VOLUME85, NUMBER 12 PHYSICAL REVIEW LETTERS 18SEPTEMBER2000Creation of Polar and Nonpolar Ultra-Long-Range Rydberg MoleculesChris H. GreeneDepartment of Physics and JILA, University of Colorado, Boulder, Colorado 80309-0440A. S. Dickinson*JILA, University of Colorado, Boulder, Colorado 80309-0440H. R. SadeghpourITAMP, Harvard-Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge, Massachusetts 02138(Received 19 May 2000)We predict the existence of a ubiquitous class of long-range molecular Rydberg states, whose Born-Oppenheimer potential curves are oscillatory in nature. These oscillations reflect the nodal structureof the atomic Rydberg state wave functions. The temperature and density of atoms in a Bose-Einsteincondensate are particularly favorable for the laser excitation of ultra-long-range vibrational bound stateslocalized at internuclear distances in the range 103–105a.u. A surprising trilobitelike class of polarhomonuclear diatomics should exhibit electric dipole moments in the kilodebye range.PACS numbers: 31.50.+w, 03.75.Fi, 33.80.RvThe creation of dilute collections of ultracold atoms inmagneto-optical traps, Bose-Einstein condensates (BECs),and optical lattices has made detailed spectroscopicinvestigation of molecules in unusual long-range statesattached to the lower-lying s 1 p dissociation thresholdspossible [1]. Rydberg molecules whose vibrational wavefunctions are shelved in shallow long-range wells (R ⬃2030 a.u.) and stem from the interaction of Rydbergmolecular potentials and the ion-pair formation potentialcurve, have been indentified in two-photon spectroscopy[2]. The spectroscopy of long-range Na2moleculestrapped in a Born-Oppenheimer potential curve whoseminimum occurs around 70 a.u., for instance, spawnedan accurate confluence of experiment and theory thatconfirmed the retardation modification of the long-rangeinteraction [3]. Other novel regimes include the frozenatomic Rydberg gas discussed independently by two dif-ferent experimental groups [4], and the ultracold neutralplasma that has received attention even more recently[5]. Rydberg states of molecules at more conventionaltemperatures have been studied extensively in contextssuch as line broadening [6].The purpose of the present Letter is to discuss how theultracold environment characteristic of modern-day Bose-Einstein condensates (BECs) should permit the efficientproduction of molecular Rydberg states whose vibrationallevels are localized at internuclear distances of order32n2Bohr radii, with n the principal quantum number. Theformation of these states does not hinge on any of thecoherence properties of a BEC, but a condensate providesparticularly favorable regimes of temperature and densityfor their observation. The vibrational binding energiesof these levels from n ⯝ 2050 fall in the GHz or MHzrange, for molecules excited out of a typical87Rb atomicBEC. To our knowledge, these molecules would be thelargest and most polar ever observed.Two qualitatively different classes of molecular Rydbergstates should be observable in an ultracold atomic gas.They are nonpolar or polar, respectively, depending onwhether the electronic orbital angular momentum is low(l # 2 for Rb) or high.Rydberg molecules of the first class (low l) are formedfrom the interaction between a Rb(nlj) Rydberg atom withlow orbital angular momentum (l & 2) and a distant Rbground state atom. These molecules possess shallow Born-Oppenheimer potential curves that oscillate as a functionof internuclear distance R. The oscillations mimic the be-havior of the Rydberg electron radial wave function, withsuccessive potential curve minima associated with succes-sive maxima of the Rydberg electron density. In regionswhere the relevant energy-dependent e2-Rb共5s兲 tripletscattering length ATis negative, wave function antinodesgenerate minima in the Born-Oppenheimer potentialcurve.Rydberg molecules of the second class (perturbed hy-drogenic states [7]), arise from the coupling of the manyquasidegenerate high-l states whose quantum defects arenegligible [7–10]. The Rydberg electron interaction withthe core electrons results in a shift of the Rydberg ener-gies from their unperturbed hydrogenic levels. Electronsin levels with large angular momenta remain sufficientlyaway from the core and their energies nearly coincide withthose of atomic hydrogen. The perturbed hydrogenic po-tentials are far deeper than those of the low-l type forcomparable n and consequently support many more boundvibrational states. Each perturbed hydrogenic state alsopossesses a permanent electric dipole moment of magni-tude D ⯝ R 212n2a.u. (Atomic units are used exceptwhere stated otherwise.) To our knowledge, this is theonly known case in which a homonuclear diatomic mole-cule is predicted to exhibit a permanent electric dipolemoment. Such a huge electric dipole moment in any2458 0031-9007兾00兾85(12)兾2458(4)$15.00 © 2000 The American Physical SocietyVOLUME85, NUMBER 12 PHYSICAL REVIEW LETTERS 18SEPTEMBER2000long-lived molecular state presents an intriguing opportu-nity for manipulation and control through the applicationof an electric field or field gradient.Throughout the present Letter, we adopt Rb2as ourprototype Rydberg molecule and demonstrate potentialcurves for states like Rb共ndj兲 1 Rb共5s兲 that display mul-tiple minima at very large internuclear separations. Theexistence of these oscillatory extrema in such potentialcurves can be understood most simply through the useof a Fermi-type pseudopotential [8,11] to characterizethe interaction between the atomic Rydberg electron anda ground state Rb(5s) atom. If៬r is the position of theRydberg electron relative to a Rb1ground state ion,and៬R is the position of the Rb(5s) atom relative to theion, then the interaction potential can be taken, in a firstapproximation, asV 共៬r,៬R兲 苷 2pAT关k共R兲兴d共៬r 2៬R兲 . (1)Here AT关k兴⬅2 tandT0共k兲兾k is the energy-dependenttriplet s-wave scattering length for electron collisionswith ground state Rb(5s) atoms, defined in terms of thetriplet s-wave phase shift dT0共k兲. The relevant electronwave number k共R兲 is defined by the kinetic energy ofthe Rydberg electron at energy ´ 苷 21兾2n2when itcollides with a perturbing atom at a distance R from theRb1ion, namely,12k2共R兲 苷 ´11兾R. The accuracy ofdiatomic potential curves obtained within the Fermi modelis improved [7,9,10] if an energy-dependent (and