Long-range Rydberg-atom-ion molecules of Rb and Cs
LLong-range Rydberg-atom-ion molecules of Rb and Cs
A. Duspayev, ∗ X. Han,
1, 2, 3
M.A. Viray, L. Ma, J. Zhao,
2, 4 and G. Raithel Department of Physics, University of Michigan, Ann Arbor, MI 48109, USA State Key Laboratory of Quantum Optics and Quantum Optics Devices,Institute of Laser Spectroscopy, Shanxi University, Taiyuan 030006, People’s Republic of China Department of Physics, Taiyuan Normal University, Jinzhong 030619, People’s Republic of China Collaborative Innovation Center of Extreme Optics,Shanxi University, Taiyuan 030006, People’s Republic of China (Dated: January 14, 2021)We propose a novel type of Rydberg dimer, consisting of a Rydberg-state atom bound to a dis-tant positive ion. The molecule is formed through long-range electric-multipole interaction betweenthe Rydberg atom and the point-like ion. We present potential energy curves (PECs) that areasymptotically connected with Rydberg nP - or nD -states of rubidium or cesium. The PECs ex-hibit deep, long-range wells which support many vibrational states of Rydberg-atom-ion molecules(RAIMs). We consider photo-association of RAIMs in both the weak and the strong optical-couplingregimes between launch and Rydberg states of the neutral atom. Experimental considerations forthe realization of RAIMs are discussed. I. INTRODUCTION
Ultralong-range Rydberg molecules (ULRM) [1, 2]have become an active field of research ever since theywere first predicted [3, 4] and experimentally observed [5,6]. There are two general types of ULRM. The first one isa bound state between ground-state and Rydberg-excitedatoms [5, 7–21]. Its binding mechanism is a result of at-tractive interaction induced due to the scattering of aRydberg electron from an atom with negative scatteringlength, and is often described by the Fermi pseudopoten-tial model [3, 22]. They exhibit a wide range of exoticproperties such as macroscopic bond length, permanentelectric dipole moment in the kilo-Debye range and un-usual electronic probability densities [7–9]. The secondtype of ULRM is formed by a pair of atoms that are bothexcited to Rydberg states, often referred to as Rydbergmacrodimers [4, 6, 23–32]. In this case, the binding mech-anism arises from the multipolar electrostatic interactionbetween the two atoms.While Rydberg-ground and Rydberg-Rydbergmolecules have been of great interest, experimental im-plementation of hybrid atom-ion systems [33, 34] also hasled into opportunities to realize a different type of ULRM- one formed due to an interaction between a cold Ryd-berg atom and a trapped ion. Hybrid atom-ion systemsare generally attractive for novel quantum chemistry atultracold temperatures [35–37], many-body physics [38–40], quantum sensing and control [41–43] as well asquantum computing and simulations [44–46]. However,atom-ion collisions can lead to depletion of the ultracoldatomic ensemble, micromotion-induced collisions, chargetransfer and other unwanted effects [47–49]. Amongmethods to prevent these processes, a tight confinementof atoms and ions in individual traps at sufficiently ∗ [email protected] R r e Rydberg electronneutral atomionx z (a)(b) (c) H -5/2 H -3/2 H -1/2 H H H H g Ω -5/2 Ω -3/2 Ω -1/2 Ω Ω Ω H= m J = -5/2 -3/2 -1/2 1/2 3/2 5/2-3/2 -1/2 1/2 3/2 λ g Ω H -5/2 H g m g = H - molecular coupling Ω - optical coupling H -3/2 H -1/2 H H H FIG. 1. (Color online) (a) The Rydberg-atom-ion molecule(RAIM) under consideration. The molecular quantizationaxis and the internuclear distance R between the ion andthe neutral atom are both along the z -axis. The relative po-sition of the Rydberg electron in the neutral atom is r e . (b)A block-matrix visualization of the Hamiltonian in Eq 1. (c)Energy-level schematic of the RAIM and optical couplings toa J = 3 / large distances [50] and optical shielding [51, 52] havebeen suggested. The latter method is based on excitingneutral atoms colliding with ions into certain Rydbergstates that become repelled. The method also lendsitself to the preparation of Rydberg-atom-ion molecules(RAIMs).Here we investigate RAIMs between Rydberg states ofrubidium and cesium atoms and point-like positive ions.Similar to the aforementioned Rydberg macrodimers, theadiabatic potential energy curves (PECs) of RAIMs arise a r X i v : . [ phy s i c s . a t o m - ph ] J a n from non-perturbative solutions of the atom-ion interac-tion Hamiltonian. Bound vibrational states exist withinsufficiently deep wells in PECs that are not coupled toun-bound PECs. After outlining the theoretical model inSec. II, in Sec. III we calculate the PECs of the atom-ionsystems and identify series of bound vibrational states ofRAIMs near the intercepts between PECs that asymptot-ically connect with nP J Rydberg states and with hydro-genic states (states with near-zero quantum defect). Weassess the convergence of the PECs as a function of thehighest Rydberg-atom multipole order included in thecalculation. We study RAIM systems in cases of bothweak and strong optical coupling between launch andRydberg states in the neutral atom. The launch state canbe the atomic ground state or another low-lying interme-diate atomic state. Implications regarding spectroscopicmeasurement of RAIMs and coherent vibrational dynam-ics are discussed, as well as the context with optical-shielding applications [51, 52]. While RAIMs exist overa wider range of n , here we show results for Rb and CsRAIMs with Rydberg levels between n = 45 and 55, be-cause these levels are easily laser-excitable and their ion-ization electric fields [53], ∼
50 V/cm, are low enough forconvenient RAIM detection. Also, their RAIM bondinglengths and vibrational frequencies are conducive to ex-perimental study. In Sec. IV we discuss RAIMs fromseveral experimental points of view. The paper is con-cluded in Sec. V.
II. THEORETICAL MODELA. Hamiltonian and potential energy curves
The system under consideration is shown in Fig. 1(a).We consider a Rb or Cs Rydberg atom near a positive ionat an internuclear separation R . The molecular quan-tization axis ˆ z is chosen to point along R . The rela-tive position of the valence electron is r e = ( r e , θ e , φ e ).The internuclear distance R is larger than the radiusof the Rydberg atom, such that dissociation througha process Rb ∗ n> Rb + −→ Rb ∗ n< +Rb + is avoided. Thisis similar to the Le Roy radius condition [54] in Ryd-berg macrodimers [4]. Also, there is no radiation retar-dation effect [55], because the energy differences in theRydberg-atom Hilbert space correspond with transitionwavelengths much larger than R . Further, the ion is con-sidered as a positive point charge without relevant inter-nal substructure. This approximation is applicable forions such as a Rb + or Cs + , in their electronic groundstate. In these cases, level shifts of the atom-ion pair dueto dipolar ionic polarization at distances (cid:38) µ m, con-sidered here, range in the sub-Hz regime, rendering ionpolarization effects negligible. Since the ion has no inter-nal substructure, the Hilbert space is that of one Rydbergatom, as opposed to a product space of two such atoms(as in Rydberg-Rydberg molecules [4, 6, 23–32, 56]).The Rydberg-state Hilbert-space used is {| λ, m J (cid:105)} , where the index λ is a shorthand for the n , (cid:96) and J quantum numbers of the Rydberg base kets. The mag-netic quantum number m J is defined in the molecularreference frame. In the absence of strong optical cou-pling, m J is conserved due to azimuthal symmetry. Thetotal Hamiltonian of the neutral atom - ion system thenis, in the molecular frame,ˆ H ( R ) = ˆ H g + (cid:88) m J (cid:16) ˆ H m J ( R ) + (cid:110) ˆΩ m J ,g + c.c. (cid:111)(cid:17) . (1)Here, ˆ H g is the Hamiltonian of the neutral atom in thelaunch-state subspace {| λ g , m g (cid:105)} . There, λ g denotes thefixed quantum numbers n g , (cid:96) g and J g of the launch state,which is treated as an effective ground state, and m g itsmagnetic quantum number in the lab (laser) frame. Thelaunch-state subspace includes all m g = − J g , ..., J g . Thesum in Eq. 1 is taken over Rydberg-atom Hamiltoniansˆ H m J , which act on Rydberg-atom subspaces {| λ, m J (cid:105)} .Each Rydberg-atom subspace has a fixed value of m J , inthe molecular frame, and a range of λ ’s covering all Ryd-berg states with that m J and within a specified energyinterval. The optical couplings between the launch-statelevels and Rydberg states are introduced via the oper-ators ˆΩ m J ,g , the matrix elements of which are opticalexcitation Rabi frequencies between Rydberg subspace {| λ, m J (cid:105)} and launch-state subspace {| λ g , m g (cid:105)} . TheRabi frequencies depend on magnitude and polarizationof the field that couples launch-state and Rydberg mani-folds, and on the geometry of the excitation laser relativeto the molecular reference frame.In our calculations we use launch states 5 D / , 5 P / or 5 S / for Rb, and 6 D / or 6 S / for Cs, which all al-low optical excitation of field-mixed Rydberg nP J states.The hyperfine structure of the launch state is neglected,because it does not add additional insight to the physicspresented in this paper. Also, ˆ H g does not depend on R ,because the effect of the ion electric field on our launchstates is negligible. The optically coupled Rydberg-atomsubspaces {| λ, m J (cid:105)} included in Eq. 1 cover the range m J = − J g − , ..., J g + 1. The corresponding ˆ H m J canbe written as:ˆ H m J ( R ) = ˆ H Ry,m J + V int,m J ( ˆr e ; R ) , (2)where ˆ H Ry,m J is the unperturbed Hamiltonian of themanifold of Rydberg states with magnetic quantum num-ber m J in the molecular frame. For the quantum de-fects of the atomic energies and the fine structure cou-pling constants in ˆ H Ry,m J we use previously publishedvalues [53, 57]. The operator ˆ V int,m J denotes the multi-pole interaction between the Rydberg atom and the ion,which is m J -conserving and is described below.In Fig. 1(b), we show a block-matrix representationof the Hamiltonian in Eq. 1. The structure of theHilbert space is visualized with the energy level diagramin Fig. 1(c). The block-diagonal structure evident inFig. 1(b) is due to the azimuthal symmetry of the multi-polar interaction between the Rydberg electron and theion within the molecular reference frame.The PECs without optical coupling or with weak opti-cal coupling follow from diagonalizations of Eq. 2 withinthe subspaces {| λ, m J (cid:105)} for fixed m J . The procedure isperformed on a dense grid of the internuclear separation R , and for all desired choices of m J . The interactionˆ V int,m J is [29, 56, 58] (in atomic units): V int,m J ( ˆr e ; R ) = − l max (cid:88) l =1 (cid:114) π l + 1 ˆ r le R l +1 Y l (ˆ θ e , ˆ φ e ) (3)Here l is the multipole order of the Rydberg atom, the po-sition operator ˆr e is the relative position of the Rydbergelectron in the neutral atom, and Y l (ˆ θ e , ˆ φ e ) are spheri-cal harmonics that depend on the angular position of theRydberg electron. The basis is truncated in energy bysetting lower and upper limits of the effective quantumnumber, n min ≤ n eff ≤ n max , typically covering a rangeof n max − n min between 5 and 10. The interaction in Eq. 3conserves m J , with representations in different subspaces {| λ, m J (cid:105) (cid:12)(cid:12) m J = const } being different (which is why weuse a subscript m J on ˆ V int,m J ). The adiabatic molec-ular states of the RAIMs without optical coupling are | k, m J (cid:105) = (cid:80) λ c λ,m J ,k | λ, m J (cid:105) , with conserved m J , andthe index k serving as a counter for PECs of the RAIMstates.The sum over the multipole order l in Eq. 3starts at 1, because the Rydberg atom has no elec-tric monopole moment, and is truncated at a maxi-mal order l max . The selection rules for the matrix el-ements (cid:104) n (cid:48) , (cid:96) (cid:48) , J (cid:48) , m (cid:48) J | ˆ V int,m J | n, (cid:96), J, m J (cid:105) are | (cid:96) (cid:48) − (cid:96) | = l max , l max − , ..., | J (cid:48) − J | ≤ l max and m (cid:48) J = m J . Since all atomic states with (cid:96) < n are in-cluded in the Hilbert space anywayS, a larger l max onlyincreases the number of non-zero interaction matrix ele-ments, but does not increase the dimension of the statespace. Here we use l max = 6, which is sufficient for con-vergence at a practical level.Since the quantization axis is given by the internuclearseparation vector, mixing of different m J states only oc-curs via coherent, laser-induced electric-dipole couplingthrough the launch-state subspace {| λ g , m g (cid:105)} . The op-tical coupling matrix elements are contained in the off-diagonal Ω m J -blocks in Fig. 1(b) and are visualized byarrows in Fig. 1(c). In the following Sub-sections, weconsider the distinct cases of weak and strong opticalcoupling.In the regime of weak optical coupling, discussed inSecs. II B (theory) and III B (results), the Rabi frequen-cies Ω between launch-state levels and Rydberg states arenegligible against Rydberg- and launch-state decay (andpossibly other sources of decoherence). In this case, co-herences and optically-induced splittings between PECsand RAIM states with different m J are negligible. Thisregime corresponds to conditions suitable for spectro-scopic measurements aimed at experimental observationof RAIMs and resolving their vibrational states.In the regime of strong optical coupling, discussed inSecs. II C (theory) and III D (results), laser intensities are high enough that the optically induced couplings betweenmanifolds of Rydberg states with different m J are notnegligible, or may even become dominant at certain inter-nuclear separations R and molecular energies. One thenfinds modifications of the PECs and changes in the molec-ular dynamics. The strong-optical-coupling regime cov-ers applications such as optical shielding [51, 52]. Also,the study of coherent vibrational wavepackets can requirelaser excitation pulses that are in that regime.It is, generally, noted that the multipolar interaction ofa Rydberg atom with an ion is less complex than that be-tween a Rydberg-atom pair [29, 56, 58]. Since ionic multi-poles beyond the monopole are negligible, the calculationresembles that of a Stark-effect calculation for a Rydbergatom in an electric field, with the addition of a large num-ber of non-dipolar matrix elements that account for theinteraction of the atom with the inhomogeneity of the ionelectric field. The Hilbert space remains, however, a rel-atively small single-atom space, as opposed to a productspace of two atoms [29, 56, 58]. As a consequence, in thetheory of RAIMs there are no significant concerns abouterrors caused by basis-size truncation. B. Weak optical coupling
If the elements of the six Ω m J -blocks in Fig. 1(b),which represent the optical Rabi frequencies from thelaunch state, are lower than the decay- and decoher-ence rates of the system, we separately diagonalize thesix H m J -blocks in Fig. 1(b) to obtain the PECs andadiabatic molecular states for the desired m J -value(s).The calculation yields the PEC energies, W k,m J ( R ), ofthe molecular states | k, m J (cid:105) , and their excitation rates, T k,m J . The latter are calculated using Fermi’s goldenrule: T k,m J ( R ) = 2 π (cid:126) J g (cid:88) m g = − J g ρ p ( m (cid:48) J ) S ( k, m J , m g ) , (4)which depend on the PEC indices k and m J , the internu-clear separation R , and other quantities as follows. Weassume that the laser bandwidth is the dominant broad-ening source. In this case, the energy density of states, ρ , is approximately given by the inverse of the FWHMenergy bandwidth of the excitation laser (for all PECs).The p ( m g ) are statistical weights for the atom popula-tions in the magnetic sublevels of the launch state, m g ,which are given in the lab (excitation-laser) frame of ref-erence. The signal strengths S ( k, m J , m g ) are related tothe optical transition matrix elements A ( k, m J , m g , θ ) = (cid:68) k, m J (cid:12)(cid:12)(cid:12) ˆ D y ( θ ) ˆΩ (cid:12)(cid:12)(cid:12) λ g , m g (cid:69) . (5)Here, ˆ D y ( θ ) is the operator used for rotating from thelaboratory frame, defined by the excitation-laser geome-try, into the molecular frame, defined by the internuclearaxis. Eq. 5 can be explicitly written as: A = (cid:88) λ (cid:88) ˜ m c ∗ λ,m J ,k d ( J ) m J , ˜ m ( θ ) (cid:104) λ, ˜ m | E L ˆ (cid:15) · ˆ r e | λ g , m g (cid:105) (6)with the excitation-laser electric field, E L , and its polar-ization vector in the lab frame, ˆ (cid:15) . The first sum is goingover the base Rydberg states | λ, m J (cid:105) , with the shorthand λ for the Rydberg-state quantum numbers n , (cid:96) and J ,and the second over a Rydberg-state magnetic quantumnumber ˜ m in the lab frame. The electric-dipole matrixelement is between a Rydberg state | λ, ˜ m (cid:105) and a launch-state, | λ g , m g (cid:105) , with both magnetic quantum numbers ˜ m and m g in the lab frame. The rotation from the lab intothe molecular frame is affected by elements of reducedWigner rotation matrices, d ( J ) m J , ˜ m .The optical transition matrix elements A ( k, m J , m g , θ )given by Eq. 6 depend on θ through the rotation opera-tor matrix elements d ( J ) m J , ˜ m ( θ ), which rotate the opticallyexcited Rydberg level | λ, ˜ m (cid:105) from the lab into the molecu-lar frame. The excitation signal strength S ( k, m J , m g ) ofthe PEC for | k, m J (cid:105) , from the launch-state level | λ g , m g (cid:105) then is a weighted average of Eq. 6 over the alignmentangle θ of the molecules relative to the lab frame: S ( k, m J , m g ) ∝ (cid:90) π P ( θ ) | A ( k, m J , m g , θ ) | d θ (7)where P ( θ ) is the probability weighting function. For anisotropic atomic sample with random molecular align-ment: P ( θ ) = sin( θ )2 . (8)In the presented excitation signal-strength calculationsfor the case of weak optical coupling, we assume a sam-ple with random molecular alignment, and with equalpopulations of p ( m g ) = 1 / (2 J g + 1) in all magnetic sub-levels of the launch state. Equations (4) to (8) yield theexcitation rates T k,m J ( R ) of the PECs for | k, m J (cid:105) . C. Strong optical coupling
In the case of strong optical coupling, the Rabi fre-quencies contained in the Ω m J -blocks in Fig. 1(b) be-come larger than decay and decoherence of launch andRydberg levels. Strong coupling is, for instance, requiredfor optical shielding [51, 52], where the passage behaviorof the Rydberg atom - ion system through optically in-duced anti-crossings has to be adiabatic and coherent forthe shielding to be effective. In the cases studied in thepresent paper, the optical Rabi frequencies necessary forstrong coupling are (cid:38)
10 MHz.In the strong-coupling case, the matrix elements in theΩ m J -blocks are given by the electric dipole couplings (cid:104) λ, m J | ˆΩ m J ,g | λ g , m g (cid:105) = (cid:80) ˜ m d ( λ ) m J , ˜ m ( θ ) (cid:104) λ, ˜ m | E L ˆ (cid:15) · ˆ r e | λ g , m g (cid:105) (9) with m J in the molecule frame, and m g and ˜ m in thelab frame. The optical coupling has up to three compo-nents ( σ + / − and π ), corresponding to different values of˜ m − m g . The matrix elements are now explicitly includedin the Ω m J -blocks of the total Hamiltonian in Fig. 1(b).Due to the rotation, generally all matrix elements in theΩ m J -blocks are different from zero. In effect, all Rydbergstates that correspond with the different H m J -blocks be-come coupled to each other in second order of the op-tical interaction. Hence, in the regime of strong opticalcoupling the entire Hamiltonian in Fig. 1(b) has to bediagonalized.In our calculation, the Hamiltonian is expressed in aninteraction picture, with a number of 2 J g + 1 dressed [59]launch-state levels, | λ g , m g (cid:105) , intersecting with RydbergPECs, | k, m J (cid:105) , at energies and internuclear separationsthat depend on the excitation-laser detuning from theRydberg nP J -levels. The diagonalization of the fullHamiltonian of Fig. 1(b) yields optically perturbed andmutually coupled PECs of all m J ’s at once. Generally,the PECs belonging to different m J ’s become coherentlymixed in second order of the optical coupling via thelaunch-state sublevels. The mixing tends to be fairly lo-calized in R and energy (see Fig. 4 below), because theoptical couplings are typically much smaller than the en-ergy variation of the PECs. III. RESULTS AND DISCUSSIONA. Determination of l max In Figs. 2 (a) and (b), we show RAIM PECs calculatedwithin the Rydberg subspace {| λ, m J (cid:105) (cid:12)(cid:12) . ≤ n eff ≤ . (cid:96) = 0 , , ..., n − J = (cid:96) ± m J = 1 / } of Rb as a function of the internuclear separation, R , for l max = 1 and 6 in Eq. 3, respectively. This subspaceis centered around the atomic states 45 P J . For l max =1, the plot is identical with the diagram of Stark statesof Rb in a homogeneous electric field, E , given by theion’s electric field at the Rydberg-atom center, E = 1 /R (in atomic units). For both values of l max , the PECsexhibit well-defined minima in the intersection regionsbetween the ion-electric-field-mixed 45 P J -PECs comingin from R = 1 . µ m (marked in Fig. 2), and a fan ofmany slightly curved, approximately parallel PECs in thelower-left halves of the plots. For l max = 1, the latter areidentical with the set of linear Stark states [53] of the n = 42 manifold of Rb. The absence of narrow anti-crossings in the lower potential well (at the lower tips ofthe arrows) indicates the existence of bound RAIM statesthat are stable against decay via non-adiabatic couplingto unbound PECs. The observed potential minima aredeep enough to support several tens of vibrational states(see Sec. III D).Comparing Figs. 2 (a) and (b), we observe a signifi-cant influence of l max on the calculated PECs. This isdue to the strong inhomogeneity of the ion electric field, R (μm) ∆ W ∆ W ∆ R5 D , l max = 1 l max ∆ R ( μ m ) ∆ W ( M H z ) (a) (b) (c) (d) P / P / P / P / P / P / W ( c m - ) launch state (e) R E L P J , m J R (μm)R (μm) ∆ W ∆ R5 P , l max = 6 D , l max = 6 ∆ R FIG. 2. Rydberg-atom-ion molecules (RAIMs) for a case in rubidium. (a) PECs as a function of internuclear separation, R ,calculated for l max = 1 in Eq. 3. The plotted PECs are relative to the atomic ionization potential. The width of the redbackdrop lines is proportional to the single-atom PA rate for the excitation geometry shown in panel (e) and details explainedin the text. The PA rate on the outer slope of the lower PEC well is ≈ . × s − . (b) Same as panel (a), but with l max = 6.Deviations from panel (a) are due to higher-order multipolar effects associated with the inhomogeneity of the ion’s electricfield. The PA rate on the outer slope of the lower PEC well is ≈ . × s − . (c) Same as panel (b), but with the PA ratescalculated for excitation from the 5 P / launch state. The PA rate on the outer slope of the lower PEC well is ≈ . × s − .(d) Effect of l max in Eq. 3 on the PEC fine-structure splitting, ∆ W , and the differential internuclear separation, ∆ R , as definedin panels (b) and (c), between the two most prominent minima in the relevant PECs. Here, ∆ W and ∆ R do not depend onlaunch state. (e) Qualitative energy-level diagram and excitation-laser geometry used for the PA-rate calculations in panels(a)-(c). which necessitates the inclusion of several higher-ordermultipole contributions in Eq. 3. Several qualitative dif-ferences between the results for l max = 1 in Fig. 2(a)and l max = 6 in Fig. 2(b)) are apparent. Including moreterms in Eq. 3 leads to an increase in ∆ W , defined asthe splitting between the PECs that asymptotically con-nect with the nP J fine-structure components at the loca-tion of the relevant PEC potential minima. The minimaof the PEC wells that asymptotically connect with the45 P / and 45 P / atomic states occur at different inter-nuclear distances, separated by a difference ∆ R , as seenin Fig. 2(b). In Fig 2(d) we show the convergence of∆ W and ∆ R as a function of the number of multipoleorders included. One can see that the inclusion of the l = 2 term is particularly important; this is because thecoupling between the 45 P J fine-structure components is l = 1-forbidden but l = 2-allowed. Indeed, the energyscale of the matrix elements in Eq. 3 between nP / and nP / states near the relevant PEC potential minima ison the same order of magnitude as the fine structuresplitting between these states, and therefore generatessignificant level repulsion. Thus, the quadrupolar term, l = 2, is particularly important to include, because itcauses level repulsion between the 45 P J fine-structurecomponents. This has a large effect on both ∆ W and∆ R . Successively higher orders have increasingly smallereffects, as seen in Fig 2(d). In our further calculations weset l max = 6, allowing for sub-MHz and sub-nm accuracyin the energies and locations of the minima on the PECsurfaces, respectively. B. Weak optical coupling
After confirming that the emergence of the RAIMs isrobust against l max in Eq. 3, we discuss how one couldprobe it optically. We first consider the excitation ofRb atoms in the regime of weak optical coupling, as de-scribed in Sec. II. The atoms are initially in the launchstate 5 D / (Figs. 2 (a) and (b)) or 5 P / (Fig. 2(c)), andare located at a distance R from a Rb + ion. The photo-association (PA) laser is assumed to be linearly polarizedand to have an intensity of 10 W/m , corresponding toa field amplitude of 868 × V/m. The angle θ be-tween E L and the molecular axis is random, as visual-ized in the inset in Fig. 2(e), and the launch atoms areevenly distributed over the magnetic sub-states | λ g , m g (cid:105) .The width of the red backdrop lines behind the PECs inFig. 2(a)-(b) is proportional to the average PA rate fora single atom, calculated by means of Eqs. 4-8. The en-ergy density of states for resonant PA is ρ = 1 / ( √ πhσ L ),with a full width at half maximum of the laser spectralintensity vs frequency of σ L √ P -character, while the rates aresmall on the inner slopes of the wells, where the adia-batic states resemble linear Stark states with small low- (cid:96) character. The electric-dipole selection rules for thelaser excitation then largely explain the distribution ofoscillator strength over the PECs. The essential physicshere is similar to that of the Stark effect in alkali Ryd-berg atoms with large quantum defects (Rb and Cs) [60].Since the PA rates in the PEC wells are strongly lopsidedtowards the outer slopes, almost all PA of vibrationalRAIM states will occur near the outer turning points ofthe wells. This fact is expected to enable initializationof vibrational RAIM wavepackets at the outer turningpoint via pulsed PA (see Sec. IV B).Further examination of Figs. 2 (a) and (b) shows thatfor l max = 6 the PA rates into the PECs asymptoticallyconnected with the 45 P J states become redistributed be-tween the J = 1 / l max = 1. This qualitative difference in behavior resultsfrom the fact that the l = 2-terms in Eq. 3 strongly mixthe fine-structure components and thereby tend to aver-age out their oscillator strengths. If the l = 2-terms wereleft out, such as in Fig. 2(a), the calculation fails to ac-count for the redistribution of oscillator strength betweenthe nP J fine-structure pairs.Since the ion electric field mixes some S and D -character into the PECs that asymptotically connectwith the nP J states, PA of RAIMs from 5 P / as a launchstate also is fairly efficient. This is shown in Fig. 2(c),where we employed Eqs. 4-8 to compute the PA ratesfrom 5 P / . For the PA laser intensity and other con-ditions identical with those in Fig. 2(b), for the 5 P / launch state we expect a PA rate that is about 1/5 ofthat for the 5 D / launch state. Comparing Figs. 2 (b)and (c), it is further noted that the lopsidedness of thePA rates between the inner and outer slopes in the PECwells is greater for the 5 P / launch state than it is for5 D / . Finally, since the decay rate of 5 P / is about tentimes higher than that of 5 D / , when using 5 P / as alaunch state one would consider two-photon, off-resonantPA of RAIMs in order to reduce or eliminate decay anddecoherence from the 5 P / -decay. C. Comparison of Rb and Cs
Before we proceed to discuss the strong optical cou-pling regime, in this section we provide a comparisonbetween Rb and Cs. We calculate the PECs of RAIMsnear the 54 P J and 55 P J Rydberg states of Rb and Cs(Figs. 3(a)) and (Fig. 3(b), respectively. This consider-ation allows us to underscore similarities and differencesbetween the two different atomic species. A key find-ing here is that the RAIM bound states are expectedto be observable near nP J Rydberg states in both Rbor Cs. Due to the similarities in the non-integer partsof the quantum defects, both cases exhibit PEC minimaat similar locations within the spectra. Hence, similarRAIMs should exist in both Rb and Cs.We find several minor differences, which result fromthe different quantum defects and fine-structure couplingconstants of Rb and Cs. First, the depth ∆ W of thelower wells, marked by the dashed circles in Fig. 3, rel-ative to the next-higher PECs is slightly deeper in Rb R (μm) R (μm) W ( c m - ) Rb 54 P J Cs 55 P J (a) (b) ∆ W FS ∆ W ∆ W FS ∆ W JJ JJ
FIG. 3. PECs relative to the atomic ionization potentials asa function of internuclear distance R , for rubidium (a) andcesium (b). Both atomic species exhibit similar RAIMs, withmild variations in potential depth, fine-structure splitting andbond length. than in Cs (by ∼ . µ m) is slightly smallerthan that of Rb (2 . µ m). Further, the splittings ∆ W F S between the close-by pairs of PEC wells for | m J | = 1 / | m J | = 3 / W F S -marks in Fig. 3), which is due to the largerfine-structure coupling in Cs. Finally, the curvature ofthe circled PEC well in Cs is less than that in Rb, indi-cating stronger coupling and level repulsion between the P -like and the linear-Stark-state-like PECs in Cs thanin Rb. This means that in Cs non-adiabatic decay ofbound vibrational states into dissociating states on thePECs that resemble linear Stark states is less likely in Csthan it may be in Rb. D. Strong optical coupling
In certain applications, one seeks optical couplingstrengths that exceed the decay and dephasing rates ofthe involved launch and Rydberg states, such as, for in-stance, in blue shielding [51, 52] and in short-pulse ex-citation of RAIM wavepackets. When the launch stateis strongly coupled to the Rydberg state, such that Rabifrequencies exceed decay and dephasing rates, we utilizea dressed-atomic-state picture [59] to calculate PECs. Inthis case, a diagonalization of the full Hamiltonian inFig. 2(b) is performed, because all levels are coherentlycoupled to each other.In Fig. 4(a), we show the results for RAIM PECsof Rb as a function of internuclear distance R for acase of strong optical coupling between the launch states { (cid:12)(cid:12) D / , m g (cid:11) } and the RAIM states. The PA laser is σ − -polarized, has a field amplitude of E L = 1 × V/m, andpropagates at an angle of θ = 45 ◦ along the internuclearaxis. This is a case in which all Ω-couplings visualized inFig. 1(c), between all pairs of m J and m g , are allowed.The RAIM system is, in general, driven into a coher- R (μm) R (μm) W ( c m - ) W ( c m - ) W ( M H z ) P / P / (b)(c) (a) d a r k s t a t e s θ = E L R σ – FIG. 4. Rydberg-atom-ion molecules (RAIMs) of rubidium inthe case of the strong optical coupling. (a) Dressed launch-state and RAIM PECs as a function of internuclear distance R , for a fixed atom-field detuning of 5.957 GHz relative tothe field-free atomic 45 P / resonance. The system is drivenby σ − -polarized light propagating at θ = 45 ◦ with respectto the molecular quantization axis as shown in the inset andwith electric-field amplitude E L = 1 × V/m. (b) Mag-nified view of the region of intersection between the dressedPECs of the launch-state and the RAIM PEC that asymptot-ically connects with the 45 P / state. (c) A magnified view ofthe lowest PEC minimum corresponding to molecular boundstates. The lowest 18 vibrational states are shown. ent superposition of states from different m J subspaces.The m J -mixing is, however, effective only in the immedi-ate vicinity of the crossings between dressed launch-statelevels and RAIM PECs, such as within the small red boxin Fig. 4(a). Outside these regions, the optical couplingis ineffective, and the PECs are not m J -mixed.In Fig. 4, we consider a case in which a J g = 3 / ←→ J = J g − / P J states.In this case, the presence of dark (uncoupled) states isinevitable. In Fig. 4(b), we provide a zoom-in of therelevant intersection region marked by the red box inFig. 4(a). The magnification clearly shows that there arefour optically coupled levels and two dark states. Theuncoupled states exist for any choice of θ . E. Vibrational states
The wells within PECs are typically several GHzdeep, such as in Fig. 4(a), and support many vibra-tional states. This makes RAIMs attractive for laser-spectroscopic study and for time-dependent wave-packetstudies.In Fig. 4(c), we show a magnified region of the lowerPEC well associated with 45 P J , outlined by the greenbox in Fig. 4(a), together with the squares of the vibra-tional wavefunctions for the lowest 18 vibrational states.The vibrational level structure is that of a perturbedharmonic oscillator in which the vibrational frequency intervals follow f n = f + α ν + α ν ... , with the low-est frequency interval f , the index ν = 0 , , ... , andonly the lowest two dispersion terms shown. Fitting thelowest ten levels in Fig. 4(c), one finds f = 18 . α = − .
61 MHz and α = 0 .
020 MHz for the first- andsecond-order dispersion coefficients. This finding couldbe tested, in the future, via laser spectroscopy.This structure also lends itself to studies of vibrationalwavepackets of RAIMs, including collapse and quantumrevivals of wavepackets [61, 62]. We use the results inFig. 4(c) to estimate the dephasing and revival times forvibrational wavepackets near the bottom of the potentialwell. Neglecting second- and higher-order dispersion, thedephasing time is ∼ . / ( N | α | ) and the revival time ≈ / | α | , with the above lowest-order dispersion coefficient α , and with the number of coherently excited states N .For wavepackets ranging in bandwidth from N = 5 to 10vibrational states, the dephasing time ranges from aboutabout 300 ns to 700 ns, while the bandwidth-independentrevival time is ≈ . µ s. These times are shorter than theRAIM lifetime, which should scale with the Rydberg-atom decay time (estimated at ∼ µ s), and shouldtherefore be observable. A detailed theoretical study onthe dynamics of vibrational RAIM wavepackets can bean interesting topic for future investigations. F. Collisions on Rydberg-atom-ion PECs
The RAIM formalism for PECs in the strong opticalcoupling regime, described in the present paper, also di-rectly applies to an analysis of certain optical shieldingmethods for atom-ion collisions. With the following ex-amples we exhibit the close relationship between RAIMsand shielding of atom-ion collisions via strong opticalcoupling to Rydberg states.
R (μm) R (μm) W ( c m - ) Rb (a) (b) J Cs op t i c a l c oup l i ng op t i c a l c oup l i ng FIG. 5. (a) Soft potential barrier for Rb atom-ion collisions,affected by strong optical coupling of the atomic ground stateto Rydberg PECs slightly below the 45 P J states. The bar-rier is ∼ h ×
100 MHz high. (b) Soft potential barrier forCs atom-ion collisions, affected by strong optical coupling ofthe atomic ground state to blue-shifted PECs with 45 D / -character. The barrier is ∼ h × As one example, lowering the dressed-launch-state de-tuning in Fig. 4(a) from -61.36 cm − to -61.393 cm − ,using a launch state of 5 S / , a laser field of E L =4 × V/m, θ = 45 ◦ , and linear polarization, resultsin the PEC landscape shown in Fig. 5 (a). In a cold-collision scenario, the configuration introduces a soft en-ergy barrier at about 1.3 µ m that is about h ×
100 MHzhigh. In cold-atom systems, the barrier can be used toprevent deep, inelastic collisions between Rb atoms andions, as described in great detail in [51, 52] for a lowerRb nP J Rydberg level.In Cs, efficient shielding of inelastic atom-ion collisionscan likely be achieved using the blue-shifting m J = 1 / nD / , which is fairly unique amongalkali Rydberg atoms. In Fig. 5 (b) we employ the RAIM-PEC formalism to find the PECs for collisions between Csatoms and ions with the coupling laser tuned above theatomic 45 D / resonance. The launch (dressed) atomicstate 6 S / is tuned to -60.52 cm − , the laser field islinearly polarized and has an amplitude of E L = 5 × V/m, and θ = 45 ◦ . The configuration introduces acomparatively tall, soft collisional barrier that is locatedat about 1.7 µ m and is about h × IV. EXPERIMENTAL CONSIDERATIONS
In the following we discuss two possible approaches tostudy RAIMs. The first, and arguably easier, methodis to perform high-precision spectroscopy measurementswith quasi-continuous narrow-band ( ≤ A. Laser spectroscopy with narrow-band lasers
In this subsection, we discuss experimental methodsto prepare RAIMs of Rb or Cs via high-resolution laserspectroscopy with quasi-continuous lasers. For Rb, somedetails are visualized in Fig. 6(a). The 5 S / → nP J ground-to-Rydberg direct transition is less frequentlyused than multi-photon schemes, as it would requirea laser with a wavelength of ∼
300 nm. A methodof preparing Rb atoms in Rydberg P -states with anall-infra-red laser system is to first excite atoms viaa low-lying D -state into Rydberg P -states. For in-stance, in Rb the following three-step excitation canbe used: 5 S / → P / at 795 nm, 5 P / → D / at 762 nm, 5 D / → nP J at 1260 nm. These threewavelengths can be attained with conventional diode Rydberg atom preparation ion preparation Rb n P J D P S (a) Cs D P S (b) Rydbergatom preparationion preparation n P J photoionization level FIG. 6. (Color online) Energy level diagram (not to scale)with relevant wavelengths for the experimental realizationof RAIM with Rb (a) and Cs (b). The lower stages of theRydberg-atom transitions are driven off-resonantly. See textfor details. lasers and without frequency-doubling, and the Rydberg-excitation stage has relatively large excitation matrix el-ements (see Fig. 2). The atoms are photo-associated intoRAIMs by seeding the atom cloud with ions that havea kinetic energy low enough to avoid line broadening bymore than a fraction of the vibrational energy level spac-ing. Due to ion recoil, the photo-ionization (PI) energyin Rb should be within about 100 GHz to 1 THz abovethe PI threshold in order to be able to resolve vibrationalstates with spacings as in Fig. 4(c). This corresponds towavelengths of 0.5 nm to 5 nm less than the Rb 5 D / PI wavelength of 1251.52 nm.A corresponding level diagram and excitation schemefor Cs is displayed in Fig. 6(b). Two-photon excitationat 885 nm, utilized in previous experiments [63, 64], canbe used to prepare Cs atoms in the 6 D / state. Then,a laser at 1144 nm can be used to photo-associate nP J RAIMs of Cs. At the same time, the excitation regioncan be seeded with low-energy ions by PI of atoms inthe 6 P / state, accessed via the 852-nm Cs D2 line, anda 508-nm PI laser. To keep the ion kinetic energy lowenough to be able to resolve vibrational levels, the upper-laser wavelength should be between 0.2 and 2 nm belowthe 6 P / PI threshold, which is at 508.28 nm.The ions must generally be prepared at a low enoughdensity to avoid Coulomb acceleration and expansion be-fore and during the photo-association pulse. Other op-tions that could be explored to reduce ion energy includeusing an ion trap to prepare laser-cooled ions. Ion veloc-ities in the range (cid:46) . µ s, leading to interaction-time broadening of less than10 MHz, which is sufficiently low to resolve vibrationalstructure.Figure 2(b) suggests RAIM production rates on theorder of 10 s − , which, for PA time windows in therange of hundreds of nanoseconds, suggests a RAIM pro-duction probability in the range of 3%. Using an ex-perimental cycle rate of 100 Hz [17], assuming severaltens of ions embedded in a dense atomic vapor, and es-timating an ion detection efficiency of 30%, the countrate would be several 10 s − . Considering that the sig-nal would be virtually free of background counts fromdistant atomic lines, laser-spectroscopic RAIM measure-ments appear quite feasible.It is worth noting that mixed-species RAIMs betweena Rb or a Cs Rydberg atom and any point-like positiveion have structures similar to the ones considered in thispaper, with different vibrational spacings due to the dif-ferent effective masses. RAIMs between negative ionsand Rydberg atoms have noticeably different PECs, astheir Rydberg-electron charge distributions are distortedaway from the ion, into regions of diminishing electricfield, as opposed to towards the ion, into regions of in-creasing electric field. B. Vibrational wavepackets
In Section III E, we have provided estimates for de-phasing and recurrence times of vibrational wavepack-ets consisting of 5-10 vibrational levels of 45 P J RAIMs.This number of vibrational levels is large enough to formreasonably well-localized wavepackets, and it is smallenough to avoid the effect of higher-order dispersionterms in the vibrational energy-level series. As seenin Fig. 3(c), such a wavepacket would cover a spectralbandwdith of 100-200 MHz. Its realization requires atransform-limited laser pulse with a duration of about5-10 ns, corresponding to the aforementioned spectralbandwidth. A shorter (longer) laser pulse leads to theexcitation of more (fewer) vibrational levels and shorter(longer) wavepacket dephasing times. Considering onlylowest-order dispersion, the revival time is independent ofthe number of vibrational levels, and the revival remainsstrong even for large numbers of coherently excited vi-brational states.We concentrate on Rb RAIMs to further discuss therelevant requirements for the laser. The excitationscheme for Rb RAIMs suggested in Sec. IV A would re-quire either continuous-wave or pulsed lower-transitionlasers and a high-intensity 1260-nm laser pulse. Unfor-tunately, there are no convenient choices for 1260 nmtransform-limited pulsed lasers with pulse durations inthe range of 5-10 ns. An alternative approach wouldbe to excite the atoms directly from the 5 S / groundstate into nP Rydberg state. In this case, the excitation pulse would be centered at a wavelength of 297 nm, whichcould be achieved by frequency-doubling of a pulsed Rho-damine 6G dye laser pumped with a 532-nm Nd:YAGlaser. The frequency-doubling could be achieved, for in-stance, with a beta barium borate or a lithium niobatecrystal.The wavepacket dynamics can be probed using pump-probe schemes, the most basic version of which wouldbe to apply two identical PA pulses with a variabledelay time and to measure the RAIM yield vs delaytime [65, 66]. Additional experiments, e.g., using quan-tum gas microscopy [31, 32], could be employed forspatio-temporal mapping of the RAIM vibrational wave-functions. Alternatively, the wavefunctions could alsobe studied in correlation measurements. In this scheme,the RAIMs would be field-ionized and imaged with asingle-ion-resolving imaging systems. Similar methodshave been used to study correlations in both Rydbergatom and ion plasma systems [67, 68].
V. CONCLUSION
We have shown through PEC calculations that it ispossible to form Rydberg atom-ion molecules that arebound by way of multipolar interactions between the twoparticles. We have provided detailed results on severalcases in which a Rb or Cs neutral-atom launch stateis optically coupled to Rydberg-state manifolds. Wehave also discussed potential experimental realizations ofRAIMs, differences in forming said molecules with rubid-ium versus with cesium, and the possibility of preparingvibrational wavepackets within these molecules. Modernstate-of-the-art experimental setups with ultracold Ryd-berg atoms and trapped ions already possess all necessarycomponents to form and observe these molecules. It maybe anticipated that the observation of RAIMs will opena new direction in the field of ultralong-range Rydbergmolecules and many-body physics.
ACKNOWLEDGMENTS
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