CHAPTER TWO Ultracold Hybrid Atom–Ion Systems R. Cot^ e1 University of Connecticut, Storrs, CT, United States 1Corresponding author: e-mail address: [email protected] Contents 1. Introduction 68 2. Atom–Ion Interactions 70 2.1 Homonuclear Case 70 2.2 Heteronuclear Case 73 3. Scattering Processes 75 3.1 Resonant Processes 75 3.2 Nonresonant Processes 83 4. Transport Properties 84 4.1 Diffusion 84 4.2 Mobility 86 4.3 Hole Mobility 88 5. Tuning Interactions: Hyperfine and Zeeman Interactions 91 5.1 Multichannel Scattering 91 5.2 Identical Nuclei 94 6. Isotopic Effects 99 6.1 Theory 100 6.2 Results for a Few Examples 102 6.3 Tuning Scattering with Magnetic Fields 103 7. Charges in a BEC 106 7.1 Ion in a BEC 106 7.2 Rydberg Electron in a BEC 111 8. Conclusions 119 Acknowledgments 119 References 119 Abstract Ultracold atomic samples doped with charged particles is a nascent field marrying two usually well-separated fields, namely trapped ions and ultracold atoms. Since the orig- inal proposals over 15 years ago, the initially slow pace has given way to rapid progress. In this chapter, we review some of the concepts relevant to this hybrid field, ranging from resonant charge transfer to the effect of isotope shifts, and the role of hyperfine Advances in Atomic, Molecular, and Optical Physics, Volume 65 # 2016 Elsevier Inc. 67 ISSN 1049-250X All rights reserved. http://dx.doi.org/10.1016/bs.aamop.2016.04.004 68 R. Cot^ e and Zeeman interactions in obtaining Feshbach resonances allowing control of the scattering processes taking place. The next frontier, charges in a Bose–Einstein conden- sate, is also introduced and discussed. 1. INTRODUCTION Many recent developments in Atomic, Molecular, and Optical Phys- ics have been triggered by the ability to reach ultralow temperatures, which allowed probing degenerate atomic gases (Dalfovo et al., 1999; Giorgini et al., 2008). In addition, the control of interactions in ultracold gases, eg, via Feshbach resonances (Chin et al., 2010; Kohler€ et al., 2006), made it possible to investigate a variety of many-body phenomena, such as molec- ular BEC (Greiner et al., 2003; Jochim et al., 2003; Xu et al., 2003) or BEC- BCS crossover (Altmeyer et al., 2007a,b; Du et al., 2009; Greiner et al., 2005; Kosˇtrun and Cot^ e, 2006; Partridge et al., 2005; Shin et al., 2008; Zhang and Leggett, 2009). This level of control paved the way to studies of few-body systems very sensitive to the details of the interactions, such as exotic three-body Efimov states (Kraemer et al., 2006), while mixtures (eg, 6Li– 7Li (Khaykovich et al., 2003), Li–Cs (Deiglmayr et al., 2009; Kraft et al., 2006; Mudrich et al., 2002), K–Rb (Damski et al., 2003; Ferlaino et al., 2006; Modugno et al., 2001; Simoni et al., 2003), Rb–Cs (Holmes et al., 2004;Kerman et al., 2004a,b; Sage et al., 2005), or NaCs (Haimberger et al., 2004, 2006)) have allowed studies of fermion–boson sys- tems and ultracold polar molecules. A nascent effort on ultracold samples containing ions is occurring in which resonances should be important. Although few results exist in these systems (Ciampini et al., 2002; Cot^ e, 2000a; Cot^ e and Dalgarno, 2000; Grier et al., 2009; Makarov et al., 2003;Smith et al., 2014; Zhang et al., 2009b), new studies are taking place, eg, on Yb (Cetina et al., 2012; Grier et al., 2009; Karpa et al., 2013), Rb with Yb+ (Lamb et al., 2012; Ratschbacher et al., 2012; Sayfutyarova et al., 2013;Zipkes et al., 2010a,b)orCa+ (Hall et al., 2013) or Ba+ (Krych et al., 2011; Schmid et al., 2010). As originally suggested by Makarov et al. (2003) and Smith et al. (2005), mixing atoms and ions requires hybrid traps, as shown in Fig. 1. In addition to their rele- vance to cold plasmas (Hahn, 2002; Mazevet et al., 2002; Robicheaux and Hanson, 2002), and ultracold Rydbergs (Anderson et al., 1998, 2002; Mourachko et al., 1998; Robinson et al., 2000), novel applications using the ion’s charge have been proposed, such as scanning tunneling Ultracold Hybrid Atom–Ion Systems 69 Fig. 1 Diagram of the hybrid trap apparatus. (a) A Na MOT (orange (gray in the print version)) is formed concentric with an ion cloud (gray) inside a segmented linear Paul trap (LPT) with six 589-nm MOT beams (yellow (light gray in the print version)) and a pair of anti-Helmholtz coils (exterior to the chamber). A 405-nm beam (blue (light gray in the print version)) aligned colinearly with one of the MOT beams is used for REMPI. Fluo- rescence measurements of the MOT can be made with a photomultiplier tube (PMT) or a CMOS camera. An electrically biased mesh is placed between the LPT and the Channeltron electron multiplier (CEM), which is used for ion detection. Right panels: CMOS camera image, without false coloring, of the smaller, denser, and colder type-I MOT (b) and the larger, warmer type-II MOT (c). The inner edges of the LPTs end-seg- ment electrodes can be seen in the corners of the image. Adapted from Sivarajah, I., Good- man, D.S., Wells, J.E., Narducci, F.A., Smith, W.W., 2012. Evidence of sympathetic cooling of Na+ ions by a Na magneto-optical trap in a hybrid trap. Phys. Rev. A 86, 063419. doi:10.1103/PhysRevA.86.063419 and Goodman, D.S., Wells, J.E., Kwolek, J.M., Blumel,€ R., Narducci, F.A., Smith, W.W., 2015. Measurement of the low-energy Na+-Na total collision rate in an ion-neutral hybrid trap. Phys. Rev. A 91, 012709. doi:10.1103/ PhysRevA.91.012709. microscopy of ultracold atoms (Kollath et al., 2007; Sherkunov et al., 2009), or a quantum gate using 87Rb+135Ba+ (Doerk et al., 2010). Exotic objects due to the capture of atoms around ions (Cot^ e et al., 2002), or atom-like mol- ecules with multiple atoms orbiting a heavy ion (Gao, 2010) were also predicted. In this chapter, we give an overview of ultracold samples doped with charged particles. We start by describing the atom–ion interaction, paying special attention to its long-range behavior. We follow by a survey of the principal processes taking place during the scattering of an atom and an 70 R. Cot^ e ion. We focus our attention first on atom and ion being from the same ele- ment, and then for different isotopes. We briefly mention the case of differ- ent elements, since it is outside the scope of this chapter. We describe the case of resonant and quasi-resonant charge transfer and how Feshbach res- onances can help tuning the scattering processes. We finally discuss the case of charges in a Bose–Einstein condensate (BEC), considering an ion or Rydberg electrons in a BEC. 2. ATOM–ION INTERACTIONS The interaction between an ion and another particle depends on the nature of that particle. For example, if it is another ion (atomic or molecu- lar), the leading long-range interaction behaves as 1/R, while a neutral par- ticle with a permanent electric dipole moment (eg, a polar molecule) leads to a 1/R2 long-range interaction. In this chapter, we focus our discussion on the case of an ion interacting with a nonpolar neutral object, for which the interaction scales as 1/R4 at large separation. Although a few experiments are starting to combine ultracold atoms with molecular ions, most experiments in ultracold samples involve ions and atoms. We discuss the interaction potential for two different cases: processes where both the ion and the atom are from the same element, and those where they are different. 2.1 Homonuclear Case In homonuclear systems X+X+, the charge can on either center, leading to an even or odd electronic wave function under symmetrization (inversion symmetry through the midpoint between the two nucleus, ie, a gerade (g) or ungerade (u) state. The multiplicity depends on the system considered. If X is an alkali metal element (Li, Na, K, Rb, Cs, or Fr), the electronic spin is 1 and 0 for X and X+, respectively, while it is 0 and 1 respectively for 2 2 alkaline earth metals (Be, Mg, Ca, Sr, Ba, or Ra): in both cases, the total electronic spin is S 1, leading to doublet molecular potentials. More com- ¼ 2 plex elements could lead to higher multiplicity, such as Cr and Cr+, with 5 1 3 11 electronic spin of 3 and , respectively, giving S , ,…, , and doublet, 2 ¼ 2 2 2 quartet, … , deca-doublet molecular curves. Since most atoms and ions used in ultracold experiments are alkali or alkaline earth metals (because of the existence of simple cycling transitions Ultracold Hybrid Atom–Ion Systems 71 to cool them), we discuss their case in what follows. As mentioned above, they have similar behaviors, ie, a single active electron for alkali metals, or 2 + hole for alkaline earth metals, giving two molecular potential curve Σg and 2 + 2 + 2 + Σu . The electronic ground state is X Σg for alkali metals, and X Σu for + + alkaline earth metals. Fig. 2 shows potential curves for Li2 , Na2 and + + Rb2 (left column), while the right column depicts those for Be2 , and + + Ca2 . We also include the curves for Yb2 , a rare-earth atom with an atomic structure similar to that of alkaline earth atoms. We note that the g/u order is inverted between the two class of elements. Usually, the ground state is much deeper than the upper state, though the existence of a double-well + + for Be2 and Ca2 masks the shallow long-range well that appears for all.