Synthetic dimension-induced conical intersections in Rydberg molecules
SSynthetic dimension-induced conical intersections in Rydberg molecules
Frederic Hummel, ∗ Matthew T. Eiles, and Peter Schmelcher
1, 3 Zentrum für Optische Quantentechnologien, Fachbereich Physik,Universität Hamburg, Luruper Chaussee 149, 22761 Hamburg, Germany Max-Planck-Institut für Physik komplexer System, Nöthnitzer Str. 38, 01187 Dresden, Germany The Hamburg Centre for Ultrafast Imaging, Universität Hamburg,Luruper Chaussee 149, 22761 Hamburg, Germany (Dated: February 9, 2021)We observe a series of conical intersections in the potential energy curves governing both thecollision between a Rydberg atom and a ground-state atom and the structure of Rydberg molecules.By employing the electronic energy of the Rydberg atom as a synthetic dimension we circumventthe von Neumann-Wigner theorem. These conical intersections can occur when the Rydberg atom’squantum defect is similar in size to the electron-–ground-state atom scattering phase shift divided by π , a condition satisfied in several commonly studied atomic species. The conical intersections have anobservable consequence in the rate of ultracold l -changing collisions of the type Rb ( nf ) +Rb (5 s ) → Rb ( nl > +Rb (5 s ) . In the vicinity of a conical intersection, this rate is strongly suppressed, andthe Rydberg atom becomes nearly transparent to the ground-state atom. The Born-Oppenheimer approximation is a corner-stone of chemical and molecular physics. It providesus with the adiabatic separation of the fast electronicfrom the slow vibrational motion, resulting in adiabaticpotential energy surfaces (PES) determined by the elec-tronic structure for a given nuclear geometry [1]. WhenPES become degenerate at a conical intersection (CI),non-adiabatic interaction effects are important and theBorn-Oppenheimer-based intuition developed in molec-ular physics breaks down [2–4]. CI occur frequently inlarger molecules with many vibrational degrees of free-dom, for example in the nucleobases [5], and play a keyrole in photosynthesis [6]. CI are responsible for ultra-fast radiationless decay mechanisms on the femtosecondtime scale [2, 7, 8]. A controlled environment to studyCI can be provided by external optical fields [9, 10] orby ultracold interacting Rydberg systems [11, 12]. Di-atomic molecules provide here an exception. Generally,the von Neumann-Wigner non-crossing theorem forbidsthe crossing of potential energy curves (PEC) in diatomicsystems, which are determined by a single vibrational pa-rameter, the internuclear coordinate R [13].Excited electronic states are key players in the ap-pearance of CI and a particular diatomic system, thecollisional complex consisting of a Rydberg atom anda ground-state atom, has attracted significant interestsince nearly the advent of quantum physics [14–18]. Theinteraction between these two atoms, as described by theFermi pseudopotential, is primarily determined by the s -wave electron-atom scattering phase shift δ s [14, 15].The resulting Born-Oppenheimer PEC can be labeled bythe principal quantum number n and angular momentumquantum number l of the Rydberg atom. These PEC canbe sufficiently attractive to support bound states, knownas ultra-long-range Rydberg molecules [19–22]. At posi-tive energies, they are responsible for the collisional dy-namics between the two highly asymmetric atomic part- ners. Just as the characteristics of a Rydberg atom aresmoothly varying polynomial functions of n , so too arethe typical molecular properties, for example the poten-tial depths, dipole moments, and bond lengths.For this reason, it is illustrative to imagine that thisdiatomic system evolves along a two-dimensional poten-tial energy surface , where the principal quantum number n plays the role of an additional synthetic dimension. Inmany other contexts, the introduction of synthetic di-mensions provides a means to control the dimensionalityof a system by mimicking additional degrees of freedom.Synthetic dimensions have been realized in optical lat-tices [23, 24], and have applications in the study of gaugefields [25], quantum simulation [26], and photonics [27].In the present work, we utilize the synthetic dimen-sion n to circumvent the non-crossing theorem, and indoing so, we show that CI can appear in the Rydberg–ground-state atom PES. These CI occur when the Ry-dberg atom is initially in a state | nl (cid:105) whose fractionalquantum defect µ l is similar in size to δ s /π , which is typ-ically satisfied in states with more than d -wave angularmomentum l . We highlight results for Rb, where l = 3 ,since interest in such a state has recently grown due toits importance for the preparation of circular Rydbergstates in the scope of quantum simulation and quantumcomputing based on neutral-atom platforms [28–31].Similar to how CI in more traditional molecular sys-tems provide fast radiationless decays between electronicstates, the synthetic CI can facilitate fast dynamics inRydberg–ground-state atom collisions. In particular,we demonstrate that they dramatically suppress the l -changing collision rate, which is otherwise a dominantprocess in this collision. This provides a clear experi-mental observable, heralding the presence of syntheticCI in several different atomic species.The interaction between the two atoms is given by a r X i v : . [ phy s i c s . a t o m - ph ] F e b Figure 1. (a) Illustration of the atomic collision and the cor-responding adiabatic (solid, red V + and blue V − ) and dia-batic (dashed, purple t and gray q ) PEC for different prin-ciple quantum numbers n ∈ { , , , } (increasing n isplotted with increasing opacity). At large internuclear dis-tances, the difference of the potentials ∆ is determined bythe quantum defect of the state with angular momentum l .The axes are rescaled in order to shift the curves onto thesame energy scale. A semi-classical ground-state atom propa-gating on V − can either adiabatically follow the potential andpick up orbital angular momentum to form a trilobite state,the electronic density of which is shown at the bottom, or hopto the potential V + and continue with an f -state electronicorbital depicted at the top. (b) The eigenvector components ψ − corresponding to V − and (c) the non-adiabatic couplingmatrix element P with varying (scaled) R for different n . Fermi’s pseudopotential, V ( r , R ) = 2 πa ( k ) δ ( r − R ) , (1)where a ( k ) = tan δ s ( k ) k = a (0) + πα k is the low-energy s -wave scattering length, α is the ground-state atoms’spolarizability, and the wave number k is determined semi-classically by k − R = − n . Higher partial waves canbe neglected because the electron energy is close to zeroat the large internuclear distances considered here. Since V ( r , R ) is a weak perturbation to the Coulomb potential,we calculate the PEC by diagonalizing V ( r , R ) within arestricted basis. This includes only the initial state | nl (cid:105) ,whose energy we take as the reference energy, and thedegenerate manifold of hydrogenic l > l states with van-ishingly small quantum defects. These are blue-detunedby a shift ∆ from the initial state. The l < l states canbe ignored since they are far detuned, ∆ l