Electrically Dressed Ultralong-Range Polar Rydberg Molecules
EElectrically Dressed Ultralong-Range Polar Rydberg Molecules
Markus Kurz and Peter Schmelcher
1, 2 Zentrum f¨ur optische Quantentechnologien, Luruper Chaussee 149, 22761 Hamburg, Universit¨at Hamburg, Germany The Hamburg Centre for Ultrafast Imaging, Luruper Chaussee 149, 22761 Hamburg, Universit¨at Hamburg, Germany (Dated: November 15, 2018)We investigate the impact of an electric field on the structure of ultralong-range polar diatomicRubidium Rydberg molecules. Both the s -wave and p -wave interactions of the Rydberg electron andthe neutral ground state atom are taken into account. In the presence of the electric field the angulardegree of freedom between the electric field and the internuclear axis acquires vibrational characterand we encounter two-dimensional oscillatory adiabatic potential energy surfaces with an antiparallelequilibrium configuration. The electric field allows to shift the corresponding potential wells insuch a manner that the importance of the p -wave interaction can be controlled and the individualwells are energetically lowered at different rates. As a consequence the equilibrium configurationand corresponding energetically lowest well move to larger internuclear distances for increasingfield strength. For strong fields the admixture of non-polar molecular Rydberg states leads to thepossibility of exciting the large angular momentum polar states via two-photon processes from theground state of the atom. The resulting properties of the electric dipole moment and the vibrationalspectra are analyzed with varying field strength. PACS numbers: 31.50.-x, 33.20.Tp, 33.80.Rv
I. INTRODUCTION
Ultracold atomic and molecular few- and many-bodysystems offer a unique platform for a detailed under-standing and analysis of fundamental quantum proper-ties. The preparation and control of such systems inspecific quantum states offer many opportunities for ex-ploring elementary quantum dynamical processes. Ex-perimentally, one can control the external motion of theatoms by designing and switching between almost arbi-trarily shaped traps [1–3], and the strength of the in-teraction among the atoms can be tuned by magneticor optical Feshbach resonances [4–6]. A striking newspecies are the weakly bound ultralong-range diatomicmolecules composed of a ground state and a Rydbergatom whose existence has been predicted theoreticallymore than a decade ago [7] and which have been dis-covered experimentally only recently [8]. The molec-ular Born-Oppenheimer potential energy curves, whichare responsible for the atomic binding, show for thesespecies a very unusual oscillatory behavior with manylocal minima. The latter can be understood intuitivelyand modeled correspondingly as the interaction of a neu-tral ground state atom with the Rydberg electron of thesecond atom. In a first approximation, the interactionbetween the two constituents is described by a s -wavescattering dominated Fermi-pseudopotential [9, 10]. Theequilibrium distance for these molecular states is of theorder of the size of the Rydberg atom and the vibrationalbinding energies are in the MHz to GHz regime for prin-cipal quantum numbers n ≈ −
40 depending on thetype of states. More specifically, low-angular momen-tum non-polar states and large angular momentum polarstates, so-called trilobites, have been predicted [7].Theypossess electric dipole moments in the range of 1Debye(low- (cid:96) )[11] up to 1kDebye (high- (cid:96) )[7] in the polar case. The large electric dipole moment of the latter makes themaccessible for electric field manipulation which opens theinteresting possibility for the external control of molecu-lar degrees of freedom. Beyond the s -wave interactions, p -wave scattering has been shown [12] to lead to a class ofshape-resonance-induced long-range molecular Rydbergstates.The impact of magnetic fields on these ultralong-rangemolecules has been studied in [13] where it has in par-ticular been shown that the magnetic field provides anangular confinement turning a rotational degree of free-dom into a vibrational one and yields, with increas-ing strength, a monotonic lowering of the magnitude ofthe electric dipole moment. Polyatomic ultralong-rangemolecules formed of a Rydberg atom and several groundstate perturber, such as collinear triatomic species, canbe constructed by taking the diatomic wave function as abasic unit and constructing the corresponding symmetry-adapted orbitals [14]. Recently the formation of Rydbergtrimers and excited dimers bound by internal quantumreflection [15] have been observed experimentally and an-alyzed in detail theoretically [15]. Moreover, it it hasbeen shown how the electric field of a Rydberg electroncan bind a polar molecule to form a giant ultralong-rangestable triatomic molecule [16–18] which can consequentlybe controlled by applying external electric fields. Com-bining electric and magnetic fields in a crossed field con-figuration the existence and properties of so-called giantdipole ultralong-range molecules have been shown [19].Opposite to the above-mentioned Rydberg molecules thisspecies has no open radiative decay channels.In spite of the diversity of works focusing on ultralong-range molecules an original investigation of the impact ofexternal electric fields specifically on the polar trilobitestates is missing. Such an investigation is particularlydesirable due to the strong sensitivity of these Rydberg a r X i v : . [ phy s i c s . a t o m - ph ] J u l molecules with respect to the external field which pro-vides a handle on the control of their properties on asingle molecule basis but also for their interactions inpotential many-body systems. For these reasons we per-form in this work a study of the impact of an electricfield on the structure and dynamics of high- (cid:96) ultralong-range diatomic Rubidium molecules. We hereby proceedas follows. Section II provides a formulation of the prob-lem presenting the working Hamiltonian and a discus-sion of the underlying interactions. Our analysis goesbeyond the s -wave approximation and takes into accountthe next order p -wave term of the Fermi-pseudopotential.Section III and IV contain our methodology and a dis-cussion of the effects of the p -wave contribution, respec-tively. In section V we analyze the evolution of the topol-ogy of the potential energy surfaces (PES) with chang-ing electric field. The resulting PES show a stronglyoscillatory behavior with bound states in the MHz andGHz regime. With increasing field strength the diatomicmolecular equilibrium distance shifts substantially in arange of the order of thousand Bohr radii. We analyzethe behavior of the corresponding electric dipole momentthereby achieving molecular states with a dipole momentup to several kDebye. Based on these properties and the s -wave admixture via the external electric field a prepa-ration scheme for high- (cid:96) polar molecular electronic statesvia a two photon excitation process is presented. Finally,we provide an analysis of the vibrational spectra whichexhibit spacings of the order of several MHz. II. MOLECULAR HAMILTONIAN ANDINTERACTIONS
We consider a highly excited Rydberg atom interact-ing with a ground state neutral perturber atom (we willfocus on the Rb atom here) in a static and homoge-neous electric field. The Hamiltonian treating the Rbionic core and the neutral perturber as point particles isgiven by (if not stated otherwise, atomic units will beused throughout) H = P M + H el + V n,e ( r , R ) , (1) H el = H + Er , H = p m e + V l ( r ) , (2)where ( M, P , R ) denote the atomic Rb mass and the rel-ative momentum and position of the neutral perturberwith respect to the ionic core. ( m e , p , r ) indicate the cor-responding quantities for the Rydberg electron. The elec-tronic Hamiltonian H el consists of the field-free Hamil-tonian H and the usual Stark term of an electron in astatic external E -field. V l ( r ) is the angular momentum-dependent one-body pseudopotential felt by the valenceelectron when interacting with the ionic core. For low-lying angular momentum states the electron penetratesthe finite ionic Rb + -core which leads to a (cid:96) -dependenceof the interaction potential V l ( r ) due to polarization and scattering effects [20]. Throughout this work we choosethe direction of the field to coincide with the z-axis of thecoordinate system, i.e. E = E e z . Finally, the interatomicpotential V n , e for the low-energy scattering between theRydberg electron and the neutral perturber is describedas a so-called Fermi-pseudopotential V n,e ( r , R ) = 2 πA s [ k ( R )] δ ( r − R ) (3)+ 6 πA p [ k ( R )] ←−∇ δ ( r − R ) −→∇ . (4)Here we consider the triplet ( S = 1) scattering of the elec-tron from the spin- ground state alkali atom. Suppres-sion of singlet scattering events can be achieved by an ap-propriate preparation of the initial atomic gas. In eq. (4) A s [ k ( R )] = − tan( δ ( k )) /k and A p [ k ] = − tan( δ ( k )) /k denote the energy-dependent triplet s - and p -wave scat-tering lengths, respectively, which are evaluated from thecorresponding phase shifts δ l ( k ) , l = 0 ,
1. The kinetic en-ergy E kin = k / k / /R − / n . The behavior of theenergy-dependent phase shifts δ l as functions of the ki-netic energy E kin is shown in Fig. 1. III. METHODOLOGY
In order to solve the eigenvalue problem associatedwith the Hamiltonian (1) we adopt an adiabatic ansatzfor the electronic and heavy particle dynamics. We writethe total wave function as Ψ( r , R ) = ψ ( r ; R ) φ ( R ) andobtain within the adiabatic approximation[ H + Er + V n,e ( r , R )] ψ i ( r ; R ) = (cid:15) i ( R ) ψ i ( r ; R ) , (5)( P M + (cid:15) i ( R )) φ ik ( R ) = E ik φ ik ( R ) , (6)where ψ i describes the electronic molecular wave func-tion in the presence of the neutral perturber for a givenrelative position R and φ ik determines the rovibrationalstate of the perturber. To calculate the potential en-ergy surface (cid:15) i ( R ) we expand ψ ( r ; R ) in the eigenba-sis of H , i.e. ψ i ( r ; R ) = (cid:80) nlm C ( i ) nlm ( R ) χ nlm ( r ) with H χ nlm ( r ) = ε nl χ nlm ( r ) , χ nlm ( r ) = R nl ( r ) Y lm ( θ, φ ).For l ≥ l min = 3 we neglect all quantum defects, i.e. H is identical to the hydrogen problem. Finally, we have tosolve the following eigenvalue problem( ε nl − (cid:15) ( R )) C nlm + (cid:88) n (cid:48) l (cid:48) m (cid:48) C n (cid:48) l (cid:48) m (cid:48) ( E (cid:104) nlm | z | n (cid:48) l (cid:48) m (cid:105) δ mm (cid:48) + (cid:104) nlm | V n,e ( r , R ) | n (cid:48) l (cid:48) m (cid:48) (cid:105) ) = 0 , (7)for which we use standard numerical techniques for thediagonalization of hermitian matrices. Throughout thiswork we mainly focus on the high- (cid:96) n = 35 manifoldwhich provides the trilobite states in case of zero electricfield [7]. To ensure convergence we vary the number ofbasis states finally achieving a relative accuracy of 10 − for the energy. For the n = 35 trilobite manifold we used,in addition to the degenerate n = 35 , l ≥ s, d, p quantum defectsplit states due to their energetically closeness. This ba-sis set contains 1225 states in total.From eqs. (5) and (6) we already deduce some symme-try properties of the states ψ, φ and the energies (cid:15) . If P r , R , E denotes the generalized parity operator that trans-forms ( r , R , E ) → ( − r , − R , − E ) we have [ H, P r , R , E ] =[ V n , e ( r , R ) , P r , R , E ] = 0. This means that the states Ψ , ψ and φ are parity (anti)symmetric and the PES fulfill (cid:15) ( R ; E ) = (cid:15) ( − R ; − E ). In addition, the PES possess anazimuthal symmetry, e.g. the vector defining the inter-nuclear axis can, without loss of generality, be chosento lie in the x − z -plane. In the absence of any field,the PES depend exclusively on the internuclear distance R . However, if a field is present, the PES are cylindricalsymmetric, which means they also depend on the angleof inclination θ between the field vector and the internu-clear axis, e. g. (cid:15) ( R ) = (cid:15) ( R, θ ). IV. DISCUSSION OF THE P-WAVECONTRIBUTION
In several previous works the interaction between theRydberg electron and the neutral perturber has beenmodeled to consist exclusively of a s -wave scattering po-tential [7, 13]. In this case, a single potential curve splitsaway from the n − l min degenerate high- l manifold form-ing a strongly oscillating Born-Oppenheimer potentialenergy surface with a depth of around − . × GHz /n providing rovibrational states with a level spacing of ap-proximately 100MHz and a permanent electric dipole mo-ment of 1kDebye. However, it is an important fact thatthe p -wave scattering length A p possesses a resonanceat E kin = 24 . a for a Rydberg electron in a n = 35 state.The effect of the p -wave contribution and its resonantbehavior on adiabatic potential curves has been studiedin detail in several works, see [12, 21]. It has been shownthat due to this interaction additional PES split awayfrom the degenerate manifold leading to avoided cross-ings with the pure s -wave dominated potential curve andeven energetically lower s, p and d -states. These avoidedcrossings cause a dramatic change in the topology of thePES leading to novel molecular dynamics.According to [21, 22], for large radial distances theoverall behavior of s and p -wave scattering dominatedPES can be well described by the Borodin and Kazanskymodel [22] (cid:15) nl ( R ) = − n − δ l ( k ( R )) /π ) ≈ − n − δ l ( k ( R )) πn . (8)The energy curves are roughly determined by the corre-sponding phase shifts for the electron-neutral atom scat-tering, although the individual oscillations necessary forthe existence of stable molecular states cannot be de-scribed by the simple equation (8). It can be shown that FIG. 1. (Color online) Energy dependent triplet phase shifts δ and δ for e − − Rb(5 s ) scattering. For E kin = 24 . δ = π/
2, i.e. the (cubed) energy dependent p -wave scattering length A p ( k ) = − tan( δ ( k )) /k possesses aresonance at this energy. This can be clearly seen in the inset. the positions of the crossing point of both PES R ( n )c andthe position of the minimum of the s -wave dominatedPES R ( n )m obey the ratio R ( n )c R ( n )m = E (m) kin + 1 / n E (c) kin + 1 / n (9)with E (c)kin = 7 . E (m)kin = 7 . . ≤ R ( n )c /R ( n )m < n in the absence of the external field. Thismeans that the p -wave contribution is crucial for the over-all topology of the PES. V. POTENTIAL ENERGY SURFACES
The mechanism underlying the oscillating behaviourof the potential energy surfaces for ultralong-rangemolecules composed of a Rydberg atom plus neutralground state atom is the following. The neutral atomis to a good approximation point-like and its interactionwith the Rydberg atom probes the highly excited elec-tronic wave function locally in space, meaning that thehighly oscillatory character of the Rydberg wave functionis mapped onto the potential energy surface. This holdsboth for the absence and presence of an electric field.For the non-polar low-angular momentum states whichare states that are splitted from the (degenerate) hydro-genic manifold by a sizable quantum defect [7] the oscil-lations of the potential energy typically amount to manyMHz. The 38 s state shown in Fig. (3a) on a GHz scaleand in the corresponding inset (ii) enlarged on a MHzscale, is such a non-polar state. Its oscillatory behaviouris weaker than the oscillatory behaviour of the polar trilo-bite or p-wave states which are in the GHz regime, seealso Fig. (3a) (many bound vibrational states exist inboth cases). This is due to the energetical lowering inthe framework of the mixing of the many high-angularmomentum states available to form the trilobite state.Indeed, it has been shown in ref.([7]) that for a pure s -wave interaction potential of the Rydberg electron andthe neutral perturber the trilobite PES is given by (cid:15) ( R ) = − n + 12 A s [ k ( R )] n − (cid:88) l = l min (2 l + 1) R nl ( R ) , (10)where R nl denote the hydrogenic radial functions [7].Let us now focus on the case of the presence of anelectric field specifically on the regime E = 0 − Vm .The dissociation limits correspond to the atomic statesRb(5 s )+Rb( n = 35 , l ≥
3) where l is used as a label inthe presence of the electric field. In Fig. 2 we presentthe PES for the electrically dressed polar trilobite statefor E = 150 Vm and 300 Vm as a function θ and R . Asmentioned in section III in the absence of an electricfield the potential curves are independent of θ . For afinite field strength, this spherical symmetry is brokenwhich is clearly seen in Fig. 2. For all field strengthsthe potential minimum is taken for the antiparallel fieldconfiguration θ = π . This is reasonable because the ex-ternal electric fields forces the electron density to alignin its direction which leads to a higher density in thenegative z -direction. The electric field therefore turns arotational degree of freedom θ to a vibrational one. Asthe field strength increases a stronger confinement of theangular motion is achieved and the corresponding equi-librium distance R eq increases substantially.In Fig. (3a) we show intersections through the PES forthe 9th-15th excitation for a field strength of 300 Vm for θ = π . In addition we present two insets. Inset (i) in thisfigure shows the high- (cid:96) field-free potential curves. Dueto the p -wave interaction a single potential curves splitsaway from the degenerate manifold causing an avoidedcrossing in the region of R = 1400 a − a . In addi-tion, the inset also shows the potential curves providedby the Borodin-Kazansky approximation (see eq. (8)). Inthe main figure the lowest potential curve is the one be-longing to the 38 s quantum defect split state. This statepossesses a very weak oscillatory behavior in the MHzregime. This can be clearly seen in inset (ii) of Fig. (3a).In general, this state is much less affected by the elec-tric field compared to the PES arising from the zero fieldhigh- (cid:96) degenerate manifold. This is reasonable since theatomic 38s state does not possess a substantial electricdipole moment in the presence of the field. Therefore itspotential curve hardly shifts with increasing electric fieldstrength from its field-free value of − . n =35 , l ≥ FIG. 2. (Color online) Two-dimensional potential energy sur-faces for the electrically dressed polar trilobite states for E =150 Vm (a) and 300 Vm (b). We observe a potential minimumat θ = π . An increase of the electric field goes along with astronger confinement of the angular motion and an increase ofthe diatomic equilibrium distance R eq . Thus, the electric fieldstabilizes the s -wave dominated molecular states. in Fig. (3a)) we obtain potential curves with a stronglyoscillatory structure in the many hundred MHz to GHzregime. It is important to note that the p -wave interac-tion dominated PES (red curve in Fig. (3a)) and the s -wave interaction dominated PES (blue curve in Fig. (3a))exhibit an avoided crossing which is crucial for the stabil-ity of the corresponding vibrational states. For zero fieldthis avoided crossing happens to be comparatively closeto the global minimum of the corresponding ’trilobite’PES (see (i) in Fig. (3a)) and might influence extendedvibrational states. For increasing field strength howeverthis avoided crossing is increasingly separated from theglobal equilibrium distance, as we shall discuss in moredetail below.In Fig. (3b) we show intersections for θ = π through thePES of the electrically dressed polar states for differentfield strengths E = 150 , , and 450 Vm . We observehow the potential curve is globally shifted with increas-ing electric field strength. For E ≥ Vm (not shown inFigure) this trilobite PES experiences avoided crossingswith the potential curve belonging to the 38 s state. Inaddition, we see that the overall topology of the PES donot change with varying E -field. In particular, the num-ber of minima and their positions remain approximatelyconstant with increasing field strength. However, thediatomic equilibrium distance R eq (which is the globalminimum in the range 1550 a ≤ R ≤ a ) changesstrongly as E varies. This means that the electric fieldcauses a ’spatial weight’ to the PES. In contrast, theregion of avoided crossing of the s -/ p -wave dominatedpotential curves is hardly affected by the applied electricfield and remains in the interval 1500 a to 1700 a . Thismeans that low-lying vibrational molecular states in thewell around the global minimum are shifted away fromthe region of the avoided crossing. As a consequence theimportance of the p -wave interaction is decreased signif-icantly for higher field strength and the PES are deter-mined mainly by the s -wave interaction. To be morespecific we show the dependence of R eq as a function of E in the inset in Fig. 4. We see a plateau-like structurewith steps at the field strengths 100 , , Vm wherethe value of R eq sharply changes. This structure sim-ply reflects the depicted effect of the electric field on thePES, i.e. by varying the electric field one changes theenergetically position of the different potential wells inthe oscillating PES, which leads to abrupt changes of theglobal equilibrium position R eq . Figs. (3a,b) also demon-strate that with increasing field strength the avoidedcrossing between and the s -wave interaction dominatedstates remains (approximately) localized in coordinatespace whereas the energetically low-lying potential wellswith bound vibrational states and in particular the onebelonging to the global equilibrium position are loweringin energy and are consequently well-separated from thisavoided crossing. In conclusion, the electric field repre-sents an excellent tool to control the energetic positionsand depths of the individual wells and to avoid destabi-lizing avoided crossing. VI. ELECTRIC DIPOLE MOMENT
In ref.[7] the authors reported on large electric dipolemoments of ultralong-range polar Rydberg molecules ofthe order of kDebye. The zero-field permanent dipolemoment for these species scales according to the semi-classical expression D el = R eq − n /
2. In Fig. 4 we showthe absolute value of the electric dipole moment along
FIG. 3. (Color online) (a) Intersections through the two-dimensional PES for θ = π for the 9th-15th excitation for E = 300 Vm . For the two lowest PES arising from the high- (cid:96) de-generate manifold a strongly oscillatory behavior is visible. Theregion of avoided crossing is clearly visible at R ∼ − a .In addition, we present the potential curves provided by theBorodin-Kazansky model given by eq. (8). The inset (i) showsthe field-free trilobite and first p -wave PES. The inset (ii) showsthe 38 s split PES which is oscillatory in the MHz regime andhardly affected by the electric field. (b) Same as in (a) but withvarying E = 150 ,
300 and 450 Vm . The diatomic equilibrium dis-tance R eq is moving away from the region of the avoided crossingat R = 1500 a − a . the z -axis as a function of the field strength D el ( E ) = | (cid:90) d r ψ ∗ ( r ; R eq , E ) zψ ( r ; R eq , E ) | = (cid:114) π | (cid:88) nn (cid:48) ll (cid:48) m C ∗ n (cid:48) l (cid:48) m C ∗ nlm (cid:90) drr R n (cid:48) l (cid:48) ( r ) R nl ( r ) × (cid:90) d Ω Y ( ϑ, ϕ ) Y ∗ l (cid:48) m ( ϑ, ϕ ) Y lm ( ϑ, ϕ ) | . (11)The integration over the angular degrees of freedomprovides ∆ l = ± D el increases up tovalues of around 4kDebye. As for R eq we see a sharpstep structure, i.e. for field strengths at approximately100 , , Vm its values suddenly increase in stepsof roughly 500 Debye. In Fig. 4 we also show a com-parison between the exact result calculated accordingto eq. (11) (blue data points) and the semi-classicalapproximation (green data points). For low electricfields the agreement is quite well, but differs withincreasing field strength up to a deviation of around10%. The semiclassical result therefore certainly allowsfor a qualitative description of the behavior of D el .For E > Vm we find an unexpected decrease of FIG. 4. (Color online) The electric dipole moment as a functionof the electric field E (blue points). For comparison we showa semi-classical prediction (green points). The inset shows thebehavior of the equilibrium distance R eq with varying electricfield strength. D el . This feature can be understood if one analyzes thefield-dependent spectrum of coefficients for the electroniceigenvector ψ ( r ; R eq , E ) = (cid:80) i C i ( E ) χ i ( r ). In Fig. 5 weshow the distribution | C i | for E = 300 Vm and 600 Vm .For E = 300 Vm the spectrum is dominated by basisstates from the n = 35 , l ≥ i = 1217 , ..., E = 600 Vm thesituation has changed in the sense that now the maincontribution is provided by the 38 s state. This canbe understood by the fact that the considered PES isapproaching the 38 s PES with increasing field strength.The latter is however barely affected by the electricfield. For E = 600 Vm the PES involve avoided crossingswhich causes the high- (cid:96) dressed trilobite PES to acquirea major contribution from the 38 s state. This finiteadmixture has two important consequences: • Due to the ∆ l = ± s stateonly acquires a contribution to the integral (11)via the 37 p state. However, the coefficient of thelatter state is negligibly small. This causes thedecrease of D el for large field strengths as seen inFig. 4. • The finite 38 s admixture provides us with the pos-sibility to prepare high- (cid:96) Rydberg molecules via atwo-photon process. This goes beyond the three-photon preparation scheme suggested in [7] ( l min =3 for the field-free case). For E ≥ Vm the trilo-bite state acquires a major l = 0 contributionwhich makes it accessible for a two-photon tran-sition scheme. The same mechanism has been re-ported recently in the analysis of ultralong-rangepolyatomic Rydberg molecules formed by a polarperturber [17]. Field-free high- (cid:96) molecular statescan then in principle be accessed via an additionaladiabatic switching of the electric field back to thezero value. VII. ROVIBRATIONAL STATES
Because of the azimuthal symmetry of the PESwe introduce cylindric coordinates ( ρ, Z, φ ) for theirparametrization (cid:15) ( R ) = (cid:15) ( ρ, Z ). For the rovibrationalwavefunctions we choose the following ansatz φ kνm ( R ) = F kνm ( ρ, Z ) √ ρ exp( imϕ ) , m ∈ Z , ν ∈ N . (12)With this we can write the rovibrational Hamiltonian ineq. (6) as H rv = − M ( ∂ ρ + ∂ Z ) + m − / M ρ + (cid:15) ( ρ, Z ) . (13)We solved the corresponding Schr¨odinger equation fordifferent azimuthal quantum numbers m using a fourthorder finite difference method.In Fig. 6 we provide the energies of the eleven lowest vi-brational ( m = 0) states living in the trilobite PES forvarying field strength. In order to obtain a normalizedview of the spectrum the corresponding energy of theminimum of the PES has been subtracted. In generalwe observe a slight increase of the level spacing with in-creasing field strength. The increase is due to the an en-hanced angular confinement of the rovibrational motionfor strong fields. For E = 100 ,
200 and 385 Vm however weencounter a dip in the rovibrational level spacing. Thelatter corresponds to the case of crossover of the equilib-rium positions between neighboring wells and thereforean accompanying relocation of the corresponding rovi-brational wavefunctions. This leads to enhanced tun-neling probabilities between neighboring wells and there-fore an increased level density. In the inset of Fig. 6 we FIG. 5. (Color online) Spectrum of coefficients of the electroniceigenvector ψ ( r ; R eq , E ) at E = 300 Vm (a) and 600 Vm (b). Forincreasing field strength the eigenstates gain a finite admixtureof the quantum defect split states. For 600 Vm we clearly see amajor contribution provided by the 38 s state. show the offset corrected potential curves for E = 300 Vm (blue curve) and E = 380 Vm (green curve) ( θ = π ). For E = 300 Vm the potential curve possesses a global mini-mum at R = 1939 a and two local minima at R = 1750 a and 2182 a with an offset of 200MHz. Bound states inthe middle well with energies higher than 200MHz cantunnel into these wells, whereby their level spacing is re-duced. For increasing field strengths the right potentialwell is shifted downwards. This enhances the tunnelingprobabilities of states with energies less than 200MHz,which correspondingly leads to a denser spectrum.In Fig. (7a,b) we present (scaled) probability densities | F kνm ( ρ, z ) | for m = 0 for the vibrational ground state( ν = 0) and the second excitation for E = 300 Vm .The equilibrium distance for the PES is located at Z = − a , ρ = 0. The m = 0 ground statedistribution is characterized by a deformed Gaussian profile that is localized at Z = − a and ρ =72 a . In Z, ρ -direction the density distribution pos-sesses an extension of approximately 50 a and 100 a ,respectively. The density profile for the second excita-tion ( ν = 2) shows three separate Gaussian like den-sity peaks with increasing intensity located at ( Z, ρ ) =( − a , a ) , ( − a , a ) and ( − a , a )with an extension of around (25 a , 30 a ), (25 a , a )and (75 a , a ) in the Z, ρ -directions, respectively.
FIG. 6. (Color online) Shown are the eleven lowest vibrationalenergies as a function of the field strength E . The dips around E = 100 ,
200 and 385 Vm are caused by the change of potentialwells determining the diatomic equilibrium distance R eq . In theinset we show the offset corrected potential curves for E = 300 Vm and E = 380 Vm ( θ = π ). For E = 300 Vm bound states in themiddle well with energies larger than 200MHz can tunnel intothe neighbored potential wells. This causes a reduction of theirlevel spacings. For E = 380 Vm we nearly get a double potentialwell and states with energies less than 200MHz possess a highertunneling probability. Correspondingly, this leads to a denserspectrum. VIII. CONCLUSIONS
The recent spectacular experiments [8, 11, 15] prepar-ing, detecting and probing some of the important proper-ties of non-polar ultralong-range Rydberg molecules haveopened the doorway towards a plethora of possibilities tocreate new exotic species where atoms, molecules or evenclusters and mesoscopic quantum objects might be boundto electronic Rydberg systems. It is therefore of crucialimportance to learn how the properties of these Rydbergmolecules can be tuned finally leading to a control of thestructure and potentially dynamics of these systems. Theprimary choice are here external fields in particular dueto the susceptibility of the weakly bound Rydberg elec-trons. In the present work we have therefore exploredthe changes the polar high angular momentum trilobite
FIG. 7. (Color online) Scaled probability densities | F kνm ( ρ, z ) | for rovibrational wavefunctions. Both wavefuntions belong tothe trilobite PES for E = 300 Vm with an azimuthal quan-tum number m = 0. In (a) we observe a deformed Gaus-sian like density profile for the groundstate ( ν = 0) centeredat Z = − a and ρ = 72 a . In Z / ρ -direction the den-sity distribution has an extension of approximately 50 a /100 a .In (b) we show the density profile for the second excited state( ν = 2). This density profile provided three peaks at ( Z, ρ ) =( − a , a ) , ( − a , a ) and ( − a , a ). states experience if they are exposed to an electric fieldof varying strength. Taking into account s - and p -waveinteractions it turns out that the electric field providesus with a unique know to control the topology of the adi-abatic potential energy surface. First of all, the angulardegree of freedom between the electric field and internu-clear axis is converted from a rotational to a vibrationaldegree of freedom thereby rendering the field-free poten-tial energy curve into a two-dimensional potential energysurface. It turns out that the global equilibrium position is always the antiparallel configuration of these two axes.The sequence of potential wells with increasing radialcoordinate, i.e. the oscillatory behavior of the potentialis changed dramatically in the presence of the field. Inparticular we encounter an overall lowering of the en-ergy accompanied by a subsequently crossover of the en-ergetically order of the individual wells. Consequently,the equilibrium distance and the lowest vibrational statesare systematically shifted to larger internuclear distances.The p -wave split state which, due to its resonant behav-ior, lowers dramatically in energy with decreasing inter-nuclear distance and therefore crosses the polar trilobitestate close to its equilibrium distance in the zero-fieldcase, can now with increasing field strength be systemat-ically shifted away from the energetically lowering equi-librium distance and corresponding well. In such a waythe respective stability of the ground and many excitedvibrational states of the polar trilobite state is guaran-teed. For strong fields the interaction of the latter statewith non-polar (quantum defect split) states, which arevery weakly polarized in the presence of the field, leads toa strong admixture of, in our specific case, s -wave char-acter to the polar high angular momentum states. Asa consequence, a two-photon excitation process startingfrom the ground state of the two-atom system should besufficient to efficiently excite these states and probe theircharacter. The electric dipole moment, which is steadilyincreasing with increasing electric field strength startingfrom zero-field, does, due to the above admixture, de-crease in the strong field regime.To obtain an even richer topology of the potential energysurfaces of the trilobite states the combination of staticelectric and magnetic field would be the next step. In thiscase both rotational degrees of freedom in field-free spacewill, in general, turn into vibrational modes rendering thefield-free potential energy curve a three-dimensional po-tential energy surface. Depending on the configuration,such as parallel or crossed fields, the remaining symme-tries might even lead to controllable crossings or avoidedcrossings of the surfaces which will be the subject of afuture investigation. IX. ACKNOWLEDGMENT
We thank the Initial Training Network COHERENCEof the European Union FP7 framework for financial sup-port. In addition, we thank Michael Mayle and IgorLesanovsky for helpful discussions and suggestions. Oneof the authors (P.S.) acknowledges the hospitality andmany fruitful discussions in particular with H.R. Sadegh-pour at the Institute for Theoretical Atomic Molecularand Optical Physics at the Harvard Smithsonian Centerfor Astrophysics in Cambridge, USA. [1] C.J. Pethick and H. Smith, Bose-Einstein Condensationin Dilute Gases, Cambridge University Press (2008)[2] R. Grimm, M. Weidem¨uler, and Y.B. Ovchinnikov, Adv.At. Mol. Opt. Phys. , 95 (2000)[3] R. Folman, P. Kr¨uger, J. Schmiedmayer, J. Denschlagand C. Henkel, Adv. At. Mol. Opt. Phys. , 263 (2002)[4] T. K¨ohler, K. G´oral, and P.S. Julienne, Rev. Mod. Phys. , 1311 (2006)[5] I. Bloch, J. Dalibard, and W. Zwerger, Rev. Mod. Phys. , 885 (2008)[6] C. Chin, R. Grimm, P.S. Julienne, E. Tiesinga, Rev.Mod. Phys. , 1225 (2010)[7] C. H. Greene, A.S. Dickinson, and H.R. Sadeghpour,Phys. Rev. Lett. , 2458 (2000)[8] V. Bendkowsky, B. Butscher, J. Nipper, J.P. Shaffer, R.L¨ow, and T. Pfau, Nature , 1005 (2009)[9] E. Fermi, Nuovo Cimento, , 157 (1934)[10] A. Omont, J. Phys. (Paris) , 1343 (2011)[11] W. Li, T. Pohl, J. M. Rost, S.T. Rittenhouse, H.R.Sadeghpour, J. Nipper, B. Butscher, J.B. Balewski, V.Bendkowsky, R. L¨ow, and T. Pfau, Science , 1110(2011)[12] E.L. Hamilton, C.H. Greene and H.R. Sadeghpour,J.Phys.B , L199 (2002) [13] I. Lesanovsky, H.R. Sadeghpour, and P. Schmelcher, J.Phys. B , L69 (2006)[14] I.C.H. Liu and J.M. Rost, Eur.Phys.J. D , 65 (2006)[15] V. Bendkowsky et al , Phys. Rev. Lett. , 163201(2010)[16] S.T. Rittenhouse, H.R. Sadeghpour, Phys. Rev. Lett. , 243002 (2010)[17] S.T. Rittenhouse, M. Mayle, P. Schmelcher and H.R.Sadeghpour, J. Phys. B , 184005 (2011)[18] M. Mayle, S.T. Rittenhouse, P. Schmelcher and H.R.Sadeghpour, Phys.Rev.A , 43001 (2012)[20] T.F. Gallagher, Rydberg Atoms , Cambridge Monographson Atomic, Molecular and Chemical Physics, 1994[21] A.A. Khuskivadze, M.I. Chibisov, I.I. Fabrikant, Phys.Rev. A, , 042709 (2002)[22] V.M. Borodin, A.K. Kazansky, J. Phys. B , 971 (1992)[23] S. Jochim, M. Bartenstein, A. Altmeyer, G. Hendl, S.Riedl, C. Chin, J. Hecker-Denschlag, and R. Grimm, Sci-ence , 2101 (2003)[24] S. Knoop, F. Ferlaino, M. Mark, M. Berninger, H.Sch¨obel, H.-C. N¨agerl, and R. Grimm, Nature Physics5