polar perturber as well as on its position the subject of ongoing studies. R. Figure 3(b) gives the s-wave electron contribution, In figure 4(a-c), we provide snapshots of additional |b(d )|2, as a function of the intermolecular separation BOpotentialsresponsibleformolecularbinding,namely, 0 distance for each of the adiabatic channel functions cor- for d = 0.30, 0.55, and 0.60 a.u., respectively. As the 0 responding to the potentials shown in figure 3(a). From dipole moment of the polar perturber increases, the L figure 3(b) it can be seen that near the minima of the and R wells are pulled down through the s-wave Ryd- outerwells, theelectronicstateofthegiantmoleculehas berg threshold. At d = 0.30 a.u., the well structure 0 a significant s-wave character, approximately 30−40%. is not perturbed at all by the s-wave potential. For Thisimpliesthepossibilitythatthesemoleculescouldbe d = 0.55, the potential wells are strongly distorted by 0 formed in a simple two photon excitation scheme similar the presence of the lower threshold, while for d = 0.60 0 to that used in the experimental realization of homonu- a.u. the outermost well structures have passed through clear Rydberg molecules [8]. The n = 25 wells shown the s-wave threshold and correspondingly have a much in figure 3(a) are deep enough to support bound vibra- smaller s-wave contribution. Figures 4(d-f) show the s- tional states, the radial wave functions of the first two wave electron contribution for each potential. For 0.5 6 interactioncreatesaseriesofBOpotentialswithanoscil- -20 1 lating series of wells that reflect the Rydberg oscillations -30 -40 in the electron wave function. By extending the work (a) (d) -50 in [15] to beyond the degenerate perturbation theory, we -60 n0 have shown that the small, but finite f-wave quantum o z)-20 ati1 defect changes little the behaviour of the resulting giant R) (GH---543000 (b) e popul (e) moBleyciunlcelsu.dinglowerelectron-angularmomentumstates, U(-60 av0 anewsetofpotentialsattachedtothe(n+3)sRydberg w -20 s-1 level appear. These added potentials have no significant -30 influence on electronic state of the Rydberg molecule for -40 -50 (c) (f) smaller subcritical dipole moments, d0 (cid:46) 0.4 a.u. For -60 0 nearly critical dipole moments, d0 ≈ 0.6 a.u., a series of 600 900 1200 600 900 1200 avoided and level crossings is formed between the poten- R (a.u.) R (a.u.) tials formed from the degenerate n(l > 2) manifold and the(n+3)s-wavepotentials. Theseavoidedcrossingslend significants-wavecharactertotheRydbergelectronstate opening the possibility of creating the giant polyatomic FIG. 4: The variations of the BO potential curves with the strength of the polar molecule dipole moment d . (a) Rydberg molecules using standard two-photon Rydberg 0 d =0.30 a.u., (b) d =0.55 a.u., and (c) d =0.60 a.u. The excitationschemes. Theseriesofavoidedandlevelcross- 0 0 0 respective s-wave contributions to the molecular wave func- ings create a complex structure of couplings between the tions are shown in panels (d-f). A Λ-doublet parity splitting various BO potentials. Extensions of this work beyond of ∆=1.85×10−7 a.u. is considered. a simple two-state polar molecule to incorporate rigid rotor-type polar molecules, such as KRb, poses a significant challenge. The added angular behaviour of the a.u.(cid:46) d (cid:46) 0.6 a.u. the s-wave character at the position 0 polar molecule will create a set of two-dimensional po- of the outermost wells varies approximately 10−40%, tential surfaces which couple the rotational behaviour of enough to give a fairly large Rabi frequency for the cre- the polar molecule to the vibrational state of the giant ation of Rydberg molecules in a standard two photon Rydbergmolecule. Creatingandexaminingthisintricate excitation scheme. energy landscape will be the subject of future work. VI. CONCLUSIONS AND OUTLOOK VII. ACKNOWLEDGEMENTS Inthispaperwehaveexaminedtheformationofgiant, polyatomic Rydberg molecules consisting of a Λ-doublet The authors would like to thank T. V. Tscherbul for polarmoleculeandaRydbergatom. Weprovidedamore help in calculating the Λ-doublet molecule parameters completedescriptionofthepolarmoleculemodelinitially used in this work. 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